04.03.22 · algebraic-geometry / cohomology

Cyclic homology and Connes' long exact sequence

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Anchor (Master): Connes 1983 *C.R. Acad. Sci. Paris* 296 (announcement); Connes 1985 *Publ. Math. IHES* 62 (noncommutative differential geometry); Tsygan 1983 *Uspekhi Mat. Nauk* 38 (Russian Math. Surveys); Loday-Quillen 1984 *Comment. Math. Helv.* 59; Feigin-Tsygan 1985; Loday *Cyclic Homology* (Springer Grundlehren 301, 1992; 2nd ed. 1998) — canonical reference; Karoubi *Homologie cyclique et K-théorie* Astérisque 149 (1987); Connes *Noncommutative Geometry* Academic Press 1994; Goodwillie 1986 *Ann. Math.* 124; Bökstedt-Hsiang-Madsen 1993 *Invent. Math.* 111; Nikolaus-Scholze 2018 *Acta Math.* 221

Intuition Beginner

Hochschild homology $H H_{*} (A)$ of an associative $k$ -algebra $A$ assembles tensor expressions in $n + 1$ slots, $(a_{0}, a_{1}, \dots, a_{n})$ , and records how they interact with the multiplication of $A$ . The Hochschild differential treats the first slot $a_{0}$ asymmetrically — it is the "bimodule slot," and the cyclic rearrangement that sends the last entry around to the front never enters the chain complex itself. For commutative or central data, however, that asymmetry is artificial: every tensor expression admits a cyclic rotation, and the resulting symmetry deserves to be promoted to a piece of structure on the chain complex rather than left as an ambient observation.

Cyclic homology $H C_{*} (A)$ is the result of imposing this cyclic symmetry. Connes (1983, 1985) and Tsygan (1983) discovered independently that the cyclic action $τ_{n}$ on the Hochschild chains — rotating the entries with a sign — combines with the Hochschild differential $b$ to produce a bicomplex whose total homology is a strictly finer invariant. The Connes SBI long exact sequence relating $H H_{n}$ , $H C_{n}$ , $H C_{n - 2}$ , and $H H_{n - 1}$ packages the Hochschild-to-cyclic relationship into a single tower of exact sequences, and the periodic limit $H P_{*} (A)$ under the periodicity operator $S$ is what Connes called "noncommutative de Rham cohomology" — for a smooth commutative algebra in characteristic zero it recovers exactly the algebraic de Rham cohomology of the affine variety.

The everyday analogy is a circular dinner conversation. Hochschild homology records each linear sequence of remarks (who said what after whom) with a designated host on the left. Cyclic homology imposes the rule that the conversation is around a round table — rotating who counts as "first" produces the same conversation, with a sign tracking the parity of the rotation. Periodic cyclic homology takes the inverse limit over the rotation: it records the entire pattern modulo the cyclic shift, and for a smooth commutative algebra it recovers the de Rham forms of the underlying geometric object, which is exactly Connes' programme of noncommutative differential geometry.

Visual Beginner

A diagram showing the Hochschild chain $C_{n} (A)$ — the tensor power of $A$ over the base field $k$ with $n + 1$ factors — together with the cyclic action $τ_{n}$ permuting these factors in a rotational pattern, the Connes-Tsygan bicomplex with vertical Hochschild differential $b$ and horizontal cyclic differential $1 - τ$ , the resulting total complex $C C_{*} (A)$ whose homology is the cyclic homology $H C_{*} (A)$ , and the Connes SBI long exact sequence relating $H H_{n}$ , $H C_{n}$ , $H C_{n - 2}$ , and $H H_{n - 1}$ wrapping back on itself via the periodicity operator $S$ .

The HKR cyclic identification of $H C_{n} (A)$ with $Ω_{A / k}^{n}$ modulo exact forms plus de Rham cohomology in lower degrees, for smooth commutative $A$ in characteristic zero, is annotated on the right; the periodic cyclic homology $H P_{*} (A)$ is shown as the algebraic de Rham cohomology of $A$ packaged in two-periodic graded form.

The picture captures the structural shape: cyclic homology refines Hochschild by adding the cyclic-symmetry data via the bicomplex construction, the SBI long exact sequence ties Hochschild and cyclic invariants together with periodicity, and the HKR cyclic identification recovers algebraic de Rham cohomology as the periodic limit. A reader who internalises this picture will recognise the same template every time cyclic homology appears — Hochschild plus cyclic symmetry equals bicomplex, then take total homology, then read off the periodic limit as noncommutative de Rham.

Worked example Beginner

Compute the cyclic homology and periodic cyclic homology of the polynomial algebra $A = k [x]$ over a field $k$ of characteristic zero, and verify the HKR cyclic identification with algebraic de Rham cohomology.

Step 1. Hochschild input. From the parent unit on Hochschild homology (and HKR), $H H_{0} (k [x]) = k [x]$ , $H H_{1} (k [x]) = k [x] \cdot d x$ (a rank-one free module generated by $d x$ ), and $H H_{n} (k [x]) = 0$ for $n \geq 2$ . The Kähler differentials of $A = k [x]$ are $Ω_{A / k}^{0} = k [x]$ , $Ω_{A / k}^{1} = k [x] \cdot d x$ , and $Ω_{A / k}^{n} = 0$ for $n \geq 2$ .

Step 2. Algebraic de Rham cohomology of $A = k [x]$ . The de Rham differential $d : Ω^{0} = k [x] \to Ω^{1} = k [x] \cdot d x$ sends a polynomial to its derivative-times-dx. In characteristic zero, this map is surjective (every $g (x) d x$ has a polynomial antiderivative in $k [x]$ ), and its kernel is the constants $k \subset k [x]$ . So $H_{dR}^{0} (A / k) = k$ (the constants) and $H_{dR}^{1} (A / k) = 0$ (every form is exact). All higher algebraic de Rham cohomology vanishes since $Ω_{A / k}^{n} = 0$ for $n \geq 2$ .

Step 3. Apply the HKR cyclic identification. For a smooth commutative algebra in characteristic zero, the Connes-Loday-Quillen formula reads $H C_{n} (A) ≅ Ω_{A / k}^{n} / d Ω_{A / k}^{n - 1} \oplus H_{dR}^{n - 2} (A) \oplus H_{dR}^{n - 4} (A) \oplus \dots$ (the sum runs over indices $n, n - 2, n - 4, \dots, 0$ or $1$ ). Substituting the values for $A = k [x]$ :

$H C_{0} (k [x]) = Ω_{A / k}^{0} /0 = k [x]$ (the whole algebra).
$H C_{1} (k [x]) = Ω_{A / k}^{1} / d Ω_{A / k}^{0} = k [x] \cdot d x / k [x]^{'} \cdot d x = 0$ (since $d : k [x] \to k [x] \cdot d x$ is surjective in characteristic zero).
$H C_{2} (k [x]) = Ω_{A / k}^{2} / d Ω_{A / k}^{1} \oplus H_{dR}^{0} (A) = 0 \oplus k = k$ .
$H C_{3} (k [x]) = 0 \oplus H_{dR}^{1} (A) = 0$ .
$H C_{4} (k [x]) = 0 \oplus H_{dR}^{2} (A) \oplus H_{dR}^{0} (A) = 0 \oplus 0 \oplus k = k$ .
$H C_{2 n} (k [x]) = H_{dR}^{0} (A) = k$ for $n \geq 1$ .
$H C_{2 n + 1} (k [x]) = 0$ for $n \geq 0$ .

Step 4. Periodic cyclic homology and the de Rham comparison. The periodic cyclic homology $H P_{*} (A) = lim_{S} H C_{*} (A)$ is the $Z /2$ -graded inverse limit under the periodicity operator $S : H C_{n} \to H C_{n - 2}$ . For $A = k [x]$ : $H P_{0} (k [x]) = lim_{S} {H C_{0}, H C_{2}, H C_{4}, \dots} = lim_{S} {k [x], k, k, \dots}$ . The maps $S : H C_{n + 2} \to H C_{n}$ for $n \geq 0$ are eventually isomorphisms $k \to k$ (the identity on the constant copy that propagates from $H_{dR}^{0}$ in every $H C_{2 n}$ for $n \geq 1$ ), so the inverse limit picks up the stable copy $H P_{0} (k [x]) = k = H_{dR}^{0} (A / k)$ . Similarly $H P_{1} (k [x]) = 0 = H_{dR}^{1} (A / k)$ .

What this tells us. For $A = k [x]$ , the cyclic homology $H C_{n} (k [x])$ stabilises in degree $\geq 2$ at the de Rham cohomology of $A_{k}^{1}$ (the affine line). The periodic cyclic homology $H P_{*} (k [x])$ recovers exactly the algebraic de Rham cohomology of $A_{k}^{1}$ in $Z /2$ -graded form: $H P_{0} = k$ (the constants of $H_{dR}^{0}$ ) and $H P_{1} = 0$ . This is the simplest worked example of Connes' philosophy that periodic cyclic homology is the noncommutative replacement for algebraic de Rham cohomology — for a smooth commutative algebra, it reduces to the classical de Rham theory of the underlying affine variety, and the SBI long exact sequence makes the reduction transparent.

Check your understanding Beginner

Exercise (easy, multiple choice).

The cyclic action $τ_{n}$ on the Hochschild chain $C_{n} (A)$ in the Connes-Tsygan construction takes a tensor expression with entries $(a_{0}, a_{1}, \dots, a_{n})$ and produces:

A. The rotation with entries $(a_{n}, a_{0}, a_{1}, \dots, a_{n - 1})$ , with no sign B. The rotation with entries $(a_{n}, a_{0}, a_{1}, \dots, a_{n - 1})$ , scaled by $(- 1)^{n}$ C. The opposite rotation with entries $(a_{1}, a_{2}, \dots, a_{n}, a_{0})$ , with no sign D. The reversal with entries $(a_{0}, a_{n}, a_{n - 1}, \dots, a_{1})$ , scaled by $(- 1)^{n (n - 1) /2}$

Hint

The cyclic action rotates the last entry around to become the first, with a sign tracking the parity of the rotation. The sign convention ensures that $τ_{n}^{n + 1}$ equals the identity, matching the algebraic parity of the rotation.

Answer

B. The cyclic action takes entries $(a_{0}, a_{1}, \dots, a_{n})$ to the rotation $(a_{n}, a_{0}, a_{1}, \dots, a_{n - 1})$ scaled by $(- 1)^{n}$ . The sign $(- 1)^{n}$ is essential: it ensures that $τ_{n}^{n + 1}$ equals the identity automorphism of $C_{n} (A)$ (so $τ_{n}$ has order dividing $n + 1$ in $Aut (C_{n})$ ), and it makes the cyclic differential $1 - τ$ compatible with the Hochschild differential $b$ in the sense that they form a bicomplex. Option A is the same rotation without the sign, which gives the wrong compatibility with $b$ . Option C rotates the wrong direction (first to last instead of last to first). Option D is the reversal involution, which is a different operation appearing in the bar-complex study but not the cyclic action.

Exercise (easy, true-false).

True or false: cyclic homology $H C_{*} (A)$ and Hochschild homology $H H_{*} (A)$ are the same invariant for every associative $k$ -algebra $A$ .

Hint

The Connes SBI long exact sequence reads $\dots \to H H_{n} \to H C_{n} \to H C_{n - 2} \to H H_{n - 1} \to \dots$ . If $H C$ and $H H$ were the same invariant, this sequence would force $H C_{n - 2} = 0$ for all $n$ — which would make cyclic homology vanish identically and is false in general. For $A = k$ , for example, $H C_{2 n} (k) = k$ for all $n \geq 0$ .

Answer

False. Cyclic homology is a strict refinement of Hochschild homology: the cyclic symmetry data on the tensor factors produces additional invariants beyond the Hochschild groups. For the base field $A = k$ , all higher Hochschild groups vanish ( $H H_{n} (k) = 0$ for $n \geq 1$ ), but cyclic homology has $H C_{2 n} (k) = k$ for every $n \geq 0$ and $H C_{2 n + 1} (k) = 0$ , reflecting the cyclic action of $Z / (n + 1)$ on the one-dimensional chain $C_{n} (k) = k$ .

The Connes SBI long exact sequence packages the Hochschild-to-cyclic comparison; if the two invariants agreed, the periodicity operator $S$ would be zero, which it is not. Periodic cyclic homology $H P_{*} (A) = lim_{S} H C_{*} (A)$ is yet a finer refinement, recovering the algebraic de Rham cohomology for smooth commutative algebras in characteristic zero.

Exercise (easy, multiple choice).

For a smooth commutative $k$ -algebra $A$ in characteristic zero, the periodic cyclic homology $H P_{*} (A)$ is canonically identified with:

A. The Hochschild homology $H H_{*} (A)$ in even degrees B. The Kähler differentials $Ω_{A / k}^{*}$ as a $Z$ -graded module C. The algebraic de Rham cohomology $H_{dR}^{*} (A / k)$ in $Z /2$ -graded form D. The algebraic $K$ -theory $K_{*} (A)$

Hint

Connes' philosophy is that periodic cyclic homology is the noncommutative replacement for algebraic de Rham cohomology. For a smooth commutative algebra in characteristic zero, the HKR cyclic identification reduces $H P_{*} (A)$ to $⨁_{i} H_{dR}^{i} (A / k)$ with the parity grading.

Answer

C. For a smooth commutative $k$ -algebra $A$ in characteristic zero, the HKR cyclic identification (Connes 1985 IHES 62, Loday-Quillen 1984 Comment. Math. Helv. 59) gives $H P_{n} (A) ≅ ⨁_{i \equiv n (mod 2)} H_{dR}^{i} (A / k)$ . This is the algebraic de Rham cohomology of the affine variety $Spec (A)$ packaged in $Z /2$ -graded form (one term for even total degree, one term for odd total degree). The result is the precise sense in which periodic cyclic homology is Connes' "noncommutative de Rham cohomology."

Option A is wrong because $H P$ in even degrees is the periodic limit of $H C$ under $S$ , not the Hochschild homology directly. Option B names the underived Kähler differentials, which are the input to the HKR identification but not the same as $H P$ . Option D names algebraic $K$ -theory, which is related to cyclic homology via the cyclotomic trace but is a strictly different invariant.

Formal definition Intermediate+

Let $k$ be a field of characteristic zero and let $A$ be an associative $k$ -algebra with unit. The Hochschild chain complex $C_{*} (A) = (C_{n} (A) = A^{\otimes (n + 1)}, b)$ with the alternating-sum differential $b = \sum_{i = 0}^{n} (- 1)^{i} d_{i}$ (where $d_{i}$ multiplies the $i$ -th and $(i + 1)$ -th adjacent factors, with the cyclic wrap $d_{n} (a_{0} \otimes \dots \otimes a_{n}) = a_{n} a_{0} \otimes a_{1} \otimes \dots \otimes a_{n - 1}$ ) is the foundational input; see 04.03.20 for the bar-resolution construction and the identification $H H_{n} (A) = H_{n} (C_{*} (A))$ .

Definition (cyclic action). The cyclic action $τ_{n} : C_{n} (A) \to C_{n} (A)$ on the Hochschild chain in degree $n$ is the $k$ -linear map $$ \tau_n(a_0 \otimes a_1 \otimes \cdots \otimes a_n) := (-1)^n a_n \otimes a_0 \otimes a_1 \otimes \cdots \otimes a_{n-1}. $$ The sign $(- 1)^{n}$ is essential: it makes $τ_{n}^{n + 1} = id$ as a $k$ -linear automorphism of $C_{n} (A)$ , so $τ_{n}$ generates an action of the cyclic group $Z / (n + 1)$ on the chain $C_{n} (A)$ .

Definition (auxiliary operators). Let $b^{'} : C_{n} (A) \to C_{n - 1} (A)$ be the truncated Hochschild differential $b^{'} := \sum_{i = 0}^{n - 1} (- 1)^{i} d_{i}$ (omitting the cyclic-wrap term). Let $N : C_{n} (A) \to C_{n} (A)$ be the norm operator $N := \sum_{i = 0}^{n} τ_{n}^{i} = 1 + τ_{n} + τ_{n}^{2} + \dots + τ_{n}^{n}$ (the sum over the cyclic group). Let $s : C_{n} (A) \to C_{n + 1} (A)$ be the extra degeneracy $s (a_{0} \otimes \dots \otimes a_{n}) := 1 \otimes a_{0} \otimes \dots \otimes a_{n}$ (inserting the unit of $A$ in the first slot).

Definition (Connes-Tsygan bicomplex). The Connes-Tsygan bicomplex $C C_{**} (A)$ has $C C_{p, q} (A) := C_{q} (A)$ in bidegree $(p, q)$ for $p \geq 0$ , with horizontal differential $1 - τ : C C_{p, q} \to C C_{p - 1, q}$ in odd columns and $N : C C_{p, q} \to C C_{p - 1, q}$ in even columns, and vertical differential $b$ in odd columns and $- b^{'}$ in even columns. The bicomplex is depicted as a half-plane bicomplex extending infinitely to the right. The compatibility identities $b (1 - τ) = (1 - τ) b^{'}$ and $b^{'} N = N b$ ensure that the bicomplex differentials anticommute, so the total complex $$ CC_n(A) := \bigoplus_{p + q = n} CC_{p, q}(A), \qquad d_{\mathrm{tot}} := b + (1 - \tau) \text{ or } -b' + N \text{ (depending on column parity)} $$ is a well-defined chain complex. The cyclic homology of $A$ is $$ HC_n(A) := H_n(\mathrm{Tot}, CC_{**}(A)). $$

Definition (Connes' $(b, B)$ -bicomplex). The equivalent Connes coboundary $B : C_{n} (A) \to C_{n + 1} (A)$ is $B := (1 - τ_{n + 1}) s N_{n}$ . The $(b, B)$ -bicomplex has $C_{q} (A)$ in bidegree $(p, q)$ for $p \geq 0$ , with vertical differential $b$ and horizontal differential $B$ . The total complex of the $(b, B)$ -bicomplex computes the same cyclic homology $H C_{n} (A)$ , and the equivalence between the Connes-Tsygan bicomplex and the $(b, B)$ -bicomplex is one of the central technical results of cyclic-homology theory (Loday-Quillen 1984, Loday Cyclic Homology §2.5).

Definition (periodicity operator). The periodicity operator $S : H C_{n} (A) \to H C_{n - 2} (A)$ is the degree $- 2$ map induced by the column shift in the Connes-Tsygan bicomplex: erasing the leftmost two columns of $C C_{**}$ shifts the total complex by $2$ , and the inclusion of the truncated bicomplex into the original induces the periodicity map $S$ on cyclic homology.

Definition (Connes SBI long exact sequence). For every associative $k$ -algebra $A$ in characteristic zero, there is a long exact sequence $$ \cdots \to HH_n(A) \xrightarrow{I} HC_n(A) \xrightarrow{S} HC_{n-2}(A) \xrightarrow{B} HH_{n-1}(A) \xrightarrow{I} HC_{n-1}(A) \to \cdots $$ called the Connes SBI long exact sequence, where $I$ is the inclusion of Hochschild as the first column of the cyclic bicomplex, $S$ is the periodicity operator described above, and $B$ is the Connes coboundary connecting cyclic to Hochschild in shifted degree. The sequence is the long exact sequence of the short exact sequence of bicomplexes $0 \to (C C_{**})_{p \leq 0} \to C C_{**} \to (C C_{**})_{p \geq 2} [2] \to 0$ (the column-truncation sequence).

Definition (periodic cyclic homology). The periodic cyclic homology of $A$ is the inverse limit $$ HP_n(A) := \varprojlim_S HC_{n + 2k}(A) $$ under the periodicity operator $S$ , taken as $k \to \infty$ . Since $S$ shifts degree by $- 2$ , the limit $H P$ is $Z /2$ -graded: $H P_{0}$ collects the even-degree limit and $H P_{1}$ collects the odd-degree limit. Equivalently, $H P_{*} (A)$ is the homology of the fully extended bicomplex $C C_{**}^{per} (A)$ obtained by allowing $p$ to range over all integers (positive and negative), with the same column data.

Definition (HKR cyclic identification; Connes 1985 IHES 62, Loday-Quillen 1984). For a smooth commutative $k$ -algebra $A$ in characteristic zero, the cyclic homology and periodic cyclic homology are $$ HC_n(A) \cong \Omega^n_{A/k} / d\Omega^{n-1}{A/k} \oplus H^{n-2}{\mathrm{dR}}(A/k) \oplus H^{n-4}{\mathrm{dR}}(A/k) \oplus \cdots $$ $$ HP_n(A) \cong \bigoplus{i \equiv n \pmod 2} H^i_{\mathrm{dR}}(A/k), $$ where the sums run over indices $n, n - 2, n - 4, \dots$ until they hit $0$ or $1$ , and $H_{dR}^{*} (A / k) = H^{*} (Ω_{A / k}^{*}, d)$ is the algebraic de Rham cohomology of $A$ . The HKR cyclic identification is the central comparison between the noncommutative cyclic invariants and the classical commutative de Rham invariants.

Counterexamples to common slips

The cyclic action $τ_{n}$ carries the sign $(- 1)^{n}$ . Dropping the sign produces an action of $Z / (n + 1)$ as a $k$ -linear group action, but the resulting horizontal differential $1 - τ$ no longer anticommutes with the Hochschild differential $b$ , and the bicomplex construction fails.
Cyclic homology $H C_{*} (A)$ is not the same as the homology of the Hochschild complex modulo the cyclic action ( $A^{\otimes (n + 1)} / (1 - τ)$ ). The naive quotient computes only $H C_{0}$ correctly; higher cyclic homology requires the bicomplex with all the auxiliary operators ( $b$ , $b^{'}$ , $1 - τ$ , $N$ ).
The periodicity operator $S : H C_{n} \to H C_{n - 2}$ has degree $- 2$ . Writing $S$ as a degree $+ 2$ map (a common notational slip) inverts the SBI sequence direction and produces wrong long exact sequences.
For a commutative algebra in positive characteristic, the HKR cyclic identification fails (just as the underlying HKR theorem fails). The correct replacement involves the derived de Rham complex of Illusie and the conjugate filtration; see 04.03.21 for the analogous discussion at the Hochschild level.
The Connes SBI sequence has many sign-convention variants in the literature. Loday Cyclic Homology and Weibel Introduction to Homological Algebra use compatible conventions; Connes' original 1985 IHES paper uses a cohomological version (cyclic cohomology $H C^{*}$ instead of homology). Sign-convention slips do not change the structural content but produce noncommuting diagrams if mixed across sources.

Key theorem with proof Intermediate+

Theorem (Connes' SBI long exact sequence; Connes 1985 Publ. Math. IHES 62, Tsygan 1983, Loday-Quillen 1984 Comment. Math. Helv. 59). Let $k$ be a field of characteristic zero and let $A$ be an associative $k$ -algebra with unit. There is a natural long exact sequence $$ \cdots \to HH_n(A) \xrightarrow{I} HC_n(A) \xrightarrow{S} HC_{n-2}(A) \xrightarrow{B} HH_{n-1}(A) \xrightarrow{I} HC_{n-1}(A) \to \cdots $$ relating Hochschild homology $HH_(A) $an d cy c l i c h o m o l o g y$ HC_*(A) $, w i t h$ I $t h e in c l u s i o n o f H oc h sc hi l d a s t h e f i r s t co l u mn o f t h ecy c l i c bi co m pl e x,$ S $t h e p er i o d i c i t y o p er a t or o f d e g r ee$ -2 $, an d$ B $t h e C o nn esco b o u n d a r y o f d e g r ee$ -1 $. F u r t h er m or e, t h e bi co m pl e xu n d er l y in g cy c l i c h o m o l o g y s a t i s f i es t h eco m p a t ibi l i t y i d e n t i t i es$ b(1 - \tau) = (1 - \tau) b' $an d$ b' N = N b$, ensuring that the total complex is well-defined.*

Proof. The argument has three steps. Step 1 verifies the compatibility identities for the bicomplex differentials. Step 2 constructs the short exact sequence of bicomplexes whose long exact sequence is the SBI sequence. Step 3 identifies the connecting homomorphism with the Connes coboundary $B$ via the $(b, B)$ -bicomplex reformulation.

Step 1 — Bicomplex compatibility identities. The Connes-Tsygan bicomplex uses two distinct vertical differentials: the full Hochschild differential $b = \sum_{i = 0}^{n} (- 1)^{i} d_{i}$ in odd columns and the truncated $b^{'} = \sum_{i = 0}^{n - 1} (- 1)^{i} d_{i}$ in even columns. The horizontal differentials are $1 - τ$ in odd columns and the norm $N$ in even columns. The bicomplex requires four compatibility identities:

(i) $b^{2} = 0$ (standard Hochschild).

(ii) $(b^{'})^{2} = 0$ (the truncated complex is also a complex, with explicit contracting homotopy $s : C_{n} \to C_{n + 1}$ satisfying $b^{'} s + s b^{'} = id$ , making $(C_{*}, b^{'})$ acyclic in positive degrees).

(iii) $b (1 - τ) = (1 - τ) b^{'}$ . Direct computation: writing out $b (1 - τ) - (1 - τ) b^{'} = b - b τ - (b^{'} - τ b^{'})$ , the differences $b - b^{'}$ and $b τ - τ b^{'}$ collapse using the explicit formula for the cyclic-wrap face map $d_{n}$ and the cyclic action $τ$ ; standard exercise in cyclic algebra (Loday Cyclic Homology §2.5).

(iv) $b^{'} N = N b$ . Similar direct computation using the symmetry of the norm $N = \sum_{i} τ^{i}$ under the cyclic action; the proof relies on the fact that $τ$ acts on the truncated and full complexes with the same algebraic shifts up to a sign that gets absorbed in the norm.

The compatibility identities (iii) and (iv) ensure that the bicomplex differentials anticommute, so $d_{tot}^{2} = 0$ on the total complex $Tot C C_{**} (A) = ⨁_{p + q = n} C C_{p, q} (A)$ .

Step 2 — Short exact sequence of bicomplexes. Define the column-truncation short exact sequence $$ 0 \to CC^{(2)}{}(A)[2] \to CC_{}(A) \to (CC{}(A))^{(0)} \to 0 $$ where $CC^{(2)}_{}(A) $i s t h es u b - bi co m pl e x o b t ain e d b y * * r e m o v in g t h e f i r s ttw oco l u mn s * * (o n l y co l u mn s$ p \ge 2 $), s hi f t e d in d e g r ee b y$ +2 $t oco m p e n s a t e, an d$ (CC_{}(A))^{(0)} $i s t h e q u o t i e n t co n t ainin g o n l y co l u mn s$ p = 0, 1$ (the first two columns). The short exact sequence of total complexes is $$ 0 \to \mathrm{Tot}, CC^{(2)}_{}(A)[2] \to \mathrm{Tot}, CC_{}(A) \to \mathrm{Tot}, (CC_{}(A))^{(0)} \to 0. $$

The associated long exact sequence reads $$ \cdots \to HC_{n-2}(A) \xrightarrow{S} HC_n(A) \to H_n(\mathrm{Tot}, (CC_{**})^{(0)}) \to HC_{n-3}(A) \to \cdots $$ where $S$ is the degree- $+ 2$ inclusion induced by the column shift; here we recover the periodicity operator. (Equivalently, working with $S : H C_{n} \to H C_{n - 2}$ of degree $- 2$ , the corresponding cohomological direction in the SBI sequence is the one given in the theorem statement; the two conventions differ only by a relabelling of indices.)

Step 3 — Identifying the quotient with Hochschild homology. The quotient bicomplex $(C C_{**} (A))^{(0)}$ consists of the first two columns of the cyclic bicomplex: the column $p = 0$ is $(C_{*} (A), b)$ (the standard Hochschild complex), and the column $p = 1$ is $(C_{*} (A), b^{'})$ (the truncated complex). The horizontal differential $1 - τ : (C_{*}, b^{'}) \to (C_{*}, b)$ connects them, but the truncated complex $(C_{*}, b^{'})$ is acyclic (Step 1 part (ii)), so the total homology of the two-column sub-bicomplex collapses to $$ H_n(\mathrm{Tot}, (CC_{**}(A))^{(0)}) \cong HH_n(A) = H_n(C_*(A), b). $$

Substituting into the long exact sequence and using the identification $H_{n} (Tot (C C_{**})^{(0)}) = H H_{n} (A)$ , the connecting homomorphism is the Connes coboundary $B : H C_{n - 2} (A) \to H H_{n - 1} (A)$ . The full SBI sequence reads $$ \cdots \to HH_n(A) \xrightarrow{I} HC_n(A) \xrightarrow{S} HC_{n-2}(A) \xrightarrow{B} HH_{n-1}(A) \to \cdots $$ where:

$I$ is the inclusion of Hochschild into cyclic, induced by the inclusion of the first column $p = 0$ of the cyclic bicomplex as the standard Hochschild complex.
$S$ is the periodicity operator of degree $- 2$ , induced by the column-shift in the bicomplex (or equivalently the connecting homomorphism in the column-truncation short exact sequence).
$B$ is the Connes coboundary of degree $- 1$ , given explicitly by $B = (1 - τ) s N$ on the chain level, where $s$ is the extra degeneracy and $N$ is the norm.

The naturality of the SBI sequence in $A$ follows from the naturality of all the constituent operators ( $b$ , $b^{'}$ , $τ$ , $N$ , $s$ ) under $k$ -algebra homomorphisms. $□$

Bridge. The Connes SBI long exact sequence builds the bridge from the noncommutative Hochschild framework to the cyclic-homology framework that refines it, and the foundational reason the construction works is the existence of a cyclic action $τ_{n}$ on the Hochschild chain $C_{n} (A) = A^{\otimes (n + 1)}$ compatible with the Hochschild differential $b$ in the precise sense given by the bicomplex compatibility identities. The bridge is the column-truncation short exact sequence of bicomplexes, whose long exact sequence packages the Hochschild-to-cyclic comparison and identifies the connecting homomorphism with the Connes coboundary $B$ . Putting these together, the SBI sequence identifies cyclic homology as the periodic refinement of Hochschild homology, with the periodicity operator $S$ governing the recursive structure that defines the periodic limit $H P_{*} (A) = lim_{S} H C_{*} (A)$ .

This pattern appears again in 04.03.20 (Hochschild homology and cohomology), where the Hochschild chain complex is constructed and the SBI sequence is forward-referenced as the cyclic refinement; in 04.03.21 (HKR theorem), where the Hochschild identification with Kähler differentials lifts via SBI to the cyclic identification with the algebraic de Rham cohomology in the periodic limit; and forward to topological cyclic homology $TC$ (Bökstedt-Hsiang-Madsen 1993, Nikolaus-Scholze 2018), where the cyclic refinement is upgraded to the spectrum-level cyclotomic structure that serves as the receiver of the cyclotomic trace from algebraic $K$ -theory. The central insight is that cyclic homology is the universal noncommutative refinement of Hochschild homology that incorporates the cyclic-symmetry data, and the SBI long exact sequence is the algebraic shadow of the $S^{1}$ -equivariant structure on the topological Hochschild homology spectrum.

Exercises Intermediate+

Exercise 1 (easy, short-answer).

Verify that the cyclic action $τ_{n} (a_{0} \otimes a_{1} \otimes \dots \otimes a_{n}) = (- 1)^{n} a_{n} \otimes a_{0} \otimes \dots \otimes a_{n - 1}$ on $C_{n} (A) = A^{\otimes (n + 1)}$ satisfies $τ_{n}^{n + 1} = id$ as a $k$ -linear automorphism. Conclude that $τ_{n}$ generates an action of the cyclic group $Z / (n + 1)$ on $C_{n} (A)$ .

Hint

Compute $τ_{n}^{n + 1}$ by tracking both the rotation of the tensor factors and the cumulative sign $(- 1)^{n}$ at each application of $τ_{n}$ . The rotation $τ_{n}^{n + 1}$ returns each factor to its original position, and the cumulative sign is $((- 1)^{n})^{n + 1} = (- 1)^{n (n + 1)} = + 1$ since $n (n + 1)$ is always even.

Answer

Apply $τ_{n}$ exactly $n + 1$ times. After one application: $τ_{n} (a_{0} \otimes \dots \otimes a_{n}) = (- 1)^{n} a_{n} \otimes a_{0} \otimes \dots \otimes a_{n - 1}$ . After two applications: $τ_{n}^{2} = (- 1)^{2 n} a_{n - 1} \otimes a_{n} \otimes a_{0} \otimes \dots \otimes a_{n - 2}$ . By induction, after $k$ applications $τ_{n}^{k} (a_{0} \otimes \dots \otimes a_{n}) = (- 1)^{k n} a_{n - k + 1} \otimes a_{n - k + 2} \otimes \dots \otimes a_{n} \otimes a_{0} \otimes \dots \otimes a_{n - k}$ (indices read modulo $n + 1$ ).

For $k = n + 1$ : the rotation $(a_{0} \otimes \dots \otimes a_{n}) \mapsto (a_{0} \otimes \dots \otimes a_{n})$ returns to the identity (every factor has cycled through $n + 1$ positions and is back at its origin), and the cumulative sign is $(- 1)^{(n + 1) n} = (- 1)^{n (n + 1)}$ . Since $n (n + 1)$ is the product of two consecutive integers, it is always even, so $(- 1)^{n (n + 1)} = + 1$ . Hence $τ_{n}^{n + 1} = id$ as a $k$ -linear automorphism of $C_{n} (A)$ .

The cyclic action therefore factors through the quotient $Z / (n + 1) = Z / ⟨ τ_{n}^{n + 1} ⟩$ , and $τ_{n}$ generates a $Z / (n + 1)$ -action on $C_{n} (A)$ . The sign $(- 1)^{n}$ is exactly what is needed to make this work: dropping it would still give a $Z / (n + 1)$ action as a permutation of tensor factors, but the chain-complex compatibility with the Hochschild differential $b$ requires the signed version. Rubric: full credit for the induction, the rotation tracking, the sign computation, and the conclusion that $τ_{n}$ generates a $Z / (n + 1)$ -action.

Exercise 2 (easy, short-answer).

Compute the cyclic homology $H C_{*} (k)$ of the base field $A = k$ over itself, with $char k = 0$ . Use the result to verify the Connes SBI long exact sequence in this case.

Hint

For $A = k$ , the Hochschild chain in each degree is $C_{n} (k) = k^{\otimes (n + 1)} = k$ (one-dimensional). The Hochschild differential $b : C_{n} \to C_{n - 1}$ on a one-dimensional chain is multiplication by the alternating sum of $1$ 's, which depends on parity. The cyclic action $τ_{n}$ acts as $(- 1)^{n}$ on the one-dimensional chain (signed identity). Compute the bicomplex carefully.

Answer

For $A = k$ , the Hochschild chain in each degree is $C_{n} (k) = k^{\otimes (n + 1)} = k$ (the tensor power of $k$ over itself is $k$ ). The Hochschild differential $b : C_{n} (k) \to C_{n - 1} (k)$ collapses each of the $n + 1$ face maps $d_{i}$ to the multiplication $k \otimes k \to k$ in the corresponding slot, which is the identity. The alternating sum $b = \sum_{i = 0}^{n} (- 1)^{i} = 1$ if $n$ is even and $0$ if $n$ is odd. Hence $b : C_{n} (k) \to C_{n - 1} (k)$ is the identity $k \to k$ for $n$ even and the zero map for $n$ odd. Reading off Hochschild homology: $H H_{0} (k) = k$ , $H H_{n} (k) = 0$ for $n \geq 1$ (already shown in 04.03.20).

Cyclic homology: the cyclic action $τ_{n}$ acts on the one-dimensional chain $C_{n} (k) = k$ as the scalar $(- 1)^{n}$ (multiplication by the sign $(- 1)^{n}$ ). Hence $1 - τ_{n}$ acts as $1 - (- 1)^{n} = 0$ for $n$ even, and $1 - (- 1)^{n} = 2$ for $n$ odd. The norm $N = \sum_{i = 0}^{n} τ_{n}^{i} = \sum_{i = 0}^{n} (- 1)^{in}$ : this equals $n + 1$ if $n$ is even (all signs $+ 1$ ) and $0$ if $n$ is odd (alternating signs sum to $0$ ).

Substituting into the Connes-Tsygan bicomplex with the operators above:

For $n$ even: vertical $b = id$ , horizontal $1 - τ = 0$ (between adjacent columns at this row), and norm $N = n + 1$ .
For $n$ odd: vertical $b = 0$ , horizontal $1 - τ = 2$ , and norm $N = 0$ .

The total complex $Tot C C_{**} (k)$ has $k$ -modules in each bidegree $(p, q)$ with the differentials above. Computing the homology directly (working through the bicomplex with the alternating zero/non-zero structure): $H C_{2 n} (k) = k$ for $n \geq 0$ and $H C_{2 n + 1} (k) = 0$ for $n \geq 0$ .

Verification of SBI for $A = k$ : substituting into $\dots \to H H_{n} (k) \to H C_{n} (k) \to H C_{n - 2} (k) \to H H_{n - 1} (k) \to \dots$ :

$n = 0$ : $H H_{0} = k \to H C_{0} = k \to H C_{- 2} = 0 \to H H_{- 1} = 0$ . The map $H H_{0} \to H C_{0}$ is an isomorphism (both are $k$ , sent by $I$ ).
$n = 2$ : $H H_{2} = 0 \to H C_{2} = k S H C_{0} = k B H H_{1} = 0$ . The map $S : H C_{2} \to H C_{0}$ is an isomorphism $k \to k$ , consistent with the exactness $im (I) = H C_{0} = ker (S followed by next stage)$ — but careful: at $H C_{0} = k$ , the map $H C_{0} \to H H_{- 1} = 0$ is zero, so $ker = H C_{0}$ , and we need $im (S : H C_{2} \to H C_{0}) = H C_{0}$ , i.e., $S$ surjective. The map $S$ is multiplication by an isomorphism scalar (modulo the bicomplex structure), so this holds.

The SBI sequence is consistent in this base case, with the periodicity operator $S$ realising the periodic $k \to k$ behaviour in even degrees and the vanishing in odd degrees. Rubric: full credit for the chain computation, the bicomplex differentials, the cyclic homology values, and the SBI verification.

Exercise 3 (medium, short-answer).

Verify the worked-example computation of $H C_{*} (k [x])$ over a field $k$ of characteristic zero using the HKR cyclic identification $H C_{n} (A) ≅ Ω_{A / k}^{n} / d Ω_{A / k}^{n - 1} \oplus H_{dR}^{n - 2} (A) \oplus H_{dR}^{n - 4} (A) \oplus \dots$ . Compute $H C_{0}$ , $H C_{1}$ , $H C_{2}$ , $H C_{3}$ , and $H C_{n}$ for $n \geq 4$ .

Hint

For $A = k [x]$ : $Ω^{0} = k [x]$ , $Ω^{1} = k [x] \cdot d x$ , $Ω^{n} = 0$ for $n \geq 2$ . The de Rham differential $d : Ω^{0} \to Ω^{1}$ is $f \mapsto f^{'} d x$ , surjective in characteristic zero. Hence $H_{dR}^{0} (k [x]) = k$ (the constants), $H_{dR}^{n} (k [x]) = 0$ for $n \geq 1$ .

Answer

Compute each $H C_{n} (k [x])$ via the HKR cyclic identification, using $Ω^{0} = k [x]$ , $Ω^{1} = k [x] \cdot d x$ , $Ω^{n} = 0$ for $n \geq 2$ , $H_{dR}^{0} (k [x]) = k$ , $H_{dR}^{n} (k [x]) = 0$ for $n \geq 1$ .

$H C_{0} (k [x]) = Ω^{0} / d Ω^{- 1} = Ω^{0} /0 = k [x]$ (the whole algebra; the empty sum on the right contributes nothing).
$H C_{1} (k [x]) = Ω^{1} / d Ω^{0} = k [x] \cdot d x / d (k [x]) = 0$ , since $d : k [x] \to k [x] \cdot d x$ is surjective in characteristic zero (every $g (x) d x$ has an antiderivative).
$H C_{2} (k [x]) = Ω^{2} / d Ω^{1} \oplus H_{dR}^{0} (k [x]) = 0/0 \oplus k = k$ .
$H C_{3} (k [x]) = Ω^{3} / d Ω^{2} \oplus H_{dR}^{1} (k [x]) = 0 \oplus 0 = 0$ .
$H C_{4} (k [x]) = 0 \oplus H_{dR}^{2} (k [x]) \oplus H_{dR}^{0} (k [x]) = 0 \oplus 0 \oplus k = k$ .
In general for $n \geq 2$ : $H C_{n} (k [x]) = ⨁_{i \geq 0, n - 2 - 2 i \geq 0} H_{dR}^{n - 2 - 2 i} (k [x])$ . The de Rham cohomology contributes only in degree $0$ , giving $H_{dR}^{0} = k$ if $n$ is even (so $n - 2 - 2 i = 0$ for $i = (n - 2) /2$ ) and $0$ if $n$ is odd.

Therefore: $H C_{2 n} (k [x]) = k$ for $n \geq 1$ , $H C_{2 n + 1} (k [x]) = 0$ for $n \geq 0$ , and $H C_{0} (k [x]) = k [x]$ .

Periodic cyclic homology: $H P_{0} (k [x]) = lim_{S} {H C_{0}, H C_{2}, H C_{4}, \dots} = lim {k [x], k, k, \dots} = k = H_{dR}^{0} (k [x])$ (the inverse limit picks up the stable copy $k$ ). Similarly $H P_{1} (k [x]) = 0 = H_{dR}^{1} (k [x])$ . This confirms Connes' principle: $H P_{*} (k [x])$ recovers the algebraic de Rham cohomology of $A_{k}^{1}$ in $Z /2$ -graded form. Rubric: full credit for each $H C_{n}$ , the inverse-limit computation, and the de Rham comparison.

Exercise 4 (medium, short-answer).

Compute the cyclic homology $H C_{*} (k [x_{1}, \dots, x_{d}])$ for the polynomial algebra in $d$ variables over a field $k$ of characteristic zero, using the HKR cyclic identification and the algebraic de Rham cohomology $H_{dR}^{*} (A_{k}^{d}) = k$ in degree zero and zero elsewhere.

Hint

For $A = k [x_{1}, \dots, x_{d}]$ , the Kähler differentials $Ω_{A / k}^{n}$ are free $A$ -modules of rank $(n d)$ for $0 \leq n \leq d$ and zero for $n > d$ . The algebraic de Rham cohomology $H_{dR}^{n} (A_{k}^{d}) = k$ if $n = 0$ and $0$ otherwise (the affine space is contractible algebraically in characteristic zero — Grothendieck's theorem on algebraic de Rham cohomology).

Answer

For $A = k [x_{1}, \dots, x_{d}]$ in characteristic zero, the algebraic de Rham cohomology is $H_{dR}^{0} (A) = k$ (the constants, since the de Rham differential is surjective onto exact forms in characteristic zero) and $H_{dR}^{n} (A) = 0$ for $n \geq 1$ (Grothendieck's algebraic Poincaré lemma in characteristic zero: affine space has vanishing algebraic de Rham cohomology in positive degrees).

Substituting into the HKR cyclic formula $H C_{n} (A) ≅ Ω_{A / k}^{n} / d Ω_{A / k}^{n - 1} \oplus H_{dR}^{n - 2} (A) \oplus H_{dR}^{n - 4} (A) \oplus \dots$ :

$H C_{0} (A) = Ω^{0} /0 = k [x_{1}, \dots, x_{d}] = A$ (the whole polynomial algebra).
For $n \geq 1$ : the first summand $Ω^{n} / d Ω^{n - 1}$ is the cokernel of the de Rham differential, which by the algebraic Poincaré lemma equals $H_{dR}^{n} (A) = k$ for $n = 0$ (already used) and $0$ for $n \geq 1$ . So $Ω^{n} / d Ω^{n - 1} = 0$ for $n \geq 1$ . The remaining summands are $H_{dR}^{n - 2} (A) \oplus H_{dR}^{n - 4} (A) \oplus \dots$ , which contribute $k$ if and only if some $n - 2 j = 0$ , i.e., $n$ is even.

Therefore: $H C_{0} (A) = k [x_{1}, \dots, x_{d}]$ , $H C_{2 n} (A) = k$ for $n \geq 1$ , $H C_{2 n + 1} (A) = 0$ for $n \geq 0$ .

Periodic cyclic homology: $H P_{0} (A) = lim_{S} {H C_{0}, H C_{2}, H C_{4}, \dots} = k = H_{dR}^{0} (A)$ , $H P_{1} (A) = 0 = H_{dR}^{*} (A)$ in odd degrees. The result is independent of $d$ (the dimension of the affine space) because the algebraic de Rham cohomology of affine space vanishes in positive degrees: $Spec (k [x_{1}, \dots, x_{d}]) = A_{k}^{d}$ is algebraically contractible in characteristic zero. Rubric: full credit for the de Rham input, the HKR cyclic computation, the dimension-independence observation, and the periodic identification.

Exercise 5 (medium, short-answer).

State the Connes-Loday-Quillen HKR cyclic identification for a smooth commutative $k$ -algebra $A$ in characteristic zero, and use it to compute $H P_{*} (O (E))$ for $E$ an elliptic curve over $k$ (or, equivalently, for $A = O_{aff} (E ∖ {O})$ the affine coordinate ring of $E$ minus a point — note this is a different question from the projective $H P_{*} (E)$ ).

Hint

For the elliptic curve $E$ over $k$ in characteristic zero, the algebraic de Rham cohomology of the affine open $E ∖ {O}$ (the elliptic curve minus a marked point) is $H_{dR}^{0} = k$ , $H_{dR}^{1} = k^{2 g} \cdot additional = k^{2}$ (related to the Weierstrass differentials but on the punctured curve), and higher de Rham vanishes. (The exact dimensions depend on whether the puncture is included; this exercise is informal regarding the exact dimensions and tests structural understanding of the HP formula rather than the precise count.)

Answer

Connes-Loday-Quillen HKR cyclic identification. For a smooth commutative $k$ -algebra $A$ in characteristic zero (Connes 1985 IHES 62, Loday-Quillen 1984 Comment. Math. Helv. 59): $$ HC_n(A) \cong \Omega^n_{A/k} / d\Omega^{n-1}{A/k} \oplus H^{n-2}{\mathrm{dR}}(A/k) \oplus H^{n-4}{\mathrm{dR}}(A/k) \oplus \cdots, $$ where $H^*{\mathrm{dR}}(A/k) $i s t h e a l g e b r ai c d e R ham co h o m o l o g y, d e f in e d a s t h eco h o m o l o g y o f t h e d e R ham co m pl e x$ (\Omega^*_{A/k}, d) $, an d t h es u m r u n so v er$ n, n - 2, n - 4, \ldots $u n t i l i t hi t s$ 0 $or$ 1$. The periodic limit is $$ HP_n(A) \cong \bigoplus_{i \equiv n \pmod 2} H^i_{\mathrm{dR}}(A/k). $$

Application to elliptic curve. Let $E$ be a smooth projective elliptic curve over $k$ (with $char k = 0$ ). The algebraic de Rham cohomology of $E$ is concentrated in degrees $0, 1, 2$ : $H_{dR}^{0} (E / k) = k$ , $H_{dR}^{1} (E / k) = k^{2}$ (Weierstrass differentials of the first and second kind), $H_{dR}^{2} (E / k) = k$ (by Serre duality and the elliptic-curve canonical bundle being $O_{E}$ ).

For the scheme-theoretic periodic cyclic homology of $E$ (using the scheme-theoretic extension of the HKR cyclic identification, Caldararu 2003 Adv. Math. 194 and Weibel 1996): $$ HP_0(E) = H^0_{\mathrm{dR}}(E) \oplus H^2_{\mathrm{dR}}(E) = k \oplus k = k^2, \qquad HP_1(E) = H^1_{\mathrm{dR}}(E) = k^2. $$ Total dimension $\sum dim H P_{*} (E) = 4 = \sum_{i} dim H_{dR}^{i} (E)$ , recovering the entire algebraic de Rham cohomology of the elliptic curve in $Z /2$ -graded form.

The result matches the topological prediction: $H P_{*} (E)$ is the algebraic shadow of the de Rham cohomology of $E (C) ≅ S^{1} \times S^{1}$ (the complex torus), with $H^{*} (S^{1} \times S^{1}; k) = k \oplus k^{2} \oplus k$ in degrees $0, 1, 2$ collapsing to $k^{2} \oplus k^{2}$ in $Z /2$ -graded form. Rubric: full credit for the HKR cyclic identification statement, the de Rham input for the elliptic curve, the periodic cyclic homology computation, and the comparison with the topological torus.

Exercise 6 (medium, short-answer).

Define the periodic cyclic homology $H P_{*} (A)$ as the inverse limit $lim_{S} H C_{*} (A)$ under the periodicity operator $S$ , and explain why $H P_{*}$ is $Z /2$ -graded. Compute $H P_{*} (A)$ for $A = k$ (the base field).

Hint

The periodicity operator $S : H C_{n} \to H C_{n - 2}$ has degree $- 2$ . The inverse limit under $S$ stabilises the ${H C_{n}, H C_{n + 2}, H C_{n + 4}, \dots}$ tower indexed by parity. Hence $H P_{*}$ has only two independent components: $H P_{0} = lim_{S} {H C_{0}, H C_{2}, H C_{4}, \dots}$ and $H P_{1} = lim_{S} {H C_{1}, H C_{3}, H C_{5}, \dots}$ .

Answer

Definition. The periodic cyclic homology is $H P_{n} (A) := lim_{k} H C_{n + 2 k} (A)$ under the periodicity operator $S : H C_{n + 2 k} \to H C_{n + 2 k - 2}$ . The inverse limit is taken over the tower $\dots S H C_{n + 4} S H C_{n + 2} S H C_{n}$ . Equivalently, $H P_{*} (A)$ is the homology of the fully extended bicomplex $C C_{**}^{per} (A)$ obtained by allowing the column index $p$ to range over all integers (positive and negative), with the same column data as the half-plane cyclic bicomplex.

Why $Z /2$ -graded. The periodicity operator $S$ shifts degree by $- 2$ , so the inverse limit identifies $H C_{n}$ with $H C_{n - 2}$ in the stable regime. The independent components of $H P_{*}$ are therefore indexed by the parity of $n$ : $$ HP_0(A) = \varprojlim_S {HC_0, HC_2, HC_4, \ldots}, \qquad HP_1(A) = \varprojlim_S {HC_1, HC_3, HC_5, \ldots}. $$ All higher even and odd indices are forced to equal $H P_{0}$ and $H P_{1}$ respectively by the periodicity of $S$ , so the $Z$ -graded structure collapses to a $Z /2$ -graded structure with two components.

Computation for $A = k$ . From Exercise 2 (or 04.03.20 Exercise 8), $H C_{2 n} (k) = k$ for $n \geq 0$ and $H C_{2 n + 1} (k) = 0$ for $n \geq 0$ . The periodicity maps $S : H C_{2 n + 2} (k) \to H C_{2 n} (k)$ are isomorphisms $k \to k$ for all $n \geq 0$ (the period- $2$ pattern is stable from degree $0$ onwards). Hence: $$ HP_0(k) = \varprojlim_S {HC_0, HC_2, HC_4, \ldots} = \varprojlim_S {k, k, k, \ldots} = k, $$ $$ HP_1(k) = \varprojlim_S {HC_1, HC_3, HC_5, \ldots} = \varprojlim_S {0, 0, 0, \ldots} = 0. $$ Comparison with the de Rham picture: for $A = k$ (the base field as a $k$ -algebra), the Kähler differentials are $Ω_{k / k}^{0} = k$ and $Ω_{k / k}^{n} = 0$ for $n \geq 1$ , so $H_{dR}^{0} (k / k) = k$ and $H_{dR}^{n} (k / k) = 0$ for $n \geq 1$ . The HKR cyclic identification predicts $H P_{0} (k) = H_{dR}^{0} (k) = k$ and $H P_{1} (k) = 0$ , matching the direct computation. Rubric: full credit for the definition, the $Z /2$ -grading explanation, the explicit limit computation, and the de Rham comparison.

Exercise 7 (hard, short-answer).

State the Loday-Quillen-Tsygan theorem identifying the primitive part of the Lie-algebra homology $H_{*} (gl (A); k)$ of the infinite general linear Lie algebra $gl (A) = lim_{n} gl_{n} (A)$ with the cyclic homology $H C_{*- 1} (A)$ of the associative algebra $A$ . Use it to compute the cyclic homology of $A = k$ and verify against Exercise 2.

Hint

The Loday-Quillen-Tsygan theorem (Loday-Quillen 1984 Comment. Math. Helv. 59, Tsygan 1983 Uspekhi 38) states $Prim H_{*} (gl (A); k) ≅ H C_{*- 1} (A)$ for $A$ an associative $k$ -algebra in characteristic zero, where $Prim$ denotes the primitive part of the cocommutative Hopf algebra $H_{*} (gl (A); k)$ . For $A = k$ , $gl (k) = lim_{n} gl_{n} (k)$ is the infinite general linear Lie algebra, whose Lie-algebra homology is computed in terms of cyclic homology via this theorem.

Answer

Loday-Quillen-Tsygan theorem (Loday-Quillen 1984 Comment. Math. Helv. 59, 565-591; Tsygan 1983 Uspekhi Mat. Nauk 38, 217-218). Let $A$ be an associative $k$ -algebra in characteristic zero. The Lie-algebra homology $H_{*} (gl (A); k)$ of the infinite general linear Lie algebra $gl (A) = lim_{n} gl_{n} (A)$ (with the limit taken via the upper-left-corner inclusion $gl_{n} ↪ gl_{n + 1}$ ) is a cocommutative graded Hopf algebra under the direct-sum operation $\oplus$ on matrices. The primitive part (in the Hopf-algebra sense — the elements $x$ with $Δ x = x \otimes 1 + 1 \otimes x$ ) is canonically identified with the cyclic homology of $A$ shifted by $1$ : $$ \mathrm{Prim}, H_*(\mathfrak{gl}(A); k) \cong HC_{-1}(A), $$ with the full Hopf-algebra structure recovered as the free graded-symmetric algebra on the primitive part: $H_(\mathfrak{gl}(A); k) \cong \mathrm{Sym}, HC_{*-1}(A) $(M i l n or - M oor e f or coco mm u t a t i v e g r a d e d H o p f a l g e b r a s in c ha r a c t er i s t i cz er o) . T h e p r oo f u ses t h e ba r co m pl e x o f$ \mathfrak{gl}(A)$, the symmetric-group decomposition of the tensor algebra, and Connes' cyclic-action data.

Application to $A = k$ . For $A = k$ (the base field), $gl (k) = lim_{n} gl_{n} (k)$ is the infinite general linear Lie algebra over $k$ . The classical computation of $H_{*} (gl (k); k)$ is the stable Lie-algebra homology of $gl$ , which by the Milnor-Moore theorem (1965 Ann. Math. 81) factors as the symmetric algebra on the primitive part. The primitive part is generated by elements in odd cohomological degree (the Chern characters), giving $Prim H_{*} (gl (k); k) = k ⟨ p_{1}, p_{3}, p_{5}, \dots ⟩$ with one generator $p_{2 j + 1}$ in degree $2 j + 1$ for each $j \geq 0$ . By LQT, this primitive part equals $H C_{*- 1} (k)$ , so $H C_{2 j} (k) = k$ for $j \geq 0$ (the generator $p_{2 j + 1}$ in degree $2 j + 1$ corresponds to $H C_{2 j} (k) = k$ after shifting by $- 1$ ) and $H C_{2 j + 1} (k) = 0$ for $j \geq 0$ .

This matches the direct computation from Exercise 2 / 04.03.20: $H C_{2 n} (k) = k$ for $n \geq 0$ and $H C_{2 n + 1} (k) = 0$ for $n \geq 0$ . The LQT theorem provides an independent path to the same answer via stable Lie-algebra homology, illustrating the bridge between cyclic homology and the cohomology of infinite-dimensional matrix Lie algebras. Rubric: full credit for the LQT statement, the Hopf-algebra primitive interpretation, the stable Lie-algebra computation, and the verification against the direct cyclic-homology calculation.

Exercise 8 (hard, short-answer).

State the cyclotomic-trace framework relating algebraic $K$ -theory to topological cyclic homology (Bökstedt-Hsiang-Madsen 1993 Invent. Math. 111, Nikolaus-Scholze 2018 Acta Math. 221), and explain the role of periodic cyclic homology $H P_{*}$ as the rational-degeneration target via Goodwillie's theorem (1986 Ann. Math. 124).

Hint

The cyclotomic trace is a natural transformation $K (A) \to TC (A)$ from algebraic $K$ -theory to topological cyclic homology. Goodwillie's theorem (1986) is the rational result: rationally, the relative K-theory of a nilpotent extension equals the relative negative cyclic homology, and the cyclotomic trace recovers this comparison. The chain of refinements is $H H \to H C \to H C^{-} \to H P \to THH \to TC$ — each refinement adds more structure.

Answer

Cyclotomic-trace framework. For an $E_{\infty}$ -ring spectrum $A$ (or an ordinary associative $k$ -algebra), the topological Hochschild homology $THH (A)$ is the spectrum-level lift of Hochschild homology: $THH (A) := A \land_{A \land A^{op}}^{L} A$ in the $\infty$ -category of $A$ -bimodule spectra. $THH (A)$ carries a canonical $S^{1}$ -action (the cyclic-symmetry lifted to the spectrum level), and the further cyclotomic-spectrum structure (Bökstedt-Hsiang-Madsen 1993 Invent. Math. 111, reformulated by Nikolaus-Scholze 2018 Acta Math. 221 in terms of genuine $S^{1}$ -equivariant homotopy theory) packages the $Z / p$ -Tate-fixed-point data for all primes $p$ . The topological cyclic homology $TC (A)$ is the universal cyclotomic-spectrum-valued invariant; in modern terms, $TC (A)$ is the limit of $THH (A)$ along the cyclotomic structure maps.

The cyclotomic trace is the natural transformation $K (A) \to TC (A)$ from algebraic $K$ -theory to topological cyclic homology. The trace was constructed in stages: Bökstedt (1985) defined $THH$ and the trace $K \to THH$ ; Bökstedt-Hsiang-Madsen 1993 promoted the trace to land in $TC$ using the cyclotomic structure; Dundas-Goodwillie-McCarthy (2013) extended the trace to all connective ring spectra; Nikolaus-Scholze 2018 rebuilt the entire framework via genuine equivariant homotopy theory.

Goodwillie's theorem and the rational target. Goodwillie 1986 Ann. Math. 124 proved the rational comparison: for a nilpotent extension $A \to A_{0}$ (where the kernel is a nilpotent ideal), the rational relative algebraic K-theory $K_{*} (A, A_{0})_{Q}$ is canonically isomorphic to the relative negative cyclic homology $H C_{*}^{-} (A, A_{0})_{Q}$ . The proof uses the rational version of the cyclotomic trace and the fact that rationally, topological cyclic homology $TC_{*} (A)_{Q}$ is computed by the negative cyclic homology $H C_{*}^{-} (A)$ (the fixed points of $S$ acting on the periodic complex, dual to the periodic cyclic homology $H P_{*}$ ).

The chain of refinements relating these invariants is $$ HH_*(A) \xrightarrow{I} HC_*(A) \xrightarrow{} HC^-*(A) \xrightarrow{} HP*(A), $$ with $H C^{-}$ the negative cyclic homology (the $S$ -fixed-point version) and $H P$ the periodic cyclic homology (the $S$ -inverted version), connected by the norm map $Nm : H C^{-} \to H P$ whose cofibre is the cyclic homology $H C$ .

Periodic cyclic homology $H P_{*} (A)$ plays the role of the rational degeneration target in this picture: it is what one obtains by inverting the periodicity operator $S$ , and for a smooth commutative $k$ -algebra $A$ in characteristic zero, $H P_{*} (A)$ recovers the algebraic de Rham cohomology of $Spec (A)$ in $Z /2$ -graded form. Rationally, the cyclotomic trace $K (A)_{Q} \to TC (A)_{Q}$ factors through $H C_{*}^{-} (A)_{Q}$ (Goodwillie 1986), and for smooth $A$ in characteristic zero this further refines to the de Rham comparison via the HKR cyclic identification. The integral version (Bökstedt-Hsiang-Madsen, Nikolaus-Scholze) requires the cyclotomic-spectrum framework and is the input to modern $K$ -theory computations: Hesselholt-Madsen on $K (Z_{p})$ (1997 Topology 36), the Bhatt-Morrow-Scholze theory of prismatic cohomology (2019 Publ. Math. IHES 129), and the chromatic-homotopy-theoretic understanding of arithmetic. Rubric: full credit for the cyclotomic-trace statement, the Goodwillie rational comparison, the chain of refinements $H H \to H C \to H C^{-} \to H P$ , and the role of $H P$ as the rational target.

Lean formalization Intermediate+

lean_status: none — Mathlib has the KahlerDifferential package, the TensorProduct R M N underived tensor product, the partial RingTheory.HochschildCohomology skeleton, and basic chain-complex infrastructure in Mathlib.Algebra.Homology, but the full apparatus needed to state and prove the Connes' SBI long exact sequence and the HKR cyclic identification is not yet assembled. The intended formalisation reads schematically:

import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.RingTheory.TensorProduct
import Mathlib.RingTheory.Kaehler.Basic

namespace Codex.HomologicalAlgebra.CyclicHomology

variable (k : Type*) [Field k] [CharZero k]
variable (A : Type*) [Ring A] [Algebra k A]

/-- The cyclic action τ_n on the Hochschild chain C_n(A) = A^⊗(n+1). -/
noncomputable def cyclicAction (n : ℕ) :
    Codex.HomologicalAlgebra.Hochschild.HochschildChain k A n →ₗ[k]
      Codex.HomologicalAlgebra.Hochschild.HochschildChain k A n := sorry

/-- The norm operator N = Σ τ^i. -/
noncomputable def normOperator (n : ℕ) :
    Codex.HomologicalAlgebra.Hochschild.HochschildChain k A n →ₗ[k]
      Codex.HomologicalAlgebra.Hochschild.HochschildChain k A n := sorry

/-- The Connes coboundary B = (1 - τ) s N. -/
noncomputable def connesCoboundary (n : ℕ) :
    Codex.HomologicalAlgebra.Hochschild.HochschildChain k A n →ₗ[k]
      Codex.HomologicalAlgebra.Hochschild.HochschildChain k A (n + 1) := sorry

/-- The Connes-Tsygan bicomplex CC_{**}(A). -/
noncomputable def connesBicomplex :
    HomologicalComplex₂ (ModuleCat k) (ComplexShape.up ℕ) (ComplexShape.up ℕ) := sorry

/-- Cyclic homology HC_n(A) := H_n(Tot CC_{**}(A)). -/
noncomputable def HC (n : ℕ) : Type* := sorry

/-- The Connes SBI long exact sequence:
    ... → HH_n(A) → HC_n(A) → HC_{n-2}(A) → HH_{n-1}(A) → ...  -/
theorem connes_SBI :
    ∀ (n : ℕ), True  -- placeholder; the LES requires the long-exact-sequence
                     -- machinery on bicomplexes
    := sorry

/-- Periodic cyclic homology HP_n(A) := lim_S HC_*(A), ℤ/2-graded. -/
noncomputable def HP (n : Fin 2) : Type* := sorry

/-- HKR cyclic identification: for smooth commutative A in char 0,
    HC_n(A) ≅ Ω^n_{A/k}/dΩ^{n-1} ⊕ H^{n-2}_dR ⊕ H^{n-4}_dR ⊕ ...  -/
theorem HKR_cyclic [CommRing A] [Algebra k A] [Algebra.FormallySmooth k A] (n : ℕ) :
    True  -- placeholder; the de Rham cohomology side requires the algebraic
          -- de Rham complex package
    := sorry

/-- Periodic HKR: HP_n(A) ≅ ⊕_{i ≡ n (2)} H^i_dR(A/k). -/
theorem HKR_periodic [CommRing A] [Algebra k A] [Algebra.FormallySmooth k A] :
    True := sorry

end Codex.HomologicalAlgebra.CyclicHomology

The proof gap is substantive at multiple levels. The Hochschild-homology API itself is not in Mathlib (as noted in 04.03.20, lean_status: none), so the cyclic-action data and the SBI sequence currently have nothing to attach to on the Hochschild side. The construction of the cyclic action $τ_{n}$ with the sign $(- 1)^{n}$ requires the explicit cyclic-group representation on $A^{\otimes (n + 1)}$ together with the compatibility identities $b (1 - τ) = (1 - τ) b^{'}$ and $b^{'} N = N b$ , each of which requires the truncated bar complex $(C_{*}, b^{'})$ and the norm operator $N$ as auxiliary data.

The Connes-Tsygan bicomplex is a half-plane bicomplex with two distinct vertical differentials (full Hochschild $b$ in odd columns, truncated $b^{'}$ in even columns) and two distinct horizontal differentials ( $1 - τ$ in odd columns, $N$ in even columns); Mathlib's HomologicalComplex and HomologicalComplex₂ infrastructure supports general bicomplexes but the specific cyclic bicomplex is not packaged. The Connes SBI long exact sequence requires the long-exact-sequence-of-bicomplexes machinery applied to the column-truncation short exact sequence, currently absent. The periodic cyclic homology $H P_{n} (A) = lim_{S} H C_{n} (A)$ as the inverse limit under the periodicity operator $S$ requires the inverse-limit construction on chain complexes together with the recognition that the result is $Z /2$ -graded.

The HKR cyclic identification with algebraic de Rham cohomology is gated on both the underlying HKR theorem (04.03.21, also lean_status: none) and the algebraic de Rham complex of a commutative $k$ -algebra, currently not packaged in Mathlib in the form needed. Topological cyclic homology $TC (A)$ and the cyclotomic trace $K (A) \to TC (A)$ require the cyclotomic-spectrum structure on $THH (A)$ , the genuine $S^{1}$ -equivariant homotopy theory of Nikolaus-Scholze 2018 Acta Math. 221, and the modern $\infty$ -categorical reformulation — all far beyond the current Mathlib infrastructure. Each component is formalisable in principle but requires substantial coordinated infrastructure that the Mathlib homological-algebra, derived-category, and equivariant-homotopy projects have not yet completed as of 2026.

Advanced results Master

Theorem (Connes' SBI long exact sequence; Connes 1985 Publ. Math. IHES 62). Let $k$ be a field of characteristic zero and let $A$ be an associative $k$ -algebra with unit. There is a natural long exact sequence $$ \cdots \to HH_n(A) \xrightarrow{I} HC_n(A) \xrightarrow{S} HC_{n-2}(A) \xrightarrow{B} HH_{n-1}(A) \xrightarrow{I} HC_{n-1}(A) \to \cdots, $$ the Connes SBI sequence, where $I$ is the inclusion of Hochschild as the first column of the cyclic bicomplex, $S$ is the periodicity operator of degree $- 2$ , and $B$ is the Connes coboundary of degree $- 1$ .

The SBI sequence is the algebraic shadow of the $S^{1}$ -equivariant homotopy fibre sequence relating $THH$ , $TC$ , and their related cyclotomic invariants. Connes' original 1985 paper Noncommutative differential geometry ^{[source pending]} introduced the cyclic-cohomology version of SBI together with the bicomplex framework and the $(b, B)$ -formulation; Tsygan 1983 Uspekhi 38 ^{[source pending]} gave the independent cyclic-homology construction in the context of Lie-algebra homology of matrix Lie algebras; Loday-Quillen 1984 Comment. Math. Helv. 59 ^{[source pending]} proved the foundational comparison with the homology of $gl (A)$ that became known as the Loday-Quillen-Tsygan theorem. The modern treatment with full operadic and $\infty$ -categorical infrastructure is in Loday Cyclic Homology (Springer Grundlehren 301, 1992; 2nd ed. 1998) ^{[source pending]} and Nikolaus-Scholze 2018 Acta Math. 221 ^{[source pending]}.

Theorem (HKR cyclic identification for smooth commutative algebras; Connes 1985 IHES 62, Loday-Quillen 1984). Let $k$ be a field of characteristic zero and let $A$ be a smooth commutative $k$ -algebra of finite type. Then $$ HC_n(A) \cong \Omega^n_{A/k} / d\Omega^{n-1}{A/k} \oplus H^{n-2}{\mathrm{dR}}(A/k) \oplus H^{n-4}{\mathrm{dR}}(A/k) \oplus \cdots, $$ and the periodic limit is $$ HP_n(A) \cong \bigoplus{i \equiv n \pmod 2} H^i_{\mathrm{dR}}(A/k), $$ where $H^{\mathrm{dR}}(A/k) = H^(\Omega^{A/k}, d) $i s t h e a l g e b r ai c d e R ham co h o m o l o g y o f$ A$.*

The HKR cyclic identification is the precise sense in which periodic cyclic homology is Connes' "noncommutative de Rham cohomology." The result lifts the Hochschild HKR identification $H H_{n} (A) ≅ Ω_{A / k}^{n}$ (04.03.21) to the cyclic level via the SBI long exact sequence: the Hochschild summand is replaced by its cokernel $Ω^{n} / d Ω^{n - 1}$ under the de Rham differential, and the recursive structure $H C_{n} = H H_{n} / im - from - H C_{n - 2}$ from SBI produces the chain of de Rham summands. The periodic limit collapses the recursive structure into the periodic de Rham cohomology in $Z /2$ -graded form. For a smooth proper $k$ -scheme $X$ in characteristic zero, the scheme-theoretic version reads $H P_{n} (X) ≅ ⨁_{i \equiv n (mod 2)} H_{dR}^{i} (X / k)$ (Weibel 1996 Proc. AMS 124, Caldararu 2003 Adv. Math. 194), recovering the entire algebraic de Rham cohomology of $X$ in periodic form.

Theorem (Loday-Quillen-Tsygan theorem; Loday-Quillen 1984 Comment. Math. Helv. 59, Tsygan 1983 Uspekhi 38). Let $A$ be an associative $k$ -algebra in characteristic zero. The Lie-algebra homology $H_(\mathfrak{gl}(A); k) $o f t h e in f ini t e g e n er a l l in e a r L i e a l g e b r a$ \mathfrak{gl}(A) = \varinjlim_n \mathfrak{gl}n(A)$ is a cocommutative graded Hopf algebra whose primitive part is canonically* $$ \mathrm{Prim}, H(\mathfrak{gl}(A); k) \cong HC_{-1}(A), $$ and the full Hopf algebra is the free graded-symmetric algebra on the primitive part: $H_(\mathfrak{gl}(A); k) \cong \mathrm{Sym}, HC_{-1}(A)$.

The Loday-Quillen-Tsygan theorem is the bridge from cyclic homology to the cohomology of stable matrix Lie algebras, and it provides an independent foundational role for cyclic homology that motivated Tsygan's 1983 introduction of the theory. The proof uses the bar complex of $gl (A)$ , the symmetric-group decomposition of the tensor algebra $T^{*} (A)$ via the Schur-Weyl correspondence, and Connes' cyclic-action data; the result is a precise instance of the Milnor-Moore theorem (1965 Ann. Math. 81) for cocommutative graded Hopf algebras in characteristic zero, with cyclic homology playing the role of the primitive generators. The theorem has been generalised in many directions: to operadic settings (Loday Cyclic Homology Ch. 13), to the equivariant context (Feigin-Tsygan 1985), and to the derived-stack and $\infty$ -categorical framework (Toën-Vezzosi 2011).

Theorem (Goodwillie's rational K-theory comparison; Goodwillie 1986 Ann. Math. 124). For a nilpotent extension of $k$ -algebras $A \to A_{0}$ (the kernel is a nilpotent ideal) in characteristic zero, the rational relative algebraic K-theory is canonically isomorphic to the relative negative cyclic homology: $$ K_*(A, A_0)\mathbb{Q} \cong HC^-{*-1}(A, A_0)_\mathbb{Q}. $$ The isomorphism is induced by the rationalised cyclotomic trace, and it identifies negative cyclic homology as the rational receiver of algebraic K-theory.

Goodwillie's theorem is the foundational rational result connecting algebraic K-theory and cyclic homology, and it organises the rational structure of $K$ -theory in terms of the more computable cyclic-homology invariants. The proof uses the rational version of the cyclotomic trace (Goodwillie 1985 Topology 24, 1986 Ann. Math. 124 ^{[source pending]}) and the fact that rationally, the periodic cyclic homology $H P$ and negative cyclic homology $H C^{-}$ together with the norm map $Nm : H C^{-} \to H P$ recover the topological cyclic homology $TC$ . The integral version is the cyclotomic-trace framework of Bökstedt-Hsiang-Madsen 1993 ^{[source pending]} and its modern reformulation by Nikolaus-Scholze 2018 ^{[source pending]}.

Theorem (topological cyclic homology and the cyclotomic trace; Bökstedt-Hsiang-Madsen 1993 Invent. Math. 111). For an $E_{\infty}$ -ring spectrum $A$ , the topological Hochschild homology $THH (A) := A \land_{A \land A^{op}}^{L} A$ is the spectrum-level lift of Hochschild homology. $THH (A)$ carries a canonical cyclotomic-spectrum structure, and the topological cyclic homology $TC (A)$ is the universal cyclotomic-spectrum-valued invariant. The cyclotomic trace $K (A) \to TC (A)$ is the natural transformation from algebraic K-theory to topological cyclic homology, and is the foundational tool of trace-method computations of algebraic K-theory.

The cyclotomic-trace framework is the integral spectrum-level lift of Goodwillie's rational theorem. Bökstedt 1985 ^{[source pending]} constructed $THH$ and the trace $K \to THH$ using the framework of functors with smash product; Bökstedt-Hsiang-Madsen 1993 Invent. Math. 111 ^{[source pending]} promoted the trace to land in $TC$ using the cyclotomic structure on $THH$ encoded by the $Z / p$ -Tate fixed points for all primes $p$ . Dundas-Goodwillie-McCarthy 2013 ^{[source pending]} extended the trace to all connective ring spectra, proving that the trace is an equivalence on relative K-theory of nilpotent extensions integrally (not just rationally — the integral Goodwillie theorem). The Nikolaus-Scholze 2018 Acta Math. 221 ^{[source pending]} reformulation rebuilds the cyclotomic structure via genuine $S^{1}$ -equivariant homotopy theory, presenting $TC$ as the limit of a clean $\infty$ -categorical diagram and identifying the cyclotomic trace with a universal property. The framework is the input to modern $K$ -theory computations: Hesselholt-Madsen 1997 Topology 36 on $K (Z_{p})$ via $TC$ , the Bhatt-Morrow-Scholze 2019 Publ. Math. IHES 129 theory of prismatic cohomology, and the chromatic-homotopy-theoretic understanding of arithmetic.

Theorem (Nikolaus-Scholze reformulation of topological cyclic homology; Nikolaus-Scholze 2018 Acta Math. 221). Topological cyclic homology $TC (A)$ is canonically the limit $$ \mathrm{TC}(A) = \mathrm{eq}!\left(\mathrm{THH}(A)^{hS^1} \rightrightarrows \prod_p \mathrm{THH}(A)^{tS^1}_p\right), $$ the equaliser of two natural maps from the homotopy $S^{1}$ -fixed points to the Tate construction over all primes. The cyclotomic trace $K (A) \to TC (A)$ is the universal map from algebraic K-theory to a cyclotomic-spectrum-valued invariant.

The Nikolaus-Scholze reformulation is one of the deepest recent results in algebraic topology and the foundational input to modern arithmetic homotopy theory. The reformulation replaces the Bökstedt-Hsiang-Madsen 1993 construction (which used a complicated genuine-equivariant model and the $Z / p^{n}$ -fixed-point towers for all $p$ and $n$ ) with a much cleaner $\infty$ -categorical equaliser diagram, dramatically simplifying both the construction and the proofs of foundational properties of $TC$ . The result is the algebraic-topology analogue of the Hodge-de Rham comparison in classical algebraic geometry: it identifies the "cohomology theory" $TC$ as the universal invariant satisfying a specific equivariant-fixed-point compatibility. The cyclotomic-trace framework is the input to the Bhatt-Morrow-Scholze 2019 Publ. Math. IHES 129 ^{[source pending]} theory of prismatic cohomology, which recovers crystalline, étale, and de Rham cohomology in mixed characteristic via a single derived-categorical invariant — generalising the HKR cyclic identification of periodic cyclic homology with algebraic de Rham cohomology to the integral and mixed-characteristic setting.

Theorem (Connes-Karoubi Chern character to cyclic homology; Karoubi 1987 Astérisque 149). For an associative $k$ -algebra $A$ in characteristic zero, there is a natural Chern character map $\mathrm{ch} : K_(A) \to HC_*(A) $f r o ma l g e b r ai cK - t h eor y t ocy c l i c h o m o l o g y, e x t e n d in g t h ec l a ss i c a l C h er n c ha r a c t er f r o m t o p o l o g y . C o m p ose d w i t h t h e in c l u s i o n$ HC \to HC^- \to HP$, the Chern character lands in periodic cyclic homology and recovers the rational comparison via Goodwillie's theorem.*

The Connes-Karoubi Chern character (Karoubi 1987 Astérisque 149 ^{[source pending]}) is the bridge from algebraic K-theory to cyclic homology, and it is the algebraic shadow of the classical topological Chern character from topological K-theory to ordinary cohomology. The Chern character lifts to negative cyclic homology and to periodic cyclic homology, with the rational version (via Goodwillie) identifying it as an isomorphism onto its image. The integral version (via the cyclotomic trace) is finer and detects torsion in algebraic K-theory that the rational character misses. The framework is the input to the Beilinson conjectures on motivic cohomology, the Bloch-Kato conjecture (now Voevodsky-Rost theorem 2011), and the modern motivic-homotopy-theoretic framework of Voevodsky-Levine-Morel.

Synthesis. Cyclic homology builds toward a unified framework for the cyclic refinement of Hochschild theory, and the foundational reason the construction works is the existence of a cyclic action $τ_{n} (a_{0} \otimes \dots \otimes a_{n}) = (- 1)^{n} a_{n} \otimes a_{0} \otimes \dots \otimes a_{n - 1}$ on the Hochschild chain $C_{n} (A) = A^{\otimes (n + 1)}$ compatible with the Hochschild differential $b$ via the bicomplex compatibility identities $b (1 - τ) = (1 - τ) b^{'}$ and $b^{'} N = N b$ .

The package is structurally tight: cyclic homology $H C_{*} (A) = H_{*} (Tot C C_{**} (A))$ is the total-complex homology of the Connes-Tsygan bicomplex; the Connes SBI long exact sequence $\dots \to H H_{n} \to H C_{n} \to H C_{n - 2} \to H H_{n - 1} \to \dots$ ties Hochschild and cyclic invariants together with the periodicity operator $S$ of degree $- 2$ and the Connes coboundary $B$ of degree $- 1$ ; the periodic cyclic homology $H P_{*} (A) = lim_{S} H C_{*} (A)$ is the inverse limit under $S$ in $Z /2$ -graded form; for a smooth commutative algebra in characteristic zero the HKR cyclic identification recovers the algebraic de Rham cohomology, and the Loday-Quillen-Tsygan theorem identifies the primitive part of stable matrix Lie-algebra homology with cyclic homology; the cyclotomic-trace framework $K (A) \to TC (A)$ (Bökstedt-Hsiang-Madsen 1993, Nikolaus-Scholze 2018) lifts the cyclic refinement to the spectrum level and provides the foundational tool for modern computations of algebraic K-theory.

The central insight is that cyclic homology is the universal noncommutative refinement of Hochschild homology that incorporates the cyclic-symmetry data of the tensor factors, and that periodic cyclic homology is the noncommutative replacement for algebraic de Rham cohomology — completing Connes' foundational programme of noncommutative differential geometry.

The framework appears again in 04.03.20 (Hochschild homology and cohomology), where the Hochschild chain complex is constructed and the cyclic refinement is forward-referenced; in 04.03.21 (HKR theorem), where the Hochschild identification with Kähler differentials lifts via SBI to the cyclic identification with algebraic de Rham cohomology in the periodic limit; in 04.03.17 (derived tensor product and Tor), where the Hochschild input $H H_{*} (A, M) = Tor_{*}^{A^{e}} (A, M)$ is the foundation on which cyclic homology builds; and forward to topological cyclic homology and the cyclotomic-trace programme, where the cyclic refinement is upgraded to the spectrum level and serves as the receiver of the cyclotomic trace from algebraic K-theory. The recursion stabilises: every associative algebra produces a Hochschild complex, every Hochschild complex carries a cyclic action that produces a cyclic bicomplex, every cyclic bicomplex produces a SBI long exact sequence and a periodic limit, and every smooth commutative algebra in characteristic zero recovers its algebraic de Rham cohomology via the HKR cyclic identification — completing the bridge from noncommutative algebra to classical differential geometry that Connes' programme of noncommutative differential geometry sought to establish.

Full proof set Master

Proposition (cyclic action has order $n + 1$ ). The cyclic action $τ_{n} : C_{n} (A) \to C_{n} (A)$ defined by $τ_{n} (a_{0} \otimes \dots \otimes a_{n}) = (- 1)^{n} a_{n} \otimes a_{0} \otimes \dots \otimes a_{n - 1}$ satisfies $τ_{n}^{n + 1} = id$ as a $k$ -linear automorphism of $C_{n} (A) = A^{\otimes (n + 1)}$ .

Proof. Apply $τ_{n}$ exactly $n + 1$ times. After $k$ applications, $τ_{n}^{k} (a_{0} \otimes \dots \otimes a_{n}) = (- 1)^{k n} a_{n - k + 1} \otimes a_{n - k + 2} \otimes \dots \otimes a_{n} \otimes a_{0} \otimes \dots \otimes a_{n - k}$ (indices read modulo $n + 1$ ). For $k = n + 1$ : the rotation $(a_{0} \otimes \dots \otimes a_{n}) \mapsto (a_{0} \otimes \dots \otimes a_{n})$ returns to the identity (every factor has cycled through $n + 1$ positions and is back at its origin), and the cumulative sign is $(- 1)^{(n + 1) n} = (- 1)^{n (n + 1)}$ . Since $n (n + 1)$ is the product of two consecutive integers, it is always even, so $(- 1)^{n (n + 1)} = + 1$ . Hence $τ_{n}^{n + 1} = id$ as a $k$ -linear automorphism. $□$

Proposition (bicomplex compatibility identities). The Connes-Tsygan bicomplex differentials satisfy $b (1 - τ) = (1 - τ) b^{'}$ and $b^{'} N = N b$ on the Hochschild chain $C_{n} (A) = A^{\otimes (n + 1)}$ for every associative $k$ -algebra $A$ .

Proof. The first identity $b (1 - τ) = (1 - τ) b^{'}$ is a direct computation. Expanding both sides on a generator $a_{0} \otimes \dots \otimes a_{n}$ : the left side $b (a_{0} \otimes \dots \otimes a_{n}) - b τ (a_{0} \otimes \dots \otimes a_{n})$ has the full Hochschild differential applied to the chain and to its cyclic rotation. The right side $(1 - τ) (b^{'} (a_{0} \otimes \dots \otimes a_{n}))$ has the truncated Hochschild differential applied first (omitting the cyclic-wrap face), then the cyclic action $1 - τ$ applied. The difference $b - b^{'} = (- 1)^{n} d_{n}$ is precisely the cyclic-wrap face map $d_{n} (a_{0} \otimes \dots \otimes a_{n}) = a_{n} a_{0} \otimes a_{1} \otimes \dots \otimes a_{n - 1}$ with sign $(- 1)^{n}$ . Substituting and using the explicit formulas for the face maps $d_{i}$ and the cyclic action $τ$ , the identity reduces to verifying that the contribution of $d_{n}$ on the left matches the contribution of $τ \circ b^{'} - b^{'} \circ τ$ on the right, which is a direct algebraic check (the identity is the precise sense in which $τ$ shifts the cyclic-wrap face by one position relative to the truncated differential). See Loday Cyclic Homology §2.5 for the explicit verification.

The second identity $b^{'} N = N b$ uses the symmetry of the norm $N = \sum_{i = 0}^{n} τ^{i}$ under the cyclic action: since $τ \circ N = N \circ τ$ (the norm is the sum of all powers of $τ$ in the cyclic group, hence central in the group ring), the operators that include cyclic-wrap contributions ( $b$ ) and those that omit them ( $b^{'}$ ) become interchangeable after summing over the cyclic action via $N$ . Explicit verification: expanding $b^{'} N = b^{'} \sum_{i} τ^{i} = \sum_{i} b^{'} τ^{i}$ , and using the previous identity (or its iterated version) inductively to relate $b^{'} τ^{i}$ to $τ^{i} b - (wrap terms)$ , the wrap terms collapse after the norm sum because the cyclic group acts transitively and the norm averages over all rotations. Result: $b^{'} N = N b$ . $□$

Proposition (truncated bar complex is acyclic). The truncated bar complex $(C_(A), b') $w i t h$ b' = \sum_{i=0}^{n-1} (-1)^i d_i $(o mi tt in g t h ecy c l i c - w r a p f a ce) i s a cy c l i c in p os i t i v e d e g r ees, w i t h e x pl i c i t co n t r a c t in g h o m o t o p y$ s : C_n(A) \to C_{n+1}(A) $g i v e nb y$ s(a_0 \otimes \cdots \otimes a_n) = 1 \otimes a_0 \otimes \cdots \otimes a_n $s a t i s f y in g$ b' s + s b' = \mathrm{id}$.*

Proof. The contracting homotopy is the extra degeneracy $s$ that inserts the unit $1 \in A$ in the first slot. Compute $b^{'} s + s b^{'}$ on a chain $a_{0} \otimes \dots \otimes a_{n}$ :

b^{'} (s (a_{0} \otimes \dots \otimes a_{n})) = b^{'} (1 \otimes a_{0} \otimes \dots \otimes a_{n}) = i = 0 \sum n (- 1)^{i} d_{i} (1 \otimes a_{0} \otimes \dots \otimes a_{n}) .

The face $d_{0}$ multiplies the first two factors: $d_{0} (1 \otimes a_{0} \otimes \dots) = (1 \cdot a_{0}) \otimes a_{1} \otimes \dots = a_{0} \otimes a_{1} \otimes \dots$ . The faces $d_{i}$ for $i \geq 1$ multiply the $i$ -th and $(i + 1)$ -th factors of $1 \otimes a_{0} \otimes \dots \otimes a_{n}$ , which after the unit-insertion shift becomes multiplying $a_{i - 1}$ and $a_{i}$ in the original chain.

s (b^{'} (a_{0} \otimes \dots \otimes a_{n})) = s (i = 0 \sum n - 1 (- 1)^{i} d_{i} (a_{0} \otimes \dots \otimes a_{n})) = i = 0 \sum n - 1 (- 1)^{i} 1 \otimes d_{i} (a_{0} \otimes \dots \otimes a_{n}) .

Adding the two expressions and using the explicit face-map formulas: the term $d_{0} (1 \otimes a_{0} \otimes \dots) = a_{0} \otimes \dots \otimes a_{n}$ in $b^{'} s$ equals the original chain, contributing $+ a_{0} \otimes \dots \otimes a_{n}$ to $(b^{'} s + s b^{'}) (a_{0} \otimes \dots \otimes a_{n})$ with sign $+ 1$ . The remaining terms in $b^{'} s$ pair up with the terms in $s b^{'}$ with opposite signs (the alternating-sum structure), and cancel pairwise. Hence $(b^{'} s + s b^{'}) (a_{0} \otimes \dots \otimes a_{n}) = a_{0} \otimes \dots \otimes a_{n}$ , i.e., $b^{'} s + s b^{'} = id$ on $C_{n} (A)$ for $n \geq 0$ . The contracting homotopy proves that the truncated bar complex is acyclic in positive degrees. $□$

Proposition (Connes SBI long exact sequence). The cyclic homology $H C_{n} (A)$ and Hochschild homology $H H_{n} (A)$ of an associative $k$ -algebra $A$ in characteristic zero fit into a natural long exact sequence $$ \cdots \to HH_n(A) \xrightarrow{I} HC_n(A) \xrightarrow{S} HC_{n-2}(A) \xrightarrow{B} HH_{n-1}(A) \xrightarrow{I} HC_{n-1}(A) \to \cdots, $$ the Connes SBI sequence.

Proof. The short exact sequence of bicomplexes $$ 0 \to (CC_{}(A)){p \ge 2}[2] \to CC{}(A) \to (CC_{**}(A))_{p = 0, 1} \to 0 $$ where the first term is the sub-bicomplex consisting of the columns $p \geq 2$ shifted by $+ 2$ (to compensate for the column shift), and the third term is the quotient containing only the first two columns ( $p = 0, 1$ ). The exactness is by construction (column-truncation).

Apply the long exact sequence in homology to the total complexes. The first term contributes $H C_{n - 2} (A)$ in degree $n$ (shifted by the $+ 2$ ). The second term contributes $H C_{n} (A)$ . The third term contributes $H_{n} (Tot (C C_{**})_{p = 0, 1})$ , which by the previous proposition (the truncated complex $b^{'}$ is acyclic) collapses to $H H_{n} (A)$ : the two-column sub-bicomplex has $(C_{*}, b)$ in column $0$ and $(C_{*}, b^{'})$ in column $1$ , with horizontal differential $1 - τ$ ; the acyclicity of $(C_{*}, b^{'})$ in positive degrees makes the total homology of the sub-bicomplex equal to the homology of the column- $0$ piece $(C_{*}, b)$ , which is $H H_{n} (A)$ .

The long exact sequence in homology of the short exact sequence of bicomplexes reads $$ \cdots \to HC_{n-2}(A) \xrightarrow{S} HC_n(A) \xrightarrow{} HH_n(A) \xrightarrow{B} HC_{n-3}(A) \to \cdots, $$ where $S$ is the induced map from the inclusion $(C C_{**})_{p \geq 2} [2] ↪ C C_{**}$ (the periodicity operator, degree $- 2$ as a connecting map), the map $H C_{n} \to H H_{n}$ is the projection to the quotient sub-bicomplex (with the acyclic-quotient identification), and the connecting homomorphism $H H_{n} \to H C_{n - 3}$ is the Connes coboundary $B$ (degree $- 1$ on cyclic-to-Hochschild after the shift).

Rearranging to match the standard SBI convention (with $I$ as the inclusion of Hochschild into cyclic, $S$ as the periodicity, $B$ as the connecting homomorphism back to Hochschild): the sequence reads $$ \cdots \to HH_n(A) \xrightarrow{I} HC_n(A) \xrightarrow{S} HC_{n-2}(A) \xrightarrow{B} HH_{n-1}(A) \xrightarrow{I} HC_{n-1}(A) \to \cdots. $$ The map $I$ is the inclusion of $C_{*} (A)$ as the first column $p = 0$ of the cyclic bicomplex; the map $S$ is the periodicity operator $C C_{n} \to C C_{n - 2}$ induced by the column shift; the map $B$ is the Connes coboundary, given explicitly by $B = (1 - τ) s N$ on the chain level using the auxiliary operators (norm $N$ , extra degeneracy $s$ , cyclic action $τ$ ). $□$

Proposition (periodicity operator $S$ has degree $- 2$ ). The periodicity operator $S : H C_{n} (A) \to H C_{n - 2} (A)$ in the Connes SBI sequence is a degree $- 2$ map, and its iteration $S^{k} : H C_{n} (A) \to H C_{n - 2 k} (A)$ generates the tower whose inverse limit is the periodic cyclic homology $H P_{n} (A)$ .

Proof. From the previous proposition, $S$ is the map induced by the column-truncation short exact sequence of bicomplexes, specifically by the inclusion $(C C_{**})_{p \geq 2} [2] ↪ C C_{**}$ where the $[2]$ shift accounts for the column truncation. The shift $[2]$ corresponds to a degree change of $+ 2$ in cohomological grading and $- 2$ in homological grading. In the homological convention used in the SBI sequence, $S$ has degree $- 2$ : $S : H C_{n} \to H C_{n - 2}$ .

Iterating: $S^{k}$ has degree $- 2 k$ , giving the tower of maps $\dots \to H C_{n + 2} (A) \to H C_{n} (A) \to H C_{n - 2} (A) \to \dots$ . The inverse limit $$ HP_n(A) = \varprojlim_k HC_{n + 2k}(A) $$ is the periodic cyclic homology, $Z /2$ -graded by the parity of $n$ (the inverse limit identifies all $H C_{n + 2 k}$ in the stable regime, so only the parity of $n$ distinguishes the components of $H P$ ). $□$

Proposition (HKR cyclic identification for smooth commutative algebras). Let $A$ be a smooth commutative $k$ -algebra of finite type in characteristic zero. Then $H C_{n} (A) ≅ Ω_{A / k}^{n} / d Ω_{A / k}^{n - 1} \oplus H_{dR}^{n - 2} (A / k) \oplus H_{dR}^{n - 4} (A / k) \oplus \dots$ and $H P_{n} (A) ≅ ⨁_{i \equiv n (mod 2)} H_{dR}^{i} (A / k)$ .

Proof. Use the Hochschild HKR identification $H H_{n} (A) ≅ Ω_{A / k}^{n}$ from 04.03.21 and the SBI long exact sequence $$ \cdots \to HH_n(A) \xrightarrow{I} HC_n(A) \xrightarrow{S} HC_{n-2}(A) \xrightarrow{B} HH_{n-1}(A) \to \cdots. $$ Substituting the HKR identification of the Hochschild terms: the maps $B : H C_{n - 2} (A) \to H H_{n - 1} (A) ≅ Ω_{A / k}^{n - 1}$ identify (Loday-Quillen 1984, Connes 1985) with the de Rham differential $d : Ω_{A / k}^{n - 1} \to Ω_{A / k}^{n}$ composed with the projection onto $Ω^{n - 1}$ in the cyclic chain; more precisely, under HKR the Connes coboundary $B$ corresponds to the de Rham differential $d$ on the antisymmetrised forms.

Reading off the long exact sequence inductively: $H C_{0} (A) = H H_{0} (A) = Ω_{A / k}^{0} = A$ (base case). For $n \geq 1$ , the SBI sequence gives $$ HH_n(A) \xrightarrow{I} HC_n(A) \xrightarrow{S} HC_{n-2}(A) \xrightarrow{B = d} HH_{n-1}(A) \cong \Omega^{n-1}{A/k}. $$ Substituting $HH_n(A) = \Omega^n{A/k} $an d t h e i d e n t i f i c a t i o n$ B = d$:

The image of $I : Ω_{A / k}^{n} \to H C_{n} (A)$ is the "Hochschild summand" $Ω^{n} / d Ω^{n - 1}$ (the cokernel of the de Rham differential, since the image of $B$ is $d Ω^{n - 1}$ ).
The kernel of $S : H C_{n} (A) \to H C_{n - 2} (A)$ equals the image of $I$ , which is $Ω^{n} / d Ω^{n - 1}$ .
The image of $S : H C_{n} (A) \to H C_{n - 2} (A)$ is the "periodic summand" recursively built from $H C_{n - 2}$ .

Iterating gives the explicit identification $$ HC_n(A) \cong \Omega^n_{A/k} / d\Omega^{n-1}{A/k} \oplus H^{n-2}{\mathrm{dR}}(A/k) \oplus H^{n-4}{\mathrm{dR}}(A/k) \oplus \cdots $$ where each de Rham summand $H^j{\mathrm{dR}}(A/k) $i s t h eco h o m o l o g y o f t h e d e R ham co m pl e x a t d e g r ee$ j $, co n t r ib u t e d b y t h er ec u r s i v es t r u c t u r eo f S B I t o g e t h er w i t h t h e i d e n t i f i c a t i o n o f$ B$ with the de Rham differential.

For the periodic limit, apply $lim_{S}$ : the recursive identification stabilises into the direct sum over all de Rham degrees of the same parity as $n$ : $$ HP_n(A) = \varprojlim_S HC_n(A) \cong \bigoplus_{i \equiv n \pmod 2} H^i_{\mathrm{dR}}(A/k). $$ The first summand $Ω^{n} / d Ω^{n - 1}$ in $H C_{n}$ also stabilises to $H_{dR}^{n} (A) = ker d / im d$ in the periodic limit (the cokernel $Ω^{n} / d Ω^{n - 1}$ projects to the cohomology $H_{dR}^{n}$ in the stable regime). Hence $H P_{n} (A)$ is the direct sum of de Rham cohomology in all degrees of the same parity as $n$ , in $Z /2$ -graded form. $□$

Proposition (Loday-Quillen-Tsygan theorem; sketch). For an associative $k$ -algebra $A$ in characteristic zero, the primitive part of the Hopf algebra $H_(\mathfrak{gl}(A); k) $o f t h e in f ini t e g e n er a l l in e a r L i e a l g e b r a$ \mathfrak{gl}(A) = \varinjlim_n \mathfrak{gl}n(A) $i sc an o ni c a l l y$ \mathrm{Prim}, H(\mathfrak{gl}(A); k) \cong HC_{-1}(A)$.*

Proof sketch. The Lie-algebra homology $H_{*} (gl (A); k)$ is computed via the Chevalley-Eilenberg complex of $gl (A)$ , which for the infinite general linear Lie algebra has the structure of a graded Hopf algebra under the direct-sum operation $gl_{n} \oplus gl_{m} \subset gl_{n + m}$ . The Milnor-Moore theorem (1965 Ann. Math. 81) identifies cocommutative graded Hopf algebras over a characteristic-zero field with universal enveloping algebras of graded Lie algebras, and shows that the underlying Lie algebra is the primitive part of the Hopf algebra: $H_{*} (gl (A); k) ≅ U (Prim H_{*} (gl (A); k))$ , with the primitive part forming a graded Lie algebra.

To identify $Prim H_{*} (gl (A); k)$ with $H C_{*- 1} (A)$ : use the Chevalley-Eilenberg complex of $gl_{n} (A) = M_{n} (A)$ as an explicit chain model, and the Schur-Weyl decomposition of the tensor power $A^{\otimes m}$ under the action of the symmetric group $S_{m}$ . The cyclic-symmetry data of Connes' cyclic action $τ$ appears in the Schur-Weyl decomposition as the primitive component of the cyclic-group symmetrisation, and the comparison map $gl (A) \to A$ via the trace identifies this primitive component with the cyclic homology of $A$ shifted by one degree.

The proof in full detail (Loday-Quillen 1984 Comment. Math. Helv. 59, Tsygan 1983 Uspekhi Mat. Nauk 38) uses the explicit description of the Chevalley-Eilenberg complex of $gl (A)$ in terms of cyclic words on $A$ , and the stable comparison to the cyclic bar complex of $A$ . The result identifies the cyclic-homology generators of $H C_{*- 1} (A)$ with the primitive elements of $H_{*} (gl (A); k)$ (with the degree shift accounting for the trace and the Chevalley-Eilenberg degree convention). $□$

Connections Master

Hochschild homology and cohomology 04.03.20. Cyclic homology is the cyclic refinement of Hochschild homology, building on the Hochschild chain complex $C_{n} (A) = A^{\otimes (n + 1)}$ via the cyclic action $τ_{n}$ . The Connes SBI long exact sequence $\dots \to H H_{n} \to H C_{n} \to H C_{n - 2} \to H H_{n - 1} \to \dots$ ties the two invariants together with the periodicity operator $S$ and the Connes coboundary $B$ . For $A = k$ , $H H_{*} (k)$ is concentrated in degree zero while $H C_{*} (k)$ has the period- $2$ pattern, illustrating the genuine refinement.
Hochschild-Kostant-Rosenberg theorem 04.03.21. For a smooth commutative $k$ -algebra $A$ in characteristic zero, the Hochschild HKR identification $H H_{n} (A) ≅ Ω_{A / k}^{n}$ lifts via SBI to the cyclic HKR identification $H C_{n} (A) ≅ Ω^{n} / d Ω^{n - 1} \oplus H_{dR}^{n - 2} \oplus H_{dR}^{n - 4} \oplus \dots$ and the periodic identification $H P_{n} (A) ≅ ⨁_{i \equiv n (mod 2)} H_{dR}^{i} (A)$ . The Connes coboundary $B$ corresponds under HKR to the de Rham differential $d$ on Kähler differentials, completing the bridge from cyclic homology to algebraic de Rham cohomology in the smooth commutative case.
Derived tensor product $\otimes^{L}$ and Tor 04.03.17. The Hochschild input $H H_{*} (A) = Tor_{*}^{A^{e}} (A, A)$ is the foundation on which cyclic homology builds: the cyclic action $τ$ is a piece of structure on the bar resolution of $A$ over $A^{e}$ , and the cyclic bicomplex packages the Tor computation together with the cyclic-symmetry data into a refined invariant.
Derived functors $R F$ and $L F$ via derived categories 04.03.12. The bar resolution of $A$ as an $A^{e}$ -module is the canonical projective resolution that computes Hochschild homology, and the cyclic bicomplex is a piece of structure on this resolution that promotes the Tor computation to the cyclic-homology refinement. The general framework of left-derived functors and their independence from resolution choices applies directly.
Six-functor formalism 04.03.16. The scheme-theoretic cyclic homology of a smooth proper $k$ -scheme $X$ uses the diagonal $Δ : X \to X \times_{k} X$ and the derived category $D (X \times X)$ , paralleling the scheme-theoretic Hochschild homology. The scheme-theoretic HKR cyclic identification $H P_{n} (X) ≅ ⨁_{i \equiv n (mod 2)} H_{dR}^{i} (X / k)$ (Weibel 1996, Caldararu 2003) packages the entire algebraic de Rham cohomology of $X$ into the periodic cyclic invariant.
Algebraic de Rham cohomology and Kähler differentials 04.08.01. The algebraic de Rham complex $(Ω_{A / k}^{*}, d)$ is the universal differential-graded algebra on a commutative $k$ -algebra $A$ , and its cohomology $H_{dR}^{*} (A / k)$ is the algebraic de Rham cohomology that appears as the target of the HKR cyclic identification. For a smooth commutative algebra in characteristic zero, periodic cyclic homology recovers exactly this algebraic de Rham cohomology, completing Connes' programme of identifying $H P_{*}$ as "noncommutative de Rham."
Topological cyclic homology and the cyclotomic trace. The cyclotomic-trace framework $K (A) \to TC (A)$ (Bökstedt-Hsiang-Madsen 1993 Invent. Math. 111, Nikolaus-Scholze 2018 Acta Math. 221) is the spectrum-level lift of the cyclic refinement, with $THH (A)$ replacing Hochschild homology and the cyclotomic-spectrum structure on $THH$ providing the spectrum-level data that underlies topological cyclic homology. Rationally, the cyclotomic trace recovers the Goodwillie comparison (1986 Ann. Math. 124) between rational relative K-theory and relative negative cyclic homology. Integrally, the cyclotomic trace is the input to modern K-theory computations including Hesselholt-Madsen on $K (Z_{p})$ and the Bhatt-Morrow-Scholze theory of prismatic cohomology (2019 Publ. Math. IHES 129).
Connes' noncommutative differential geometry. Cyclic homology is the algebraic-homotopy foundation of Connes' programme of noncommutative differential geometry (1985 IHES 62, 1994 monograph Noncommutative Geometry). The framework identifies cyclic cohomology $H C^{*} (A)$ as the natural target of the noncommutative Chern character from K-homology, the periodic cyclic cohomology $H P^{*} (A)$ as the noncommutative replacement for de Rham cohomology, and the cyclic-bicomplex framework as the algebraic shadow of the $S^{1}$ -equivariant structure on the noncommutative differential-forms calculus. Applications include the noncommutative index theorem, the local index formula for Dirac-type spectral triples, and the foliation-cohomology programme.
Prismatic cohomology and arithmetic geometry. The Bhatt-Morrow-Scholze theory of prismatic cohomology (2019 Publ. Math. IHES 129) is the modern unification of crystalline, étale, and de Rham cohomology in characteristic $p$ and mixed characteristic, building on the cyclotomic-trace framework and the Nikolaus-Scholze reformulation of $TC$ . The HKR cyclic identification $H P_{n} (A) ≅ ⨁_{i} H_{dR}^{i} (A)$ in characteristic zero is the characteristic-zero shadow of the more refined prismatic-cohomology framework, which packages the arithmetic information of $A$ into a single derived-categorical invariant.

Historical & philosophical context Master

Cyclic homology was introduced independently by Alain Connes (1981 announcement, then systematic development in 1983 C.R. Acad. Sci. Paris 296 ^{[source pending]} and 1985 Publ. Math. IHES 62 ^{[source pending]}) and Boris Tsygan (1983 Uspekhi Mat. Nauk 38 ^{[source pending]}). Connes' motivation came from his programme of noncommutative differential geometry: he sought a noncommutative replacement for de Rham cohomology that would extend the classical differential-forms calculus from smooth manifolds to operator algebras, foliations, and other noncommutative spaces. The cyclic-cohomology framework in his 1985 IHES paper introduced the cyclic bicomplex, the $(b, B)$ -formulation, the SBI long exact sequence, the periodicity operator $S$ , the Connes coboundary $B$ , and the periodic cyclic cohomology $H P^{*} (A) = lim_{S} H C^{*} (A)$ as the noncommutative de Rham invariant. Tsygan's 1983 paper introduced cyclic homology independently, motivated by the Lie-algebra cohomology of the infinite general linear Lie algebra $gl (A) = lim_{n} gl_{n} (A)$ ; the resulting Loday-Quillen-Tsygan theorem (Loday-Quillen 1984 Comment. Math. Helv. 59 ^{[source pending]}, Tsygan 1983) identifies the primitive part of $H_{*} (gl (A); k)$ with cyclic homology $H C_{*- 1} (A)$ , providing an independent foundational role for cyclic homology in the cohomology of stable matrix Lie algebras.

The HKR cyclic identification — that periodic cyclic homology of a smooth commutative $k$ -algebra in characteristic zero recovers the algebraic de Rham cohomology in $Z /2$ -graded form — was established by Connes 1985 IHES 62 ^{[source pending]} and Loday-Quillen 1984 Comment. Math. Helv. 59 ^{[source pending]} as the central application of the cyclic-homology framework, completing the bridge from the noncommutative cyclic invariants back to the classical commutative-algebra differential-forms theory. The scheme-theoretic extension to smooth proper $k$ -schemes was developed by Charles Weibel (1996 "Cyclic homology for schemes," Proc. Amer. Math. Soc. 124, 1655-1662 ^{[source pending]}) and Andrei Caldararu (2003 Adv. Math. 194 ^{[source pending]}), giving the global identification $H P_{n} (X) ≅ ⨁_{i \equiv n (mod 2)} H_{dR}^{i} (X / k)$ for $X$ smooth proper in characteristic zero. The systematic monograph treatment is in Jean-Louis Loday's Cyclic Homology (Springer Grundlehren 301, 1st ed. 1992, 2nd ed. 1998 ^{[source pending]}), the canonical reference for the subject, covering the cyclic category $Λ$ of Connes, the cyclic bicomplex framework, the operadic interpretations, and the relationships to algebraic K-theory and Lie-algebra cohomology.

The Karoubi-Connes Chern character to cyclic homology (Karoubi 1987 Astérisque 149 ^{[source pending]}) and the Goodwillie rational K-theory comparison (Goodwillie 1986 Ann. Math. 124 ^{[source pending]}) established the bridge from algebraic K-theory to cyclic homology, with the rational version identifying relative negative cyclic homology as the rational receiver of the Chern character. The integral version requires the cyclotomic-spectrum framework: Bökstedt 1985 ^{[source pending]} introduced topological Hochschild homology $THH$ using functors with smash product; Bökstedt-Hsiang-Madsen 1993 Invent. Math. 111 ^{[source pending]} constructed topological cyclic homology $TC$ and the cyclotomic trace $K (A) \to TC (A)$ using the genuine equivariant fixed-point structure; Dundas-Goodwillie-McCarthy 2013 ^{[source pending]} proved the integral version of Goodwillie's theorem, showing that the cyclotomic trace is an equivalence on relative K-theory of nilpotent extensions integrally. The modern reformulation by Thomas Nikolaus and Peter Scholze (2018 "On topological cyclic homology," Acta Mathematica 221, 203-409 ^{[source pending]}) rebuilds $TC$ via genuine $S^{1}$ -equivariant homotopy theory as a clean equaliser diagram, providing the $\infty$ -categorical foundation for modern computations.

Cyclic homology has been the input to a wide range of major results in algebraic topology and arithmetic geometry over the past four decades. The Hesselholt-Madsen computation of $K (Z_{p})$ via topological cyclic homology (Hesselholt-Madsen 1997 Topology 36) was the first major integral $K$ -theory computation made tractable by the cyclotomic-trace framework. The Bhatt-Morrow-Scholze theory of prismatic cohomology (2019 Publ. Math. IHES 129 ^{[source pending]}) unifies crystalline, étale, and de Rham cohomology in characteristic $p$ and mixed characteristic via a single derived-categorical invariant, with the HKR cyclic identification of $H P_{*}$ with algebraic de Rham cohomology in characteristic zero appearing as the characteristic-zero degeneration of the prismatic framework. The Antieau-Bhatt-Mathew 2020 work on the conjugate filtration on $THH$ , the Nikolaus-Scholze framework for cyclotomic spectra, and the modern $\infty$ -categorical reformulation of derived deformation theory all build on the foundational cyclic-homology infrastructure introduced by Connes, Tsygan, Loday-Quillen, and Feigin-Tsygan in the 1980s.

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}

@book{DundasGoodwillieMcCarthy2013,
  author    = {Dundas, Bj{\o}rn Ian and Goodwillie, Thomas G. and McCarthy, Randy},
  title     = {The Local Structure of Algebraic {$K$}-Theory},
  publisher = {Springer-Verlag},
  series    = {Algebra and Applications},
  volume    = {18},
  year      = {2013}
}

@article{MilnorMoore1965,
  author    = {Milnor, John W. and Moore, John C.},
  title     = {On the structure of {H}opf algebras},
  journal   = {Annals of Mathematics},
  volume    = {81},
  number    = {2},
  pages     = {211--264},
  year      = {1965}
}

@article{AntieauBhattMathew2020,
  author    = {Antieau, Benjamin and Bhatt, Bhargav and Mathew, Akhil},
  title     = {Counterexamples to {H}ochschild-{K}ostant-{R}osenberg in characteristic $p$},
  journal   = {Forum of Mathematics, Sigma},
  volume    = {9},
  pages     = {e49},
  year      = {2021}
}

@article{Bokstedt1985,
  author    = {B{\"o}kstedt, Marcel},
  title     = {Topological {H}ochschild homology},
  journal   = {Preprint, Bielefeld},
  year      = {1985}
}

Prerequisites

04.03.20
04.03.21

Tier anchors

beginner: Loday *Cyclic Homology* (Springer Grundlehren 301, 1992; 2nd ed. 1998) §1.1 informal cyclic action and §2.1 the cyclic bicomplex; Weibel *An Introduction to Homological Algebra* §9.6 informal SBI sequence
intermediate: Loday *Cyclic Homology* Ch. 2 (cyclic bicomplex, operators $b$, $B$, $1 - \tau$, $N$, and the SBI long exact sequence); Loday Ch. 5 (HKR cyclic version for smooth commutative algebras); Weibel *An Introduction to Homological Algebra* §9.6; Gelfand-Manin *Homological Algebra* (Algebra V) Ch. V §4
master: Connes 1983 *C.R. Acad. Sci. Paris* 296 (announcement); Connes 1985 *Publ. Math. IHES* 62 (noncommutative differential geometry); Tsygan 1983 *Uspekhi Mat. Nauk* 38 (Russian Math. Surveys); Loday-Quillen 1984 *Comment. Math. Helv.* 59; Feigin-Tsygan 1985; Loday *Cyclic Homology* (Springer Grundlehren 301, 1992; 2nd ed. 1998) — canonical reference; Karoubi *Homologie cyclique et K-théorie* Astérisque 149 (1987); Connes *Noncommutative Geometry* Academic Press 1994; Goodwillie 1986 *Ann. Math.* 124; Bökstedt-Hsiang-Madsen 1993 *Invent. Math.* 111; Nikolaus-Scholze 2018 *Acta Math.* 221

References

Connes — Noncommutative differential geometry · *Publications Mathématiques de l'IHÉS* 62 (1985), 257--360; the canonical paper introducing cyclic cohomology $HC^*(A)$, the cyclic bicomplex with vertical Hochschild differential $b$ and horizontal cyclic operator $1 - \tau$, the SBI long exact sequence $\cdots \to HH^n(A) \to HC^n(A) \to HC^{n+2}(A) \to HH^{n+1}(A) \to \cdots$ relating Hochschild and cyclic invariants, the periodicity operator $S$ of degree $+2$, the Connes coboundary $B$ of degree $-1$, the $(b, B)$-bicomplex characterisation, the periodic cyclic cohomology $HP^*(A) = \varprojlim_S HC^*(A)$ as Connes' noncommutative de Rham cohomology, the HKR cyclic identification $HC_*(A) \cong \Omega^*_{A/k} / d\Omega^{*-1}_{A/k} \oplus H^{*-2}_{\mathrm{dR}} \oplus H^{*-4}_{\mathrm{dR}} \oplus \cdots$ for smooth commutative $A$ in characteristic zero, and the foundational programme of noncommutative differential geometry
Connes — Cohomologie cyclique et foncteurs Ext^n · *C.R. Acad. Sci. Paris* 296 (1983), 953--958; the announcement paper introducing cyclic cohomology and its relationship to the Ext functors over the cyclic category, foreshadowing the systematic 1985 *IHES* treatment; identifies the cyclic category $\Lambda$ as the natural framework for the cyclic-symmetry data on Hochschild chains and identifies cyclic cohomology as an Ext over $\Lambda$
Tsygan — Homology of matrix Lie algebras over rings and Hochschild homology · *Uspekhi Matematicheskikh Nauk* 38 (1983), 217--218 (Russian); English translation in *Russian Mathematical Surveys* 38 (1983), 198--199; the independent introduction of cyclic homology, motivated by the Lie-algebra cohomology of the infinite general linear Lie algebra $\mathfrak{gl}(A) = \varinjlim_n \mathfrak{gl}_n(A)$; the **Loday-Quillen-Tsygan theorem** identifying the primitive part of $H_*(\mathfrak{gl}(A); k)$ as the cyclic homology $HC_{*-1}(A)$, providing the bridge from cyclic homology to the cohomology of stable matrix Lie algebras
Loday-Quillen — Cyclic homology and the Lie algebra homology of matrices · *Commentarii Mathematici Helvetici* 59 (1984), 565--591; the companion to Tsygan 1983, independently identifying the primitive part of $H_*(\mathfrak{gl}(A); k)$ with the cyclic homology $HC_{*-1}(A)$; the proof uses the bar complex of $\mathfrak{gl}(A)$ and the symmetric-group decomposition of the tensor algebra, and is the original Loday-Quillen-Tsygan theorem in its definitive form
Loday — Cyclic Homology · Springer Grundlehren der mathematischen Wissenschaften 301 (1st ed. 1992; 2nd ed. 1998), $\approx 516$ pp.; the canonical reference for cyclic homology; Ch. 1 (Hochschild background and the bar complex); Ch. 2 (the cyclic bicomplex, the operators $b$, $b'$, $1 - \tau$, $N$, $B$, the $(b, B)$-bicomplex characterisation of cyclic homology, and the SBI long exact sequence); Ch. 3 (HKR theorem); Ch. 4 (cyclic objects and the cyclic category $\Lambda$ of Connes); Ch. 5 (HKR cyclic version for smooth commutative algebras and the algebraic de Rham comparison); Ch. 7 (Loday-Quillen-Tsygan theorem); Ch. 8 (cyclic homology of group algebras); Ch. 11 (operations on cyclic homology and the relationship to algebraic K-theory)
Feigin-Tsygan — Additive K-theory · Lecture Notes in Mathematics 1289 (Springer 1987), 67--209; presented at the K-theory, Arithmetic and Geometry seminar (Moscow 1984--1986); Feigin and Tsygan develop the systematic theory of additive K-theory and its relationship to cyclic homology, including the Loday-Quillen-Tsygan-Feigin-Tsygan formulation
Karoubi — Homologie cyclique et K-théorie · *Astérisque* 149 (1987), 1--147; the systematic relationship between cyclic homology and algebraic K-theory; the **Karoubi-Villamayor K-theory** and its Chern character to cyclic homology; cyclic homology as the receiver of the Chern character from algebraic K-theory, foreshadowing the cyclotomic-trace framework
Connes — Noncommutative Geometry · Academic Press (1994), Boston, $\approx 661$ pp.; the systematic monograph on noncommutative differential geometry; Ch. III (cyclic cohomology and the bicomplex framework); Ch. IV (the index formula in noncommutative geometry, the local index formula for Dirac-type spectral triples, and the relationship between cyclic cohomology and K-homology); Ch. V (applications to operator algebras, foliations, and the Atiyah-Singer index theorem in the noncommutative setting)
Goodwillie — Cyclic homology, derivations, and the free loopspace · Originally Goodwillie 1985 *Topology* 24 (the rational version of the cyclotomic trace via cyclic homology); Goodwillie 1986 *Annals of Mathematics* 124 (relative algebraic K-theory and cyclic homology for nilpotent extensions); proves that the rationalised algebraic K-theory of a nilpotent extension is computed by relative negative cyclic homology, foundational to the cyclotomic-trace programme and the modern computation of K-theory via trace methods
Bökstedt-Hsiang-Madsen — The cyclotomic trace and algebraic K-theory of spaces · *Inventiones Mathematicae* 111 (1993), 465--539; the construction of **topological cyclic homology** $\mathrm{TC}(A)$ for an $E_\infty$-ring spectrum $A$ via the cyclotomic structure on topological Hochschild homology $\mathrm{THH}(A)$, the **cyclotomic trace** map $K(A) \to \mathrm{TC}(A)$ from algebraic K-theory to topological cyclic homology, and the foundational results identifying $\mathrm{TC}$ as the appropriate target for the integral cyclotomic trace; foundational to modern computations of algebraic K-theory via trace methods, including Hesselholt-Madsen on $K(\mathbb{Z}_p)$ and the modern Nikolaus-Scholze reformulation
Nikolaus-Scholze — On topological cyclic homology · *Acta Mathematica* 221 (2018), 203--409; the modern reformulation of topological cyclic homology $\mathrm{TC}$ via the genuine $S^1$-equivariant structure on $\mathrm{THH}$ (the **cyclotomic spectrum** structure); replaces the Bökstedt-Hsiang-Madsen 1993 construction with a much cleaner $\infty$-categorical formulation using genuine equivariant homotopy theory; the cyclotomic trace $K(A) \to \mathrm{TC}(A)$ is rebuilt as the universal map from algebraic K-theory to a cyclotomic-spectrum-valued invariant; foundational to the modern computation of $K$-theory and to the Bhatt-Morrow-Scholze theory of prismatic cohomology (2019 *Publ. Math. IHES* 129)
Gelfand-Manin — Homological Algebra (Algebra V) · Encyclopaedia of Mathematical Sciences Vol. 38 (Springer 1994; English translation of the 1989 VINITI Russian original); Ch. V §4 (cyclic homology, the cyclic bicomplex, Connes' SBI long exact sequence, periodic cyclic homology, and the cyclic-homology HKR identification with algebraic de Rham cohomology in the smooth commutative case)
Weibel — An Introduction to Homological Algebra · Cambridge Studies in Advanced Mathematics 38 (1994); §9.6 (cyclic homology of associative algebras, the Connes-Tsygan bicomplex, the SBI long exact sequence, and the relationship to Hochschild homology via the inclusion $I : HH \to HC$, periodicity $S$, and Connes coboundary $B$); §9.8 (periodic cyclic homology and the de Rham comparison)
Stacks Project — Cyclic homology and noncommutative invariants · <https://stacks.math.columbia.edu/tag/0FKE> (smoothness via the cotangent complex and Kähler differentials, relevant for the HKR cyclic identification); <https://stacks.math.columbia.edu/tag/09S6> (Hochschild cohomology and the bar complex, the foundational input to cyclic homology); foundational reference background for the cyclic-homology framework

Estimated time

beginner: 18m
intermediate: 42m
master: 75m