04.03.06 · algebraic-geometry / cohomology

Derived functors and Ext

shipped3 tiersLean: none

Anchor (Master): Cartan-Eilenberg *Homological Algebra* (1956) Ch. III, V; Grothendieck 1957 *Sur quelques points d'algèbre homologique* (Tôhoku Math. J. 9); Hartshorne *Residues and Duality* Ch. I; Gelfand-Manin *Methods of Homological Algebra* Ch. III

Intuition [Beginner]

A functor between categories of modules or sheaves often fails to preserve short exact sequences. Take the global-sections functor $Γ$ on sheaves over a topological space: given a short exact sequence of sheaves $0 \to A \to B \to C \to 0$ , the induced sequence on global sections $0 \to Γ A \to Γ B \to Γ C$ is exact on the left, but the rightmost map $Γ B \to Γ C$ need not be surjective. The cokernel of that final map is where global-gluing information is lost — and recovering it systematically is what derived functors do.

Why bother? Because the failure of right-exactness is not a defect to apologise for, it is the carrier of geometric information. The missing surjection is measured by a new group $R^{1} Γ C$ , and that group is exactly the first cohomology of $C$ . The same machine that produces sheaf cohomology produces Ext groups, Tor groups, group cohomology, and the universal-coefficient correction terms — one construction, many applications.

The intuition for Ext is that it parametrises extensions of one module by another. An extension is a short exact sequence $0 \to A \to E \to C \to 0$ with prescribed ends $A$ and $C$ ; the set of such sequences modulo a natural equivalence forms an abelian group called $Ext^{1} (C, A)$ . When $Ext^{1}$ vanishes, every extension splits and $E = A \oplus C$ . When it does not, the nonzero elements are the genuinely non-split extensions.

Visual [Beginner]

A schematic shows a short exact sequence of modules at the top, and the truncated four-term exact sequence under a left-exact functor below, with the failure of the final arrow to be surjective measured by the first derived functor on the right. A second panel shows a non-split extension of $C$ by $A$ — the module $E$ wraps around $A$ inside it without being a direct sum — as a representative element of $Ext^{1} (C, A)$ .

Worked example [Beginner]

Compute $Ext_{Z}^{1} (Z /2, Z)$ , the group that classifies extensions of the integers by $Z /2$ .

Step 1. Build an explicit short resolution. The cyclic group $Z /2$ is presented by the multiplication-by- $2$ map: $0 \to Z 2 Z \to Z /2 \to 0$ . This is a projective resolution of $Z /2$ because $Z$ is a free abelian group and hence projective.

Step 2. Apply the contravariant functor $Hom_{Z} (-, Z)$ to the resolution. The map $2 : Z \to Z$ becomes the map $2 : Hom_{Z} (Z, Z) \to Hom_{Z} (Z, Z)$ , which after the identification $Hom_{Z} (Z, Z) = Z$ is multiplication by $2$ on the integers.

Step 3. Take cohomology of the resulting two-term complex $Z 2 Z$ . The kernel is $0$ (multiplication by $2$ is injective on $Z$ ); this is $Ext^{0} = Hom_{Z} (Z /2, Z) = 0$ . The cokernel is $Z /2 Z$ ; this is $Ext_{Z}^{1} (Z /2, Z) = Z /2$ .

Step 4. Interpret. The group $Ext_{Z}^{1} (Z /2, Z) = Z /2$ has two elements. The zero element corresponds to the split extension $0 \to Z \to Z \oplus Z /2 \to Z /2 \to 0$ . The nonzero element corresponds to the genuinely non-split extension $0 \to Z 2 Z \to Z /2 \to 0$ , the same short exact sequence we started with, now reinterpreted as an extension class.

What this tells us. The single non-split extension of $Z /2$ by $Z$ is the multiplication-by- $2$ sequence itself, and it represents the nontrivial class in a two-element group. The cyclic group of order four $Z /4$ contains $Z /2$ as a subgroup with quotient $Z /2$ , but that is an extension of $Z /2$ by $Z /2$ , not by $Z$ — those classes live in $Ext_{Z}^{1} (Z /2, Z /2) = Z /2$ , where the nonzero class is exactly $Z /4$ as an extension. Ext, computed from a resolution, sees these extension classes directly.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $A$ and $B$ be abelian categories and $F : A \to B$ an additive functor.

Definition (left exact, right exact, exact). $F$ is left exact if for every short exact sequence $0 \to A \to B \to C \to 0$ in $A$ the sequence $0 \to F A \to F B \to F C$ is exact in $B$ . $F$ is right exact if the symmetric statement holds: $F A \to F B \to F C \to 0$ is exact. $F$ is exact if both hold, equivalently if $F$ preserves short exact sequences.

Definition (injective object). An object $I$ of $A$ is injective if the functor $Hom_{A} (-, I)$ is exact, equivalently if every monomorphism $A ↪ B$ and every morphism $A \to I$ extends to a morphism $B \to I$ .

Definition (enough injectives). The category $A$ has enough injectives if every object $A$ admits a monomorphism $A ↪ I$ into some injective object $I$ .

Definition (injective resolution). An injective resolution of an object $A \in A$ is an exact sequence $$ 0 \to A \to I^0 \xrightarrow{d^0} I^1 \xrightarrow{d^1} I^2 \to \cdots $$ with each $I^{n}$ injective. The cochain complex $I^{∙} : I^{0} \to I^{1} \to \dots$ (without the augmentation $A \to I^{0}$ ) is the injective cochain complex attached to the resolution.

Definition (right-derived functor). Let $F : A \to B$ be a left-exact additive functor and assume $A$ has enough injectives. For an object $A$ , choose an injective resolution $A \to I^{∙}$ . The $n$ -th right-derived functor of $F$ at $A$ is $$ (R^n F)(A) := H^n\big( F(I^\bullet) \big), $$ the $n$ -th cohomology of the cochain complex obtained by applying $F$ term-by-term to $I^{∙}$ .

The definition is well-posed: any two injective resolutions of $A$ are chain-homotopy equivalent, $F$ takes chain-homotopy equivalences to chain-homotopy equivalences, and cohomology is invariant under chain homotopy. Hence $(R^{n} F) (A)$ depends only on $A$ up to canonical isomorphism, and a morphism $A \to A^{'}$ in $A$ induces a morphism $(R^{n} F) (A) \to (R^{n} F) (A^{'})$ functorial in $A$ . For $n = 0$ , left-exactness of $F$ identifies $(R^{0} F) (A) = F (A)$ canonically.

Definition (Ext as a derived functor). Let $R$ be a ring and $A$ , $C$ left $R$ -modules. The category $R -Mod$ of $R$ -modules is abelian with enough injectives. The functor $Hom_{R} (C, -) : R -Mod \to Ab$ is left exact and additive. The $n$ -th Ext group is $$ \mathrm{Ext}^n_R(C, A) := R^n \mathrm{Hom}_R(C, -)(A) = H^n\big( \mathrm{Hom}_R(C, I^\bullet) \big), $$ where $A \to I^{∙}$ is any injective resolution. Equivalently, $Ext_{R}^{n} (C, A) = R^{n} Hom_{R} (-, A) (C)$ by computing via a projective resolution of $C$ in place of an injective resolution of $A$ ; the two definitions agree by the balanced-functor property recorded in the Key theorem below.

Definition ( $Ext$ sheaves on a ringed space). Let $(X, O_{X})$ be a ringed space and $F$ , $G$ sheaves of $O_{X}$ -modules. The local-Hom functor $H o m_{O_{X}} (F, -)$ from $O_{X}$ -modules to $O_{X}$ -modules is left exact and additive, and the category of $O_{X}$ -modules has enough injectives (Hartshorne III.2.2). The $n$ -th local Ext sheaf is $$ \mathit{Ext}^n_{\mathcal{O}X}(\mathcal{F}, \mathcal{G}) := R^n \mathcal{H}om{\mathcal{O}X}(\mathcal{F}, -)(\mathcal{G}), $$ computed by applying $\mathcal{H}om{\mathcal{O}X}(\mathcal{F}, -) $t o aninj ec t i v er eso l u t i o n$ \mathcal{G} \to \mathcal{I}^\bullet $an d t ak in g co h o m o l o g y o f t h er es u l t in g co m pl e x o f s h e a v es . T h e * * g l o ba l E x t g r o u p * * i s$ \mathrm{Ext}^n{\mathcal{O}X}(\mathcal{F}, \mathcal{G}) := R^n \mathrm{Hom}{\mathcal{O}_X}(\mathcal{F}, -)(\mathcal{G}) $, w i t h t h e g l o ba l$ \mathrm{Hom}$ in place of the local one.

Counterexamples to common slips

A projective resolution of $C$ computes $Ext_{R}^{n} (C, A)$ via $Hom_{R} (-, A)$ ; an injective resolution of $A$ computes the same group via $Hom_{R} (C, -)$ . The balanced property says the two answers agree, but the resolutions need not be of the same length, and an isomorphism between them is not generally explicit.
The local-Hom functor $H o m_{O_{X}} (F, -)$ on sheaves and the global-Hom functor $Hom_{O_{X}} (F, -)$ have different right-derived functors. The local $Ext^{n}$ is a sheaf; the global $Ext^{n}$ is an abelian group. They are linked by the local-to-global spectral sequence $H^{p} (X, Ext^{q} (F, G)) \Rightarrow Ext^{p + q} (F, G)$ (Hartshorne III.6.4).
Existence of enough injectives in $A$ is essential for the definition of $R^{n} F$ via injective resolutions. The category of finitely-generated modules over a noetherian ring does not have enough injectives in general; one passes to all modules to build the resolution, then restricts.
$Ext_{R}^{1} (C, A)$ is canonically in bijection with equivalence classes of one-fold extensions $0 \to A \to E \to C \to 0$ under the Baer-sum group structure (Yoneda 1960). Two extensions are equivalent if there is a commutative ladder with the identity on $A$ and $C$ and an isomorphism on the middle term. The Yoneda interpretation does not pre-suppose enough projectives or enough injectives, and it extends to higher Ext as equivalence classes of $n$ -fold extensions.

Key theorem with proof [Intermediate+]

Theorem (existence of right-derived functors; Grothendieck Tôhoku §2.2 ^[pending]). Let $A$ be an abelian category with enough injectives, $B$ an abelian category, and $F : A \to B$ a left-exact additive functor. There exists a sequence of additive functors $$ R^n F : \mathcal{A} \to \mathcal{B}, \quad n \ge 0, $$ with $R^{0} F = F$ canonically, such that for every short exact sequence $0 \to A \to B \to C \to 0$ in $A$ there is a long exact sequence $$ 0 \to FA \to FB \to FC \xrightarrow{\delta} R^1 F (A) \to R^1 F (B) \to R^1 F (C) \xrightarrow{\delta} R^2 F (A) \to \cdots $$ natural in the short exact sequence, where each connecting morphism $δ : R^{n} F (C) \to R^{n + 1} F (A)$ is uniquely determined by the construction.

Proof. The argument has four steps. Step 1: choose an injective resolution and define $R^{n} F$ . Step 2: show independence of the resolution. Step 3: construct the connecting morphism $δ$ via a horseshoe lemma. Step 4: verify exactness of the long sequence.

Step 1: definition. For each $A \in A$ , the assumption that $A$ has enough injectives produces an injection $A ↪ I^{0}$ into an injective. Iterating on the cokernel $I^{0} / A ↪ I^{1}$ , on $I^{1} / I^{0} ↪ I^{2}$ , and so on, yields an injective resolution $A \to I^{∙}$ . Apply $F$ term-by-term to the complex $I^{∙}$ (without the augmentation) to obtain a cochain complex $F (I^{∙}) : F (I^{0}) \to F (I^{1}) \to F (I^{2}) \to \dots$ in $B$ . Define $(R^{n} F) (A) := H^{n} (F (I^{∙}))$ , the $n$ -th cohomology of this complex. For $n = 0$ , the kernel $ker (F (I^{0}) \to F (I^{1}))$ is $F (A)$ by left exactness of $F$ applied to the sequence $0 \to A \to I^{0} \to I^{0} / A \to 0$ , so $R^{0} F = F$ .

Step 2: independence. Suppose $A \to I^{∙}$ and $A \to J^{∙}$ are two injective resolutions. A morphism of resolutions over the identity on $A$ exists because $I^{0}$ is injective (lifting the composite $A \to J^{0}$ along $A \to I^{0}$ ), and the construction iterates degree by degree using injectivity of the $I^{n}$ . The induced morphism $I^{∙} \to J^{∙}$ is unique up to chain homotopy by the standard chain-homotopy lifting argument: a second lift differs from the first by a null-homotopic map $I^{∙} \to J^{∙}$ , which factors through a contraction built from injectivity. Applying $F$ to a chain-homotopy equivalence yields a chain-homotopy equivalence in $B$ , and cohomology is a chain-homotopy invariant. Hence $H^{n} (F (I^{∙})) ≅ H^{n} (F (J^{∙}))$ canonically, and $(R^{n} F) (A)$ is well-defined up to canonical isomorphism. The same lifting argument shows that a morphism $A \to A^{'}$ extends to a morphism of resolutions $I^{∙} \to I^{' ∙}$ unique up to chain homotopy, producing the functoriality of $R^{n} F$ .

Step 3: horseshoe lemma and the connecting morphism. Given a short exact sequence $0 \to A \to B \to C \to 0$ and injective resolutions $A \to I_{A}^{∙}$ and $C \to I_{C}^{∙}$ , the horseshoe lemma constructs an injective resolution $B \to I_{B}^{∙}$ with $I_{B}^{n} = I_{A}^{n} \oplus I_{C}^{n}$ as objects of $A$ (not necessarily as resolutions of $B$ via the direct-sum differential) such that the resolutions assemble into a short exact sequence of cochain complexes $$ 0 \to I_A^\bullet \to I_B^\bullet \to I_C^\bullet \to 0, $$ degreewise split. The construction is the dual of the standard horseshoe-lemma construction for projective resolutions; the splitting in $A$ uses that each $I_{A}^{n}$ is injective and hence the rightward arrow $I_{A}^{n} \to I_{B}^{n}$ admits a section. Applying $F$ to this short exact sequence of complexes preserves degreewise splitness, and a degreewise-split short exact sequence of cochain complexes induces a long exact sequence in cohomology with connecting morphism $δ$ defined by the snake lemma in each degree.

Step 4: long exact sequence. The connecting morphism $δ : (R^{n} F) (C) \to (R^{n + 1} F) (A)$ is the snake-lemma connecting morphism for the short exact sequence of complexes $0 \to F (I_{A}^{∙}) \to F (I_{B}^{∙}) \to F (I_{C}^{∙}) \to 0$ . Exactness of the long sequence at every node is the standard snake-lemma corollary for a short exact sequence of cochain complexes. Naturality in the short exact sequence is the naturality of the snake-lemma construction with respect to morphisms of short exact sequences. At $n = 0$ , the long exact sequence begins with $0 \to F (A) \to F (B) \to F (C) δ R^{1} F (A) \to \dots$ , with the first three terms identified with $R^{0}$ of $A, B, C$ respectively via $R^{0} F = F$ , and the connecting morphism producing the first derived contribution. $□$

Bridge. The construction here builds toward 04.04.01 (Riemann-Roch theorem for curves), where the long exact sequence in $H^{p}$ for short exact sequences of sheaves is the engine that propagates the dimension formula from twists of $O$ on $P^{1}$ to line bundles on an arbitrary smooth projective curve; without right-derived functors and the long exact sequence, Riemann-Roch is an unmotivated dimension count. The bridge appears again in 04.03.04 (cohomology of line bundles on projective space), where the closed-form dimension table for $H^{p} (P^{n}, O (d))$ is obtained by Čech cohomology, which is itself a special case of the derived-functor construction once Cartan's comparison theorem identifies the two. Putting these together, the universal feature is that the right-derived functor of a left-exact $F$ is the unique extension of $F$ to a $δ$ -functor exact on short exact sequences; the construction via injective resolutions is one realisation, the Yoneda extension-classes description for $Ext$ is another. The bridge to Ext is direct: $Ext_{R}^{n} (C, A) = R^{n} Hom_{R} (C, -) (A)$ is the special case where $A = R -Mod$ , $B = Ab$ , and $F = Hom_{R} (C, -)$ , and the long exact sequence becomes the long exact sequence of Ext in either variable. The local-to-global spectral sequence $H^{p} (X, Ext^{q}) \Rightarrow Ext^{p + q}$ on a ringed space 04.03.01 is the next layer.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Compute $Ext_{Z}^{n} (Z / m, Z / k)$ for all $n \geq 0$ and arbitrary positive integers $m, k$ . Express your answer in terms of $g cd (m, k)$ .

Hint

Use the projective resolution $0 \to Z m Z \to Z / m \to 0$ and apply $Hom_{Z} (-, Z / k)$ . The kernel and cokernel of multiplication by $m$ on $Z / k$ have orders equal to $g cd (m, k)$ .

Answer

Apply $Hom_{Z} (-, Z / k)$ to the projective resolution $0 \to Z m Z \to Z / m \to 0$ . Using $Hom_{Z} (Z, Z / k) = Z / k$ , the resolution becomes the two-term complex $Z / k m Z / k$ , where the arrow is multiplication by $m$ on $Z / k$ .

The kernel of multiplication by $m$ on $Z / k$ is the subgroup of elements $a \in Z / k$ with $ma \equiv 0 (mod k)$ , namely the cyclic subgroup of order $g cd (m, k)$ generated by $k / g cd (m, k)$ . The cokernel is $Z / k$ modulo the image of multiplication by $m$ , which is the cyclic subgroup of order $k / g cd (m, k)$ generated by $g cd (m, k)$ in $Z / k$ ; the cokernel is then cyclic of order $g cd (m, k)$ .

So $Ext_{Z}^{0} (Z / m, Z / k) = Hom_{Z} (Z / m, Z / k) = Z / g cd (m, k)$ and $Ext_{Z}^{1} (Z / m, Z / k) = Z / g cd (m, k)$ . For $n \geq 2$ , the resolution has zeros in those degrees, so $Ext^{n} = 0$ . Note the symmetry $Ext_{Z}^{1} (Z / m, Z / k) = Ext_{Z}^{1} (Z / k, Z / m)$ , a special case of the balanced-functor property. Rubric: full credit for the order $g cd (m, k)$ in degrees $0$ and $1$ , vanishing above; partial credit for degree $1$ alone.

Exercise 4 (medium, symbolic).

Show that for any abelian group $A$ , the group $Ext_{Z}^{1} (Q, A)$ is a divisible abelian group, and compute it when $A = Z$ .

Hint

$Q$ has an injective resolution $0 \to Q \to Q \to 0$ (taking $Q$ as its own injective hull) — but this is not a projective resolution of $Q$ . Use a projective resolution of $Q$ instead, or the fact that $Ext^{1} (Q, A)$ classifies extensions of $Q$ by $A$ .

Answer

$Q$ is a flat $Z$ -module but not projective. A projective resolution can be built explicitly from the colimit $Q = lim_{n} \frac{1}{n !} Z$ ; the result is that $Ext_{Z}^{1} (Q, A) = A / A$ , where $A$ is a completion construction depending on the divisibility of $A$ .

For $A = Z$ : $Ext_{Z}^{1} (Q, Z) = Z / Z$ , the profinite integers modulo the integers, an uncountable divisible torsion-free group of cardinality continuum. The divisibility follows from the projective-resolution computation: $Q$ is divisible, so for every prime $p$ , multiplication by $p$ on the resolution is invertible, hence multiplication by $p$ on $Ext^{1} (Q, A)$ is surjective for every $A$ and every prime $p$ , which means $Ext^{1} (Q, A)$ is divisible.

The Yoneda interpretation: $Ext_{Z}^{1} (Q, Z) = Z / Z$ records the extensions of $Q$ by $Z$ ; these are the torsion-free abelian groups of rank $1$ with a chosen subgroup isomorphic to $Z$ , and the divisible quotient $Z / Z$ encodes the type-theoretic invariant of such a group. Rubric: full credit for the divisibility argument and identification with $Z / Z$ ; partial credit for divisibility alone.

Exercise 5 (medium, short-answer).

Prove the balanced-functor property of $Ext$ : if $A = R -Mod$ has enough projectives and enough injectives, then computing $Ext_{R}^{n} (C, A)$ via a projective resolution of $C$ and the functor $Hom_{R} (-, A)$ yields the same result as computing via an injective resolution of $A$ and the functor $Hom_{R} (C, -)$ .

Hint

Apply the bifunctor $Hom_{R} (-, -)$ to a projective resolution of $C$ and an injective resolution of $A$ simultaneously, producing a double complex. Show that both spectral sequences of the double complex collapse on the same total cohomology.

Answer

Take a projective resolution $P_{∙} \to C$ and an injective resolution $A \to I^{∙}$ . Form the double complex $E^{p, q} = Hom_{R} (P_{- p}, I^{q})$ with horizontal differential induced by $P_{∙}$ and vertical differential induced by $I^{∙}$ . The associated total complex has two spectral sequences.

First spectral sequence. Take cohomology in the $I$ -direction first. For each fixed $p$ , the row $q \mapsto Hom_{R} (P_{- p}, I^{q})$ is the complex obtained by applying $Hom_{R} (P_{- p}, -)$ to $I^{∙}$ . Since $P_{- p}$ is projective, $Hom_{R} (P_{- p}, -)$ is exact, so the cohomology equals $Hom_{R} (P_{- p}, H^{q} (I^{∙})) = Hom_{R} (P_{- p}, A)$ in degree $0$ and zero in positive degree. The $E_{1}$ page collapses to a single row at $q = 0$ , and the spectral sequence converges to the cohomology of the complex $Hom_{R} (P_{∙}, A)$ , i.e. to $Ext_{R}^{n} (C, A)$ computed via the projective resolution of $C$ .

Second spectral sequence. Take cohomology in the $P$ -direction first. For each fixed $q$ , the column $p \mapsto Hom_{R} (P_{- p}, I^{q})$ is the complex obtained by applying $Hom_{R} (-, I^{q})$ to $P_{∙}$ . Since $I^{q}$ is injective, $Hom_{R} (-, I^{q})$ is exact, so the cohomology equals $Hom_{R} (H_{- p} (P_{∙}), I^{q}) = Hom_{R} (C, I^{q})$ in degree $0$ and zero in negative degree. The $E_{1}$ page collapses to a single column at $p = 0$ , and the spectral sequence converges to the cohomology of the complex $Hom_{R} (C, I^{∙})$ , i.e. to $Ext_{R}^{n} (C, A)$ computed via the injective resolution of $A$ .

Both spectral sequences converge to the cohomology of the same total complex, so the two computations of $Ext_{R}^{n} (C, A)$ agree. Rubric: full credit for the double-complex setup and the two-spectral-sequence collapse argument; partial credit for the double complex without the collapse.

Exercise 6 (medium, short-answer).

State and prove the long exact sequence in $Ext$ in the second variable: for a short exact sequence $0 \to A \to B \to C \to 0$ in $R -Mod$ and a fixed module $M$ , exhibit the long exact sequence $$ 0 \to \mathrm{Hom}_R(M, A) \to \mathrm{Hom}_R(M, B) \to \mathrm{Hom}_R(M, C) \to \mathrm{Ext}^1_R(M, A) \to \mathrm{Ext}^1_R(M, B) \to \mathrm{Ext}^1_R(M, C) \to \mathrm{Ext}^2_R(M, A) \to \cdots $$

Hint

This is the long exact sequence of the right-derived functor $R^{n} Hom_{R} (M, -)$ applied to the short exact sequence in $R -Mod$ . Use the existence theorem from the Key theorem section.

Answer

The functor $F = Hom_{R} (M, -)$ from $R -Mod$ to $Ab$ is additive and left exact. The category $R -Mod$ has enough injectives. By the Key theorem (existence of right-derived functors), the short exact sequence $0 \to A \to B \to C \to 0$ produces a long exact sequence $$ 0 \to F A \to F B \to F C \to R^1 F (A) \to R^1 F (B) \to R^1 F (C) \to R^2 F (A) \to \cdots $$ where $R^{n} F = R^{n} Hom_{R} (M, -) = Ext_{R}^{n} (M, -)$ by definition of Ext. Substituting yields the long exact sequence in the statement.

The connecting morphism $δ : Hom_{R} (M, C) \to Ext_{R}^{1} (M, A)$ has a concrete Yoneda interpretation: given $f : M \to C$ , pull back the short exact sequence $0 \to A \to B \to C \to 0$ along $f$ to obtain $0 \to A \to B \times_{C} M \to M \to 0$ , an extension of $M$ by $A$ . The class of this pulled-back extension in $Ext_{R}^{1} (M, A)$ is $δ (f)$ . The recipe extends to higher $δ$ via the Yoneda-product structure on Ext. Rubric: full credit for the derivation from the Key theorem plus the Yoneda interpretation of $δ$ ; partial credit for the long-exact-sequence statement without the connecting morphism description.

Exercise 7 (hard, short-answer).

Let $X$ be a scheme and $F$ , $G$ quasi-coherent sheaves on $X$ . State the local-to-global Ext spectral sequence and use it to derive the special case $Ext_{O_{X}}^{n} (F, G) = H^{n} (X, H o m_{O_{X}} (F, G))$ when $F$ is locally free of finite rank.

Hint

The local-to-global spectral sequence reads $E_{2}^{p, q} = H^{p} (X, Ext_{O_{X}}^{q} (F, G)) \Rightarrow Ext_{O_{X}}^{p + q} (F, G)$ . For $F$ locally free of finite rank, $Ext^{q}$ vanishes for $q > 0$ .

Answer

The local-to-global Ext spectral sequence is the Grothendieck spectral sequence for the composition $Hom_{O_{X}} (F, -) = Γ (X, -) \circ H o m_{O_{X}} (F, -)$ . The composition is left exact, both factors are left exact, and the local-Hom functor takes injective $O_{X}$ -modules to $Γ$ -acyclic objects (Hartshorne III.6.3 — the local Hom of an injective is flasque, which is acyclic for global sections). The Grothendieck spectral sequence applies: $$ E_2^{p, q} = H^p(X, \mathit{Ext}^q_{\mathcal{O}X}(\mathcal{F}, \mathcal{G})) \Rightarrow \mathrm{Ext}^{p + q}{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}). $$

For $F$ locally free of finite rank, the local-Hom $H o m_{O_{X}} (F, -)$ is an exact functor on $O_{X}$ -modules: locally $F ∣_{U} ≅ O_{X} ∣_{U}^{\oplus r}$ , and $H o m (O_{X}^{\oplus r}, -) = (-)^{\oplus r}$ , which is exact. So $Ext_{O_{X}}^{q} (F, G) = 0$ for $q \geq 1$ , and the $E_{2}$ page concentrates in the single row $q = 0$ . The spectral sequence collapses on $E_{2}$ , giving $$ \mathrm{Ext}^n_{\mathcal{O}X}(\mathcal{F}, \mathcal{G}) = E_2^{n, 0} = H^n(X, \mathcal{H}om{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})). $$ A special case: $F = O_{X}$ , $H o m (O_{X}, G) = G$ , and $Ext_{O_{X}}^{n} (O_{X}, G) = H^{n} (X, G)$ — sheaf cohomology emerges as Ext into the structure sheaf. Rubric: full credit for the spectral sequence, the vanishing of $Ext^{q}$ for locally free $F$ , and the collapse argument; partial credit for the spectral sequence statement alone.

Exercise 8 (hard, short-answer).

State the Yoneda interpretation of $Ext^{1}$ : every element of $Ext_{R}^{1} (C, A)$ corresponds to an equivalence class of short exact sequences $0 \to A \to E \to C \to 0$ , with the zero class being the split extension. Verify the bijection on the example $R = Z$ , $A = Z /2$ , $C = Z /2$ .

Hint

Two extensions $0 \to A \to E \to C \to 0$ and $0 \to A \to E^{'} \to C \to 0$ are equivalent if there is a map $E \to E^{'}$ making the obvious diagram with identities on $A$ and $C$ commute. By the five lemma the map is an isomorphism. The Yoneda group structure is given by Baer sum.

Answer

Yoneda interpretation. For modules $A, C$ over a ring $R$ , the set $Yext^{1} (C, A)$ of equivalence classes of short exact sequences $0 \to A \to E \to C \to 0$ (where two extensions are equivalent if connected by a map of extensions over the identities on $A$ and $C$ ) carries an abelian group structure by Baer sum: the sum of $ξ : 0 \to A \to E \to C \to 0$ and $ξ^{'} : 0 \to A \to E^{'} \to C \to 0$ is constructed as $(ξ + ξ^{'}) := 0 \to A \to (E \times_{C} E^{'}) / Δ_{A} \to C \to 0$ , where $Δ_{A} = {(a, - a) : a \in A}$ is the antidiagonal. The zero element is the split extension $0 \to A \to A \oplus C \to C \to 0$ . The canonical bijection $Yext^{1} (C, A) ≅ Ext_{R}^{1} (C, A)$ (Yoneda 1960) sends an extension to the connecting morphism applied to $1_{C} \in Hom_{R} (C, C)$ .

Verification on $R = Z$ , $A = C = Z /2$ . From Exercise 3 with $m = k = 2$ , $Ext_{Z}^{1} (Z /2, Z /2) = Z /2$ . The two extension classes are: (i) the split extension $0 \to Z /2 \to Z /2 \oplus Z /2 \to Z /2 \to 0$ (zero class); (ii) the non-split extension $0 \to Z /2 \to Z /4 \to Z /2 \to 0$ where the middle term is the cyclic group of order $4$ and the first map sends $1 \mapsto 2 \in Z /4$ (nontrivial class). These are the two abelian groups of order $4$ , both realised as extensions of $Z /2$ by $Z /2$ , and the bijection $Yext^{1} (Z /2, Z /2) ≅ Z /2$ is the natural identification. The Baer sum of $Z /4$ with itself is the split extension, consistent with the order- $2$ structure of the group. Rubric: full credit for the Yoneda statement and the two-extension verification; partial credit for the statement alone.

Advanced results [Master]

Theorem (Yoneda extension interpretation; Yoneda 1960 ^[pending]). For modules $A, C$ over a ring $R$ and integer $n \geq 1$ , the group $Ext_{R}^{n} (C, A)$ is canonically in bijection with the set of equivalence classes of $n$ -fold exact sequences $$ 0 \to A \to E_n \to E_{n-1} \to \cdots \to E_1 \to C \to 0 $$ under the equivalence relation generated by morphisms of $n$ -fold sequences fixing the ends, with the abelian group structure given by Baer-sum-like splice composition. The composition $Ext^{m} (B, A) \times Ext^{n} (C, B) \to Ext^{m + n} (C, A)$ , sending a pair of sequences to their concatenation at $B$ , makes the bigraded family $⨁_{n} Ext_{R}^{n} (-, -)$ into the Yoneda product, and the $Ext$ -algebra of a module $M$ is $\mathrm{Ext}^_R(M, M)$ with Yoneda product as multiplication.*

The Yoneda product is graded-commutative on modules over a commutative ring up to sign, and on a finite-dimensional algebra over a field $k$ , the $Ext$ -algebra $Ext_{R}^{*} (k, k)$ controls the homotopy theory of the module category in the sense of Koszul duality and derived Morita theory.

Theorem (universal $δ$ -functor characterisation). Let $A$ be an abelian category with enough injectives and $F : A \to B$ a left-exact additive functor. The collection ${R^{n} F}_{n \geq 0}$ is a universal $δ$ -functor extending $F$ : for any other $δ$ -functor ${T^{n}}$ extending $F$ (so $T^{0} = F$ and ${T^{n}}$ produces long exact sequences from short exact sequences), there is a unique morphism of $δ$ -functors ${T^{n}} \to {R^{n} F}$ . Equivalently, ${R^{n} F}$ is effaceable: for every object $A$ and every $n \geq 1$ , there is a monomorphism $A ↪ A^{'}$ such that $R^{n} F (A) \to R^{n} F (A^{'})$ is zero. Effaceability characterises ${R^{n} F}$ uniquely among $δ$ -functors extending $F$ .

The effaceability characterisation, originally due to Grothendieck in the Tôhoku paper, is the abstract framework that lets one compute $R^{n} F$ on a particular object $A$ by replacing $A$ with an $F$ -acyclic resolution rather than an injective resolution: any resolution $A \to J^{∙}$ with $R^{n} F (J^{p}) = 0$ for $n \geq 1$ and every $p$ computes $(R^{n} F) (A) = H^{n} (F (J^{∙}))$ . Flasque resolutions on a topological space are $Γ$ -acyclic; fine resolutions on a smooth manifold are $Γ$ -acyclic for the de Rham functor; injective resolutions are universally acyclic by construction.

Theorem (Ext via flat / projective resolutions and Tor duality). Let $R$ be a commutative ring and let $C$ , $A$ be $R$ -modules. The Tor groups $Tor_{n}^{R} (C, A)$ are defined as the left-derived functors of the tensor product, computable via a projective (or flat) resolution of either argument. The pairing $$ \mathrm{Ext}^n_R(C, A) \otimes \mathrm{Tor}^R_n(C, A^\vee) \to \mathrm{Ext}^0_R(A, A^\vee) \cong \mathrm{Hom}R(A, A^\vee) $$ is more delicate: in the setting of finitely-generated modules over a Gorenstein ring, the Auslander-Reiten formula relates the two via the Nakayama functor and exhibits a duality between $\mathrm{Ext}^(-, R) $an d$ \mathrm{Tor}^R(-, R) $. O na r e g u l a r r in g, t h e l oc a l co h o m o l o g y$ H^i_\mathfrak{m}(M) $a t ama x ima l i d e a l$ \mathfrak{m} $i sr e l a t e d t o$ \mathrm{Ext}^_R(R/\mathfrak{m}, M)$ by the local-duality theorem.*

Theorem (local-to-global Ext spectral sequence; Hartshorne III.6.4 ^[pending]). Let $(X, O_{X})$ be a ringed space and $F, G$ sheaves of $O_{X}$ -modules. There is a convergent spectral sequence $$ E_2^{p, q} = H^p(X, \mathit{Ext}^q_{\mathcal{O}X}(\mathcal{F}, \mathcal{G})) \Rightarrow \mathrm{Ext}^{p + q}{\mathcal{O}X}(\mathcal{F}, \mathcal{G}), $$ *the Grothendieck spectral sequence for the composition $\mathrm{Hom}{\mathcal{O}X}(\mathcal{F}, -) = \Gamma(X, -) \circ \mathcal{H}om{\mathcal{O}_X}(\mathcal{F}, -)$.*

When $F$ is locally free of finite rank, $Ext^{q} (F, G) = 0$ for $q \geq 1$ and the spectral sequence degenerates: $Ext_{O_{X}}^{n} (F, G) = H^{n} (X, F^{\lor} \otimes G)$ , where $F^{\lor} = H o m (F, O_{X})$ . The structure-sheaf case $F = O_{X}$ gives $Ext_{O_{X}}^{n} (O_{X}, G) = H^{n} (X, G)$ , identifying sheaf cohomology as Ext into the structure sheaf.

Theorem (Serre duality via Ext; Serre 1955, Hartshorne III.7). Let $X$ be a smooth projective variety of dimension $n$ over a field $k$ with canonical sheaf $ω_{X}$ . For any coherent sheaf $F$ on $X$ and integer $0 \leq i \leq n$ , there is a natural perfect pairing of finite-dimensional $k$ -vector spaces $$ \mathrm{Ext}^i_{\mathcal{O}_X}(\mathcal{F}, \omega_X) \times H^{n - i}(X, \mathcal{F}) \to H^n(X, \omega_X) \cong k. $$ When $F$ is locally free, the local-to-global spectral sequence identifies $Ext^{i} (F, ω_{X}) = H^{i} (X, F^{\lor} \otimes ω_{X})$ , and the pairing becomes the more familiar $H^{i} (X, F^{\lor} \otimes ω_{X}) \times H^{n - i} (X, F) \to k$ . The Ext formulation is the correct one for general coherent (possibly non-locally-free) sheaves.

Theorem (Ext and the derived category). Let $A$ be an abelian category and $D (A)$ its derived category. For objects $A, C \in A$ and integer $n \geq 0$ , $$ \mathrm{Ext}^n_{\mathcal{A}}(C, A) = \mathrm{Hom}_{D(\mathcal{A})}(C, A[n]), $$ where $A [n]$ is the object $A$ placed in cohomological degree $- n$ and shifts in the triangulated structure. The Yoneda product becomes composition of morphisms in $D (A)$ with appropriate shifts, and the long exact sequence in Ext is the long exact sequence attached to a distinguished triangle in $D (A)$ .

The derived-category formulation, due to Verdier in his thesis and developed in Hartshorne Residues and Duality ^[pending] and Gelfand-Manin ^[pending], is the modern setting in which Ext is most naturally studied. Many statements that appear ad hoc in the abelian-category framework — the Yoneda product, the local-to-global spectral sequence, Serre duality — become functorial constructions in the derived category. Grothendieck-Verdier duality on a proper morphism of schemes is the global form of Serre duality and lives natively in $D (A)$ .

Synthesis. The derived-functor construction is the universal mechanism for repairing the failure of an additive functor to be exact. The central insight is that a left-exact functor $F : A \to B$ extends uniquely to a sequence of functors ${R^{n} F}_{n \geq 0}$ characterised by the existence of long exact sequences attached to short exact sequences and by the universal-effaceability property, and the extension is computed concretely via injective resolutions. The bridge between the abstract characterisation (universal $δ$ -functor) and the concrete construction (cohomology of $F (I^{∙})$ for $I^{∙}$ an injective resolution) is the chain-homotopy invariance of cohomology together with the lifting property of injectives. Three apparently distinct constructions — derived functors of $Hom$ (Ext), derived functors of $\otimes$ (Tor), derived functors of $Γ$ (sheaf cohomology) — fit into one framework, and the framework extends through the local-to-global spectral sequence to the sheaf-theoretic local-Hom functor and, in the derived category, to the Yoneda product. Putting these together, the central computational fact is that for a left-exact $F$ with enough injectives, $R^{n} F$ is exact on short exact sequences in $A$ at the cost of exactness in $n$ , and the long exact sequence transports information across $n$ . This bridge appears again in 04.04.01 (Riemann-Roch theorem for curves) where the long exact sequence in cohomology of a short exact sequence of sheaves drives the dimension count, and in 04.03.05 (Serre vanishing and finiteness) where the derived-functor formalism on a projective scheme produces the finite-dimensionality and asymptotic vanishing that anchor the cohomology of coherent sheaves.

The construction generalises in two directions. First, to Tor via projective resolutions in place of injective ones, with the same long-exact-sequence formalism and the same universal-effaceability characterisation; the pair $(Ext, Tor)$ encodes the obstruction to a balanced bifunctor being exact in either variable separately. Second, to the derived category $D (A)$ , where $R^{n} F$ becomes the $n$ -th cohomology of a triangulated functor $R F : D (A) \to D (B)$ and where Ext becomes graded-Hom in the triangulated structure. The derived-category extension is the framework in which Grothendieck-Verdier duality, the six-functor formalism on schemes, and the derived-category formulation of Hodge theory all live; the foundational identity $Ext^{n} (C, A) = Hom_{D (A)} (C, A [n])$ is the bridge between the abelian-category construction and its triangulated home. The synthesis is that derived functors and Ext are not technical conveniences; they are the carriers of the cohomological information that distinguishes algebraic geometry, commutative algebra, and topology from their classical, exact-functor predecessors.

Full proof set [Master]

Proposition (universal $δ$ -functor characterisation), proof. Let ${T^{n}}_{n \geq 0}$ be a $δ$ -functor with $T^{0} = F$ and let ${R^{n} F}$ be the right-derived functors. We construct a morphism of $δ$ -functors ${T^{n}} \to {R^{n} F}$ by induction on $n$ .

At $n = 0$ , the identity $T^{0} = F = R^{0} F$ is the base case. Suppose by induction that morphisms $ϕ^{k} : T^{k} \to R^{k} F$ are defined for $0 \leq k \leq n$ and commute with the connecting morphisms attached to every short exact sequence. To construct $ϕ^{n + 1}$ at an object $A$ , embed $A$ in an injective: $0 \to A \to I \to Q \to 0$ exact with $I$ injective. The associated long exact sequences read $$ \cdots \to T^n(I) \to T^n(Q) \xrightarrow{\delta_T} T^{n+1}(A) \to T^{n+1}(I) \to \cdots $$ and $$ \cdots \to R^n F(I) \to R^n F (Q) \xrightarrow{\delta_R} R^{n+1} F (A) \to R^{n+1} F (I) \to \cdots $$ The right-derived functors satisfy $R^{k} F (I) = 0$ for $k \geq 1$ on any injective $I$ (an injective $I$ admits the resolution $I \to I \to 0 \to \dots$ , and the derived functors vanish above degree $0$ ). Hence $δ_{R} : R^{n} F (Q) \to R^{n + 1} F (A)$ is surjective (and an isomorphism for $n \geq 1$ , where $R^{n} F (I) = 0$ on both sides).

For $n = 0$ : $δ_{T} : F (Q) \to T^{1} (A)$ and $δ_{R} : F (Q) \to R^{1} F (A)$ both have source $F (Q)$ identifying via $ϕ^{0} = id$ . Define $ϕ^{1}$ at $A$ by sending $δ_{T} (q) \mapsto δ_{R} (ϕ^{0} (q)) = δ_{R} (q)$ . This is well-defined: if $δ_{T} (q) = δ_{T} (q^{'})$ , then $q - q^{'}$ is in the image of $F (I) \to F (Q)$ , and applying $δ_{R}$ to the same difference gives zero by exactness of the lower sequence at $R^{1} F (A)$ . So $ϕ^{1} (A)$ is a well-defined homomorphism.

For $n \geq 1$ : $δ_{R}$ is an isomorphism, so $ϕ^{n + 1} (A) := δ_{R} \circ ϕ^{n} \circ δ_{T}^{- 1}$ is well-defined on $T^{n + 1} (A)$ via the surjective $δ_{T}$ (the kernel is in $T^{n} (I)$ , which maps via $ϕ^{n}$ to $R^{n} F (I) = 0$ ). Naturality in $A$ and compatibility with connecting morphisms follow from the naturality of all involved snake-lemma constructions.

Uniqueness: any other morphism $ψ^{n + 1}$ of $δ$ -functors must satisfy $ψ^{n + 1} \circ δ_{T} = δ_{R} \circ ψ^{n}$ on $T^{n} (Q)$ ; if $ψ^{n} = ϕ^{n}$ , then $ψ^{n + 1} \circ δ_{T} = δ_{R} \circ ϕ^{n} = ϕ^{n + 1} \circ δ_{T}$ , and since $δ_{T}$ is surjective, $ψ^{n + 1} = ϕ^{n + 1}$ . $□$

Proposition ( $R^{n} F (I) = 0$ for $I$ injective and $n \geq 1$ ), proof. Let $I$ be injective in $A$ . The resolution $I \to I$ (with $I$ in degree zero and zeros above) is an injective resolution of $I$ , since $I$ injects into itself by the identity, and the quotient is zero, which injects vacuously into zero, and so on. Applying $F$ gives the cochain complex with $F (I)$ in degree zero and zero in positive degree. Cohomology in positive degree is zero, so $R^{n} F (I) = 0$ for $n \geq 1$ . $□$

Proposition (long exact sequence in Ext, second variable), proof. Let $0 \to A \to B \to C \to 0$ be a short exact sequence in $R -Mod$ , and let $M \in R -Mod$ be fixed. Choose an injective resolution $A \to I_{A}^{∙}$ , $B \to I_{B}^{∙}$ , $C \to I_{C}^{∙}$ assembled via the horseshoe lemma into a short exact sequence of complexes $0 \to I_{A}^{∙} \to I_{B}^{∙} \to I_{C}^{∙} \to 0$ that is degreewise split. Apply the functor $Hom_{R} (M, -)$ degreewise: degreewise splitness implies that the resulting sequence of cochain complexes $0 \to Hom_{R} (M, I_{A}^{∙}) \to Hom_{R} (M, I_{B}^{∙}) \to Hom_{R} (M, I_{C}^{∙}) \to 0$ is itself degreewise short exact in $Ab$ . The snake-lemma long exact sequence in cohomology then produces the long exact sequence in $Ext_{R}^{n} (M, -)$ stated. Naturality follows from the naturality of the horseshoe lemma and the snake lemma. $□$

Proposition ( $Ext_{R}^{1} (C, A)$ classifies extensions; Yoneda 1960), proof sketch. Define a map $Yext^{1} (C, A) \to Ext_{R}^{1} (C, A)$ as follows: an extension $ξ : 0 \to A \to E \to C \to 0$ produces a connecting morphism $δ_{ξ} : Hom_{R} (C, C) \to Ext_{R}^{1} (C, A)$ in the long exact sequence for $Hom_{R} (C, -)$ . The image $δ_{ξ} (1_{C})$ is the class of $ξ$ .

For the inverse, take an element $α \in Ext_{R}^{1} (C, A)$ . Choose a projective resolution $\dots \to P_{1} \to P_{0} \to C \to 0$ of $C$ . The Ext class $α$ corresponds to a homomorphism $P_{1} \to A$ killing the image of $P_{2} \to P_{1}$ , defined up to maps factoring through $P_{0} \to P_{1}$ . Form the pushout of $P_{1} \to A$ along $P_{1} \to P_{0}$ : this is a module $E$ with a map $A \to E$ (the second pushout leg) and an induced surjection $E \to C$ . The kernel of $E \to C$ is the image of $A \to E$ , and the resulting sequence $0 \to A \to E \to C \to 0$ is the extension assigned to $α$ .

Equivalence of extensions corresponds to equivalence of representing homomorphisms $P_{1} \to A$ up to the boundary, so the construction descends to the bijection. The Baer sum on $Yext^{1}$ corresponds to addition in $Ext_{R}^{1} (C, A)$ by direct computation on the pushout description. $□$

Connections [Master]

Sheaf cohomology 04.03.01. Sheaf cohomology $H^{n} (X, F)$ on a topological space or scheme is the right-derived functor $R^{n} Γ$ of the global-sections functor $Γ : Sh (X) \to Ab$ , computed via injective resolutions in the category of sheaves of abelian groups (or $O_{X}$ -modules in the ringed-space setting). Every fact about sheaf cohomology — long exact sequence, finite-dimensionality on a projective scheme, vanishing in degrees above the dimension — is a special case of the derived-functor formalism developed here. The bridge from local Ext to sheaf cohomology runs through the local-to-global spectral sequence and the identification $Ext_{O_{X}}^{n} (O_{X}, G) = H^{n} (X, G)$ .
Coherent sheaf 04.06.02. Coherent sheaves on a noetherian scheme form an abelian category with enough injectives (in the category of all $O_{X}$ -modules), and the local Ext sheaves $Ext_{O_{X}}^{n} (F, G)$ for coherent $F, G$ are themselves coherent (Hartshorne III.6.8). The Ext groups parametrise extensions of one coherent sheaf by another and are the geometric carriers of obstruction theory: deformation problems land in $Ext^{1}$ and obstructions to extending deformations land in $Ext^{2}$ . This is the foundation of moduli theory of sheaves.
Cohomology of line bundles on projective space 04.03.04. The explicit cohomology table $dim H^{p} (P^{n}, O (d))$ is the foundational example of a right-derived-functor computation. The intermediate vanishing $H^{p} = 0$ for $0 < p < n$ illustrates the derived-functor framework, and the Serre-duality pairing $H^{n} (P^{n}, O (d)) ≅ H^{0} (P^{n}, O (- d - n - 1))^{*}$ is the Ext-duality $Ext^{n} (O (d), ω_{P^{n}}) ≅ H^{0} (O (d))^{*}$ in disguise.
Sheaf 04.01.01. The category of sheaves of abelian groups on a topological space and the category of $O_{X}$ -modules on a ringed space are the source categories where the derived-functor formalism is most heavily applied in algebraic geometry. The existence of enough injectives in both categories (Grothendieck 1957) is what makes the construction work; without it, derived functors would require either Čech resolutions (which require finite covers) or fine / soft / flasque resolutions (which work on paracompact spaces).
Higher direct images and base change 04.03.07. The relative form of the present unit's derived-functor construction. For a morphism $f : X \to S$ , the right-derived functors $R^{q} f_{*} : Sh (X) \to Sh (S)$ of the left-exact pushforward $f_{*}$ are computed by exactly the injective-resolution recipe given here, and the local-to-global Ext spectral sequence built in this unit is one specialisation of the Grothendieck composite-functor spectral sequence that 04.03.07 develops in its general form. The two spectral sequences are siblings: the Leray spectral sequence $H^{p} (S, R^{q} f_{*} F) \Rightarrow H^{p + q} (X, F)$ comes from $Γ_{X} = Γ_{S} \circ f_{*}$ with $f_{*}$ taking injectives to flasque (hence $Γ_{S}$ -acyclic) sheaves, and the local-to-global Ext spectral sequence $H^{p} (X, Ext^{q} (F, G)) \Rightarrow Ext^{p + q} (F, G)$ developed here comes from $Ext^{∙} = Γ_{X} \circ H o m (F, -)$ with $H o m$ of an injective flasque. The cohomology-and-base-change theorem in 04.03.07 is the relative form of the Ext-and-base-change comparison and depends on the same flat-tensor-product preservation of finite-free-module cohomology that the universal-coefficient theorem records in the absolute case.
Hypercohomology of a complex of sheaves 03.04.14. The derived-functor formalism developed here for a single sheaf or single object extends to complexes of sheaves. Hypercohomology is the right-derived functor $R^{n} Γ : D^{+} (Sh (X)) \to Ab$ of global sections, applied to a bounded-below complex $K^{∙}$ instead of a single $F$ , with the resolution upgraded from a single injective resolution $F \to I^{∙}$ to a Cartan-Eilenberg resolution $K^{∙} \to I^{∙, ∙}$ — a double complex of injectives whose total complex is quasi-isomorphic to $K^{∙}$ . The two hypercohomology spectral sequences (filtration-by-rows and filtration-by-columns of the double complex) are exactly the analogues of the local-to-global Ext spectral sequence built here, with the input upgraded from a single $Ext^{q}$ functor to a chain $H^{q} (K^{∙})$ . Every concrete sheaf-theoretic spectral sequence — Leray, de Rham, Dolbeault, Hodge-to-de-Rham, Frölicher — is a hypercohomology spectral sequence for some specific complex of sheaves. The unit 03.04.14 is the direct extension of the present unit's machinery from $D^{+} (A)$ -objects in a single position to bounded-below complexes, with the de Rham theorem $H^{n} (X, Ω_{X}^{∙}) = H^{n} (X, \underline{R})$ and Deligne cohomology as its load-bearing applications.

Historical & philosophical context [Master]

The systematic development of derived functors begins with the 1956 Princeton monograph Homological Algebra by Henri Cartan and Samuel Eilenberg ^[pending]. Cartan and Eilenberg defined Ext and Tor as derived functors of $Hom$ and $\otimes$ respectively, using projective resolutions of modules over a ring, and established the universal-coefficient theorem in topology as a Tor-and-Ext consequence of the derived-functor formalism. The book unified disparate constructions — Ext groups arising in the universal coefficient theorem, the Tor groups arising in flat-base-change for tensor products, the cohomology of groups defined by Eilenberg-MacLane, and the cohomology of Lie algebras defined by Chevalley-Eilenberg — under a single framework.

Alexander Grothendieck's 1957 Tôhoku paper Sur quelques points d'algèbre homologique (Tôhoku Mathematical Journal 9, 119--221) ^[pending] extended the formalism from module categories to arbitrary abelian categories with enough injectives, and applied it to sheaf cohomology on topological spaces and schemes. Grothendieck's contribution was twofold: the axiomatic abelian-category framework (with the AB-axiom hierarchy for various exactness conditions) and the proof that the category of sheaves of $O_{X}$ -modules on a ringed space has enough injectives. The latter is the technical foundation of all derived-functor cohomology in algebraic geometry, and the Tôhoku paper is universally regarded as the founding document of modern homological algebra.

The Yoneda interpretation of $Ext^{n}$ as equivalence classes of $n$ -fold extensions, developed by Nobuo Yoneda in On Ext and exact sequences (J. Fac. Sci. Univ. Tokyo 1960) ^[pending], gives an alternative construction free of resolutions: $Ext^{n}$ is defined directly from the abelian-category data of $A$ without choosing injective or projective resolutions, and the Yoneda composition $Ext^{m} \otimes Ext^{n} \to Ext^{m + n}$ provides the algebra structure that subsequent developments (Koszul duality, $A_{\infty}$ -algebra structures, derived Morita theory) exploit. The Yoneda framework requires neither enough projectives nor enough injectives and is therefore the correct one for categories where resolutions are unavailable.

The derived-category formulation, developed by Jean-Louis Verdier in his 1967 thesis (published 1996 as Des catégories dérivées des catégories abéliennes, Astérisque 239), reformulates derived functors as triangulated functors between derived categories, and rephrases $Ext^{n} (C, A)$ as $Hom_{D (A)} (C, A [n])$ in the derived category. Verdier's setup is the framework in which Grothendieck-Verdier duality on a proper morphism of schemes (the global form of Serre duality), the six-functor formalism on schemes, and the modern formulation of Hodge theory and étale cohomology all live. Robin Hartshorne's Residues and Duality (Springer LNM 20, 1966) ^[pending] is the canonical reference for the derived-category approach to algebraic geometry duality theory, and Sergei Gelfand and Yuri Manin's Methods of Homological Algebra (Springer 1988, 2nd ed. 2003) ^[pending] gives the modern textbook treatment of derived categories and triangulated structures, with derived functors and Ext as primary examples.

Bibliography [Master]

@book{CartanEilenberg1956,
  author    = {Cartan, Henri and Eilenberg, Samuel},
  title     = {Homological Algebra},
  publisher = {Princeton University Press},
  series    = {Princeton Mathematical Series},
  volume    = {19},
  year      = {1956}
}

@article{GrothendieckTohoku1957,
  author    = {Grothendieck, Alexander},
  title     = {Sur quelques points d'alg{\`e}bre homologique},
  journal   = {T{\^o}hoku Mathematical Journal},
  volume    = {9},
  year      = {1957},
  pages     = {119--221}
}

@article{Yoneda1960,
  author    = {Yoneda, Nobuo},
  title     = {On Ext and exact sequences},
  journal   = {Journal of the Faculty of Science, University of Tokyo, Section I},
  volume    = {8},
  year      = {1960},
  pages     = {507--576}
}

@book{HartshorneAG,
  author    = {Hartshorne, Robin},
  title     = {Algebraic Geometry},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {52},
  year      = {1977}
}

@book{HartshorneResiduesDuality,
  author    = {Hartshorne, Robin},
  title     = {Residues and Duality},
  publisher = {Springer-Verlag},
  series    = {Lecture Notes in Mathematics},
  volume    = {20},
  year      = {1966}
}

@book{Weibel1994,
  author    = {Weibel, Charles A.},
  title     = {An Introduction to Homological Algebra},
  publisher = {Cambridge University Press},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {38},
  year      = {1994}
}

@book{GelfandManin2003,
  author    = {Gelfand, Sergei I. and Manin, Yuri I.},
  title     = {Methods of Homological Algebra},
  publisher = {Springer-Verlag},
  series    = {Springer Monographs in Mathematics},
  edition   = {2},
  year      = {2003}
}

@book{VerdierDerivedCategories,
  author    = {Verdier, Jean-Louis},
  title     = {Des cat{\'e}gories d{\'e}riv{\'e}es des cat{\'e}gories ab{\'e}liennes},
  publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
  series    = {Ast{\'e}risque},
  volume    = {239},
  year      = {1996}
}

@article{EilenbergMacLane1942,
  author    = {Eilenberg, Samuel and MacLane, Saunders},
  title     = {Group extensions and homology},
  journal   = {Annals of Mathematics},
  volume    = {43},
  year      = {1942},
  pages     = {757--831}
}

@article{Buchsbaum1955,
  author    = {Buchsbaum, David A.},
  title     = {Exact categories and duality},
  journal   = {Transactions of the American Mathematical Society},
  volume    = {80},
  year      = {1955},
  pages     = {1--34}
}

Prerequisites

04.03.01
04.06.02
04.01.01

Tier anchors

beginner: Weibel *An Introduction to Homological Algebra* Ch. 1 informal — modules and the failure of Hom to preserve exactness
intermediate: Hartshorne *Algebraic Geometry* §III.1 (derived functors), §III.6 (Ext); Weibel Ch. 2 and §3.3
master: Cartan-Eilenberg *Homological Algebra* (1956) Ch. III, V; Grothendieck 1957 *Sur quelques points d'algèbre homologique* (Tôhoku Math. J. 9); Hartshorne *Residues and Duality* Ch. I; Gelfand-Manin *Methods of Homological Algebra* Ch. III

References

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Hartshorne — Algebraic Geometry · §III.1 derived functors of left-exact functors via injective resolutions; §III.2 cohomology of sheaves as $R^i\Gamma$; §III.6 Ext groups and Ext sheaves on a ringed space
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Cartan-Eilenberg — Homological Algebra · Princeton University Press 1956; Ch. III (projective and injective modules), Ch. V (derived functors), Ch. VI (Ext and Tor)
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Grothendieck — Sur quelques points d'algèbre homologique · Tôhoku Mathematical Journal 9 (1957) 119--221; §1.10 abelian categories, §2 derived functors via injective resolutions in an abelian category with enough injectives, §3 application to sheaf cohomology
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Weibel — An Introduction to Homological Algebra · Cambridge Studies in Advanced Mathematics 38, 1994; Ch. 2 derived functors, §3.3 Ext via Yoneda, Ch. 10 derived categories
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Gelfand-Manin — Methods of Homological Algebra · Springer 2003; Ch. III derived functors and triangulated categories; the modern derived-category perspective on Ext
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Yoneda — On Ext and exact sequences · J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960) 507--576; originator of Ext as equivalence classes of $n$-fold extensions

Reviewer

TBD

Estimated time

beginner: 18m
intermediate: 45m
master: 75m