04.03.18 · algebraic-geometry / cohomology

t-Structure on a triangulated category — heart and truncations

shipped3 tiersLean: none

Anchor (Master): Beilinson-Bernstein-Deligne *Faisceaux pervers* (*Astérisque* 100, 1982) §§1.3--1.4; Verdier *Catégories dérivées* (*Astérisque* 239, 1996) §3; Bondal-Kapranov 1989 *Math. USSR Izv.* 35 (representability for derived categories); Bondal-Orlov 2001 *Compositio Math.* 125 (reconstruction theorem); Gelfand-Manin *Methods of Homological Algebra* Ch. IV

Intuition Beginner

A triangulated category records homological data without a notion of degree. The objects of the derived category $D (A)$ of an abelian category $A$ are chain complexes up to quasi-isomorphism, and their cohomology objects sit in specific degrees, but the triangulated structure alone forgets where the cohomology lives — a shift $X [n]$ moves an object up by $n$ degrees, and the triangulated axioms treat all shifts on equal footing. A t-structure is the extra piece of data that picks out which objects are "negative-degree" and which are "positive-degree", recovering the missing notion of degree without changing the triangulated structure.

The data of a t-structure on a triangulated category $D$ is a pair of full subcategories $(D^{\leq 0}, D^{\geq 0})$ — objects concentrated in non-positive degrees and objects concentrated in non-negative degrees — satisfying three compatibility axioms. The first says the subcategories are stable under the appropriate shift; the second says there is no nontrivial homomorphism from a non-positive object to a strictly positive object; the third says every object decomposes via a truncation triangle into a non-positive part and a strictly positive part. The intersection $D^{♡} = D^{\leq 0} \cap D^{\geq 0}$ , called the heart, turns out to be an honest abelian category — the t-structure reconstructs an abelian category sitting inside the triangulated one.

The everyday analogy is a Dewey-decimal classification system overlaid on a library. The library is the triangulated category — books of every subject sit together with no inherent ordering. The t-structure is the catalogue that says which shelf each book belongs on. The heart is the reference desk: the books whose call numbers are exactly zero, the ones at the centre of the system. Different t-structures on the same library reorganise the same books differently, and the choice of t-structure encodes a choice of perspective on the same underlying homological data.

Visual Beginner

A diagram showing a horizontal axis labelled by integer degrees, with the subcategory $D^{\leq 0}$ filling the left half (objects with cohomology in non-positive degrees) and $D^{\geq 0}$ filling the right half (cohomology in non-negative degrees), with the overlap at degree $0$ being the heart. Above the diagram is the truncation triangle $τ_{\leq 0} X \to X \to τ_{\geq 1} X \to (τ_{\leq 0} X) [1]$ , showing how every object $X$ decomposes via the truncations.

The picture captures the structural shape: a t-structure splits the triangulated category into two halves whose overlap is an abelian category, and every object has a canonical truncation triangle that decomposes it relative to the cut. A reader who internalises this picture will recognise the same template every time a t-structure appears — pair of subcategories, no Hom between the strict halves, every object truncates.

Worked example Beginner

Compute the truncation functors and the heart for the standard t-structure on the derived category $D (Ab)$ of abelian groups. The result will recover the classical identification that the cohomology objects of a chain complex live in $Ab$ .

Step 1. Identify the subcategories. The standard t-structure on $D (Ab)$ has $D^{\leq 0}$ = complexes $C^{∙}$ with $H^{n} (C^{∙}) = 0$ for all $n \geq 1$ , and $D^{\geq 0}$ = complexes $C^{∙}$ with $H^{n} (C^{∙}) = 0$ for all $n \leq - 1$ . So $D^{\leq 0}$ is the complexes whose cohomology vanishes in positive degrees, and $D^{\geq 0}$ is the complexes whose cohomology vanishes in negative degrees.

Step 2. Identify the heart. The heart $D^{♡} = D^{\leq 0} \cap D^{\geq 0}$ is the complexes whose cohomology is concentrated in degree $0$ — equivalently, complexes that are quasi-isomorphic to a single abelian group placed in degree $0$ . The heart is therefore equivalent to the category $Ab$ itself, via the functor that sends an abelian group $A$ to the complex with $A$ in degree $0$ and zero elsewhere.

Step 3. Compute the truncations on a sample complex. Take $C^{∙} = \dots \to 0 \to Z \cdot 2 Z \to Z /3 \to 0 \to \dots$ with $Z$ in degrees $- 1$ and $0$ and $Z /3$ in degree $1$ . The cohomology is $H^{- 1} (C^{∙}) = ker (\cdot 2) = 0$ , $H^{0} (C^{∙}) = Z /2 \cdot Z / im (\cdot 2) \cap Z = Z /2$ (since the kernel of $Z \to Z /3$ is $3 Z$ , the cohomology in degree $0$ is $3 Z /2 Z$ which equals $Z / (g cd (2, 3)) = Z /1 = 0$ — let us simplify the example: take instead the complex $0 \to Z \cdot 2 Z \to 0$ with $Z$ in degrees $0$ and $1$ ). Cohomology: $H^{0} = Z$ (the kernel), $H^{1} = Z /2$ (the cokernel).

Step 4. Apply the truncations. The truncation $τ_{\leq 0} C^{∙}$ is the complex with $Z$ in degree $0$ and zero elsewhere (quasi-isomorphic to $H^{0} (C^{∙}) = Z$ in degree $0$ ). The truncation $τ_{\geq 1} C^{∙}$ is the complex with $Z /2$ in degree $1$ and zero elsewhere. The truncation triangle reads $Z [0] \to C^{∙} \to (Z /2) [- 1] \to Z [1]$ , where the connecting morphism encodes the extension data: the short exact sequence $0 \to Z \to Z \to Z /2 \to 0$ comes back as the boundary map $Z /2 \to Z [1]$ representing this extension class.

What this tells us. For the standard t-structure on the derived category of abelian groups, the heart is the original category of abelian groups, the truncations extract the cohomology objects in each range of degrees, and the truncation triangle records the canonical extension of $τ_{\leq 0}$ by $τ_{\geq 1}$ . The single derived-category object $C^{∙}$ carries both pieces of cohomological information at once, and the t-structure is what lets you read them off separately.

Check your understanding Beginner

Exercise (easy, multiple choice).

A t-structure on a triangulated category $D$ is which of the following?

A. A single full subcategory $D^{'} \subset D$ closed under shifts B. A pair of full subcategories $(D^{\leq 0}, D^{\geq 0})$ satisfying the BBD axioms C. A direct-sum decomposition $D = D_{1} \oplus D_{2}$ D. A symmetric monoidal structure on $D$

Hint

The standard t-structure on $D (A)$ splits complexes into two halves by where their cohomology lives.

Answer

B. A t-structure on a triangulated category $D$ is a pair of full strict subcategories $(D^{\leq 0}, D^{\geq 0})$ satisfying the three BBD axioms: stability under shift, vanishing of $Hom (D^{\leq 0}, D^{\geq 1}) = 0$ , and existence of a truncation triangle for every object. Option A captures only part of the data — a single subcategory does not produce a heart or truncations. Option C describes a different decomposition pattern (block-diagonal categories) that does not produce an abelian heart. Option D is the orthogonal symmetric-monoidal structure, which is independent of the t-structure data.

Exercise (easy, true-false).

True or false: the heart $D^{♡} = D^{\leq 0} \cap D^{\geq 0}$ of a t-structure on a triangulated category is always an abelian category.

Hint

This is one of the foundational theorems of the BBD framework — it says the t-structure axioms force the intersection to carry an abelian structure even though the ambient category is only triangulated.

Answer

True. The heart of any t-structure on a triangulated category is an abelian category, by the BBD §1.3.6 theorem. Kernels and cokernels in the heart are constructed from the truncation triangles of morphisms; the proof uses the t-structure axioms (the vanishing of $Hom (D^{\leq 0}, D^{\geq 1})$ in particular) to verify the abelian-category axioms. This is one of the most useful features of the t-structure framework — it reconstructs an abelian category from triangulated data.

Exercise (easy, multiple choice).

For the standard t-structure on $D (Ab)$ , the heart is:

A. The zero category (zero object only) B. The category $Ab$ of abelian groups in degree $0$ C. The category of all chain complexes D. The category of injective abelian groups

Hint

The heart consists of complexes whose cohomology is concentrated in degree $0$ — these are quasi-isomorphic to a single abelian group placed in degree $0$ .

Answer

B. The heart of the standard t-structure on $D (Ab)$ is equivalent to the category $Ab$ of abelian groups, via the functor $A \mapsto A [0]$ that places $A$ in degree $0$ . Option A would correspond to a degenerate t-structure with empty heart, which is not the standard one. Option C is the entire derived category, not the heart. Option D is a particular full subcategory of $Ab$ but is not the heart of the standard t-structure (the heart is all of $Ab$ , not just the injectives).

Formal definition Intermediate+

Let $D$ be a triangulated category with shift functor $[1]$ . A t-structure on $D$ is a pair $(D^{\leq 0}, D^{\geq 0})$ of full strict subcategories satisfying the three BBD axioms.

Definition (t-structure, Beilinson-Bernstein-Deligne 1982 §1.3.1). Write $D^{\leq n} := D^{\leq 0} [- n]$ and $D^{\geq n} := D^{\geq 0} [- n]$ for $n \in Z$ . A t-structure on $D$ is a pair of full strict subcategories $(D^{\leq 0}, D^{\geq 0})$ satisfying:

(Shift) $D^{\leq - 1} \subseteq D^{\leq 0}$ and $D^{\geq 1} \subseteq D^{\geq 0}$ . Equivalently, the subcategories are stable under the shifts that preserve their direction.
(Orthogonality) $Hom_{D} (X, Y) = 0$ for all $X \in D^{\leq 0}$ and $Y \in D^{\geq 1}$ . The two halves have no nontrivial morphisms from below to above.
(Truncation triangle) For every $X \in D$ , there exist objects $A \in D^{\leq 0}$ and $B \in D^{\geq 1}$ together with a distinguished triangle $A \to X \to B \to A [1]$ .

Definition (heart). The heart of the t-structure is $$ \mathcal{D}^\heartsuit := \mathcal{D}^{\le 0} \cap \mathcal{D}^{\ge 0}. $$ The BBD theorem (§1.3.6) asserts that $D^{♡}$ is an abelian category, with kernels and cokernels constructed from truncation triangles.

Definition (truncation functors). The objects $A$ and $B$ in the truncation triangle are unique up to canonical isomorphism, and the assignments $X \mapsto A$ and $X \mapsto B$ extend to functors $$ \tau_{\le 0} : \mathcal{D} \to \mathcal{D}^{\le 0}, \qquad \tau_{\ge 1} : \mathcal{D} \to \mathcal{D}^{\ge 1}, $$ called the truncation functors. They are right adjoint and left adjoint respectively to the inclusions $D^{\leq 0} ↪ D$ and $D^{\geq 1} ↪ D$ . By iteration one obtains $τ_{\leq n} := τ_{\leq 0} [n] \circ [- n]$ and $τ_{\geq n} := τ_{\geq 1} [n - 1] \circ [- (n - 1)]$ for all $n \in Z$ .

Definition (cohomological functor). The composition $$ H^0_t := \tau_{\le 0} \circ \tau_{\ge 0} = \tau_{\ge 0} \circ \tau_{\le 0} : \mathcal{D} \to \mathcal{D}^\heartsuit $$ is the $0$ -th cohomology functor of the t-structure. The two compositions agree up to canonical natural isomorphism. The $n$ -th cohomology is $H_{t}^{n} := H_{t}^{0} \circ [n] : D \to D^{♡}$ , and the family ${H_{t}^{n}}_{n \in Z}$ is a cohomological functor in the BBD sense: every distinguished triangle in $D$ produces a long exact sequence in $D^{♡}$ .

Definition (t-exact functor). Let $D, D^{'}$ be triangulated categories with t-structures and let $F : D \to D^{'}$ be a triangulated functor. $F$ is left t-exact if $F (D^{\geq 0}) \subseteq D^{' \geq 0}$ , right t-exact if $F (D^{\leq 0}) \subseteq D^{' \leq 0}$ , and t-exact if it is both. A t-exact functor restricts to a (necessarily exact) functor $D^{♡} \to D^{'♡}$ between the hearts.

Definition (standard t-structure on $D (A)$ ). For an abelian category $A$ , the standard t-structure on the derived category $D (A)$ has $$ D(\mathcal{A})^{\le 0} := { C^\bullet \in D(\mathcal{A}) : H^n(C^\bullet) = 0 \text{ for all } n \ge 1 }, $$ $$ D(\mathcal{A})^{\ge 0} := { C^\bullet \in D(\mathcal{A}) : H^n(C^\bullet) = 0 \text{ for all } n \le -1 }. $$ The heart is equivalent to $A$ , via $A \mapsto A [0]$ , and the cohomology functors $H_{t}^{n}$ on this t-structure coincide with the classical cohomology functors $H^{n} : D (A) \to A$ .

Definition (truncation triangle). For any $X \in D$ and any $n \in Z$ , the canonical truncation triangle is $$ \tau_{\le n} X \to X \to \tau_{\ge n+1} X \to (\tau_{\le n} X)[1], $$ distinguished by construction. The truncation triangles assemble into a tower of canonical morphisms $\dots \to τ_{\leq n} X \to τ_{\leq n + 1} X \to \dots \to X$ exhibiting $X$ as the union (in an appropriate sense) of its non-positive truncations.

Counterexamples to common slips

A t-structure is not uniquely determined by its heart. The heart $D^{♡}$ is an abelian category but the original triangulated category $D$ recovers from $D^{♡}$ only in special cases (Bondal-Orlov-style reconstruction). Different t-structures on the same triangulated category can have non-equivalent hearts; the perverse and standard t-structures on $D_{c}^{b} (X)$ for a complex algebraic variety $X$ are the canonical examples.
The orthogonality axiom $Hom (D^{\leq 0}, D^{\geq 1}) = 0$ does not say morphisms between the two halves are zero — it says morphisms from non-positive to strictly positive vanish. Morphisms from $D^{\leq 0}$ to $D^{\geq 0}$ are generally nonzero, and the part landing in the heart is what produces the cohomology functor.
The truncation functors $τ_{\leq n}$ and $τ_{\geq n}$ are adjoints to the inclusions, but they are not triangulated functors in general. They preserve distinguished triangles only in restricted situations; the cohomological functor $H_{t}^{n}$ is a triangulated cohomological functor (taking distinguished triangles to long exact sequences in the heart), but the truncations themselves do not preserve triangles.
The standard t-structure on $D (A)$ exists for any abelian category $A$ , but exotic t-structures (perverse, stability conditions, tilting) require additional data and are not produced automatically from the triangulated structure. Identifying a t-structure with the standard one is incorrect outside specific contexts.

Key theorem with proof Intermediate+

Theorem (the heart is abelian; Beilinson-Bernstein-Deligne 1982 Astérisque 100 §1.3.6). Let $D$ be a triangulated category with a t-structure $(D^{\leq 0}, D^{\geq 0})$ . Then the heart $D^{♡} = D^{\leq 0} \cap D^{\geq 0}$ is an abelian category. Moreover, a sequence $0 \to A \to B \to C \to 0$ in $D^{♡}$ is short exact if and only if there is a distinguished triangle $A \to B \to C \to A [1]$ in $D$ .

Proof. The argument has five steps. First, verify that $D^{♡}$ is additive. Second, construct kernels via truncation triangles. Third, construct cokernels by the dual argument. Fourth, verify that the kernel-cokernel comparison map is an isomorphism (the AB2 axiom). Fifth, identify short exact sequences with distinguished triangles.

Step 1: additive structure. The heart $D^{♡}$ is a full subcategory of the additive category $D$ , closed under finite direct sums (since both $D^{\leq 0}$ and $D^{\geq 0}$ are closed under direct sums by stability under shift and by the orthogonality axiom). The zero object lies in $D^{♡}$ by definition. Hence $D^{♡}$ is additive.

Step 2: kernels. Given a morphism $f : A \to B$ in $D^{♡}$ , complete to a distinguished triangle $A f B \to C \to A [1]$ in $D$ . Since $A, B \in D^{\geq 0}$ , the long exact sequence of the cohomological functor $H_{t}^{0}$ applied to this triangle gives $H_{t}^{0} (C) \in D^{♡}$ , and the cone $C$ sits in $D^{\geq - 1}$ but not necessarily $D^{\geq 0}$ . Define the kernel of $f$ to be $K := H_{t}^{- 1} (C)$ , which is $τ_{\leq 0} τ_{\geq 0} C [- 1]$ . The truncation triangle on $C$ produces the morphism $K [1] \to C$ landing in $D^{\leq - 1} [1] = D^{\leq 0}$ , and rotating gives a morphism $K \to A$ that factors through $ker f$ in the sense of universal property. Verification: a morphism $g : X \to A$ in $D^{♡}$ satisfies $f g = 0$ iff $g$ factors through $K$ , by the long exact sequence and the orthogonality axiom (since $X \in D^{\leq 0}$ and the relevant cone is in $D^{\geq 1}$ ).

Step 3: cokernels. By the dual argument (truncate the same cone from above instead of below), the cokernel of $f : A \to B$ is $coker f = H_{t}^{0} (C) = τ_{\leq 0} τ_{\geq 0} C$ . The universal property of the cokernel follows by the dual long-exact-sequence and orthogonality argument.

Step 4: kernel-cokernel comparison (AB2). The image of $f : A \to B$ is canonically $coker (K \to A) = τ_{\geq 0}$ of the cone of $K \to A$ , and the coimage is $ker (B \to coker f)$ . The comparison map $coim f \to im f$ is constructed from the morphism of triangles induced by $f$ on the truncation triangles of $A$ and $B$ . The orthogonality axiom forces this comparison to be an isomorphism: the obstruction lies in a $Hom$ -group from $D^{\leq - 1}$ to $D^{\geq 1}$ , which vanishes by the orthogonality. Hence $coim f ≅ im f$ canonically, and $D^{♡}$ satisfies the AB2 axiom that every morphism has a kernel-cokernel decomposition with isomorphism between coimage and image. Combined with the additive structure from Step 1, this is the abelian-category axioms (AB1 + AB2).

Step 5: short exact sequences vs distinguished triangles. Suppose $0 \to A \to B \to C \to 0$ is short exact in $D^{♡}$ , with $A = ker (B \to C)$ and $C = coker (A \to B)$ . By the kernel construction in Step 2 and the cokernel construction in Step 3, both can be read off from a single distinguished triangle $A \to B \to C \to A [1]$ in $D$ — the same triangle exhibits both the kernel-relation $A = H_{t}^{- 1} (cone (A \to B))$ (vanishing because $H_{t}^{- 1}$ of a triangle with $A, B \in D^{♡}$ vanishes by orthogonality) and the cokernel relation $C = H_{t}^{0} (cone (A \to B))$ . Conversely, given a distinguished triangle $A \to B \to C \to A [1]$ with all three terms in $D^{♡}$ , the long exact sequence of $H_{t}^{n}$ shows that $A = ker (B \to C)$ and $C = coker (A \to B)$ in the heart, hence the sequence $0 \to A \to B \to C \to 0$ is short exact.

The biimplication between short exact sequences in $D^{♡}$ and distinguished triangles with all terms in $D^{♡}$ is the link between the abelian and triangulated structures, and it is what makes the cohomological functor $H_{t}^{n}$ assemble the t-structure cohomology into a long exact sequence in the heart for every distinguished triangle. $□$

Bridge. The t-structure builds toward an abelian category sitting inside the triangulated category, and the foundational reason the construction works is that the orthogonality axiom $Hom (D^{\leq 0}, D^{\geq 1}) = 0$ propagates through every cohomological calculation, forcing kernels and cokernels in the heart to behave compatibly with truncation triangles in $D$ . The bridge is the cohomology functor $H_{t}^{n} : D \to D^{♡}$ , which extracts the t-structure cohomology of any object as a sequence of objects in the heart, and the long exact sequence of $H_{t}^{n}$ on a distinguished triangle in $D$ recovers the classical long exact sequence in the abelian setting. Putting these together, every triangulated category equipped with a t-structure carries a built-in abelian-category extraction, and the choice of t-structure is the choice of "where to look" inside the homological data.

This pattern appears again in 04.03.11 (the derived category $D (A)$ ), where the standard t-structure recovers $A$ as the heart, identifying the abelian category one started with as a full subcategory of its derived category in degree $0$ , and in 04.03.12 (derived functors $R F$ and $L F$ ), where t-exact functors between triangulated categories with t-structures induce exact functors between the hearts, providing the abelian-category-level theorem from the derived-category construction. The central insight is that t-structures are the mechanism by which derived-category arguments produce abelian-category corollaries — every choice of t-structure is a choice of how to read the abelian content out of the triangulated content.

Exercises Intermediate+

Exercise 1 (easy, short-answer).

State the three BBD axioms for a t-structure $(D^{\leq 0}, D^{\geq 0})$ on a triangulated category $D$ and verify the shift compatibility for the standard t-structure on $D (Ab)$ .

Hint

Axioms: shift stability, orthogonality, truncation triangle. For the standard t-structure, $D^{\leq 0}$ is complexes with vanishing positive cohomology.

Answer

The three BBD axioms are: (1) $D^{\leq - 1} \subseteq D^{\leq 0}$ and $D^{\geq 1} \subseteq D^{\geq 0}$ ; (2) $Hom_{D} (X, Y) = 0$ for $X \in D^{\leq 0}$ and $Y \in D^{\geq 1}$ ; (3) every $X \in D$ fits in a distinguished triangle $A \to X \to B \to A [1]$ with $A \in D^{\leq 0}$ and $B \in D^{\geq 1}$ .

For the standard t-structure on $D (Ab)$ : $D^{\leq 0}$ = complexes with $H^{n} = 0$ for $n \geq 1$ , $D^{\leq - 1}$ = complexes with $H^{n} = 0$ for $n \geq 0$ . If $C^{∙} \in D^{\leq - 1}$ , then $H^{n} (C^{∙}) = 0$ for $n \geq 0$ , which is a stronger condition than $H^{n} = 0$ for $n \geq 1$ , so $D^{\leq - 1} \subseteq D^{\leq 0}$ . The shift compatibility is the immediate consequence: the shift $[1]$ takes a complex's cohomology in degree $n$ to degree $n - 1$ . Rubric: full credit for the three axioms plus the shift verification.

Exercise 2 (easy, short-answer).

For the standard t-structure on $D (Ab)$ , identify the truncation $τ_{\leq 0}$ and $τ_{\geq 1}$ applied to the complex $C^{∙} = 0 \to Z \cdot 3 Z \to Z /6 \to 0$ with $Z$ in degrees $- 1, 0$ and $Z /6$ in degree $1$ .

Hint

Compute the cohomology of $C^{∙}$ first, then identify what survives in each truncation range.

Answer

Cohomology of $C^{∙}$ : $H^{- 1} (C^{∙}) = ker (\cdot 3) = 0$ ; $H^{0} (C^{∙}) = ker (Z \to Z /6) / im (\cdot 3) = 6 Z /3 Z$ inside $Z$ , but better: working with the explicit differentials, $H^{0} = ker (d^{0}) / im (d^{- 1}) = 6 Z /3 Z$ in $Z$ — equivalently, the subgroup generated by the element $6 \cdot k$ modulo the subgroup generated by $3 \cdot k$ , so $H^{0} = Z /2$ via the inclusion $6 Z ↪ 3 Z$ with index $2$ ; $H^{1} (C^{∙}) = Z /6/ im (Z \to Z /6) = Z /6/ Z / (g cd (1, 6)) = 0$ (since $Z \to Z /6$ is surjective). So $H^{*} (C^{∙})$ is $Z /2$ in degree $0$ , zero elsewhere. Hence $τ_{\leq 0} C^{∙} ≅ (Z /2) [0]$ (the heart object in degree $0$ ) and $τ_{\geq 1} C^{∙} ≅ 0$ . The truncation triangle reduces to the identity triangle: $C^{∙} ≅ τ_{\leq 0} C^{∙} ≅ (Z /2) [0]$ . Rubric: full credit for the cohomology computation and the truncation identification.

Exercise 3 (medium, short-answer).

Prove that the truncation functor $τ_{\leq 0}$ is right adjoint to the inclusion $ι : D^{\leq 0} ↪ D$ .

Hint

For $A \in D^{\leq 0}$ and $X \in D$ , use the truncation triangle on $X$ and the orthogonality axiom to show $Hom_{D} (A, X) ≅ Hom_{D} (A, τ_{\leq 0} X)$ .

Answer

Given $A \in D^{\leq 0}$ and $X \in D$ , apply $Hom_{D} (A, -)$ to the truncation triangle $τ_{\leq 0} X \to X \to τ_{\geq 1} X \to (τ_{\leq 0} X) [1]$ . The resulting long exact sequence reads $$ \cdots \to \mathrm{Hom}(A, \tau_{\ge 1} X[-1]) \to \mathrm{Hom}(A, \tau_{\le 0} X) \to \mathrm{Hom}(A, X) \to \mathrm{Hom}(A, \tau_{\ge 1} X) \to \cdots $$ The terms $Hom (A, τ_{\geq 1} X [- 1])$ and $Hom (A, τ_{\geq 1} X)$ both vanish by the orthogonality axiom: $A \in D^{\leq 0}$ and $τ_{\geq 1} X \in D^{\geq 1}$ , plus $τ_{\geq 1} X [- 1] \in D^{\geq 2} \subseteq D^{\geq 1}$ by shift stability. Hence the middle map $Hom_{D} (A, τ_{\leq 0} X) \to Hom_{D} (A, X)$ is an isomorphism. Naturality in $A$ and $X$ is automatic from the naturality of the truncation triangle. This is the adjunction $τ_{\leq 0} ⊣ ι$ on the right side: $τ_{\leq 0}$ is the right adjoint to the inclusion. The dual argument gives $τ_{\geq 1}$ as left adjoint to $D^{\geq 1} ↪ D$ . Rubric: full credit for the long exact sequence, the orthogonality applications, and the adjunction identification.

Exercise 4 (medium, short-answer).

Show that for the standard t-structure on $D (A)$ (with $A$ abelian), the cohomology functor $H_{t}^{0} : D (A) \to A$ of the t-structure coincides with the classical cohomology $H^{0}$ .

Hint

Compute $τ_{\leq 0} τ_{\geq 0} C^{∙}$ explicitly: it should give a complex quasi-isomorphic to $H^{0} (C^{∙})$ placed in degree $0$ .

Answer

For a complex $C^{∙} \in D (A)$ , the truncation $τ_{\geq 0} C^{∙}$ is the canonical truncation: the complex $$ \cdots \to 0 \to \ker(d^0) / \mathrm{im}(d^{-1}) \to C^1 \to C^2 \to \cdots $$ with $H^{0} (C^{∙}) = ker (d^{0}) / im (d^{- 1})$ in degree $0$ (replacing $C^{0}$ by the cohomology group when only its image into the next term matters), or equivalently the "good truncation" $τ^{\geq 0} C^{∙}$ where everything in negative degrees is replaced by zero and the degree- $0$ term is replaced by $ker (d^{0})$ . The further truncation $τ_{\leq 0} τ_{\geq 0} C^{∙}$ takes the result down to a complex concentrated in degree $0$ , which is exactly $H^{0} (C^{∙})$ placed in degree $0$ . Hence $H_{t}^{0} (C^{∙}) = H^{0} (C^{∙}) \in A$ , identifying the t-structure cohomology with the classical cohomology. The same argument with shifts gives $H_{t}^{n} (C^{∙}) = H^{n} (C^{∙})$ for all $n \in Z$ . Rubric: full credit for the truncation computation, the identification with the classical cohomology, and the extension to all degrees.

Exercise 5 (medium, short-answer).

Verify the long exact sequence property of the cohomological functor: given a distinguished triangle $X \to Y \to Z \to X [1]$ in $D$ , derive the long exact sequence $$ \cdots \to H^{n-1}_t(Z) \to H^n_t(X) \to H^n_t(Y) \to H^n_t(Z) \to H^{n+1}_t(X) \to \cdots $$ in the heart $D^{♡}$ .

Hint

A cohomological functor on a triangulated category is by definition a functor that produces a long exact sequence from every distinguished triangle. Verify the property by checking that $H_{t}^{0} = τ_{\leq 0} τ_{\geq 0}$ is cohomological.

Answer

The cohomology functor $H_{t}^{0} = τ_{\leq 0} τ_{\geq 0}$ is constructed as a composition of truncations adjoint to inclusions, and the resulting functor is cohomological by the following argument: apply $τ_{\geq 0}$ to the distinguished triangle $X \to Y \to Z \to X [1]$ ; the result is a triangle in $D^{\geq 0}$ (since $τ_{\geq 0}$ is exact in the appropriate sense — it preserves distinguished triangles up to canonical isomorphism via the long-exact-sequence behaviour of the truncation triangles). Then apply $τ_{\leq 0}$ and pass to the heart: the long exact sequence in $D^{♡}$ reads $$ \cdots \to H^{n-1}_t(Z) \xrightarrow{\delta} H^n_t(X) \to H^n_t(Y) \to H^n_t(Z) \xrightarrow{\delta} H^{n+1}_t(X) \to \cdots $$ with the connecting morphism $δ$ coming from the rotation of the triangle in $D$ . Exactness at each term follows from the orthogonality axiom together with the truncation-triangle property: each three-term portion at $H_{t}^{n} (X), H_{t}^{n} (Y), H_{t}^{n} (Z)$ is the image of a distinguished triangle in $D^{♡}$ , which by the BBD §1.3.6 theorem (Step 5 of the proof above) is exact in the abelian category $D^{♡}$ . Rubric: full credit for the long-exact-sequence verification and the connecting-morphism construction.

Exercise 6 (medium, short-answer).

Show that the degenerate t-structure $(D^{\leq 0}, D^{\geq 0}) = (D, 0)$ , where $D^{\leq 0}$ is the entire category and $D^{\geq 0}$ is the zero subcategory, satisfies the BBD axioms but produces an empty heart. Identify the smallest non-degenerate example.

Hint

Orthogonality says $Hom (D^{\leq 0}, D^{\geq 1}) = 0$ . With $D^{\geq 1} = 0$ this is automatic. The failure is elsewhere — check the truncation triangle axiom on a non-zero object.

Answer

With $D^{\geq 0} = 0$ , the orthogonality axiom is satisfied vacuously (the right side of the Hom is always zero). The truncation triangle axiom, however, requires every $X \in D$ to fit in a triangle $A \to X \to B \to A [1]$ with $A \in D^{\leq 0} = D$ and $B \in D^{\geq 1} = 0$ . This forces $B = 0$ , so the triangle becomes $X \to X \to 0 \to X [1]$ , which is distinguished iff the second map is the zero morphism and the first is the identity — fine for any $X$ . So the truncation axiom is satisfied. The shift axiom asks $D^{\leq - 1} \subseteq D^{\leq 0}$ , which becomes $D \subseteq D$ — satisfied. So $(D, 0)$ technically forms a degenerate t-structure with empty heart ( $D^{♡} = D \cap 0 = 0$ ). The empty-heart degeneracy is permitted by the axioms but is uninteresting; the standard non-degenerate t-structures all have non-empty heart, and the BBD framework studies the non-degenerate case. The correct conclusion is that the degenerate pair $(D, 0)$ satisfies the formal axioms but produces no useful cohomological information; "real" t-structures require non-empty heart, which holds for the standard t-structure on $D (A)$ . Rubric: full credit for the careful axiom verification and the identification of the degeneracy.

Exercise 7 (hard, short-answer).

State the perverse t-structure on $D_{c}^{b} (X)$ for a complex algebraic variety $X$ of dimension $d$ , using the middle perversity $p (i) = - i$ (or its equivalent formulation): a constructible complex $F^{∙} \in D_{c}^{b} (X)$ lies in $^{p} D^{\leq 0}$ iff $dim supp H^{i} (F^{∙}) \leq - i$ for all $i$ . Sketch why the heart is the abelian category of perverse sheaves $Perv (X)$ .

Hint

The middle-perversity t-structure is the BBD-perverse t-structure. The heart consists of complexes that are simultaneously in $^{p} D^{\leq 0}$ and in the Verdier dual $^{p} D^{\geq 0}$ .

Answer

The perverse t-structure on $D_{c}^{b} (X)$ (the bounded derived category of constructible sheaves on $X$ with respect to a fixed stratification, with coefficients in a field $k$ ) is defined by: $$ {}^p D^{\le 0}(X) := { \mathcal{F}^\bullet \in D^b_c(X) : \dim \mathrm{supp}, \mathcal{H}^i(\mathcal{F}^\bullet) \le -i \text{ for all } i }, $$ $$ {}^p D^{\ge 0}(X) := { \mathcal{F}^\bullet \in D^b_c(X) : \dim \mathrm{supp}, \mathcal{H}^i(\mathbb{D} \mathcal{F}^\bullet) \le -i \text{ for all } i }, $$ where $D = R H o m (-, ω_{X}^{∙})$ is the Verdier dual.

The BBD axioms are verified directly: shift stability follows from the dimension-bound shift; orthogonality follows from Verdier-duality and the support-dimension inequality; the truncation triangle is constructed via the standard truncation-and-glue argument using the open-closed decomposition relative to a stratum's support.

The heart $Perv (X) :=^{p} D^{\leq 0} (X) \cap^{p} D^{\geq 0} (X)$ is the category of perverse sheaves, an abelian category by the general BBD §1.3.6 theorem. Perverse sheaves are constructible complexes that are simultaneously in both halves of the perverse t-structure, equivalently: they are middle-extension complexes from open strata, organised by the support-dimension condition and its Verdier dual. The simple objects in $Perv (X)$ are intermediate extensions $j_{! *} L [d_{Z}]$ of irreducible local systems $L$ on smooth locally closed subvarieties $Z$ of dimension $d_{Z}$ . The category $Perv (X)$ is artinian and noetherian, and it organises intersection cohomology in a single abelian setting.

Rubric: full credit for the perverse t-structure definition, the support-dimension condition and its Verdier-dual form, the identification of the heart with $Perv (X)$ , and the mention of intermediate extensions as simple objects.

Exercise 8 (hard, short-answer).

State the Beilinson reconstruction theorem: for a complex algebraic variety $X$ with the middle-perversity t-structure on $D_{c}^{b} (X)$ , there is an equivalence of triangulated categories $D^{b} (Perv (X)) ≃ D_{c}^{b} (X)$ . Sketch the consequence: the perverse t-structure recovers the full constructible derived category from its heart.

Hint

The reconstruction theorem identifies the derived category of the heart with the original triangulated category — an unusual property that depends on the perverse t-structure being "bounded" and "non-degenerate" in a specific sense.

Answer

Theorem (Beilinson 1987 Funct. Anal. Appl. 21; full treatment in BBD 1982 §3.1). For a complex algebraic variety $X$ over a field of characteristic zero (e.g., $C$ ), the natural functor $$ D^b(\mathrm{Perv}(X)) \to D^b_c(X) $$ sending a complex of perverse sheaves to its associated constructible complex is an equivalence of triangulated categories.

The equivalence is constructed using the perverse-cohomology functors $^{p} H^{n} : D_{c}^{b} (X) \to Perv (X)$ associated to the perverse t-structure: every object of $D_{c}^{b} (X)$ assembles from its perverse cohomology objects via the perverse spectral sequence, and the perverse-cohomology functors are jointly conservative on $D_{c}^{b} (X)$ . The Beilinson theorem states that this reconstruction is not just informational but is an actual equivalence of triangulated categories: $D_{c}^{b} (X)$ is the same as the bounded derived category of the abelian category $Perv (X)$ .

The consequence: the constructible derived category, which carries the cohomological information of all stratifications of $X$ , is recovered from the single abelian category $Perv (X)$ together with its derived category. This is striking — most heart-of-t-structure reconstructions fail (the derived category of a heart is generally bigger than the original triangulated category), and the perverse case succeeds because of the specific Artin-vanishing properties of constructible sheaves on algebraic varieties.

The same property fails for the standard t-structure on $D_{c}^{b} (X)$ (whose heart is the abelian category of constructible sheaves $Cons (X)$ ): the natural functor $D^{b} (Cons (X)) \to D_{c}^{b} (X)$ is fully faithful but not essentially surjective, because the standard t-structure does not detect the higher-degree constructible information that the perverse t-structure picks up.

Rubric: full credit for the theorem statement, the equivalence interpretation, the perverse-cohomology construction, and the contrast with the standard t-structure (no equivalence in the standard case).

Lean formalization Intermediate+

lean_status: none — Mathlib has the triangulated-category infrastructure and the derived-category framework, but the named t-structure package on a triangulated category is not yet assembled. The intended formalisation reads schematically:

import Mathlib.CategoryTheory.Triangulated.Basic
import Mathlib.CategoryTheory.Localization.DerivedCategory
import Mathlib.CategoryTheory.Abelian.Basic

namespace Codex.Triangulated.tStructure

variable {D : Type*} [Category D] [HasShift D ℤ] [Pretriangulated D]

/-- A t-structure on a triangulated category D is a pair of strictly full
    subcategories (D_le, D_ge) satisfying the BBD axioms. -/
structure tStructure (D : Type*) [Category D] [HasShift D ℤ]
    [Pretriangulated D] where
  D_le : Subcategory D  -- D^{≤ 0}
  D_ge : Subcategory D  -- D^{≥ 0}
  shift_le : ∀ X ∈ D_le, X⟦-1⟧ ∈ D_le  -- D^{≤ -1} ⊆ D^{≤ 0}
  shift_ge : ∀ X ∈ D_ge, X⟦1⟧ ∈ D_ge   -- D^{≥ 1} ⊆ D^{≥ 0}
  orthogonality : ∀ (X : D) (Y : D), X ∈ D_le → Y⟦-1⟧ ∈ D_ge →
    (X ⟶ Y) ≃ PUnit  -- Hom(D^{≤ 0}, D^{≥ 1}) = 0
  truncation_triangle : ∀ (X : D), ∃ (A B : D) (f : A ⟶ X) (g : X ⟶ B)
      (h : B ⟶ A⟦1⟧), A ∈ D_le ∧ B⟦-1⟧ ∈ D_ge ∧
      IsDistinguishedTriangle ⟨A, X, B, f, g, h⟩

/-- The heart of a t-structure is the intersection of the two
    subcategories; it is an abelian category. -/
def heart (t : tStructure D) : Subcategory D := t.D_le ∩ t.D_ge

/-- The truncation functor τ_{≤ 0} : D → D^{≤ 0}, right adjoint to
    the inclusion. -/
noncomputable def tauLE0 (t : tStructure D) : D ⥤ D := sorry

/-- The truncation functor τ_{≥ 1} : D → D^{≥ 1}, left adjoint to
    the inclusion. -/
noncomputable def tauGE1 (t : tStructure D) : D ⥤ D := sorry

/-- The 0-th cohomology functor of the t-structure. -/
noncomputable def H0_t (t : tStructure D) : D ⥤ D := tauLE0 t ⋙ tauGE1 t

/-- The heart of a t-structure is an abelian category. -/
theorem heart_is_abelian (t : tStructure D) :
    IsAbelian (FullSubcategory (· ∈ heart t)) := sorry

/-- The standard t-structure on the derived category of an abelian
    category A has heart equivalent to A. -/
theorem standard_t_structure_heart (A : Type*) [Category A] [Abelian A] :
    ∃ (t : tStructure (DerivedCategory A)),
      Equivalence A (FullSubcategory (· ∈ heart t)) := sorry

end Codex.Triangulated.tStructure

The proof gap is substantive. The construction of the truncation functors requires the adjunction-with-inclusion argument that goes through Brown representability or explicit construction via the truncation triangles, neither packaged in Mathlib at the triangulated-category level. The abelian-category structure on the heart requires the BBD §1.3.6 theorem, which needs the kernel-and-cokernel construction via truncation triangles plus the orthogonality-based verification of the AB2 axiom. The standard t-structure on $D (A)$ requires the equivalence between bounded-cohomology complexes and the abelian-category objects placed in degree $0$ , which is essentially the universal property of the derived category but in the t-structure formulation. The cohomology functor $H_{t}^{n}$ requires the t-exactness analysis of the truncation compositions. Each component is formalisable in principle but requires substantial coordinated infrastructure that the Mathlib derived-category and triangulated-category projects have not yet completed as of 2026.

Advanced results Master

Theorem (the heart is abelian; Beilinson-Bernstein-Deligne 1982 Astérisque 100 §1.3.6). The heart $D^{♡} = D^{\leq 0} \cap D^{\geq 0}$ of any t-structure on a triangulated category $D$ is an abelian category, with short exact sequences corresponding to distinguished triangles in $D$ all three of whose terms lie in $D^{♡}$ .

The BBD theorem is the foundational theorem of t-structure theory and the key statement that lets one extract abelian-category information from triangulated-category data. The proof, sketched in the Key Theorem section above, proceeds by constructing kernels and cokernels in the heart via the truncation triangles of morphisms, and verifying the AB2 axiom via the orthogonality of the t-structure. The theorem applies to every t-structure on every triangulated category, and is the source of every abelian-category construction in modern derived algebraic geometry, perverse-sheaf theory, and Bridgeland stability.

Theorem (cohomological functor $H_{t}^{n}$ ; BBD 1982 §1.3.13). Let $D$ be a triangulated category with t-structure. The composition $H_{t}^{0} = τ_{\leq 0} \circ τ_{\geq 0} : D \to D^{♡}$ is a cohomological functor: for every distinguished triangle $X \to Y \to Z \to X [1]$ in $D$ , the sequence $$ \cdots \to H^{n-1}_t(Z) \to H^n_t(X) \to H^n_t(Y) \to H^n_t(Z) \to H^{n+1}t(X) \to \cdots $$ *is exact in $D^{♡}$ , where $H_{t}^{n} := H_{t}^{0} \circ [n]$ . Moreover, the family ${H^n_t}{n \in \mathbb{Z}} $i s j o in tl y co n ser v a t i v eo n t h e b o u n d e d s u b c a t e g or y$ \mathcal{D}^b := {X : H^n_t(X) = 0 \text{ for } |n| \gg 0}$ if the t-structure is bounded.*

The cohomology functor is the natural way to extract abelian-category invariants from a triangulated-category object, and the long exact sequence is the standard derived-functor tool generalised to the t-structure framework. The conservativity statement for bounded t-structures says that a bounded object is detected up to isomorphism by its sequence of cohomology objects in the heart — the t-structure cohomology recovers all the homological information when the t-structure is well-behaved.

Theorem (standard t-structure on $D (A)$ ). Let $A$ be an abelian category. The standard t-structure on $D (A)$ , defined by $D (A)^{\leq 0} = {C^{∙} : H^{n} (C^{∙}) = 0 for n \geq 1}$ and $D (A)^{\geq 0} = {C^{∙} : H^{n} (C^{∙}) = 0 for n \leq - 1}$ , has heart equivalent to $A$ via the inclusion $A ↪ D (A)$ , $A \mapsto A [0]$ , and the cohomology functors $H_{t}^{n}$ coincide with the classical cohomology functors $H^{n} : D (A) \to A$ .

The standard t-structure is the canonical example, and its heart recovers the original abelian category. This is the canonical means by which an abelian category sits inside its derived category as a full subcategory, with the standard t-structure providing the t-exact realisation. The truncation functors of the standard t-structure are the canonical truncations $τ^{\leq n}, τ^{\geq n}$ on complexes (replacing degrees by zero or by the cohomology group in the cut-off degree).

Theorem (perverse t-structure on $D_{c}^{b} (X)$ ; BBD 1982 §§2.1--2.2). Let $X$ be a complex algebraic variety with a fixed stratification, and let $D_{c}^{b} (X)$ be the bounded derived category of complexes of sheaves constructible with respect to the stratification. The middle-perversity t-structure on $D_{c}^{b} (X)$ is defined by $$ {}^p D^{\le 0}(X) := {\mathcal{F}^\bullet : \dim \mathrm{supp}, \mathcal{H}^i(\mathcal{F}^\bullet) \le -i \text{ for all } i}, $$ $$ {}^p D^{\ge 0}(X) := {\mathcal{F}^\bullet : \dim \mathrm{supp}, \mathcal{H}^i(\mathbb{D} \mathcal{F}^\bullet) \le -i \text{ for all } i}, $$ where $D = R H o m (-, ω_{X}^{∙})$ is the Verdier dual. The heart $Perv (X) =^{p} D^{\leq 0} (X) \cap^{p} D^{\geq 0} (X)$ is the abelian category of perverse sheaves on $X$ .

The perverse t-structure is the foundational example of a non-standard t-structure with deep geometric content. Perverse sheaves organise the singular cohomology of algebraic varieties, intersection cohomology of singular varieties, and the support data of constructible complexes into a single abelian setting. Simple perverse sheaves on $X$ are the intermediate extensions $j_{! *} L [d_{Z}]$ of irreducible local systems $L$ on smooth locally closed subvarieties $Z \subset X$ of dimension $d_{Z}$ , classified by (stratum, irreducible local system) pairs.

Theorem (Beilinson reconstruction; Beilinson 1987 Funct. Anal. Appl. 21). Let $X$ be a complex algebraic variety. The natural functor $D^{b} (Perv (X)) \to D_{c}^{b} (X)$ from the bounded derived category of perverse sheaves to the bounded constructible derived category is an equivalence of triangulated categories.

Beilinson's reconstruction theorem is unusual: it says that the constructible derived category $D_{c}^{b} (X)$ is identified with the bounded derived category of the abelian heart $Perv (X)$ of the perverse t-structure. Most t-structures do not have this property — the derived category of the heart is generally larger than the original triangulated category. The perverse case succeeds because of the Artin-vanishing properties of constructible sheaves and the specific behaviour of the perverse cohomology functors on stratified spaces. The consequence is that all constructible cohomological information is encoded in $Perv (X)$ and recovered by its derived category, providing a powerful tool for studying mixed Hodge structures (Beilinson-Bernstein 1981; Saito's theory of mixed Hodge modules), intersection cohomology (Goresky-MacPherson 1980), and the BBD decomposition theorem.

Theorem (BBD decomposition theorem; BBD 1982 §6.2.5; Gabber 1981). Let $f : X \to Y$ be a proper morphism of complex algebraic varieties. For any pure perverse sheaf $F^{∙}$ on $X$ (e.g., $IC_{X}$ , the intersection cohomology complex), the derived direct image $Rf_ \mathcal{F}^\bullet $d eco m p oses a s a d i r ec t s u m o f (s hi f t e d) in t er m e d ia t ee x t e n s i o n so f i r r e d u c ib l e l oc a l sy s t e m so n s m oo t h l oc a l l y c l ose d s u b v a r i e t i eso f$ Y$:* $$ Rf_* \mathrm{IC}X \cong \bigoplus{i, Z, \mathcal{L}} ({}^p \mathcal{IC}_Z(\mathcal{L}))[d_i]. $$

The decomposition theorem is one of the most powerful results of the BBD framework and the perverse-sheaf formalism. It implies a vast generalisation of the hard Lefschetz theorem, the Leray spectral sequence degeneration on every page for proper maps, and the topological invariance of intersection cohomology under resolution of singularities. The proof goes through positive characteristic via the Frobenius twist (Deligne-Gabber Weil II, 1980), with the characteristic-zero statement obtained by reduction. The theorem is the input to most modern applications of perverse sheaves, including geometric representation theory (Beilinson-Bernstein localisation), the geometric Satake equivalence (Mirković-Vilonen 2007), and the Fundamental Lemma (Ngô 2010).

Theorem (Bondal-Orlov reconstruction; Bondal-Orlov 2001 Compositio Math. 125). Let $X$ be a smooth projective variety over a field with $ω_{X}$ ample or anti-ample (i.e., $X$ is Fano or canonically polarised). Then $X$ is uniquely determined up to isomorphism by the bounded derived category of coherent sheaves $D^{b} (Coh (X))$ as a $k$ -linear triangulated category. More precisely: if $X, X^{'}$ are two such varieties and there is a $k$ -linear triangulated equivalence $D^{b} (Coh (X)) ≃ D^{b} (Coh (X^{'}))$ , then $X ≅ X^{'}$ as schemes.

The Bondal-Orlov reconstruction theorem is the geometric counterpart of the t-structure framework: it identifies the variety $X$ from its derived category alone, when $ω_{X}$ is "extremal" (ample or anti-ample). The proof uses point-like objects (the structure sheaves of points are characterised as objects with specific $Ext$ -properties relative to the canonical Serre functor $S_{X} = (- \otimes ω_{X}) [dim X]$ ), and the variety $X$ is reconstructed as a moduli space of these point-like objects equipped with their derived-category morphism structure. The theorem fails for Calabi-Yau varieties ( $ω_{X} = 0$ , the Serre functor is just a shift) and for varieties with vanishing canonical bundle, where the derived category has many autoequivalences and reconstruction fails — this is the input to Kuznetsov's homological projective duality and to Bridgeland-stability-based mirror symmetry. Originator: Bondal-Kapranov 1989 Math. USSR Izv. 35 (Serre functors and the reconstruction framework); Bondal-Orlov 2001 Compositio Math. 125 (the ample / anti-ample reconstruction theorem); subsequent extensions by Kawamata 2002, Ballard 2011 (singular case), and Favero 2012.

Theorem (tilting and equivalences of derived categories; Happel 1988, Rickard 1989). A tilting object $T$ in a triangulated category $D$ (a compact generator with $Hom_{D} (T, T [n]) = 0$ for $n \neq = 0$ ) induces an equivalence $D ≃ D (End (T)^{op})$ when $D$ is suitably enhanced. The heart of the standard t-structure on $D (End (T)^{op})$ pulls back to a non-standard t-structure on $D$ called the tilted t-structure, with heart $End (T)^{op} - Mod$ .

Tilting theory produces non-standard t-structures via Morita-type equivalences and is the input to many derived-category-equivalence theorems in algebraic geometry and representation theory. The tilted t-structure has heart $End (T)^{op} - Mod$ rather than the original abelian category, illustrating how different t-structures on the same triangulated category correspond to different abelian-category presentations of the same homological data. Originator: Happel 1988 Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras (Cambridge LMS Lecture Note Series 119); Rickard 1989 J. London Math. Soc. 39 (Morita theory for derived categories).

Synthesis. The t-structure framework builds toward an extraction of abelian-category information from triangulated-category data, and the foundational reason it works is that the BBD axioms force the orthogonality $Hom (D^{\leq 0}, D^{\geq 1}) = 0$ to propagate through every cohomological calculation, making the heart $D^{♡}$ an abelian category whose short exact sequences are distinguished triangles in $D$ . The package is dual to itself: the choice of t-structure is the choice of "what abelian category to see inside the triangulated category", and the same triangulated category $D$ admits many different t-structures with non-equivalent hearts (standard vs perverse on $D_{c}^{b} (X)$ ; standard vs tilted via a tilting object; standard vs Bridgeland-stability-induced). The central insight is that triangulated categories are richer than any single abelian category — they encode the network of all abelian-category extractions via the moduli of t-structures, and the explicit choice of t-structure is the input to every concrete cohomological computation.

The formalism appears again in 04.03.11 (the derived category $D (A)$ ), where the standard t-structure recovers $A$ as the heart, identifying the original abelian category as the canonical t-structure heart inside its derived category; in 04.03.12 (derived functors $R F$ and $L F$ ), where t-exact functors between triangulated categories with t-structures induce exact functors between the hearts, providing the abelian-category-level theorem from the derived-category construction; in 04.03.16 (six-functor formalism), where the perverse t-structure on $D_{c}^{b} (X)$ for a complex algebraic variety $X$ supplies the abelian category $Perv (X)$ that organises constructible-sheaf cohomology and supports the BBD decomposition theorem; and in 04.03.10 (triangulated categories), where the axiomatic foundation of triangulated categories supplies the ambient setting on which t-structures live. The recursion stabilises: every t-structure produces an abelian category, every abelian category sits inside a triangulated category via the derived category, and the moduli space of t-structures on a fixed triangulated category (Bridgeland's stability manifold) organises the different abelian-category presentations into a single geometric object.

The synthesis is structural: every abelian-category extraction from triangulated-category data is a t-structure, every standard cohomology functor on a derived category is a $H_{t}^{n}$ of the standard t-structure, and every non-standard t-structure (perverse, tilted, Bridgeland-stability-induced) supplies a new abelian-category presentation of the same triangulated category — the same homological information seen through a different abelian lens. The t-structure framework is the universal mechanism by which derived-category arguments produce abelian-category corollaries.

Full proof set Master

Proposition (uniqueness of truncation up to canonical isomorphism). Let $D$ be a triangulated category with t-structure. Given $X \in D$ and two truncation triangles $A \to X \to B \to A [1]$ and $A^{'} \to X \to B^{'} \to A^{'} [1]$ with $A, A^{'} \in D^{\leq 0}$ and $B, B^{'} \in D^{\geq 1}$ , there exist unique isomorphisms $A ≅ A^{'}$ and $B ≅ B^{'}$ making the comparison diagram commute.

Proof. Consider the diagram

A ↓ A^{'} \to \to X ∣∣ X \to \to B ↓ B^{'} \to \to A [1] ↓ A^{'} [1]

By the TR3 axiom of the triangulated structure, given the identity on $X$ , there exist morphisms $A \to A^{'}$ and $B \to B^{'}$ completing the diagram to a morphism of triangles. The morphism $A \to A^{'}$ is unique by the orthogonality axiom: any difference $δ : A \to A^{'}$ between two candidate morphisms factors through the cone of the original morphism, which sits in $D^{\geq 0}$ (specifically in $D^{\geq 1}$ after a shift), and $Hom (A, D^{\geq 1}) = 0$ by orthogonality (with $A \in D^{\leq 0}$ ). Hence $δ = 0$ and the candidate morphism $A \to A^{'}$ is unique.

By symmetry, the morphism $A^{'} \to A$ is also unique, and the compositions $A \to A^{'} \to A$ and $A^{'} \to A \to A^{'}$ are the identities (by uniqueness applied to the identity case). Hence $A ≅ A^{'}$ canonically, and the same argument for $B$ gives $B ≅ B^{'}$ canonically. The full diagram commutes by uniqueness. $□$

Proposition (truncation functors as adjoints to inclusions). The functor $τ_{\leq 0} : D \to D^{\leq 0}$ is right adjoint to the inclusion $ι_{\leq 0} : D^{\leq 0} ↪ D$ , and the functor $τ_{\geq 1} : D \to D^{\geq 1}$ is left adjoint to the inclusion $ι_{\geq 1} : D^{\geq 1} ↪ D$ .

Proof. Apply Exercise 3's argument: for $A \in D^{\leq 0}$ and $X \in D$ , the long exact sequence of $Hom_{D} (A, -)$ on the truncation triangle $τ_{\leq 0} X \to X \to τ_{\geq 1} X \to (τ_{\leq 0} X) [1]$ has terminal terms $Hom (A, τ_{\geq 1} X [- 1])$ and $Hom (A, τ_{\geq 1} X)$ that vanish by orthogonality (since $A \in D^{\leq 0}$ and the target shifts of $τ_{\geq 1} X$ remain in $D^{\geq 1}$ ). The middle isomorphism $Hom_{D} (A, τ_{\leq 0} X) \sim Hom_{D} (ι_{\leq 0} A, X)$ is natural in $A$ (a morphism of inputs) and in $X$ (uses the natural truncation triangles), exhibiting the adjunction $ι_{\leq 0} ⊣ τ_{\leq 0}$ .

The dual argument for $τ_{\geq 1} ⊣ ι_{\geq 1}$ : for $B \in D^{\geq 1}$ and $X \in D$ , apply $Hom_{D} (-, B)$ to the truncation triangle. The terminal terms $Hom (τ_{\leq 0} X [1], B) = Hom (τ_{\leq 0} X, B [- 1])$ and $Hom (τ_{\leq 0} X, B)$ both vanish by orthogonality (since $B \in D^{\geq 1}$ and $B [- 1] \in D^{\geq 2} \subseteq D^{\geq 1}$ ). The middle isomorphism $Hom_{D} (τ_{\geq 1} X, B) \sim Hom_{D} (X, ι_{\geq 1} B)$ exhibits the adjunction. $□$

Proposition (the cohomology functor is well-defined). For any t-structure on $D$ , the two compositions $τ_{\leq 0} \circ τ_{\geq 0}$ and $τ_{\geq 0} \circ τ_{\leq 0}$ are canonically naturally isomorphic, giving a single well-defined functor $H_{t}^{0} : D \to D^{♡}$ .

Proof. Consider the canonical truncation triangles $τ_{\leq - 1} X \to τ_{\leq 0} X \to H_{t}^{0} (X)$ and the dual diagrams. The key observation: the composition $τ_{\leq 0} τ_{\geq 0}$ takes $X$ to an object in $D^{♡}$ (the intersection), and the canonical morphism $τ_{\geq 0} X \to τ_{\leq 0} τ_{\geq 0} X$ factors through the truncation $τ_{\leq 0}$ . The other composition $τ_{\geq 0} τ_{\leq 0} X$ has the canonical morphism $τ_{\leq 0} X \to τ_{\geq 0} τ_{\leq 0} X$ . Both target objects are in $D^{♡}$ , and the orthogonality axiom plus the truncation-triangle universal property produce the natural isomorphism between them.

Explicit: applying the cohomological universal property of the truncations, $τ_{\leq 0} τ_{\geq 0} X$ and $τ_{\geq 0} τ_{\leq 0} X$ both equal the unique object of $D^{♡}$ that fits into both the "bottom" truncation triangle of $τ_{\geq 0} X$ and the "top" truncation triangle of $τ_{\leq 0} X$ . The uniqueness up to canonical isomorphism follows from the uniqueness of truncation (Proposition above). $□$

Proposition (heart-restricted morphisms are abelian-category morphisms). For $A, B \in D^{♡}$ , every morphism $f : A \to B$ in $D$ has a kernel and cokernel in $D^{♡}$ , given by the cohomology of the cone: $ker (f) = H_{t}^{- 1} (cone (f))$ and $coker (f) = H_{t}^{0} (cone (f))$ .

Proof. Complete $f$ to a distinguished triangle $A f B \to C \to A [1]$ in $D$ . Apply the cohomology functor $H_{t}^{n}$ to get the long exact sequence $$ \cdots \to H^{-1}_t(C) \xrightarrow{\delta} H^0_t(A) \xrightarrow{f} H^0_t(B) \to H^0_t(C) \xrightarrow{\delta} H^1_t(A) \to \cdots $$ Since $A, B \in D^{♡}$ , $H_{t}^{0} (A) = A$ and $H_{t}^{0} (B) = B$ , and $H_{t}^{n} (A) = H_{t}^{n} (B) = 0$ for $n \neq = 0$ . Hence the long exact sequence reduces to $$ 0 \to H^{-1}_t(C) \xrightarrow{\delta} A \xrightarrow{f} B \to H^0_t(C) \to 0. $$ Exactness at $A$ means $H_{t}^{- 1} (C)$ is the kernel of $f$ as a subobject of $A$ in $D^{♡}$ , and exactness at $B$ means $H_{t}^{0} (C)$ is the cokernel of $f$ as a quotient of $B$ . Hence $ker (f) = H_{t}^{- 1} (cone (f))$ and $coker (f) = H_{t}^{0} (cone (f))$ .

The universal property of the kernel: a morphism $g : X \to A$ in $D^{♡}$ satisfies $f g = 0$ iff $g$ factors through $ker (f)$ . By the construction, $g$ factors through $H_{t}^{- 1} (cone (f))$ iff the composition $g \to A \to B$ is zero — exactly the condition $f g = 0$ . The dual universal property holds for the cokernel. $□$

Proposition (t-exact functors preserve hearts). Let $F : D \to D^{'}$ be a t-exact triangulated functor between triangulated categories equipped with t-structures. Then $F$ restricts to an exact functor $F ∣_{D^{♡}} : D^{♡} \to D^{'♡}$ between the hearts.

Proof. $F$ is t-exact means $F (D^{\leq 0}) \subseteq D^{' \leq 0}$ and $F (D^{\geq 0}) \subseteq D^{' \geq 0}$ . Hence $F (D^{♡}) = F (D^{\leq 0} \cap D^{\geq 0}) \subseteq D^{' \leq 0} \cap D^{' \geq 0} = D^{'♡}$ . The restriction $F ∣_{D^{♡}}$ is a functor between the abelian hearts; to show it is exact, take a short exact sequence $0 \to A \to B \to C \to 0$ in $D^{♡}$ . By the heart-distinguished-triangle correspondence (BBD §1.3.6), this is equivalent to a distinguished triangle $A \to B \to C \to A [1]$ in $D$ . Apply $F$ (which preserves distinguished triangles as a triangulated functor) to get a distinguished triangle $F (A) \to F (B) \to F (C) \to F (A) [1]$ in $D^{'}$ . All three terms are in $D^{'♡}$ by t-exactness, so by the heart correspondence applied in reverse, the sequence $0 \to F (A) \to F (B) \to F (C) \to 0$ is short exact in $D^{'♡}$ . Hence $F ∣_{D^{♡}}$ is exact. $□$

Connections Master

Triangulated category — Verdier axioms TR1-TR4 04.03.10. The t-structure framework sits on top of the triangulated-category framework: the BBD axioms are extra structure beyond the triangulated axioms, and the cohomology functor $H_{t}^{n}$ depends crucially on the distinguished-triangle behaviour of $D$ for its long-exact-sequence property. The triangulated-category axioms supply the ambient setting; the t-structure supplies the abelian-category extraction.
Derived category $D (A)$ — localisation at quasi-isomorphisms 04.03.11. The standard t-structure on $D (A)$ is the canonical example, with heart equivalent to $A$ in degree $0$ . The derived category and the t-structure together give the canonical means by which an abelian category sits inside its derived category. The standard t-structure is what makes the inclusion $A ↪ D (A)$ a t-exact realisation.
Derived functors $R F$ and $L F$ via derived categories 04.03.12. T-exact functors between triangulated categories with t-structures restrict to exact functors between the hearts, providing the abelian-category-level theorem from the derived-category construction. Many natural derived functors are t-exact (or have measured failure of t-exactness, recorded in the BBD §3.1 axiomatic framework), and the t-exactness analysis is the standard tool for understanding how a derived functor interacts with cohomology.
Six-functor formalism — adjunctions and base change 04.03.16. The perverse t-structure on $D_{c}^{b} (X)$ for a complex algebraic variety $X$ supplies the abelian category $Perv (X)$ that organises constructible-sheaf cohomology. The six-functor operations $f^{*}, f_{*}, f_{!}, f^{!}$ interact with the perverse t-structure in a controlled way: $f_{!}$ and $f^{*}$ are right and left t-exact respectively (in the appropriate ranges), and the BBD decomposition theorem for $R f_{*}$ of pure perverse sheaves is the central application of the t-structure framework to the six-functor formalism.
Derived tensor product $\otimes^{L}$ and Tor 04.03.17. The derived tensor product is t-exact in each variable for the standard t-structure on $D (A)$ when one factor is flat, and the recovery of Tor groups as cohomology of the derived tensor product is a $H_{t}^{n}$ -style extraction in the t-structure framework. The interaction of $\otimes^{L}$ with non-standard t-structures (perverse, tilted) is a substantial extension subject of derived algebraic geometry.
Pointer: perverse sheaves $Perv (X)$ 04.03.19. The heart of the perverse t-structure on $D_{c}^{b} (X)$ is the abelian category of perverse sheaves. The simple objects are intermediate extensions $j_{! *} L [d_{Z}]$ of irreducible local systems on smooth locally closed strata. The BBD decomposition theorem says proper-direct-images of pure perverse sheaves decompose as direct sums of (shifted) intermediate extensions, and the resulting framework is the input to intersection cohomology, Saito's mixed Hodge modules, geometric Langlands, and the Fundamental Lemma.
Sheaf cohomology 04.03.01. The standard t-structure on $D^{+} (Sh (X))$ (or $D^{+} (O_{X})$ ) for a topological space or ringed space $X$ has heart $Sh (X)$ , and the cohomology functor $H_{t}^{n} = H^{n}$ recovers the classical sheaf cohomology. The hypercohomology spectral sequence for $R Γ$ of a complex of sheaves is the spectral sequence of the t-structure cohomology of the derived global sections.
Spectral sequence of a filtered complex 04.03.14. The perverse spectral sequence $^{p} H^{p} (X,^{p} H^{q} (F^{∙})) \Rightarrow H^{p + q} (X, F^{∙})$ is the spectral sequence of the perverse t-structure cohomology, derived from the truncation tower of $F^{∙}$ in the perverse t-structure. The general theme: every t-structure produces a spectral sequence (the t-structure spectral sequence) computing cohomology via the heart objects $H_{t}^{n}$ .

Historical & philosophical context Master

The notion of a t-structure was introduced by Alexander Beilinson, Joseph Bernstein, and Pierre Deligne in their 1982 paper "Faisceaux pervers" (Astérisque 100, Société Mathématique de France) ^{[source pending]}, where they axiomatised the data of a "good" pair of subcategories of a triangulated category and proved that the heart is always an abelian category. The paper introduced the t-structure framework precisely to support the construction of perverse sheaves: the middle-perversity t-structure on the bounded constructible derived category $D_{c}^{b} (X)$ of a complex algebraic variety $X$ has heart the category $Perv (X)$ of perverse sheaves, which the BBD authors used to prove the decomposition theorem (a far-reaching generalisation of the hard Lefschetz theorem) and to develop the theory of intersection cohomology in a coherent abelian-category framework.

The triangulated-category framework underlying the BBD construction is due to Jean-Louis Verdier in his 1963 thesis Catégories dérivées (defended 1967; published as Astérisque 239, Société Mathématique de France 1996) ^{[source pending]}, where the derived category and its triangulated structure first appeared as a separate object of study. The standard t-structure on $D (A)$ for an abelian category $A$ was implicit in Verdier's thesis but was axiomatised as a "t-structure" by BBD a decade later. Sergei Gelfand and Yuri Manin's Methods of Homological Algebra (Springer 1996; 2nd ed. 2003) ^{[source pending]} Ch. IV provides the canonical textbook treatment of t-structures in the modern formulation, integrating BBD's axiomatic framework into the standard derived-category narrative. Masaki Kashiwara and Pierre Schapira's Categories and Sheaves (Springer Grundlehren 332, 2006) ^{[source pending]} Ch. 10 supplies the comprehensive modern reference, with full proofs of the BBD theorems and extensions to non-bounded derived categories.

The reconstruction theme — recovering a geometric object from its derived category — was initiated by Alexei Bondal and Mikhail Kapranov in 1989 Math. USSR Izvestiya 35 ^{[source pending]}, where they introduced Serre functors and the framework for "enhanced" triangulated categories that admit reconstruction. The Bondal-Orlov reconstruction theorem (Bondal-Orlov 2001 Compositio Mathematica 125 ^{[source pending]}) states that a smooth projective variety $X$ with $ω_{X}$ ample or anti-ample is recovered up to isomorphism from $D^{b} (Coh (X))$ as a $k$ -linear triangulated category. The theorem has been extended to the singular case (Ballard 2011), the equivariant case (Kawamata 2002), and the noncommutative case (Bondal-Van den Bergh 2003), and is the input to homological projective duality (Kuznetsov 2007) and to Bridgeland-stability-based mirror symmetry (Bridgeland 2007 Ann. Math. 166).

Beilinson's reconstruction theorem $D^{b} (Perv (X)) ≃ D_{c}^{b} (X)$ for a complex algebraic variety $X$ (Beilinson 1987 Funct. Anal. Appl. 21; full treatment in BBD 1982 §3.1) ^{[source pending]} is the t-structure-level analogue: it identifies the constructible derived category with the bounded derived category of the heart of the perverse t-structure. The unusual feature of this reconstruction is that the heart determines not just the original triangulated category up to equivalence, but the precise derived-category structure — most heart-of-t-structure reconstructions fail. The success of the perverse case rests on the Artin-vanishing properties of constructible sheaves on algebraic varieties, and is the technical input to many subsequent results in geometric representation theory, including Beilinson-Bernstein localisation (Beilinson-Bernstein 1981 C.R. Acad. Sci. Paris 292), the geometric Satake equivalence (Mirković-Vilonen 2007 Ann. Math. 166), and the Fundamental Lemma (Ngô 2010 Publ. Math. IHES 111).

The modern $\infty$ -categorical reformulation of the t-structure framework, in which a t-structure becomes a pair of full sub- $\infty$ -categories of a stable $\infty$ -category satisfying analogous axioms, appears in Jacob Lurie's Higher Algebra (2017) §1.2.1 and in the Toën-Vezzosi homotopical algebraic geometry programme. The t-structure framework is now the foundational tool in derived algebraic geometry (Toën-Vezzosi, Lurie), prismatic cohomology (Bhatt-Scholze 2018), condensed mathematics (Clausen-Scholze 2019), and the geometrisation of the local Langlands correspondence (Fargues-Scholze 2021).

Bibliography Master

@article{BBD1982,
  author    = {Beilinson, Alexander A. and Bernstein, Joseph and Deligne, Pierre},
  title     = {Faisceaux pervers},
  journal   = {Ast{\'e}risque},
  volume    = {100},
  year      = {1982},
  publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
  note      = {Proceedings of the 1981 CIRM colloquium on the analysis and topology on singular spaces}
}

@phdthesis{VerdierThesis,
  author    = {Verdier, Jean-Louis},
  title     = {Cat{\'e}gories d{\'e}riv{\'e}es et cat{\'e}gories triangul{\'e}es},
  school    = {Universit{\'e} de Paris},
  year      = {1967},
  note      = {Published as Ast{\'e}risque 239, Soci{\'e}t{\'e} Math{\'e}matique de France, 1996; preliminary version `Cat{\'e}gories d{\'e}riv{\'e}es, {\'e}tat 0' in SGA 4{$\frac{1}{2}$}, Springer LNM 569, 1977}
}

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

@book{KashiwaraSchapira,
  author    = {Kashiwara, Masaki and Schapira, Pierre},
  title     = {Categories and Sheaves},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {332},
  year      = {2006}
}

@article{BondalKapranov1989,
  author    = {Bondal, Alexei I. and Kapranov, Mikhail M.},
  title     = {Representable functors, Serre functors, and reconstructions},
  journal   = {Mathematics of the USSR-Izvestiya},
  volume    = {35},
  number    = {3},
  pages     = {519--541},
  year      = {1990},
  note      = {English translation of the Russian original from 1989; Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 53 (1989), 1183--1205}
}

@article{BondalOrlov2001,
  author    = {Bondal, Alexei I. and Orlov, Dmitri O.},
  title     = {Reconstruction of a variety from the derived category and groups of autoequivalences},
  journal   = {Compositio Mathematica},
  volume    = {125},
  number    = {3},
  pages     = {327--344},
  year      = {2001}
}

@article{Beilinson1987,
  author    = {Beilinson, Alexander A.},
  title     = {On the derived category of perverse sheaves},
  journal   = {Funktsional'nyi Analiz i Ego Prilozheniya},
  volume    = {21},
  year      = {1987},
  note      = {English translation in Functional Analysis and Its Applications; the reconstruction theorem $D^b(\mathrm{Perv}(X)) \simeq D^b_c(X)$ was announced here and proved in full in BBD 1982 §3.1}
}

@article{Bridgeland2007,
  author    = {Bridgeland, Tom},
  title     = {Stability conditions on triangulated categories},
  journal   = {Annals of Mathematics},
  volume    = {166},
  number    = {2},
  pages     = {317--345},
  year      = {2007}
}

@book{Happel1988,
  author    = {Happel, Dieter},
  title     = {Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras},
  publisher = {Cambridge University Press},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {119},
  year      = {1988}
}

@article{Rickard1989,
  author    = {Rickard, Jeremy},
  title     = {Morita theory for derived categories},
  journal   = {Journal of the London Mathematical Society},
  volume    = {39},
  number    = {3},
  pages     = {436--456},
  year      = {1989}
}

@article{BeilinsonBernstein1981,
  author    = {Beilinson, Alexander A. and Bernstein, Joseph},
  title     = {Localisation de $g$-modules},
  journal   = {Comptes Rendus de l'Acad{\'e}mie des Sciences de Paris S{\'e}rie I},
  volume    = {292},
  pages     = {15--18},
  year      = {1981}
}

@article{MirkovicVilonen2007,
  author    = {Mirkovi{\'c}, Ivan and Vilonen, Kari},
  title     = {Geometric Langlands duality and representations of algebraic groups over commutative rings},
  journal   = {Annals of Mathematics},
  volume    = {166},
  number    = {1},
  pages     = {95--143},
  year      = {2007}
}

@article{Ngo2010,
  author    = {Ng{\^o}, Bao Ch{\^a}u},
  title     = {Le lemme fondamental pour les alg{\`e}bres de Lie},
  journal   = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume    = {111},
  pages     = {1--169},
  year      = {2010}
}

@book{LurieHigherAlgebra,
  author    = {Lurie, Jacob},
  title     = {Higher Algebra},
  note      = {Book draft, available at the author's website; §1.2.1 on t-structures in stable $\infty$-categories},
  year      = {2017}
}

@misc{StacksTStructures,
  author       = {{The Stacks Project authors}},
  title        = {The Stacks Project, Tag 0FNI (t-structures on triangulated categories)},
  howpublished = {\url{https://stacks.math.columbia.edu/tag/0FNI}},
  year         = {2026}
}

Prerequisites

04.03.10
04.03.11
04.03.12

Tier anchors

beginner: Gelfand-Manin *Methods of Homological Algebra* Ch. IV §4 informal — a t-structure splits a triangulated category into negative and positive halves, and the overlap is an honest abelian category called the heart
intermediate: Gelfand-Manin *Methods of Homological Algebra* Ch. IV §4; Kashiwara-Schapira *Categories and Sheaves* Ch. 10 §1; Stacks Project Tag 0FNI
master: Beilinson-Bernstein-Deligne *Faisceaux pervers* (*Astérisque* 100, 1982) §§1.3--1.4; Verdier *Catégories dérivées* (*Astérisque* 239, 1996) §3; Bondal-Kapranov 1989 *Math. USSR Izv.* 35 (representability for derived categories); Bondal-Orlov 2001 *Compositio Math.* 125 (reconstruction theorem); Gelfand-Manin *Methods of Homological Algebra* Ch. IV

References

Beilinson-Bernstein-Deligne — Faisceaux pervers · *Astérisque* 100, Société Mathématique de France 1982; §1.3 (definition of a t-structure on a triangulated category, axioms, truncation triangles); §1.3.6 (the heart is an abelian category); §1.3.13 (cohomological functor $H^0_t$); §§2.1--2.2 (the middle-perversity t-structure on $D^b_c(X)$, recovering perverse sheaves as the heart $\mathrm{Perv}(X)$)
Verdier — Catégories dérivées · thesis defended 1967, preliminary appendix to SGA 4½ (Springer LNM 569, 1977) pp. 262--311; full thesis published as *Astérisque* 239, Société Mathématique de France 1996; §3 (truncation functors and the standard t-structure on $D(\mathcal{A})$ for an abelian category $\mathcal{A}$)
Gelfand-Manin — Methods of Homological Algebra · Springer 2nd ed. 2003; Ch. IV §4 (t-structures on a triangulated category, the heart, truncations, cohomological functors); Appendix A on BBD t-structures with worked examples
Kashiwara-Schapira — Categories and Sheaves · Springer Grundlehren 332 (2006); Ch. 10 §1 (axiomatic t-structure on a triangulated category, the heart, truncations, adjunctions); Ch. 10 §2 (cohomology functors, t-exact functors)
Bondal-Kapranov — Representable functors, Serre functors, and reconstructions · *Math. USSR Izv.* 35 (1990), 519--541 (English translation of the 1989 Russian original); enhanced triangulated categories, Serre functors, and the reconstruction problem for triangulated categories from their bounded subcategories
Bondal-Orlov — Reconstruction of a variety from the derived category · *Compositio Mathematica* 125 (2001), 327--344; the theorem: for $X$ a smooth projective variety with ample (or anti-ample) canonical bundle $\omega_X$, $X$ is recovered up to isomorphism from $D^b(\mathrm{Coh}(X))$ as a $\mathbb{C}$-linear triangulated category
Stacks Project — t-structures · <https://stacks.math.columbia.edu/tag/0FNI> (definition of a t-structure on a triangulated category and its heart); Tag 0FNJ (truncation functors); Tag 0FNL (the heart is an abelian category); Tag 0FNN (cohomological functors $H^n_t$)
Beilinson reconstruction theorem · Beilinson 1987 *Funct. Anal. Appl.* 21 (announcement); full treatment in BBD 1982 §3.1 (the equivalence $D^b(\mathrm{Perv}(X)) \simeq D^b_c(X)$ for the middle-perversity t-structure on a complex algebraic variety $X$); the perverse-cohomology decomposition recovers all constructible cohomological information

Estimated time

beginner: 18m
intermediate: 45m
master: 80m