Cohomological Functor

A cohomological functor converts distinguished triangles into long exact sequences. It is the triangulated analogue of a half-exact functor in abelian categories. The prototype is the cohomology functor $H^n$ on the derived category.

Definition

Definition5.6Cohomological Functor

An additive functor $H : \mathcal{T} \to \mathcal{A}$ from a triangulated category to an abelian category is cohomological if for every distinguished triangle $X \to Y \to Z \to X[1]$ , the induced sequence

$H(X) \to H(Y) \to H(Z)$

is exact in $\mathcal{A}$ .

RemarkLong exact sequence

By applying $H$ to the rotations of the triangle, a cohomological functor produces a long exact sequence:

$\cdots \to H(X[n]) \to H(Y[n]) \to H(Z[n]) \to H(X[n+1]) \to \cdots$

Writing $H^n(-) = H(-[n])$ , this becomes $\cdots \to H^n(X) \to H^n(Y) \to H^n(Z) \to H^{n+1}(X) \to \cdots$ .

Examples

ExampleCohomology functor H^n

$H^n : D(\mathcal{A}) \to \mathcal{A}$ is a cohomological functor. For a distinguished triangle $A \to B \to C \to A[1]$ , the sequence $H^n(A) \to H^n(B) \to H^n(C)$ is exact for each $n$ , giving the long exact sequence in cohomology.

ExampleHom functor

For any object $W$ in a triangulated category $\mathcal{T}$ , the functor $\mathrm{Hom}(W, -) : \mathcal{T} \to \mathbf{Ab}$ is cohomological. This follows from the axioms of distinguished triangles.

ExampleContravariant Hom

$\mathrm{Hom}(-, W) : \mathcal{T}^{\mathrm{op}} \to \mathbf{Ab}$ is cohomological (a homological functor in Gelfand-Manin's terminology). For a triangle $X \to Y \to Z \to X[1]$ : $\mathrm{Hom}(Z, W) \to \mathrm{Hom}(Y, W) \to \mathrm{Hom}(X, W)$ is exact.

ExampleGlobal sections as cohomological

On $D^b(\mathbf{Sh}(X))$ , the derived global sections functor $R\Gamma(X, -) : D^b(\mathbf{Sh}(X)) \to D^b(\mathbf{Ab})$ is exact (maps distinguished triangles to distinguished triangles). Composing with $H^0$ gives the cohomological functor $H^0 \circ R\Gamma = H^0(X, -)$ .

ExampleEuler characteristic is NOT cohomological

The Euler characteristic $\chi : K_0(D^b(\mathcal{A})) \to \mathbb{Z}$ is additive on triangles ( $\chi(Y) = \chi(X) + \chi(Z)$ ), but it is not a functor (it does not act on morphisms), so it is not cohomological in the strict sense.

ExampleK-theory as a cohomological functor

For a triangulated category $\mathcal{T}$ with a bounded t-structure, the K-theory functor $K_0 : \mathcal{T} \to \mathbb{Z}$ (sending objects to their class in the Grothendieck group) is additive on distinguished triangles.

ExampleExt as cohomological

$\mathrm{Ext}^n(X, -) = \mathrm{Hom}_{D(\mathcal{A})}(X, -[n]) = H^n \circ \mathrm{RHom}(X, -)$ is cohomological. For a triangle $A \to B \to C \to A[1]$ : $\cdots \to \mathrm{Ext}^n(X, A) \to \mathrm{Ext}^n(X, B) \to \mathrm{Ext}^n(X, C) \to \mathrm{Ext}^{n+1}(X, A) \to \cdots$ .

ExampleStalk functor for sheaves

For sheaves on $X$ and a point $x \in X$ , the stalk functor $(-)_x : D^b(\mathbf{Sh}(X)) \to D^b(\mathbf{Ab})$ is exact. Composing with $H^n$ gives the cohomological functor $\mathcal{F}^\bullet \mapsto H^n(\mathcal{F}^\bullet_x)$ .

ExampleHypercohomology

$\mathbb{H}^n(X, -) = H^n \circ R\Gamma(X, -)$ is a cohomological functor from $D^b(\mathbf{Sh}(X))$ to $\mathbf{Ab}$ . It converts distinguished triangles of sheaf complexes into long exact sequences.

ExampleRepresentable cohomological functors

In a triangulated category with enough structure, the Brown representability theorem states that every cohomological functor $H : \mathcal{T}^{\mathrm{op}} \to \mathbf{Ab}$ satisfying a product condition is representable: $H(-) \cong \mathrm{Hom}(-, E)$ for some $E \in \mathcal{T}$ .

ExamplePerverse cohomology

The perverse cohomology functors ${}^p H^n : D^b_c(X) \to \mathrm{Perv}(X)$ associated to the perverse t-structure are cohomological functors. They send distinguished triangles to long exact sequences of perverse sheaves.

ExampleDerived tensor product

$H^n(A \otimes^L -) : D(\mathcal{A}) \to \mathcal{A}$ is a cohomological functor for each fixed $A$ and each $n$ . This gives the long exact sequence for Tor: $\cdots \to \mathrm{Tor}_n(A, X) \to \mathrm{Tor}_n(A, Y) \to \mathrm{Tor}_n(A, Z) \to \cdots$ .

Theorem5.3Characterization via representability

In a triangulated category $\mathcal{T}$ , a functor $H : \mathcal{T} \to \mathcal{A}$ is cohomological if and only if it converts distinguished triangles to exact sequences. By rotation, this is equivalent to converting them to long exact sequences.

RemarkExact functors vs cohomological functors

An exact functor $F : \mathcal{T} \to \mathcal{T}'$ between triangulated categories sends distinguished triangles to distinguished triangles (and commutes with shift). A cohomological functor $H : \mathcal{T} \to \mathcal{A}$ to an abelian category sends them to exact sequences. The composition $H \circ F$ is cohomological.