Bounded Derived Categories

The bounded derived categories $D^b$ , $D^+$ , $D^-$ restrict attention to complexes with bounded cohomology. The bounded derived category $D^b(\mathcal{A})$ is the most commonly used variant, especially in algebraic geometry where one works with $D^b(\mathrm{Coh}(X))$ .

Definitions

Definition6.3Bounded Derived Categories

For an abelian category $\mathcal{A}$ :

$D^+(\mathcal{A})$ : full subcategory of complexes $A^\bullet$ with $H^n(A) = 0$ for $n \ll 0$ (bounded below cohomology).
$D^-(\mathcal{A})$ : complexes with $H^n(A) = 0$ for $n \gg 0$ (bounded above cohomology).
$D^b(\mathcal{A})$ : complexes with $H^n(A) = 0$ for $|n| \gg 0$ (bounded cohomology).

Each is a full triangulated subcategory of $D(\mathcal{A})$ .

RemarkBounded complexes vs bounded cohomology

$D^b(\mathcal{A})$ consists of complexes with bounded cohomology, not necessarily bounded complexes. However, if $\mathcal{A}$ has enough injectives/projectives, every object of $D^b(\mathcal{A})$ is isomorphic to a bounded complex (of injectives/projectives).

Examples

ExampleD^b(Vect_k)

In $D^b(\mathbf{Vect}_k)$ , every complex is isomorphic to $\bigoplus_n H^n[-n]$ (the direct sum of its cohomology, placed in the correct degrees). This is because every short exact sequence of vector spaces splits.

ExampleD^b(Coh(X)) for a smooth variety

$D^b(\mathrm{Coh}(X))$ for a smooth projective variety is the primary object of study in derived algebraic geometry. It has Serre duality, and its autoequivalence group is an important invariant.

ExampleD^+(Sh(X)) and sheaf cohomology

$D^+(\mathbf{Sh}(X))$ is the natural home for sheaf cohomology. The derived global sections $R\Gamma : D^+(\mathbf{Sh}(X)) \to D^+(\mathbf{Ab})$ is the total derived functor of $\Gamma$ .

ExampleD^b(R-mod) for hereditary R

If $R$ is a hereditary ring (global dimension $\leq 1$ ), every complex in $D^b(R\text{-}\mathrm{mod})$ is isomorphic to a direct sum of shifted modules. The derived category decomposes as a direct sum of copies of module categories.

ExampleD^b(Ab)

Since $\mathbb{Z}$ is a PID (hereditary), every complex in $D^b(\mathbf{Ab})$ is formal: $A^\bullet \cong \bigoplus_n H^n(A)[-n]$ . This fails for more general rings.

ExampleD^b with non-formal complexes

Over $R = k[x]/(x^2)$ , the complex $\cdots \to R \xrightarrow{x} R \xrightarrow{x} R \to \cdots$ has cohomology $k$ in every degree but is not formal (not isomorphic in $D^b$ to $\bigoplus k[-n]$ ). The $A_\infty$ -structure on cohomology detects this non-formality.

ExampleCompact objects in D(R)

The compact objects of $D(R)$ are precisely the perfect complexes: bounded complexes of finitely generated projective modules. These form $D^b(\mathrm{proj}\text{-}R) = \mathrm{Perf}(R) \subseteq D^b(R)$ .

ExampleD^b and K-theory

$K_0(D^b(\mathcal{A})) \cong K_0(\mathcal{A})$ for an abelian category $\mathcal{A}$ : the Grothendieck group of the bounded derived category equals that of the abelian category. The relation $[Y] = [X] + [Z]$ from triangles corresponds to additivity of Euler characteristic.

ExampleD^b of an abelian category with enough injectives

If $\mathcal{A}$ has enough injectives, $D^+(\mathcal{A}) \simeq K^+(\mathrm{Inj}\, \mathcal{A})$ : bounded below derived category is equivalent to the homotopy category of bounded below injective complexes. Similarly, $D^-(\mathcal{A}) \simeq K^-(\mathrm{Proj}\, \mathcal{A})$ when enough projectives exist.

ExampleTruncation functors

The canonical truncation functors $\tau_{\leq n}, \tau_{\geq n}$ on $D(\mathcal{A})$ restrict to endofunctors of $D^b(\mathcal{A})$ . They satisfy $\tau_{\leq n} \circ \tau_{\geq m} = 0$ for $m > n$ and form part of the t-structure data.

ExampleDerived equivalences preserve D^b

An exact equivalence $F : D^b(\mathcal{A}) \to D^b(\mathcal{B})$ sends bounded complexes to bounded complexes. Fourier-Mukai equivalences and tilting equivalences preserve boundedness.

ExampleEmbedding of D^b into D

$D^b(\mathcal{A}) \hookrightarrow D(\mathcal{A})$ is a full triangulated subcategory. For well-behaved categories (Grothendieck abelian categories), $D^b$ contains all the "finitary" information.

Theorem6.2Resolution model for bounded categories

If $\mathcal{A}$ has enough injectives:

$D^+(\mathcal{A}) \simeq K^+(\mathrm{Inj}\, \mathcal{A})$

If $\mathcal{A}$ has enough projectives:

$D^-(\mathcal{A}) \simeq K^-(\mathrm{Proj}\, \mathcal{A})$

These equivalences allow computation: to compute $\mathrm{Hom}_{D^+}(A, B)$ , replace $B$ by an injective resolution $I_B$ and compute $\mathrm{Hom}_{K^+}(A, I_B)$ .

RemarkLooking ahead

The bounded derived category $D^b(\mathcal{A})$ carries a natural t-structure whose heart recovers $\mathcal{A}$ . Different t-structures on $D^b(\mathcal{A})$ give different abelian categories (e.g., perverse sheaves), providing a powerful tool for constructing new abelian categories from old ones.