Derived Categories¶

Job: localize chain complexes at quasi-isomorphisms; triangulated structure on $D(\mathcal{A})$ and bounded $D^b(\mathcal{A})$; $t$-structures as a pointer to hearts and perverse sheaves.

Core¶

Complexes and homotopy¶

Let $\mathcal{A}$ be abelian. A (cochain) complex $A^\bullet$ is $\cdots\to A^n\xrightarrow{d^n}A^{n+1}\to\cdots$ with $d^{n+1}d^n=0$. Morphisms $f^\bullet:A^\bullet\to B^\bullet$ commute with $d$. Category $\operatorname{Ch}(\mathcal{A})$ (or $C(\mathcal{A})$).

Shift: $A[n]^k=A^{n+k}$, $d_{A[n]}=(-1)^n d_A$ (sign conventions vary; fix one and stay consistent).

Homology $H^n(A^\bullet)=\ker d^n/\operatorname{im}d^{n-1}$. $f$ is a quasi-isomorphism (qis) if $H^n(f)$ is iso for all $n$.

Homotopy. $f\simeq g$ if $f-g=dh+hd$ for some $h^n:A^n\to B^{n-1}$. Homotopic maps induce the same map on homology. Homotopy category $K(\mathcal{A})$: objects complexes, morphisms homotopy classes. $K$ is additive; quasi-isos still make sense.

Subcategory	Objects
$K^b$	bounded complexes ($A^n=0$ for $
$K^+$, $K^-$	bounded below / above
$K(\operatorname{Inj})$, $K(\operatorname{Proj})$	injective / projective complexes

Derived category as localization¶

Derived category $D(\mathcal{A})$: localization of $K(\mathcal{A})$ at quasi-isomorphisms (Gabriel–Zisman), or equivalently Verdier quotient by the thick subcategory of acyclic complexes.

Existence / models. If $\mathcal{A}$ has enough injectives, $D^+(\mathcal{A})\simeq K^+(\operatorname{Inj}(\mathcal{A}))$. Enough projectives $\Rightarrow D^-(\mathcal{A})\simeq K^-(\operatorname{Proj})$. Morphisms: $$ \operatorname{Hom}{D(\mathcal{A})}(X,Y) \cong \varinjlim(X',Y) $$ (roof calculus). In particular }}X} \operatorname{Hom}_{K$\operatorname{Hom}_{D(\mathcal{A})}(M,N[i])\cong\operatorname{Ext}^i_{\mathcal{A}}(M,N)$ for objects of $\mathcal{A}$ placed in degree $0$.

Moral: objects of $D(\mathcal{A})$ are complexes up to qis; a single object $M\in\mathcal{A}$ is a complex concentrated in degree $0$; resolutions become isomorphisms $P^\bullet\xrightarrow{\sim}M$ in $D$.

Triangulated structure¶

$D(\mathcal{A})$ (and $K(\mathcal{A})$) carry a triangulated category structure: shift autoequivalence $[1]$ and distinguished triangles $X\to Y\to Z\to X[1]$.

From short exact sequences of complexes. A short exact $0\to A^\bullet\to B^\bullet\to C^\bullet\to 0$ in $\operatorname{Ch}$ yields a distinguished triangle in $D$ (same objects) after using that the cone of $A\to B$ is qis to $C$.

Mapping cone. For $f:X\to Y$, $$ \operatorname{Cone}(f)^n=X^{n+1}\oplus Y^n,\qquad d=\begin{pmatrix}-d_X&0\f&d_Y\end{pmatrix}, $$ and $X\xrightarrow{f}Y\to\operatorname{Cone}(f)\to X[1]$ is distinguished (in $K$ and $D$).

Axioms (Verdier), informal:

TR	Content
TR1	identity triangles; isomorphic to distinguished are distinguished
TR2	rotate triangles
TR3	morphisms of triangles fill in
TR4	octahedral axiom (compose cones)

Consequences. Long exact sequences of Hom functors; LES of homology from any distinguished triangle; $H^0$ as cohomological functor $D\to\mathcal{A}$.

Bounded derived category $D^b(\mathcal{A})$¶

Full subcategory of complexes with $H^n=0$ for $|n|\gg 0$ (or localize $K^b$). For $\mathcal{A}=\operatorname{Coh}(X)$ or $R\text{-mod}$ f.g. modules under finiteness hyp., $D^b$ is the working universe of algebraic geometry and representation theory.

Perfect complexes $\operatorname{Perf}(R)\subset D(R\text{-Mod})$: quasi-isomorphic to bounded complexes of finite projectives. On schemes, $\operatorname{Perf}(X)$ = locally quasi-isomorphic to bounded finite locally free complexes.

Derived functors upgraded¶

Left exact $F:\mathcal{A}\to\mathcal{B}$ with enough injectives yields total right derived $RF:D^+(\mathcal{A})\to D^+(\mathcal{B})$ with $H^i(RF(M))\cong R^iF(M)$. Similarly $LF$ for right exact $F$. Composition: $R(FG)\cong RF\circ RG$ under acyclicity conditions (Grothendieck spectral sequence as degeneration of a triangle/spectral object).

$\operatorname{RHom}$ and $\otimes^L$: bi-derived functors on $D$; adjunction $\operatorname{RHom}(X\otimes^L Y,Z)\cong\operatorname{RHom}(X,\operatorname{RHom}(Y,Z))$.

$t$-structures (pointer)¶

A $t$-structure on a triangulated $\mathcal{T}$ is a pair of full subcategories $(\mathcal{T}^{\le 0},\mathcal{T}^{\ge 0})$ with: - $\mathcal{T}^{\le 0}[1]\subset\mathcal{T}^{\le 0}$, $\mathcal{T}^{\ge 0}[-1]\subset\mathcal{T}^{\ge 0}$; - $\operatorname{Hom}(X,Y)=0$ for $X\in\mathcal{T}^{\le 0}$, $Y\in\mathcal{T}^{\ge 1}$; - every $X$ fits in a triangle $X^{\le 0}\to X\to X^{\ge 1}\to$.

Heart $\mathcal{H}=\mathcal{T}^{\le 0}\cap\mathcal{T}^{\ge 0}$ is abelian; standard $t$-structure on $D(\mathcal{A})$ has heart $\mathcal{A}$. Perverse $t$-structure on $D^b_c(X)$ (constructible sheaves) has heart of perverse sheaves — middle perversity truncations. Tilting produces new hearts with the same derived category.

See it¶

Beilinson’s equivalence. $D^b(\operatorname{Coh}(\mathbb{P}^n))\simeq D^b(\operatorname{mod}\text{-}A)$ for a finite-dimensional algebra $A=\operatorname{End}(\bigoplus_{i=0}^n\mathcal{O}(i))$ (exceptional collection). Geometric derived categories become representation theory of quivers/algebras — prototype of Bondal–Orlov, mirror symmetry equivalences $D^b(X)\simeq D^b(Y)$ for non-isomorphic varieties.

Grothendieck duality. Proper $f:X\to Y$ of schemes: $Rf_*$ has right adjoint $f^!$ on derived categories; dualizing complex $\omega_Y^\bullet$ with $\operatorname{RHom}(-,\omega^\bullet)$ implementing Serre duality. Statement invisible in underived language for singular $Y$.

Check¶

Why is a quasi-isomorphism not necessarily a homotopy equivalence? Give an example in $\mathbf{Ch}(\mathbf{Ab})$.

$0\to\mathbb{Z}\xrightarrow{\times 2}\mathbb{Z}\to 0$ (degrees $0,1$) maps by qis to $0$ if replaced? Standard: resolution $P^\bullet\to M$ with $P^\bullet$ not split: $\cdots\to 0\to\mathbb{Z}\xrightarrow{\times n}\mathbb{Z}\to 0$ quasi-isomorphic to $\mathbb{Z}/n$ in degree $0$, but not homotopy equivalent to the stalk complex $\mathbb{Z}/n$ unless the extension splits in $\operatorname{Ch}$ (Homotopy equivalence would imply projectivity issues). Explicitly: multiplication by $n$ on $\mathbb{Z}[0]$ is qis onto image complex only in trivial cases — classic: cone of $\times n:\mathbb{Z}\to\mathbb{Z}$ is qis to $\mathbb{Z}/n[1]$ wait degree: cone is quasi-isomorphic to $\mathbb{Z}/n$ placed appropriately, and $\times n$ is mono not split so not homotopy equiv to zero on the kernel side. Bottom line: $H^*$ iso does not give inverse maps up to homotopy without projectivity.

Show $\operatorname{Hom}_{D(\mathbf{Ab})}(\mathbb{Z}/n,\mathbb{Z}/n[1])\cong\mathbb{Z}/n$.

Equals $\operatorname{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/n,\mathbb{Z}/n)\cong\mathbb{Z}/n$. Generator = class of nonsplit extension $0\to\mathbb{Z}/n\to\mathbb{Z}/n^2\to\mathbb{Z}/n\to 0$ (or $0\to\mathbb{Z}/n\xrightarrow{\times n}\mathbb{Z}/n^2\to\cdots$).

Pitfalls¶

$K(\mathcal{A})$ vs $D(\mathcal{A})$: homotopy equivalences vs quasi-isomorphisms; $\operatorname{Hom}_K$ can vanish while $\operatorname{Hom}_D$ does not (roofs).
Morphisms in $D$ are not chain maps — roofs $X\leftarrow X'\to Y$; hard to “write down” without resolutions.
$D^b(\operatorname{Coh}(X))$ is not the same as bounded complexes of coherents modulo qis that stay coherent without care — use h-injective models / existence theorems.
Triangulated categories lack functorial cones — enhancement (dg / $\infty$-categories) repairs this for modern work.

Refs¶

Gelfand–Manin, Methods of Homological Algebra
Weibel, An Introduction to Homological Algebra (Ch. 10)
Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry
Kashiwara–Schapira, Sheaves on Manifolds / Categories and Sheaves

Next¶

Foundations · HA index · AG cohomology · Modules · Category theory · Algebraic topology