## Notes of Frédéric Bourgeois’ lecture

Derived category of sheaves

1. Generalities on categories

The categories we shall encounter are

1. Sets.
2. Top, Diff.
3. Abelian Groups.
4. ${k}$-modules.
5. Sh${(M)}$, sheaves on ${M}$.

Examples of functors that we shall encounter are

1. Forgetful functor to Sets.
2. ${Hom(X,\cdot)}$.
3. ${\otimes^n}$
4. ${Hom_k(\cdot,M)}$.
5. Sections of a sheaf ${\Gamma}$: OpenSets${(M)\rightarrow}$ Abelian Groups.

There is a notion of morphism of functors. A functor is representable if it is ${Hom(X,\cdot)}$ for some object ${X}$.

An additive category is a category such that

1. There is an object ${0}$ such that ${Hom(0,0)=0}$.
2. All ${Hom(X,Y)}$ are abelian groups, and composition is bilinear.
3. There is a direct sum, i.e. for all ${X_1}$, ${X_2}$, an object ${Z}$ with arrows ${i_j \in Hom(X_j,Z)}$ such that ${Hom(X_j,Y)=Hom(Z,Y)\circ i_j}$ for ${j=1,2}$.
4. There is a direct product, i.e. for all ${Y_1}$, ${Y_2}$, an object ${W}$ with arrows ${k_j \in Hom(W,Y_j)}$ such that ${Hom(X,Y_j)=k_j \circ Hom(X,W)}$ for ${j=1,2}$ and all ${X}$.

Ab, ${k}$-Mod, Sh${(M)}$ are additive.

A functor ${F}$ is additive if the map ${Hom(X,Y)\rightarrow Hom(F(X),F(Y))}$ is a group homomorphism.

1.2. Complexes

If ${\mathcal{C}}$ is an additive category, the category of complexes ${C(\mathcal{C})}$ is additive as well. We denote by ${T^n}$ the shift functor that re-numbers spaces in a complex. A homotopy between chain maps ${f,g:X\rightarrow Y}$ is a morphism ${h:X\rightarrow Y[-1]}$ such that ${f-g=hd+dh}$. The intermediate category ${\mathcal{K}(\mathcal{C}))}$ is the additive category with objects complexes and morphisms homotopy classes of chain maps.

This category is merely an intermediate step.

1.3. Abelian categories

Homology cannot be defined for general additive categories, since we do not have kernels and images. For ${f\in Hom(X,Y)}$, the kernel functor ${ker(f):\mathcal{C}^0\rightarrow }$ Sets, is defined as follows :

$\displaystyle \begin{array}{rcl} ker(f)(W)=\{g\in Hom(W,X)\,;\,f\circ g=0\}. \end{array}$

If this functor is representable, this provides us with an object, also denoted by ${ker(f)}$, such that ${ker(f)(W)=Hom(W,ker(f))}$.

Similarly, given ${f\in Hom(X,Y)}$, there is a cokernel functor ${coker(f):\mathcal{C}\rightarrow}$ Sets. If this functor is representable, this provides us with an object, also denoted by ${coker(f)}$.

Remark 1 ${Z\mapsto Hom(Z,Y)/f\circ Hom(Z,X)}$ is a functor, but it is not representable, usually. So this is not the right definition for cokernel.

Indeed, in Ab, let ${f:{\mathbb Z}\rightarrow{\mathbb Z}}$ is multiplication by ${n}$, and ${Z={\mathbb Z}_n}$. Then ${Hom({\mathbb Z}_n,{\mathbb Z})=0}$ whereas ${Hom({\mathbb Z}_n,{\mathbb Z}_n)\not=0}$.

Definition 1 In an abelian category,

1. For all ${f\in Hom(X,Y)}$, ${ker(f)}$ and ${coker(f)}$ exist.
2. Let ${c:Y\rightarrow coker(f)}$ and ${k:ker(f)\rightarrow X}$ be the natural maps. Then ${coker(k)}$ and ${ker(c)}$ are canonically isomorphic.

In fact, ${f}$ has a canonical factorization ${X\rightarrow I\rightarrow Y}$ where ${I=coker(k)=ker(c)}$.

1.4. Homology

If ${\mathcal{C}}$ is an abelian category, given a complex ${X}$, for all degrees ${n}$, the differential ${d^{n-1}:X^{n-1}\rightarrow X^n}$ factors as ${a^n : X^{n-1}\rightarrow ker(d^n)}$ and ${d^{n}:X^{n}\rightarrow X^{n+1}}$ factors as ${b^{n-1} : coker(d^{n-1})\rightarrow X^{n+1}}$. One defines the homology of ${X}$ as ${H(X)^n=\ker(a^n)=coker(b^{n-1})}$.

A quasi-isomorphism between complexes is a morphism which induces and isomorphism on homology. We want to think of quasi-isomorphisms as as good as isomorphisms.

Example 1 Consider the following two complexes.

$\displaystyle \begin{array}{rcl} 0\rightarrow{\mathbb Z}\stackrel{\times 2}{\rightarrow}{\mathbb Z}\rightarrow 0,\\ 0\rightarrow 0\rightarrow {\mathbb Z}/2{\mathbb Z}\rightarrow 0, \end{array}$

There is an obvious chain map between them. This is a quasiisomorphism, but there is no homotopy from it to 0.

1.5. Derived category

The derived category of an abelian category ${\mathcal{C}}$ has for objects complexes and for morphisms a localization of chain maps mod homotopy, which we now define as the solution of a universal problem.

Definition 2 The derived category of an abelian category ${\mathcal{C}}$ is the unique category such that there exists a functor ${Q:K(\mathcal{C})\rightarrow D(\mathcal{C})}$ such that

1. ${Q(}$quasiisomorphism${)=}$isomorphism.
2. For any category ${\mathcal{D}}$, any functor ${F:K(\mathcal{C})\rightarrow\mathcal{D}}$ such that

$\displaystyle F(quasiisomorphism)=isomorphism$

factorizes through ${Q}$.

Concretely, ${D(\mathcal{C})}$ is constructed as follows. Let ${S}$ be the collection of quasiisomorphisms in ${K(\mathcal{C})}$. Define

$\displaystyle \begin{array}{rcl} Hom_{D(\mathcal{C})}(X,Y)=\{(s,f)\in (Hom_{K(\mathcal{C})}(Z,X)\cap S)\times Hom_{K(\mathcal{C})}(Z,Y)\}/\sim \end{array}$

where the equivalence relation ${(s,f)\sim(t,g)}$ roughly means that ${f\circ s^{-1}=g\circ t^{-1}}$, and precisely means that there exist quasiisomorphisms ${s,v}$ such that

$\displaystyle \begin{array}{rcl} s\circ u=t\circ v,\quad g\circ v=f\circ u. \end{array}$

The functor ${Q}$ is defined as identity on objects (i.e. complexes) and on morphisms ${f}$, by ${Q(f)=(id,f)}$.

1.6. Localizing systems

The proof of the statement included in the definition of the derived category is not straightforward.

For the definition to work, one needs the following properties of ${S}$ (say ${S}$ is a localizing system) :

1. Identity belongs to ${S}$.
2. ${S}$ is stable under composition (whenever defined).
3. Given ${f\in Hom(X,Y)}$ and a quasiisomorphism ${t\in Hom(Z,Y)}$, there exists a complex ${W}$, ${g\in Hom(W,Z)}$ and a quasiisomorphism ${s\in Hom(W,X)}$ such that ${t\circ g=f\circ s}$.
4. For all ${f,g\in Hom(X,Y)}$, existence of ${t\in S}$ such that ${t\circ f=t\circ g}$ implies existence of ${s\in S}$ such that ${f\circ s=g\circ s}$.

Proposition 3 The collection ${S}$ of quasiisomorphisms is a localizing system.

For the proof, we shall use the notion of mapping cone. Given ${f\in Hom_{C(\mathcal{C})}(X,Y)}$, define the mapping cone ${M(f)}$ as ${M(f)=X[1]\oplus Y}$ with differential ${\begin{pmatrix} d & 0 \\ f & d \end{pmatrix}}$. There is an exact sequence ${X\stackrel{f}{\rightarrow}Y\rightarrow M(f)\rightarrow X[1]}$.

Now we can prove property 3 of localizing systems. Given ${t}$ and ${f}$, let ${j=j'\circ f}$ where ${j':Y\rightarrow M(g)}$ arises in the mapping cone exact sequence for ${t}$. Let ${W=M(j)[-1]}$ and ${s:M(j)[-1]\rightarrow X}$ arise in the mapping cone exact sequence for ${j}$. Then ${s}$ is a quasiisomorphism, there is a natural map ${g:M(j)[-1]\rightarrow Z}$ making the diagram commute.

Property 4 is proven in a similar way.

1.7. Derived functors

Any functor ${F}$ from ${C(\mathcal{C})}$ to itself induces a functor ${K(F)}$ from ${K(\mathcal{C})}$ to itself, but ${K(F)(}$quasiisomorphisms${)\not=}$quasiisomorphism in general (counterexample ?).

Say a functor ${F}$ if it maps short exact sequences to short exact sequences. Say ${F}$ is left exact if it maps short exact sequences to short sequences which are exact on the left and middle.

Proposition 4 An exact functor maps quasiisomorphisms to quasiisomorphisms, and so induces a functor ${D(F)}$ on derived categories.

Indeed, ${f}$ is a quasiisomorphism iff ${M(f)}$ is acyclic. For an exact functor, ${X}$ acyclic implies ${F(X)}$ acyclic (split ${X}$ into short exact sequences using kernels and images).

So life with exact functors is easy. Unfortunately, the functors we are interested in are not exact. For instance, ${Hom_k(\cdot,M)}$ is only left exact. ${\otimes^n}$ is right exact. The section functor for a sheaf is left exact.

Definition 5 The derived functor of an additive left exact functor from ${\mathcal{C}}$ to ${\mathcal{D}}$ is a pair

1. of an exact functor ${RF}$ from ${D^+(\mathcal{C})}$ to ${D^+(\mathcal{D})}$,
2. and a morphism of functors ${\epsilon_F}$ from ${Q_{\mathcal{D}}\circ K^+(F)}$ to ${RF\circ Q_{\mathcal{C}}}$,

such that for any exact functor ${G}$ from ${D^+(\mathcal{C})}$ to ${D^+(\mathcal{D})}$ and morphism of functors ${\epsilon}$ from ${Q_{\mathcal{D}}\circ K^+(F)}$ to ${G\circ Q_{\mathcal{C}}}$, there exists a unique morphism of functors ${\eta}$ from ${RF}$ to ${G}$ such that

$\displaystyle \begin{array}{rcl} \eta\circ Q_{\mathcal{C}}\circ \epsilon_F=\epsilon. \end{array}$

Looks like a mess.

1.8. Strategy for computing derived functors.

Select a subclass of objects from ${K(\mathcal{C})}$ (typically, resolutions). If functor ${F}$ is nice on subclass, work componentwise. If subclass is large enough, its localization is equivalent to the derived category ${D(\mathcal{C})}$.