Let X,Y be topological spaces and f:X→Y a continuous map. We let Shv(X) be the category of sheaves on X, D(Shv(X)) the derived category of sheaves on X, and Db(Shv(X)) the bounded variant. Recall that D(A) for an abelian category A is constructed first by taking C(A), the category of cochains of elements of A, quotienting by chain homotopy, then quotienting by all acylic chains.
Remark: Let F∈Shv(X). Recall:
Definition: The map f induces functors between categories of sheaves, called
direct imagef∗ : Shv(X)→Shv(Y),(U↦F(U))↦(V↦F(f−1(V))),inverse imagef∗ : Shv(Y)→Shv(X),(V↦G(V))↦sh(U↦colimV⊇f(U)G(V)),direct image with compact supportf! : Shv(X)→Shv(Y),(U↦F(U))↦(V↦{s∈F(f−1(V)) : f|supp(s) is proper}).
Above we used that f:X→Y is proper if f−1(K)⊆X is compact, for every K⊆Y compact. Next, recall that a functor φ:A→B induces a functor Rφ:D(A)→D(B), called the (first) derived functor of φ, given by Rφ(A∙)=H1(φ(A)∙).
Remark: Each of the maps f∗,f∗,f! have their derived analogues Rf∗,Rf∗,Rf!, respectively. For reasons unclear, Rf! has a right adjoint, denoted Rf!:D(Shv(Y))→D(Shv(X)). This is called the exceptional inverse image.
We are now ready to define perverse sheaves.
Definition: Let A∙∈D(Shv(X)). Then:
We finish off with an example.
Example: Let X=R be a stratified space, with X0=0 the origin and X1=R∖0. Let F∈Shv(X) be an R-valued sheaf given by F(U)=inf, and define a chain complex A^\bullet in the following way:
0 \longrightarrow A^{-1} = \mathcal F \xrightarrow{ d^{-1}=\text{id} } A^0 = \mathcal F \xrightarrow{ d^0=0 } 0.
Note that for any U\subseteq \R, we have H^{-1}(A^\bullet)(U) = \ker(d^{-1})(U) = \ker(\id:\mathcal F(U)\to \mathcal F(U)) = \emptyset if 0\not\in U, and 0 otherwise. Hence \supp(H^{-1}(A^\bullet)) = \R\setminus 0, whose dimension is 1. Next, H^0(A^\bullet)(U) = \ker(d^0)(U)/\im(d^{-1})(U) = \ker(0:\mathcal F(U)\to 0)/\im(\id:\mathcal F(U)\to \mathcal F(U)) = \mathcal F(U)/\mathcal F(U) = 0, and so \dim(\supp(H^0(A^\bullet))) = 0. Note that A^\bullet is self-dual and constructible, as the cohomology sheaves are locally constant. Hence A^\bullet is a perverse sheaf.
References: Bredon (Sheaf theory, Chapter II.1), de Catalado and Migliorini (What is... a perverse sheaf?), Stacks project (Articles "Supports of modules and sections" and "Complexes with constructible cohomology")
Remark: Let F∈Shv(X). Recall:
- a section of F is an element of F(U) for some U⊆X,
- a germ of F at x∈X is an equivalence class in {s∈F(U) : U∋x}/∼x,
- s∼xt iff every neighborhood W of x in U∩V has s|W=t|W, for s∈F(U), t∈F(V),
- the support of the section s∈F(U) is supp(s)={x∈U : s≁x0},
- the support of the sheaf F is supp(F)={x∈X : Fx≠0}.
Definition: The map f induces functors between categories of sheaves, called
direct imagef∗ : Shv(X)→Shv(Y),(U↦F(U))↦(V↦F(f−1(V))),inverse imagef∗ : Shv(Y)→Shv(X),(V↦G(V))↦sh(U↦colimV⊇f(U)G(V)),direct image with compact supportf! : Shv(X)→Shv(Y),(U↦F(U))↦(V↦{s∈F(f−1(V)) : f|supp(s) is proper}).
Above we used that f:X→Y is proper if f−1(K)⊆X is compact, for every K⊆Y compact. Next, recall that a functor φ:A→B induces a functor Rφ:D(A)→D(B), called the (first) derived functor of φ, given by Rφ(A∙)=H1(φ(A)∙).
Remark: Each of the maps f∗,f∗,f! have their derived analogues Rf∗,Rf∗,Rf!, respectively. For reasons unclear, Rf! has a right adjoint, denoted Rf!:D(Shv(Y))→D(Shv(X)). This is called the exceptional inverse image.
We are now ready to define perverse sheaves.
Definition: Let A∙∈D(Shv(X)). Then:
- the ith cohomology sheaf of A∙ is Hi(A∙)=ker(di)/Im(di),
- A∙ is a constructible complex if Hi(A∙) is a constructible sheaf for all i,
- A∙ is a perverse sheaf if A∙∈Db(Shv(X)) is constructible and dim(supp(H−i(P)))⩽i for all i∈Z and for P=A∙ and P=(A∙)∨=(A∨)∙ the dual complex of sheaves.
We finish off with an example.
Example: Let X=R be a stratified space, with X0=0 the origin and X1=R∖0. Let F∈Shv(X) be an R-valued sheaf given by F(U)=inf, and define a chain complex A^\bullet in the following way:
0 \longrightarrow A^{-1} = \mathcal F \xrightarrow{ d^{-1}=\text{id} } A^0 = \mathcal F \xrightarrow{ d^0=0 } 0.
Note that for any U\subseteq \R, we have H^{-1}(A^\bullet)(U) = \ker(d^{-1})(U) = \ker(\id:\mathcal F(U)\to \mathcal F(U)) = \emptyset if 0\not\in U, and 0 otherwise. Hence \supp(H^{-1}(A^\bullet)) = \R\setminus 0, whose dimension is 1. Next, H^0(A^\bullet)(U) = \ker(d^0)(U)/\im(d^{-1})(U) = \ker(0:\mathcal F(U)\to 0)/\im(\id:\mathcal F(U)\to \mathcal F(U)) = \mathcal F(U)/\mathcal F(U) = 0, and so \dim(\supp(H^0(A^\bullet))) = 0. Note that A^\bullet is self-dual and constructible, as the cohomology sheaves are locally constant. Hence A^\bullet is a perverse sheaf.
References: Bredon (Sheaf theory, Chapter II.1), de Catalado and Migliorini (What is... a perverse sheaf?), Stacks project (Articles "Supports of modules and sections" and "Complexes with constructible cohomology")
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