Derived proper base change p X X We had proved that for f f q - - PowerPoint PPT Presentation

Derived proper base change

◮ We had proved that for X ′ X Y ′ Y

p f ′ f q

we have q∗ ◦ f! ∼ = f ′

! ◦ p∗ an isomorphism of functors.

◮ Theorem. q∗ ◦ Rf! ∼ = Rf ′

! ◦ p∗ is also an isomorphism (between

functors from D+(ShR(X)) to D+(ShR(Y ′)). ◮ The derived proper push-forward may be computed using the following kind of f!-injective resolution: a sheaf F ∈ ShR(X) is called c-soft if for any compact K ⊂ X, Γ(X; F) ։ Γ(K; (K ֒ → X)∗F) is

• surjective. One can see this as a weaker version of flabby.

◮ For a map f : X → Y , F ∈ ShR(X) is called f -soft if F|f −1(y) is c-soft for any y ∈ Y , see [KS,3.1.1] ◮ The full subcategory of f -soft sheaves is f!-injective. Moreover, p∗ takes f -soft sheaves to f ′-soft sheaves. Hence we can compute both q∗ ◦ f! and f ′

! ◦ p∗ using f -soft resolutions, and this proves that the

natural transformation q∗ ◦ Rf! → Rf ′

! ◦ p∗ is again an isomorphism.

Verdier duality

◮ For convenience, let us now work with sheaves of Z-modules, and write ShZ(X) = Sh(X). One can work with R-modules with R of finite global dimension. But Cheng-Chiang refuses to. ◮ The wonderful Verdier duality starts with a seemingly innocent abstract concept that Rf! has a right adjoint for a nice map f : X → Y (e.g. continuous function between finite-dimensional CW-complexes). ◮ In other words, there is a functor f ! : D+(Sh(Y )) → D+(Sh(X)) so that there are natural isomorphisms HomD+(Sh(Y ))(Rf!F, G) = HomD+(Sh(X))(F, f !G) for all F ∈ D+(Sh(X)), G ∈ D+(Sh(Y )).

Poincar´ e duality

HomD+(Sh(Y ))(Rf!F, G) = HomD+(Sh(X))(F, f !G)

◮ Suppose we are in a simple case for which X is a manifold and Y is a point so we identify Sh(Y ) = Ab. Let F = QX[n] and G = Q. We have HomD+(Ab)(H∗

c (X; Q)[n], Q) = HomD+(Sh(X))(QX, f !Q)

◮ We have HomD+(Ab)(H∗

c (X; Q)[n], Q) = Hom(Hn c (X; Q), Q).

◮ What is HomD+(Sh(X))(QX[n], f !Q)? One probably expects HomD+(Sh(X))(QX, f !Q) = H−n(X; f !Q) (i.e. Rf∗(f !Q)) no matter what kind of complex of sheaves f !Q is. Suppose this is true. ◮ Together, this reads Hom(Hn

c (X; Q), Q) = H−n(X; f !Z). Hmmm ...

◮ If X is orientable, we better have f !Q = QX[dim X] so that H−n(X; f !Q) = Hdim X−n(X; Q) and this is Poincar´ e duality!

Proof of Verdier duality, I

HomD+(Sh(Y ))(Rf!F, G) = HomD+(Sh(X))(F, f !G)

◮ So we would like to prove Verdier duality; the existence of the right adjoint f !. ◮ Our proof follow [KS,§3.1] and partially a note of Akhil Mathew. ◮ In particular, we want to represent the functor F → HomD+(Sh(Y ))(Rf!F, G) for any G ∈ Sh(Y ). ◮ Replacing Rf! by Rf∗ makes this impossible, because even f∗ does not commute with infinite direct sum! ◮ Suppose for the moment that every object in whatever category we work on is injective and projective. Then it looks like f !G can be the sheaf with f !G(X) = HomD+(Sh(Y ))(f!ZX, G) and in general f !G(U) = HomD+(Sh(Y ))((f |U)!ZU, G). ◮ This doesn’t make much sense. But we can derive all of them.

Proof of Verdier duality, II

Suppose for the moment that every object in whatever category we work on is injective and

• projective. Then f !G = HomD+(Sh(Y ))(f!ZX , G)?

◮ Recall that a sheaf F ∈ ShR(X) is c-soft if for any compact K ⊂ X, Γ(X; F) ։ Γ(K; (K ֒ → X)∗F) is surjective. A sheaf F is called f -soft if F|f −1(y) is c-soft for all y ∈ Y . ◮ Lemma. Suppose there is an integer n such that Rif!F = 0 for all j > n, F ∈ Sh(X). Then for any exact sequence K0 → K1 → ... → Kn → 0 in Sh(X) such that K0, K1, ..., Kn−1 are all f -soft, we have Kn is f -soft as well. ◮ Now let us suppose it is true that Rif!F = 0 for all j > n; we will prove that if X is an n-dimensional CW-complex, then this is true. ◮ Let 0 → ZX → I0 → I1 → I2 → ... → In−1 → In → ... be a flabby resolution of flat Z-modules. This can be done as follows: I0 is the direct product of all the stalks of ZX, where each stalk is realized as a skyscraper sheaf at the same point. After that I1 is the direct product of all the stalks of the cokernel ZX → I0. One checks by induction that the sheaf constructed in each step has flat (i.e. torsion-free) stalks.

Proof of Verdier duality, III

Let 0 → ZX → I0 → I1 → I2 → ... → In−1 → In → ... be a flabby resolution of flat Z-modules.

◮ Let Kn := Im(In−1 → In) so we have exact sequence 0 → ZX → I0 → ... → In−1 → Kn → 0, i.e. a resolution of ZX. ◮ Flabby ⇒ soft ⇒ c-soft ⇒ f -soft. Hence by the previous lemma, Kn is also f -soft. ◮ Write Ka = Ia for a = 0, 1, ..., n − 1. ◮ In dealing with HomD+(Sh(Y ))(Rf!F, G), we would like to compute F via 0 → F → F ⊗ K0 → ... → F ⊗ Kn → 0 This is exact by the flatness. ◮ Here let me remind that tensor product of sheaves is the sheafification of tensor product on sections, and one checks the stalk

• f tensor product is tensor product of stalks.

Proof of Verdier duality, IV

we would like to compute F via 0 → F → F ⊗ K0 → ... → F ⊗ Kn → 0

◮ We claim that each F ⊗ Ka is again f -soft. ◮ We have the resolution ... → P−1 → P0 → F → 0 by taking P0 to be the direct sum of (U → X)!ZU for each section s ∈ F(U) (so P0 → F sends 1 to s). And P−1 is the analogous for ker(P0 → F), etc. ◮ As Ka is flat, we again have ... → P−1 ⊗ Ka → P0 ⊗ Ka → F ⊗ Ka → 0. ◮ Each Pb ⊗ F can be checked to be f -soft. In particular this is the case for −n + 1 ≤ b ≤ 0. But then the lemma last page again says F ⊗ Ka is f -soft. This proves the claim. ◮ Now for G ∈ Sh(Y ), we define f !G to be the complex of sheaves given as follows: for any open j : U → X we have (f !G)−a(U) = HomSh(Y )(f!(j!j∗Ka), G). ◮ Likewise, for any bounded below complex G•, let f !G• be the complex associated to the double complex (f !Gb)−a.