Topological digression Clause Samples
Topological digression. Let us recall that for a locally compact topo- logical space X, one may define Borel–▇▇▇▇▇ homology groups HBM (X, Z) (see [Ive1986, Chapter IX]). These will make their appearance in §2.1, but now they will serve us as a motivating example of duality. Local Verdier duality [Ive1986, §VII.5] tells that if f : X Y is a contin- uous map between locally compact topological spaces of finite dimension, then there is a natural isomorphism in the derived category D+(Y) RHom(R f!F •, G•) ∼= R f∗ RHom(F •, f !G•) where F • ∈ D+(X), G• ∈ D+(Y), and f ! : D+(Y) → D+(X) is the right adjoint functor to R f! : D+(X) → D+(Y). In particular, for the projection to the point p : X → ∗ the above reads RHom(RΓc(X, F •), G•) =∼ RΓ(X, RHom(F •, p!G•)) for F D+(X) and G• D(Ab). If we take G• to be the complex consisting of a single constant sheaf Z, the object p!Z D+(X) is called the dualizing sheaf on X, and Borel–▇▇▇▇▇ homology is defined by HBM(X, Z) := H−i(RΓBM(X, Z)), RΓBM(X, Z) := RΓ(X, p!Z) ∼= RHom(RΓc(X, Z), Z). This means that Borel–▇▇▇▇▇ homology is covariantly functorial for proper maps and contravariantly functorial for inclusions of open subsets U ‹→ X:
1) a proper continuous map of locally compact topological spaces f : X Y induces a morphism RΓc(Y, Z) RΓc(X, Z), and therefore on Borel– ▇▇▇▇▇ homology we have the proper pushforward morphism RΓBM(X, Z) → RΓBM(Y, Z).
2) an inclusion of an open subset U ‹ X induces a morphism RΓc(U, Z) RΓc(X, Z), and therefore the corresponding pullback on Borel–▇▇▇▇▇ homology RΓBM(X, Z) → RΓBM(U, Z). Moreover, if U X is an open subset and Z := X U is its closed complement, then the corresponding pushforwards and pullbacks fit into a distinguished triangle RΓBM(Z, Z) → RΓBM(X, Z) → RΓBM(U, Z) → RΓBM(X, Z)[1] This is dual to the triangle RΓc(U, Z) → RΓc(X, Z) → RΓc(Z, Z) → RΓc(U, Z)[1] The cycle complex Zc(n) behaves similarly to Borel–▇▇▇▇▇ homology.
0.11.1. Fact ([Gei2010, Corollary 7.2]).
1) a proper morphism of schemes f : X → Y induces a pushforward morphism R f∗Zc (n) → Zc (n);
2) an open immersion of schemes f : U ‹→ X induces a flat pullback morphism f ∗Zc (n) → Zc (n). If U ⊂ X is an open subscheme and Z := X \ U is its closed complement, then the proper pushforward associated to Z ‹→ X and the flat pullback associated to U ‹→ X give a distinguished triangle RΓ(Zét, Zc(n)) → RΓ(Xét, Zc(n)) → RΓ(Uét, Zc(n)) → RΓ(Zét, Zc(n))[1] Chapter 1 For an arithmetic scheme X (separated, of finite type over Spec Z) and a strictly negative in...