Common use of Topological digression Clause in Contracts

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 integer n, we are going to construct certain complexes RΓW,c(X, Z(n)), following ▇▇▇▇▇ and ▇▇▇▇▇ [Mor2014, FM2016]. Here “W” stays for “Weil-étale” and “c” stays for “compact support”. The constructions are based on complexes of sheaves Zc(n) on Xét, dis- cussed in §0.11. The basic properties of motivic cohomology for arithmetic schemes are still conjectural, and in order to make sense of all our construc- tions, we will need to assume in 1.1.1 that the groups Hi (Xét, Zc(n)) are finitely generated. It is worth mentioning that the constructions in [FM2016] use other cycle complexes Z(n), mentioned in §0.11. If X has pure dimension d, then all this amounts to the renumbering (1.0.1) Zc(n) = Z(d − n)[2d], which should be taken into account when comparing formulas that will ap- pear below with the formulas from [FM2016]. We use Zc(n) instead of Z(n) precisely to avoid any references to the dimension of X (which is not as- sumed anymore to be equidimensional). Indeed, the dimensions of coho- mology groups in many formulas in [FM2016] have terms “2d”, and if one rewrites everything using (1.0.1), they magically disappear. This suggests that Zc(n) is a more natural object than Z(n) in our situation. In fact, §1.2 introduces a special definition of Z(n), motivated by [FM2016], which is unrelated to the cycle complexes. In our setting n < 0, the complex Z(n) will consist of a single étale sheaf, rather easy to define and under- stand. Both Zc(n) and Z(n) will appear in a certain arithmetic duality theorem in §1.3, which is stated as a quasi-isomorphism of complexes ^ =∼ c RΓc(Xét, Z(n)) −→ RHom(RΓ(Xét, Z (n)), Q/Z[−2]). ^ In §1.4 I take a look at RΓc(Xét, Z(n)) and related complexes. Then using the duality theorem, I define in §1.5 a morphism in the derived category D(Ab) αX,n : RHom(RΓ(Xét, Zc(n)), Q[−2]) → RΓc(Xét, Z(n)) and declare RΓfg(X, Z(n)) to be its cone: RHom(RΓ(Xét, Zc(n)), Q[−2]) αX,n RΓc(Xét, Z(n)) → RΓ (X, Z(n)) → RHom(RΓ(Xét, Zc(n)), Q[−1]) The complex RΓfg(X, Z(n)) is almost perfect in the sense of 0.3.3 (i.e. a per- fect complex modulo possible 2-torsion in arbitrarily high degrees), canoni- cal and functorial (despite being defined as a cone in the derived category). Then §1.6 is dedicated to the definition of RΓW,c(X, Z(n)). For this we will need a morphism i∞∗ : RΓfg(X, Z(n)) → RΓc(GR, X( ), (2πi)n Z), where RΓc(GR, X( ), (2πi)n Z) stays for the GR-equivariant cohomology with compact support on X( ). Then RΓW,c(X, Z(n)) will be given (sadly, up to a non-unique isomorphism in D(Ab)) by the distinguished triangle RΓW,c(X, Z(n)) → RΓfg(X, Z(n)) i∞∗ RΓc(GR , X( ), (2πi)n Z) → RΓW,c(X, Z(n))[1] The sheaf (2πi)n Z is the constant GR-equivariant sheaf on X( ), which is the image of Z(n) under the morphism α∗ from §0.7 (see 1.6.2). The existence of i∞∗ relies on a rather nontrivial argument (theorem 1.6.4). I show in §1.7 that there is a (non-canonical) splitting RΓW,c(X, Z(n)) ⊗Z Q ∼= RHom(RΓ(Xét, Zc(n)), Q)[−1] ⊕ RΓc(GR, X( ), (2πi)n Q)[−1]. Finally, §1.8 is dedicated to verifying that RΓW,c(X, Z(n)) is well-behaved with respect to open-closed decompositions of schemes U ‹ X Z. With the present definition, this cannot be shown for the complex itself, but we are going to establish a canonical isomorphism of the determinants ▇▇▇▇ RΓW,c(X, Z(n)) =∼ ▇▇▇▇ RΓW,c(U, Z(n)) ⊗Z ▇▇▇▇ RΓW,c(Z, Z(n)), which will be enough for our purposes.