In this section, we will continue the construction from the previous section and gen- eralize Proposition 4.1.3 to certain arithmetic schemes of positive relative dimension over X, where X, K, S and U have the same meaning from the previous section. Recall thatK is a global field of characteristic different from 2 and 2 is invertible on
U. Let π : Y →X be an integral, projective scheme over X of relative dimension
d > 0. Suppose Y is smooth over an open subschemeU ⊆X and its generic fiber
YK is geometrically irreducible. First we recall the following generalization of the local Tate duality [Sai89]:
Proposition 4.2.1. Kv is a non-archimedean local field of characteristic different
from 2. Given a locally constant constructible sheaf F ∈Sh(YKv,Z/2),
Hi(YKv,F)×H
2d+2−i(Y Kv,F
D)→H2d+2(Y
Kv, µ2) = Z/2 (4.2.1)
is a perfect pairing of F2 vector spaces.
When v is a complex place, the Tate cohomology groupsHbi(C, M) = 0 for any module M and i∈Z. Therefore for a complex variety π :Z →C and F ∈Sh(Z),
b
Hi(Z,F) = b
Hi(
C, Rπ∗F) = 0. Proposition 4.2.1 is trivially satisfied.
Proposition 4.2.2. Given a locally constant constructible sheaf F ∈Sh(Y,Z/2),
Hetr(YU,F)×Hc3+2d−r(YU,F∨)→Hc3+2d(YU, µ2)∼=Z/2 (4.2.2)
is a perfect pairing of F2 vector spaces.
Using Proposition 4.2.1 and 4.2.2, the following is a corollary of Proposition 4.1.3:
Corollary 4.2.3. When K is a totally imaginary number field or a global function field of characteristic different from 2, the image of the restriction homomorphism
Φ : Hetd+1(YU, µ2)→ ⊕v∈SHetd+1(YKv, µ2)
is its own orthogonal complement with respect to the non-degenerate bilinear product
⊕v∈SHetd+1(YKv, µ2) × ⊕v∈SHetd+1(YKv, µ2) → ⊕v∈SHet2d+2(YKv, µ2) δ2d+3 −→Hc3+2d(YU, µ2)∼=F2 (4.2.3)
which is the Yoneda-pairing composed with taking summations.
Remark 4.2.4. The reason we do not consider the case whenK has a real embedding is that although Proposition 4.2.2 remains valid, Proposition 4.2.1 is in general not
true for a real local field, see [Cox79]. ♦
When d = 1, V is the diagonal image Φ : H2(YU, µ2)→
P v∈SH
2(Y
Kv, µ2). By the Kummer sequence, H2(Y, µ
2) can be computed by
whereBr(Y) :=H2(Y,
Gm) is the cohomological Brauer group. An explicit descrip- tion of the map Φ would be very interesting, which we leave for further study.
In general, it is not an easy problem to determine if the Yoneda-pairing
h,i:Hd+1(YKv, µ2)×H
d+1
(YKv, µ2)→H
2d+1
(YKv, µ2) (4.2.4)
used in Corollary 4.2.3 is Euclidean or alternate, as is illustrated in the following example.
Example 4.2.5. Consider the case YKv = P
1
Kv. The Hochschild-Serre spectral se- quence is often used to calculate H∗(PK1v, µ2):
Hi(Kv, Hj(PK1v, µ2))⇒Hi+j(PK1v, µ2) (4.2.5)
where Kv denotes an algebraic closure of Kv. This spectral sequence is multiplica- tive, in the sense that there is a pairing on the E2 page:
Ep1,q1 2 ∪E p2,q2 2 →E p1+p2,q1+q2 2
which when passing toE∞is compatible with the cup product structure onH∗(PK1v, µ2).
It is easy to see that the spectral sequence 4.2.5 degenerates on the E2 page. De-
note E22,0 = {0, y} ,→ H2(P1
Kv, µ2) = {0, x, y, x+y} =:Wv. E
0,2
2 can be naturally
identified with the quotient {0,x¯} of Wv modulo the subspace{0, y}.
Since the pairing h,i on Wv is non-degenerate, the fact that y∪y ∈ E24,0 = 0
implies hx, yi= 1 is non-trivial. Thus on the quotient spaceE20,2,
The multiplicative structure onE2of 4.2.5 alone does not suffice to determine if 4.2.4
is Euclidean or alternate. In Example 5.6.5, we will prove the following theorem using techniques from an equivariant ´etale cohomology theory.
Theorem 4.2.6. The cup product pairing 4.2.4 is alternate when YKv =P
1
Kv or EKv,
where EKv is an elliptic curve with good reduction over a local field Kv with residue
characteristic different from 2.
Chapter 5
The Equivariant Construction
Ever since the 1950s, equivariant cohomology has been a powerful tool in the study of group actions on spaces. Borel defined an equivariant cohomology for the action of a compact group G on a topological space [Bor60]. In [Gro57], Grothendieck defined an equivariant sheaf cohomology for the action of a discrete group. For a finite group, the Borel construction can be generalized to actions on sheaves and it coincides with Grothendieck’s equivariant sheaf cohomology [Sti79]. In this chap- ter, we will transplant certain statements for a finite group action on a finite CW complex to an action on equivariant ´etale sheaves over a scheme. For an application to the construction of binary self-dual codes, we will mainly be concerned with the case G=Z/2.
In Section 5.1, we will follow [AP93, Chapter I] and review the set-up of a construction for Grothendieck’s equivariant cohomology. When G = Z/2, we will
use a minimal Hirsch-Brown model for an equivariantG-complex. In section 5.2, we will specialize to consider equivariant ´etale sheaves over a scheme, and recall Morin’s construction of a modified equivariant ´etale cohomology. In section 5.3, we prove a “Smith-type inequality” 5.3.1 using Theorem 5.3.3 from [Mor08]. We will call it the maximal case when 5.3.1 is an equality. In the maximal case for a Z/2 action on a scheme Y where 2 is invertible, cohomological duality statements on Y can be utilized to construct binary self-orthogonal spaces, following [Pup01]. In section 5.5 we compare the Equivariant Construction and Construction G in the previous chapter. In Example 5.5.4, the reader will find that while the two constructions give the same underlying vector spaces for a code, their product structures are not necessarily the same. In the final section 5.6, we provide some more discussions on the maximum condition 5.3.8. In particular, in Example 5.6.5 the maximum condition is met, and the deformation trick can be used to prove Theorem 4.2.6.