On the Eisenstein functoriality in cohomology for maximal parabolic subgroups

Abstract

In his paper, ’On torsion in the cohomology of locally symmetric varieties’, Peter Scholze has introduced a new, purely topological method to construct the cohomology classes on arithmetic quotients of symmetric spaces of reductive groups over $\mathbb {Q}$ originating from the cohomology of the similar quotients of Levi subgroups of maximal parabolic subgroups. We extend this construction beyond the cases he considers, and, in the complex case, to the cohomology of local systems.

Appendix : Poincaré duality for small Gorenstein rings of coefficients

For clarity of notation, we denote here by R a local Artin ring that is Gorenstein and by $k=R/\mathfrak {m}$ its residue field. The following theorem does not seem to figure in the literature, although (in fact for $R=\mathbb {Z}/N$, N being prime to the characteristic) it is well–known for the étale cohomology of varieties over algebraically closed fields (e.g. [18, Cor. 11.2]).

Theorem A1

Assume X is a manifold of dimension d.

(i) There is a perfect pairing

$$\begin{aligned} H_c^i(X,R) \times H^{d-i}(X,R) \longrightarrow R \end{aligned}$$

(ii) If X is compact, this yields a perfect pairing

$$\begin{aligned} H^i(X,R) \times H^{d-i}(X,R) \longrightarrow R. \end{aligned}$$

(This is used in the proof of Lemma 2.1).

We first recall some properties of such rings.

$$\begin{aligned} \begin{array}{rcl} \mathrm {Ext}_R^i(k,R) &{}\cong k &{}(i=0)\\ &{}=0&{}(i>0). \end{array} \end{aligned}$$

(1)

Let $\mathrm {inj}\, \dim \,M$ denote the injective dimension of a R–module M. Then

$$\begin{aligned} \mathrm {inj}\, \dim \,M =0 \Leftrightarrow \mathrm {Ext}_R^i(k,M)=0\quad (i>0). \end{aligned}$$

(2)

For (1) see [17, Thm 18.1]. For (2), [17, Lemma 1 p. 139]. In particular, in view of (1), (2) applies to $M=R$ ; this implies that

$$\begin{aligned} R\, \mathrm {is}\,\mathrm { injective}\,\mathrm { as}\,\mathrm { a}\,R\mathrm {-module.} \end{aligned}$$

(3)

Finally, let E(k) be the injective envelope of k as a R–module.

$$\begin{aligned} E(k) \cong R. \end{aligned}$$

(4)

This follows from [17, Thm. 18.4] and the fact that R is an indecomposable R–module.

Let M be a finite R–module. Then (4) implies :

$$\begin{aligned}&\mathrm {The}\,\mathrm {pairing~}M\times \mathrm {Hom}_R(M,R) \longrightarrow R\\&\quad (x,f) \longmapsto f(x) \end{aligned}$$

(5)

is non–degenerate.

See [17, Thm 18.6].

Now Theorem A1 follows easily from these facts and the next result. (Compare Weibel [30, 3.6.5].

Theorem A2

Let $P_\bullet$ be a chain complex of projective R–modules. Then, for any n, there exists an isomorphism

$$\begin{aligned} H^n(\mathrm {Hom}_R(P,R)) \cong \mathrm {Hom}_R(H_n(P),R). \end{aligned}$$

Proof

Consider the short exact sequences

$$\begin{aligned} 0 \longrightarrow Z_n \longrightarrow P_n \longrightarrow dP_n \longrightarrow 0, \end{aligned}$$

whence

$$\begin{aligned} 0 \longrightarrow \mathrm {Hom}(dP_n,R)\longrightarrow \mathrm {Hom}(P_n,R) \longrightarrow \mathrm {Hom}(Z_n,R) \longrightarrow 0 \end{aligned}$$

since $\mathrm {Ext}^1(dP_n,R)=0$ as R is injective. This defines an exact sequence of (cochain) complexes, with the dual differential. Consider the cohomology. We obtain exact sequences :

$$\begin{aligned} \cdots&\rightarrow H^{n-1} (\mathrm {Hom}(Z,R))\rightarrow H^n(\mathrm {Hom}(dP,R)) \rightarrow H^n(\mathrm {Hom}(P,R)) \rightarrow \\&\rightarrow H^n(\mathrm {Hom}(Z,R)\rightarrow H^{n+1}(\mathrm {Hom}(dP,R))\rightarrow \cdots \end{aligned}$$

and the differentials for the complexes Z and $d\,P$ being zero :

$$\begin{aligned} \cdots&\rightarrow \mathrm {Hom}(Z_{n-1},R) \rightarrow \mathrm {Hom}(dP_n,R) \rightarrow H^n(\mathrm {Hom}(P,R)) \rightarrow \\&\rightarrow \mathrm {Hom}(Z_n,R)\rightarrow \mathrm {Hom}(dP_{n+1},R))\rightarrow \cdots \end{aligned}$$

However, the exact sequence

$$\begin{aligned} 0\longrightarrow d\, P_{n+1} \longrightarrow Z_n \longrightarrow H_n(P) \longrightarrow 0 \end{aligned}$$

(6)

yields

$$\begin{aligned} 0\rightarrow \mathrm {Hom}(H_n(P),R) \rightarrow \mathrm {Hom}(Z_n,R) \rightarrow \mathrm {Hom}(dP_{n+1},R) \rightarrow 0 \end{aligned}$$

since $\mathrm {Ext}^1(H_n(P),R)=0$ ; shifting the indices by 1 in (6) we also get

$$\begin{aligned} \mathrm {Hom}(Z_{n-1},R) \longrightarrow \mathrm {Hom}(dP_n,R)\longrightarrow 0. \end{aligned}$$

Finally, the last long exact sequence implies

$$\begin{aligned} H^n(\mathrm {Hom}(P,R)) \overset{}{\underset{\approx }{\longrightarrow }} \mathrm {Hom}(H_n(P),R). \end{aligned}$$

It is now easy to derive Theorem A1. We take $P_\bullet =S_\bullet (X)$ be the simplicial chain complex of X with coefficients in R ; $\mathrm {Hom}(P_\bullet ,R)$ is the simplicial cochain complex. We have the isomorphisms

$$\begin{aligned} H^i(X,R)&\cong \mathrm {Hom}(H_i(X),R), \,\alpha \mapsto h_\alpha \\ D:H_c^i(X,R)&\cong H_{d-i}(X,R), \,\beta \mapsto D\beta , \end{aligned}$$

and we set for $\alpha \in H^i$, $\beta \in H_c^{d-i}$ :

$$\begin{aligned} <\alpha ,\beta > = h_\alpha (D\beta ) \in R. \end{aligned}$$

Property (5) now implies that the pairing is non–degenerate (in both variables).