SMITH THEORY FOR ALGEBRAIC VARIETIES

PETER SYMONDS

Abstract. We show how an approach to Smith Theory about group actions on CWcomplexes using Bredon cohomology can be adapted to work for algebraic varieties.

1. Introduction

Peter May described in [5] a version of Smith Theory based on Bredon cohomology, so it really applies to any complex of projective coefficient systems rather than just a topological space. Later Jeremy Rickard in [6] showed how to associate a complex of p-permutation modules to a group action on a variety in such a way that the cohomology of this complex is the étale cohomology of the variety. We show how to generalize this to obtain a complex of projective coefficient systems. Thus Smith Theory becomes available for algebraic varieties, even over fields of finite characteristic.

2. Coefficient Systems

A coefficient system L on a group G over a ring R is a functor from the right orbit category of G to R-modules. In more concrete terms, it consists of a collection of Rmodules L(H), one for each subgroup H ≤ G together with R-linear restriction maps res_{K}^{H} : L(H) → L(K) for each K ≤ H ≤ G and conjugation maps c_{g,H} : L(H^{g}) → L(H) for each g ∈ G and H ≤ G.

These must satisfy the identities:

(1) res_{H}^{H} = id, H ≤ G;

(2) res_{J}^{K} res_{K}^{H} = res_{J}^{H}, J ≤ K ≤ H ≤ G;

(3) c_{g_{1},^{g_{2}}H}c_{g_{2},H} = c_{g_{1}g_{2},H}, H ≤ G, g_{1}, g_{2} ∈ G;

(4) res_{^{g}K}^{^{g}H} c_{g,H} = c_{g,K} res_{K}^{H}, K ≤ H ≤ G, g ∈ G;

(5) c_{h,H} = id, H ≤ G, h ∈ H.

In particular, the conjugation maps make L(H) into a left RN_{G}(H)/H-module.

A morphism f : L → M is a collection of R-linear maps f(H) : L(H) → M(H) which commute with the res and c.

The coefficient systems on G over R form an abelian category, which we denote by CS_{R}(G). If H ≤ G there is a forgetful map Res_{H}^{G} : CS_{R}(G) → CS_{R}(H)

Examples. (1) The constant coefficient system R̄, which is just R on each evaluation

and all the maps are the identity.

(2) The fixed point coefficient V^{?}, where V is a left RG-module and the notation indicates that the evaluation on H ≤ G is the fixed point submodule V^{?}. Restriction is inclusion and conjugation is multiplication by g ∈ G. These have the important property that Hom_{CS_{R}(G)}(L, V^{?}) ≅ Hom_{RG}(L(1), V).

(3) A variation on V^{?} is V_{0}, which takes the value V on 1 and 0 elsewhere.

(4) The systems R[X^{?}], where X is a left G-set and the evaluation at H is the free R-module on the fixed point set ^{K}X. (Writing the ? on the right is confusing but traditional.)

The particular cases R[G/H^{?}] have the important property Hom_{CS_{R}(G)}(R[G/H^{?}], L) ≅ L(H). This implies that they are projective and that they provide enough projectives. Thus every projective is a summand of a sum of these.

For more information on coefficient systems see [7].

Given a set of coefficient systems I it is convenient to define add(I) to be the full subcategory of CS_{R}(G) with objects isomorphic to a summands of a coefficient system of the form L_{1} ⊕ … ⊕ L_{n}, L_{i} ∈ I.

Thus the subcategory proj(CS_{R}(G)) of finitely generated projective coefficient systems is the same as add({R[G/H^{?}] : H ≤ G}).

If X is a G-CW-complex then there is a complex of coefficient systems C[X^{?}] associated to it, in which C_{n}[X] = R[(X_{n})^{?}] where X_{n} is the G-set of n-cells in X and the boundary morphisms are defined in the usual way.

The Bredon cohomology of X with coefficients in a coefficient system L, as defined in [2], is H_{G}^{∗}(X, L) = Hom_{CS_{R}(G)}(C[X^{?}], L).

Examples. (1) H_{G}^{∗}(X, (RG)^{?}) ≅ H^{∗}(X, R), the usual CW-cohomology, (2) H_{G}^{∗}(X, R_{G}) ≅ H^{∗}(X/G, R), where R_{G} takes the value R on G and 0 elsewhere.

(3) H_{G}^{∗}(X, R̄) ≅ H^{∗}(X/G, R),

(4) More generally we can regard H_{?}^{∗}(X, R) as a coefficient system itself under the natural restriction and conjugation maps, and then we have H_{?}^{∗}(X, R̄) ≅ H^{∗}(X/?, R).

The dual concept to that of a coefficient system we term an efficient system, in which the restriction maps go in the opposite direction. E(H) is now a right N_{G}(H)-module, although we could remedy this by taking the contragredient instead of the dual. The category of efficient systems for G over R is denoted by ES_{R}(G).

If R is self-injective then applying Hom_{R}(−, R) provides a duality between the subcategories taking values in finitely generated modules.

The dual of R[G/H^{?}] is denoted by R[G/H^{?}]^{∗}. The evaluation on K ≤ G can be thought of as the functions on the fixed point set ^{K}(G/H) and the restriction maps just restrict the functions. If R is self injective then R[G/H^{?}]^{∗} is injective.

CS_{R}(G) can also be viewed as the category of modules over an R-algebra C_{R}(G) of finite rank over R, (cf. [1]). Similarly ES_{R}(G) is equivalent to the category of modules over another R-algebra E_{R}(G).

3. Varieties

From now on k is an algebraically closed field and X is a separated scheme of finite type over k.

Let A be a torsion Artin algebra and let F be a constructible sheaf of A-modules over X. Let stalks(F) denote the set of stalks of F at the k-rational points. This contains only a finite number of isomorphism classes of A-modules.

Recall that RΓ_{c}(X, F) is the complex of A-modules whose homology is étale cohomology with compact supports H_{c}^{∗}(X, F).

Our main tool will be the following result from [6]:

Theorem 3.1. (Rickard) There is a complex of modules in add(stalks(F)) of finite type, which we denote by Ω_{c}(X, F). It is well defined up to homotopy equivalence. It has the following properties:

(1) Ω_{c}(X, F) is equal to RΓ_{c}(X, F) in D(A − Mod);

(2) F ↦ Ω_{c}(X, F) is a functor from constructible sheaves of A-modules over X to K^{b}(A − mod);

(3) If f : Y → X is a finite morphism of separated schemes of finite type over k then there is an induced map Ω_{c}(X, F) → Ω_{c}(Y, f^{∗}F);

(4) If B is also a torsion Artin algebra and L is a functor add(stalks(F)) → B − mod then LΩ_{c}(X, F) ≅ Ω_{c}(X, L̃F), where L̃F denotes the sheafification of the presheaf LF.

We will apply this in the case that R = ℤ/ℓ^{n} and A = E_{R}(G).

We suppose that a finite group G acts on X with quotient variety Y = X/G and projection map ρ : X → Y. We let F = F_{X} be the sheafification of the presheaf that sends a Zariski open set U ⊆ Y in the Zariski topology to R[(π_{0}(ρ^{−1}U))^{?}]^{∗}, where π_{0}(ρ^{−1}U) is the G-set of components of ρ^{−1}U. (This extends to the étale site on X by evaluating on the image of an étale map U → X.) Then stalks(F) consists of injective modules.

Theorem 3.1 produces a complex of injective efficient systems of finite type Ω_{c}(Y, F). These complexes for different n can be pieced together in such a way that we can take the inverse limit and obtain a complex of finite type of efficient systems in add({ℤ̂_{ℓ}[G/H^{?}]^{∗} : H the stabilizer of a k-rational point}) as in [6]. The dual of this by Homℤ̂_{ℓ}(−, ℤ̂_{ℓ}) is the complex that we will denote by C[X^{?}].

Theorem 3.2. For any H ≤ G, C[X^{?}](H)^{∗} ≅ RΓ_{c}(X^{H}, ℤ̂_{ℓ}).

In other words C[X^{?}](H) is a complex whose dual has cohomology H_{c}^{∗}(X^{H}, ℤ̂_{ℓ}). We can therefore think of it as the analogue of the complex C[X^{?}] for the Bredon cohomology of a G-CW-complex.

Since C[X^{H}](1) ≅ C[X^{?}](H) our notation is justified and, after the proof is complete, we will write C[X^{H}] instead of C[X^{?}](H).

We will prove theorem 3.2 as a corollary of some more general results.

Notice that C[X^{?}] is natural with respect to group homomorphisms f : H → G for which the kernel acts trivially on X.

Let A be a set of subgroups of G closed under supergroups and conjugation. Let S_{A}X = ⋃_{J∈A} X^{J}.

Define L_{A} : CS_{R}(G) → CS_{R}(G) by taking L_{A}C to be the smallest subsystem of C that is equal to C(H) for all H ∈ A. Then

L_{A}R[G/H^{?}] = \begin{cases}R[G/H^{?}] H ∈ A \\ 0 otherwise \end{cases}
= R[(S_{A}G/H)^{?}].

So L_{A} induces a functor L_{A} : add({R[G/H^{?}]; H ≤ G}) → add({R[G/H^{?}]; H ∈ A}).

Proposition 3.3. L_{A}C[X^{?}] is homotopy equivalent to C[(S_{A}X)^{?}].

Proof. It is sufficient to prove the analogous statement for R = ℤ/ℓ^{n}.

By 3.1, L_{A}C[X^{?}]^{∗} ≅ Ω_{c}(Y, L̃_{A}F_{1}), where F_{1} is the sheafification of F′_{1} : U ↦ Γ((π_{0}(ρ^{−1}U))^{?}, R). So L̃_{A}F_{1} is the sheafification of U ↦ Γ((S_{A}π_{0}(ρ^{−1}U))^{?}, R).

Now C[(S_{A}X)^{?}]^{∗} ≅ Ω_{c}((S_{A}X)/G, F_{S_{A}X}) ≅ Ω_{c}(Y, F_{2}), where F_{2} is the sheafification of F′_{2} : U ↦ Γ((π_{0}(ρ^{−1}U ∩ S_{A}X))^{?}, R).

Inclusion of fixed points gives a map L̃_{A}F′_{1} → F′_{2}, which induces an isomorphism on the stalks and hence an isomorphism of sheaves. □

Lemma 3.4. C[X^{?}](1)^{∗} ≅ RΓ_{c}(X, R)

Proof. Again it is enough to work over ℤ/ℓ^{n}.

Considering the functor “evaluate at 1”, we find that Ω_{c}(Y, F)(1) ≅ Ω_{c}(Y, { \atop F(1)}), where { \atop F(1)} is the sheafification of U ↦ Γ(π_{0}(ρ^{−1}U), R), which is just ρ_{∗}R.

Finally Ω_{c}(Y, ρ_{∗}R) ≅ RΓ_{c}(Y, ρ_{∗}R) ≅ RΓ_{c}(X, R). □

Remark. The above lemma shows that C[X] = C[X^{?}](1) is the dual of Rickard’s complex of ℓ-permutation modules.

Proof. of 3.2. We can restrict to N_{G}(H) if necessary, by naturality under inclusions, so we may assume that H is normal in G. Let H be the set of subgroups of G containing H and apply 3.3 to obtain C[X^{H}](H) ≅ L_{H}C[X^{?}](H) = C[X^{?}](H).

Notice that C[X^{H}](H) ≅ C[X^{H}](1) by naturality under G → G/H and apply 3.4. □

A similar method will prove the following result (see [6]). We abbreviate H_{0}(G, M) by M_{G}.

Lemma 3.5. C[X]_{G} ≅ C[X/G].

Let S^{1} denote the set of non-trivial subgroups and write S = S_{S^{1}} and L = L_{S^{1}}.

Lemma 3.6. (C[X^{?}]/LC[X^{?}])^{∗} ≅ RΓ_{c}(X ∖ SX, R)_{0}.

Proof. Both sides are zero on non-trivial subgroups, so we only need to check at the trivial group.

The inclusion map C[SX] → C[X] is equivalent to LC[SX] → C[X]. It is also dual to RΓ_{c}(X, R) → RΓ_{c}(SX, R). Thus the triangles LC[SX] → C[X] → C[SX]/C[X] and RΓ_{c}(X ∖ SX, R) → RΓ_{c}(X, R) → RΓ_{c}(SX, R) are dual. □

4. Smith Theory

Various results are known collectively as Smith theory (see [3], for example), but the prototype is the theorem that if a p-group P acts on a finite dimensional CW-complex which has acyclic mod-p cohomology then the fixed point subcomplex also has acyclic mod-p cohomology. Once the case of P of order p is proved this follows by induction on the order of P.

From now on we will take R to be 𝔽_{p}, so the ℓ in the previous section becomes p. As a consequence we will need X to be defined over a field of characteristic not equal to p in order to be able to use the étale cohomology.

Lemma 4.1. We have the following identifications:

H_{G}^{∗}(X, (RG)^{?}) ≅ H_{c}^{∗}(X, R),
H_{G}^{∗}(X, (RG^{?})/(RG)_{0}) ≅ H_{c}^{∗}(SX, R)
H_{G}^{∗}(X, R_{0}) ≅ H_{c}^{∗}((X ∖ SX)/G, R).

The analogous result for G-CW-complexes is well known.

Proof. By the adjointness property of (RG)^{?},

Hom_{CS_{R}(G)}(C[X^{?}], (RG)^{?}) ≅ Hom_{RG}(C[X], RG)
≅ Hom_{R}(C[X], R)
≅ RΓ_{c}(X, R).

Notice that C[SX] ≅ LC[X], by 3.3. Because LC[X] is in add({R[G/H^{?}]; H ≠ 1) we obtain

Hom_{CS_{R}(G)}(C[X^{?}], (RG)^{?}/(RG)_{0}) ≅ Hom_{CS_{R}(G)}(LC[X^{?}], (RG)^{?})
≅ Hom_{RG}(LC[X], RG)
≅ Hom_{RG}(C[SX], RG)
≅ Hom_{R}(C[SX], R)
≅ RΓ_{c}(SX, R).

There are no non-zero homomorphisms from add({R[G/H^{?}]; H ≠ 1}) to R_{0}. Also C[X ∖ SX] is in add({R[G^{?}]}), so vanishes off the trivial group. We find that

Hom_{CS_{R}(G)}(C[X^{?}], R_{0}) ≅ Hom_{CS_{R}(G)}(C[X^{?}]/LC[X^{?}], R_{0})
≅ Hom_{CS_{R}(G)}(C[(X ∖ SX)^{?}], R_{0})
≅ Hom_{RG}(C[X ∖ SX], R)
≅ RΓ_{c}((X ∖ SX)/G, R),

by 3.5. □

May’s approach to Smith Theory considers the Bredon cohomology groups in the lemma above and uses various long exact sequences associated to a short exact sequence of coefficient systems.

Let I denote the augmentation ideal of RG. Notice that if G is a p-group, which we will denote by P, then (RP)^{?}/I_{0} ≅ (RP)^{?}/(RP)_{0} ⊕ R_{0} and the composition factors of I_{0} are all R_{0}.

Let

a_{q} = dim H_{G}^{q}(X, R_{0}) = dim H_{c}^{q}((X ∖ SX)/G, R),
b_{q} = dim H_{G}^{q}(X, (RG)^{?}) = dim H_{c}^{q}(X, R),
c_{q} = dim H_{G}^{q}(X, (RG^{?})/(RG)_{0}) = dim H_{c}^{q}(SX, R).

May proves the following result in [5] for P -CW-complexes but, since the proof uses only manipulations with Bredon cohomology and the identifications in 4.1, it is valid for separated schemes of finite type too.

Theorem 4.2. (Floyd, May) The following inequality holds for any q ≥ 0 and r ≥ 0:

a_{q} + ∑_{i=0}^{r}(|P | − 1)^{i}c_{q+i} ≤ ∑_{i=0}^{r}(|P | − 1)^{i}b_{q+i} + (|P | − 1)^{r+1}a_{q+r+1}.

In particular, if a_{i} = 0 for i sufficiently large,

a_{q} + ∑_{i≥0}(|P | − 1)^{i}c_{q+i} ≤ ∑_{i≥0}(|P | − 1)^{i}b_{q+i}.

Moreover, if a_{i}, b_{i}, c_{i} = 0 for i sufficiently large then

χ_{c}(X) = χ_{c}(SX) + |P |χ_{c}((X ∖ SX)/G).

If P is cyclic of order p, and r is even if p ≠ 2, then we can remove the factors (|P |−1), i.e.

a_{q} + ∑_{i=0}^{r} c_{q+i} ≤ ∑_{i=0}^{r} b_{q+i} + a_{q+r+1}.

Remark. (1) If X is a CW-complex then we can use ordinary cohomology instead of compactly supported cohomology provided that we also replace (X ∖ SX)/G by X/SX and take its reduced cohomology.

(2) Notice that the last line includes Illusie’s result [4] for varieties that if P acts freely on X then |P | divides χ_{c}(X). In fact, in this case, C[X] is a complex of projective RP -modules and, since P is a p-group, the modules are free.

(3) In the topological case, if we take X to be EP (the universal cover of the classifying space) and q = 0 then we recover the well-known result that the H^{i}(P, 𝔽_{p}) is nonzero in every degree.

Recall that

H_{c}^{i}(𝔸^{n}(k), 𝔽_{p}) = \begin{cases}𝔽_{p}, i = n \\ 0, otherwise \end{cases}

provided that p is not the characteristic of k.

Corollary 4.3. Suppose that X has the cohomology of an affine space 𝔸^{n} and also that if X is a CW-complex then it is finite-dimensional. Then X^{P} has the cohomology of some affine space 𝔸^{m} for some m, with n − m even if |P | ≠ 2.

Remark. (1) By taking n = 0 this includes the case that when X is mod-p acyclic

then X^{P} must also be mod-p acyclic.

(2) When X is compact a similar argument shows that if X is a mod-p homology sphere then so is X^{P}.

Proof. By induction on P we can reduce to the case when P is cyclic of order p. For P must have a normal subgroup Q of index p, and by induction X^{Q} has the cohomology of an affine space. But X^{P} = (X^{Q})^{P}.

From the last line in 4.2 with r large it follows that ∑_{i≥0} c_{i} ≤ 1. The sum can not be 0 by the Euler characteristic formula. □

We now present a more conceptual approach to these results which shows how coefficient systems can provide a very flexible tool. It is based on the following lemma:

Lemma 4.4. Any monomorphism between two projective coefficient systems in CS_{R}(P) is split.

Proof. Consider a map R[P/U^{?}] → R[P/V^{?}]. It must be zero unless U is conjugate to a subgroup of V. But then it can only be a monomorphism if |U| ≥ |V | so in fact U is conjugate to V and the map is an isomorphism.

Now suppose that we have a monomorphism f : R[P/U^{?}] → ⨁ F_{j}, where each F_{j} is an indecomposable projective, so isomorphic to some R[P/V^{?}]. The socle of R[P/U^{?}] is just the sub-system generated by ∑_{g∈P/U} gP in degree 0. One of the components of f, say f_{j} : R[P/U^{?}] → F_{j} must be non-zero on the socle, hence a monomorphism and so an isomorphism. The splitting is now projection onto F_{j} followed by (f_{j})^{−1}.

Now consider the case f : ⨁_{i∈I} E_{i} → ⨁_{j∈J} F_{j}. Let S be the set of pairs (K, g), where K ≤ I and g : ⨁_{j∈J} F_{j} → ⨁_{i∈I} E_{i} is a map such that gf restricted to ⨁_{i∈K} E_{i} is the identity. We give S a partial ordering by setting (K′, g′) ≤ (K, g) if K′ ≤ K and gf = g′f on ⨁_{i∈K′} E_{i}. Zorn’s lemma now yields a maximal element (M, m) of S.

We claim that M = I, so that m is the desired splitting. For if not, let x ∈ M ∖ I. Let π be the projection of ⨁_{i∈I} E_{i} on to E_{x} and let ι be the inclusion.

Now f(⨁_{i∈M} E_{i}) is a summand of ⨁_{j∈J} F_{j}, so (⨁_{j∈J} F_{j})/f(⨁_{i∈M} E_{i}) is projective. Let ρ : ⨁_{j∈J} F_{j} → (⨁_{j∈J} F_{j})/f(⨁_{i∈M} E_{i}) be the quotient map.

The composite ρfι : E_{x} → (⨁_{j∈J} F_{j})/f(⨁_{i∈M} E_{i}) is injective, so is split by the previous case. Let σ be a splitting. Then (M, m) ≤ (M ∪ {x}, m + σρ − πm) in S, a contradiction. □

Corollary 4.5. If C is a complex of projectives in CS_{R}(P) that is bounded above and such that C(1) is exact then C is split exact.

Proof. Evaluation at 1 detects monomorphisms so there is an easy argument based on 4.4 and induction on the number of boundary maps that can be split, starting from the left. □

Corollary 4.6. Let f : C → D be a map between two bounded complexes of projective coefficient systems in CS_{R}(P). If f(1) : C(1) → D(1) is a quasi-isomorphism, i.e. induces an isomorphism in homology, then f is a homotopy equivalence.

Proof. Apply 4.5 to the cone of f, to deduce that f is a quasi-isomorphism. Since the complexes consist of projectives, f must be a homotopy equivalence. □

Corollary 4.7. Let f : X → Y be a finite morphism of separated schemes that induces an isomorphism on étale cohomology with coefficients in R . Suppose that P acts on both X and Y and that f is equivariant. Then the induced morphism f^{P} : X^{P} → Y^{P} also induces an isomorphism on cohomology.

Remark. By considering a map from affine space to X, this yields 4.3.

Remark. It is not sufficient to consider complexes of p-permutation modules. For example, if we let C_{2} denote the cyclic group of order 2 take R = 𝔽_{2} then there is a short exact sequence R → RC_{2} → R. But this is not split.

References

[1] Bouc, S., Résolutions de foncteurs de Mackey, Proceedings of Symposia in Pure Mathematics 63 (1998), 31-83.

[2] Bredon, G.E., Equivariant Cohomology Theories, Lecture Notes in Math. 34 (1967), Springer-Verlag, Berlin.

[3] Bredon, G.E., Introduction to Compact Transformation Groups, Pure and Applied Math. 46 (1972), Academic Press, New York.

[4] Illusie, L., Théorie de Brauer et caractéristique d’Euler-Poincaré (d’après P. Deligne), Astérisque 81-82 (1981), 161-172.

[5] May, J.P., A generalization of Smith Theory, Proc. Amer. Math. Soc. 101 (1987), 728-730.

[6] Rickard, J., Finite group actions and étale cohomology, Inst. Hautes Études Sci. Publ. Math. 80 (1994), 81-94.

[7] Symonds, P., The Bredon cohomology of subgroup complexes, preprint.

Department of Mathematics, U.M.I.S.T., P.O. Box 88, Manchester M60 1QD, England

 E-mail address: Peter.Symonds@umist.ac.uk