POINCAR 'E DUALITY AND STEINBERG'S THEOREM
ON RINGS OF COINVARIANTS
W. G. DWYER AND C. W. WILKERSON
1. Introduction Let k be a field, possibly of finite characteristic, V a finite dimensional vector space over k of dimension r, and W ae Autk(V ) a finite subgroup. There is a natural action of W on the k-dual V # of V , as well as on the symmetric algebra S = S(V #). The algebra S is isomorphic to a polynomial algebra over k on r generators, and we are interested in the question of when the invariant algebra R = SW is also isomorphic to such a polynomial algebra. It is well-known (see for instance Serre [8]) that R is polynomial if W is generated by reflections and the characteristic of k is zero or prime to the order of W . In a slightly different direction, Steinberg [10] has shown that R is polynomial if k is the field of complex numbers and the quotient algebra P* = S \Omega R k satisfies Poincar'e duality (1.3). Steinberg's result was extended by Kane [4, 5] to other fields of characteristic zero, and by T.-C. Lin [6] to the case in which k is a finite field of characteristic prime to the order of W .
In this note we use elementary methods to prove Steinberg's result for fields of characteristic 0 or of characteristic prime to the order of W . This gives a new proof even in the characteristic zero case.
1.1. Theorem. Let k be a field, V an r-dimensional k-vector space, and W a finite subgroup of Autk(V ). Let S = S[V #] be the symmetric algebra on V #, and R = SW the ring of invariants of the natural action of W on S. Define P* to be the quotient algebra S \Omega R k. If the characteristic of k is zero or prime to the order of W and P* satisfies Poincar'e duality, then R is isomorphic to a polynomial algebra on r generators.
In the context of 1.1, P* is sometimes called the ring of coinvariants of W . This is a bit odd, since W acts on P* in a way which is in general nontrivial (2.1) but the terminology is well-established [5, 6, 9].
Date: January 25, 2010.
1
2 W. G. DWYER AND C. W. WILKERSON 1.2. Remark. We will follow commutative algebra conventions with respect to gradings and signs. Except in 4.4, the ring S is to be treated as a graded algebra in which the elements of V # are given grading one; S is a strictly commutative algebra. (We do not consider algebras which are graded commutative in the sense of topology.) The invariant ring R = SW inherits a grading from S, and the homomorphism R ! k which figures implicitly in the formula for P* is the unique homomorphism which sends to zero all the elements in R of strictly positive degree.
1.3. Remark. Since the homomorphisms R ! S and R ! k preserve degree (where k has gradation zero), P* = S \Omega R k has a natural grading. It is easy to see that P* is finite dimensional over k; this amounts to observing that S is finitely generated as a module over R, which follows from the fact that S is integral over R and is finitely generated as an algebra. The statement that P* satisfies Poincar'e duality (of dimension n) then means that there is a number n such that*
Pi = 0 for i > n,* dimk Pn = 1, and* for 0 <= i <= n, the product map Pi\Omega k Pn-i ! Pn is a nonsingular pairing.
Let R+ denote the ideal in R generated by elements of strictly positive degree. In a direction converse to 1.1, it is well known that if R is a polynomial algebra, then I = SR+ is a complete intersection ideal, so that P* = S/I is Gorenstein and hence (since P* is finite-dimensional over k) satisfies Poincar'e duality. See for example the books of BrunsHerzog [1] or Kane [4].
The authors would like to thank Bernd Ulrich for many useful conversations and R. J. Shank for his remarks on example 4.2.
2. Some one dimensional k[W ] submodules of P* Since |W | is a unit in k, the group algebra k[W ] is semisimple, and hence up to isomorphism there are only a finite number of irreducible k[W ]-modules. Moreover, up to isomorphism each k[W ]-module is a direct sum of these irreducible modules.
2.1. Lemma. The trivial one dimensional module 1W occurs as a summand in P* only as P0.
POINCAR'E DUALITY AND RINGS OF COINVARIANTS 3 Proof. Note that as k[W ]-modules, both S and P* have the averaging operator
AV (x) = |W |-1 X
w2W
w(x).
The projection map S ! P* commutes with AV , and so if y 2 Ps is W -invariant with preimage y0 2 S, AV (y0) projects to AV (y) = y. If y0 has degree s > 0, AV (y0) belongs to R+, which implies that AV (y0) projects to zero in P* = S/R+S. \Lambda
2.2. Proposition. Suppose that P* = S \Omega R k satisfies Poincar'e duality (1.3). Let ae = Pn as a k[W ]-module. Then ae is a one dimensional irreducible module and occurs as a summand of P* only as Pn.
Proof. The dimension and irreducibility of ae are clear. The multiplication maps
Pi \Omega k Pn-i ! Pn = ae
are W -equivariant (where W acts diagonally on the tensor product) and so induce W -isomorphisms
Pi ,= P #n-i \Omega k ae . If Pn-i contains ae as a summand, then Pi contains ae# \Omega k ae ,= 1W . By 2.1, this can occur only if i = 0, i.e., n - i = n. \Lambda
3. Proof of 1.1 We assume that P* satisfies Poincar'e duality of dimension n (1.3) and that |W | is a unit in k. We first recall an elementary lemma.
3.1. Lemma. Let T be a graded k-algebra which is concentrated in degrees >= 0 and isomorphic in degree 0 to k. Let T ! k be the homomorphism which sends all elements of positive degree to 0, and M and N two T -modules which themselves are concentrated in degrees >= 0 (more generally, are bounded below). Then
(1) M ,= 0 if and only if M \Omega T k ,= 0, (2) a homomorphism M ! N is surjective if and only if M \Omega T k !
N \Omega T k is surjective, and (3) if N is free, a homomorphism M ! N is an isomorphism if
and only if M \Omega T k ! N \Omega T k is an isomorphism.
Proof. Part (1) is clear. Part (2) then follows from an application of (1) to the cokernel of M ! N . For (3), assume that M \Omega T k ! N \Omega T k is surjective. By (2), M ! N is surjective; since N is free, the surjection can be split and M can be written as N \Phi K. Clearly K \Omega T k ,= 0, and hence by (1), K ,= 0. \Lambda
4 W. G. DWYER AND C. W. WILKERSON
Back to the proof of 1.1. It is a theorem of Serre [8] that R is a polynomial algebra if and only if S is free as a module over R. If AE is an irreducible k[W ]-module, let Si[AE] be the AE-isotypic summand of Si, i.e., the unique summand of Si which is isomorphic as a W -module to a direct sum of copies of AE. Denote by S[AE] = \Phi iSi[AE] the AE-isotypic summand of S. Each S[AE] is an R-submodule of S. Recall that ae = Pn.
3.2. Lemma. S[ae] is a free rank one R-module. Proof. Clearly S[ae] \Omega R k is the ae-isotypic part of P*, that is, the one- dimensional space Pn (2.2). It follows easily from 3.1(2) that S[ae] requires only one generator as an R-module; pick such a generator, say, ff. Since S is an integral domain, rff 6= 0 2 S[ae] for any r 6= 0 2 R. It follows that S[ae] is freely generated by ff as an R-module. \Lambda
3.3. Definition. For 0 <= k <= n, S(k) is the R-submodule of S generated over R by elements of S of degree k or less.
Note that S(0) = R and S(n) = S. Given Serre's result, 1.1 follows from the next proposition.
3.4. Proposition. For 0 <= k <= n, S(k) is a free R-module. Proof. This is clear for k = 0, since S(0) = R. Assume by induction that S(i) is a free R module for all i < k. It suffices to show then that S(k)/S(k - 1) is a free R-module. But
(S(k)/S(k - 1)) \Omega R k ,= Pk. For each i, choose a splitting of the surjection
S(i)/S(i - 1) ! Pi. Let Vi be the image of this splitting. Taking products with Vn-i yields an R-module map
(S(i)/S(i - 1)) \Omega k Vn-i ! S(n)/S(n - 1). Here the target is just S[ae], which is free over R (3.2). In fact, it is clear that the natural map S[ae] ! S(n)/S(n - 1) is surjective; injectivity follows from the fact that the representation ae of W does not appear in Pk for k < n (2.2) and hence cannot appear in S(n - 1) either. Choose a k-basis {v1, . . . , vm} of Vn-i. Each basis element vj produces, via the above multiplication pairing, a map
fj : S(k)/S(k - 1) ! S(n)/S(n - 1). Taking the sum over all j gives
S(k)/S(k - 1) ! \Phi m(S(n)/S(n - 1)).
POINCAR'E DUALITY AND RINGS OF COINVARIANTS 5 This is a map of R-modules. By Poincar'e duality in P*, it is an isomorphism after - \Omega R k. Since the target is a free R-module, it follows from 3.1(3) that the map before tensoring is an isomorphism. Thus S(k)/S(k - 1) is a free R-module and the induction is established. \Lambda
4. Examples We give two types of examples; in both, P* satisfies Poincar'e duality, but |W | is not a unit in k. In the first case, the ring of invariants is a polynomial algebra, but in the second it is not.
4.1. Example. Let k = Fp, and let W ae GL2(Fp) be generated by the matrix
A = ^1 01 1* .
A is a pseudo-reflection of order p. For S = k[x, y], we can choose generators so that AT x = x, AT y = y + x. It is not hard to see that R = SW is generated by x and v = y(y + x) . . . (y + (p - 1)x) = y(yp-1 - xp-1). The algebra P* is isomorphic to
k[y]/yp and thus satisfies Poincar'e duality.
In the above example, the group W acts trivially on P*, and so the semisimplicity needed for our proof of 1.1 dramatically fails. Nevertheless, the ring R of invariants is polynomial.
4.2. Example. Let k = Fp and let W 0 ae GL4(Fp) be generated by the
B = 2664
1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1
3775
.
Let S = k[x1, y1, x2, y2]; we can choose generators so that BT xi = xi and BT yi = yi + xi. Then R = SW 0 has generators x1, v1, x2, v2 with vi = yi(yp-1i - xp-1i ) and an additional generator z = x1y2 - x2y1. (This is more difficult to see; the argument is below.) There is an isomorphism
P* = k[y1]/(yp1) \Omega k[y2]/(yp2)
and so P* satisfies Poincar'e duality. However, R = SW is not a polynomial algebra.
The case p = 2 of this example has been treated by Smith [9].
6 W. G. DWYER AND C. W. WILKERSON 4.3. Remark. Let H ae GL4(Fp) be generated by
a = 2664
1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1
3775
, b = 2664
1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1
3775
.
Then H is isomorphic to the product W * W , where W is the group of 4.1, embedded along the diagonal in GL4(Fp). The algebra of invariants SH is k[x1, v1, x2, v2], which is isomorphic to the tensor product of two copies of the algebra of invariants from 4.1 . The ideal in S generated by x1, v1, x2, v2, z is that same as that generated by x1, v1, x2, v2; this is one way to calculate P* = S/SR+.
It remains to calculate R = SW 0. Let T be the subalgebra of S = k[x1, y1, x2, y2] generated by
{x1, v1, x2, v2, z}. It is clear that T is invariant under W 0, i.e., T ae R, and we must show that T = R. Our goal is to use as little explicit calculation as possible to accomplish this. The ring R is integrally closed for general reasons, and so it's enough to show T ! S has the proper degree on the level of fraction fields and that T is integrally closed.
We write Fr(T ), for instance, for the fraction field of T . Note to begin with that Fr(T ) properly contains Fr(SH) (where H is the group from 4.3), since z is in Fr(T ) but not in Fr(SH). The extension degree [Fr(SH), Fr(S)] is p2, and hence [Fr(T ), Fr(S)] properly divides p2. Since this latter degree is not one, it must be p. Since [Fr(R), Fr(S)] is not one and divides [Fr(T ), Fr(S)], it follows that [Fr(R), Fr(S)] also equals p and hence that Fr(R) = Fr(T ).
It remains to show that T is integrally closed in Fr(T ). For this, we use Serre's R1 and S2 criterion; see for example Matsumura [7, p. 183]. Write T as the quotient of the polynomial ring T 0 = k[a, b, c, d, e] with the surjection OE defined by
OE(a) = x1 OE(b) = v1
OE(c) = x2 OE(d) = v2 OE(e) = z
.
Since T is a domain, ker(OE) is a prime ideal. The dimension of T 0 is 5 and that of its image T is 4, so the height of ker(OE) is less than or equal to one. But T 0 is a domain and ker(OE) 6= 0, so the height of the kernel is one. Since T 0 is a UFD, any irreducible element generates a prime ideal. Since ker(OE) has height one, any irreducible element of it
POINCAR'E DUALITY AND RINGS OF COINVARIANTS 7 must be a generator. Note that
f = ep - ap-1cp-1e + bcp - apd is in ker(OE), since
xp1yp2 - xp2yp1 - xp-11 xp-12 (x1y2 - x2y1)
+ y1(yp-11 - xp-11 )xp2 - y2(yp-12 - xp-12 )xp1 = 0. The element f is irreducible, since its image is irreducible in T 0/(c). Hence f generates the prime ideal ker(OE) and T = T 0/(f ). {f } is a regular sequence of length one, so T = T 0/(f ) is Cohen-Macaulay. But T being Cohen-Macaulay is equivalent to T satisfying Serre's condition Sn for all n [7, p. 183]. Condition R1 is that for each height one idealP 2
Spec(T ), the localization TP must be regular. This remains to be checked. We first calculate the singular locus of T , using the Jacobian criterion [3, p. 306, 5.7.5]. The 1 * 1 minors of the Jacobian of f generate the ideal
j = (ap-2cp-1e, cp, ap, ap-1cp-2e, ap-1cp-1) . Its radical is generated by a and c, and the radical Q0 of the ideal in T 0 generated by f and j is (a, c, e), which has height 3. Now if P ae T is a height one prime ideal, then its preimage P0 ae T 0 is only height two. Hence P0 cannot contain Q0. That is TP is regular, since P is not in the singular locus of T , defined by Q = Image(Q0). Here we've used that T is a domain and that k = Fp is perfect.
4.4. Example. Our last example arises as the ring of invariants for the action of the Weyl group W (F4) acting on the mod 3 cohomology of the classifying space of the maximal torus of the simple compact exceptional Lie group F4 of rank four. The group W (F4) has cardinality 1152, and the vector space V on which it acts can be identified with the first homology group of the maximal torus; this has rank 4 overF
3. The algebra S = S(V #) is then the cohomology algebra of the classifying space of the torus, and in order to conform to the topological origin of this algebra, we grade it so that the elements of V # have degree 2. Toda [12] computed that R = SW (F4) is isomorphic toF
3[y1, y2, y5, y9, y12]/(f15), where the degree of yi is 4i, and
f15 = y35 + y22y1y25 - y12y31 - y9y22. Define R00 = F3[y1, y2, y9, y12]. The fraction field degree of S over R is 1152, and that of S over R00 is three times this. The quotient S \Omega R00 k is a Poincar'e algebra. Under the inclusion R ! S, y5 is contained in the S-ideal generated by y1 and y2. Hence S \Omega R k = S \Omega R00 k is a Poincar'e duality algebra, even though R is not a polynomial algebra.
8 W. G. DWYER AND C. W. WILKERSON Note that in this case W is generated by reflections of order 2 and the algebra of coinvariants satisfies Poincar'e duality, but SW still fails to be a polynomial algebra. We thank Larry Smith for telling us of this example.
4.5. Remark. In each of the counterexamples above, the ring of coinvariants is a complete intersection, not just a Gorenstein ring (for algebras like P*, being Gorenstein and finite dimensional over k is equivalent to satisfying Poincar'e duality). M. Sezer and R.J Shank [11] have shown that 4.2 generalizes to k pairs of generators, using work of Campbell and Hughes [2] to show that the ideal SR+ is generated by the {xi, ypi }. For these examples with k >= 3 pairs of generators, the rings of invariants are not Cohen-Macaulay, even though the coinvariants have Poincar'e duality. In the above examples, the coinvariants are actually complete intersections rather than just Gorenstein. It would be interesting to find purely Gorenstein examples in this context.
References [1] Bruns, Winfried and Herzog, J"urgen, Cohen-Macaulay rings, Cambridge Stud-ies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge,
1993.[2] Campbell, H. E. A. and Hughes, Ian, Vector Invariants of
U2(Fp): a proof ofa conjecture of Richman, Adv. Math. (126), 1997, pp. 1-20.
[3] Greuel, Gert-Martin and Pfister, Gerhard, A Singular introduction to com-mutative algebra, With contributions by Olaf Bachmann, Christoph Lossen
and Hans Sch"onemann, Springer-Verlag, Berlin, 2002, pages xviii+588.[4] Kane, Richard, Poincar'e duality and the ring of coinvariants, Canad. Math. Bull.,(37),1994,number 1, pp. 82-88.[5] Kane, Richard, Reflection groups and invariant theory, CMS Books in Mathematics/Ouvrages de Math'ematiques de la SMC, 5, Springer-Verlag, New York,2001. [6] Lin, Tzu-Chun, Poincar'e duality algebras and rings of coinvariants, Proceed-ings of the American Mathematical Society, (134), 2005, no. 6, pp. 1599-1604. [7] Matsumura, Hideyuki, Commutative ring theory, Cambridge Studies in Ad-vanced Mathematics,vol. 8, 2nd ed., Cambridge University Press, Cambridge,
1989, xiv+320.[8] Serre, Jean-Pierre, Groupes finis d'automorphismes d'anneaux locaux r'eguliers, Colloque d'Alg`ebre (Paris, 1967), Exp. 8, 11 pages, Secr'etariat math'ematique,Paris, 1968. [9] Smith, Larry, On a theorem of R. Steinberg on rings of coinvariants, Proc.Amer. Math. Soc., (131), no. 4, 2003, pp. 1043-1048. [10] Steinberg, Robert, Differential equations invariant under finite reflectiongroups,Trans. Amer. Math. Soc.,(112),1964, pp. 392-400. [11] Sezer, M"ufit and Shank, R. James, On the coinvariants of modular represen-tations of cyclic groups of prime order, Journal of Pure and Applied Algebra,
(205), 2006 no. 1, pp. 210-225.
POINCAR'E DUALITY AND RINGS OF COINVARIANTS 9 [12] Toda, Hirosi, Cohomology mod 3 of the classifying space BF4 of the exceptionalgroup
F4, J. Math. Kyoto Univ., vol. 13, 1973, pp. 97-115.
E-mail address: dwyer.1@nd.eduE-mail address:
cwilkers@purdue.edu