A ramification formula for Poincare series, and a hyperplane formula for modular invariants D. J. Benson and W. W. Crawley-Boevey* Mathematical Institute 24-29 St. Giles Oxford OX1 3LB Great Britain March 24, 1993 Abstract Let (A) be the coefficient of the second term in the Laurent expansion about t = 1 of the Poincare series p(A; t) of a graded ring A. For a fin* *ite extension of integrally closed domains we prove a formula relating (B), * * (A) and the differential exponent of B over A. As an application we prove a conjecture of Carlisle and Kropholler which computes for rings of modul* *ar invariants of finite groups in terms of stabilizers of hyperplanes. 1 Introduction Let A be a positively graded Noetherian domain of Krull dimension n, and suppose that A0 = K is a field.PIf M is a finitely generated graded A-module, its Poinc* *are series p(M; t) = itidim K(Mi) is a rational function of t, and the pole at t * *= 1 has order equal to the Krull dimension of M. Rational numbers deg(M) and (M) are defined by the Laurent expansion about t = 1: ! deg(M) (M) 1 p(M; t) = ________+ __________+ O __________ : (1 - t)n (1 - t)n-1 (1 - t)n-2 Now suppose that A is integrally closed and has fraction field L. Let L0=L be a finite field extension, and let B be the integral closure of A in L0. Suppose t* *hat B is ________________________________ *Supported by the SERC of Great Britain 1 graded in such a way that the inclusion A ,! B and the trace map TrL0=L: B ! A both preserve degrees. In this case we have deg(B) = |L0: L| deg(A). We prove t* *he following formula for in Theorem 3.1: X |L0: L| (A) - (B) = 1_2 vP (DB=A ) (B=P): P Here the sum runs over the homogeneous height one primes P of B. The different DB=A is the inverse of the divisorial fractional ideal D-1B=A= {x 2 L0| TrL0=L(Bx) A}; and vP (DB=A ) is the differential exponent of P over A. It is non-zero if and * *only if P is ramified. Now let V be an n-dimensional Fp-vector space, and let G be a subgroup of GL(V ) = GLn(Fp), with its natural action on the ring B = Fp[V ] = Fp[x1; : :;:* *xn] of polynomials. Let BG be the ring of invariants. One can show that deg(BG ) = 1=|* *G|. We use the above formula to prove (Theorem 4.3) the following conjecture of Car* *lisle and Kropholler [1]: 1 X (Fp[V ]G ) = _____ ((p - 1)aW + hW - 1) 2|G| W Here, the sum runs over all hyperplanes W of V , and the integers hW and aW a* *re defined by |GW | = hW paW with (hW ; p) = 1, where GW is the pointwise stabil* *izer of W in G. A hyperplane makes a non-zero contribution to this sum if and only if it is a reflecting hyperplane; namely a codimension one subspace fixed pointwis* *e by some non-trivial element of G. In characteristic zero, the formula obtained by the same method is the same* * as the usual hyperplane formula obtained from Molien's theorem. Over larger fields* * of characteristic p, the local calculation is harder, and it is not easy to state * *a clean result. 2 Graded modules Let A be a positively graded Noetherian domain of Krull dimension n with field * *of fractions L, and suppose that A0 = K is a field. If M is a finitely generated g* *raded A-module, then Mi can only be non-zero for finitely many negative values of i. * *If N is another finitely generated graded A-module, we write Hom A (M; N) for the gr* *aded module of homomorphisms, so that in degree i it consists of those homomorphisms which raise degree by i. 2 Lemma 2.1 If M and N are finitely generated graded A-modules then the inclu* *sion of the graded homomorphisms Hom A(M; N) in the ungraded homomorphisms is an isomorphism. Proof This is true when M is freely generated over A by a finite set of homoge* *neous elements, since Hom A(A; N) = N. So choose an epimorphism F ! M of graded modules, with F freely generated over A by a finite set of homogeneous elements. Then the ungraded homomorphisms fromPM to N are contained in the graded homomorphisms Hom A(F; N). If OE = iOEi is an ungraded homomorphism from M to N with OEi 2 Hom A(F; N)i, then each OEi vanishes on the kernel of F ! M, a* *nd so each OEi is in Hom A(M; N). * * 2 It follows that the modules Ext rA(M; N) may also be regarded as graded A- modules. Clearly deg and are additive over short exact sequences of modules. We de* *fine the rank of M to be the dimension over L of L A M. We write M[d] for M with a degree shift of d, so that M[d]i = Mi+d. The formula p(M[d]; t) = t-dp(M; t) * *and dimension theory prove the following lemma: Lemma 2.2 (i) deg(M[d]) = deg(M). (ii) (M[d]) = (M) - d deg(M). (iii) deg(M) = 0 if M is torsion. (iv) If M has codimension at least two (i.e., Krull dimension at most n - 2* *) then (M) = 0. 2 Lemma 2.3 We have deg(M) = rankA (M) deg(A). If M is torsion then X (M) = lengthAp(Mp) (A=p) p where the sum runs over the homogeneous height one primes p of A. Proof Both sides of both formulae are additive on short exact sequences, so we may assume that M = (A=p)[d]. The assertions now follow from Lemma 2.2. 2 . We now prove a formula for which is reminiscent of the Riemann-Roch formu* *la for curves. Theorem 2.4 Suppose that A is integrally closed. If M and N are finitely gen* *erated graded A-modules then (Hom A(M; N)) - (Ext 1A(M; N)) = rankA(M)rank A(N) (A) + rankA(M) (N) - rankA(N) (M): 3 Proof The right hand side of the equation is clearly additive on short exact s* *e- quences 0 ! M ! M0 ! M00! 0 of graded modules. To see that the same is true of the left hand side, we argue as follows. Since A is integrally closed,* * the localization at any height one prime is a Dedekind domain, and therefore has gl* *obal dimension one. Thus Ext2A(M00; N) has codimension at least two, and hence so do* *es the image of the last map in the exact sequence 0 ! Hom A (M00; N) ! Hom A (M0; N) ! Hom A (M; N) ! Ext 1A(M00; N) ! Ext1A(M0; A) ! Ext1A(M; N) ! Ext2A(M00; N): It follows that vanishes on the image of this last map, and so (Hom A(M; N)* *) - (Ext 1A(M; N)) is additive on short exact sequences as required. In a the same way, one sees that the left and right hand side of the equati* *on are additive on short exact sequences 0 ! N ! N0 ! N00! 0 in the second variable. It follows that it suffices to prove the theorem with M and N of the form (A=p)* *[d] with p a homogeneous prime ideal in A. We divide into cases according to the fo* *rms of the primes involved. First, we note that if either prime has height two or m* *ore, all terms in the equation are zero. In the case where M = A[d] we have Hom A(M; N) ~=N[-d] and Ext1A(M; N) = 0, and so by Lemma 2.2, both sides of the equation are equal to (N) + d deg(N). In the case where M = (A=p)[d], N = A[d0], and p is a height one homogeneous prime, it follows from the fact that Ap is a Dedekind domain that Ext1A(M; N)p * *and Mp have the same length over Ap. So by Lemma 2.3 we have (Ext 1A(M; N)) = rank A(N) (M). We also have Hom A(M; N) = 0, and so the equation is proved. In the case where M = (A=p)[d] and N = (A=p0)[d0] with p and p0 height one homogeneous primes, again all terms in the equation vanish unless p = p0. In this case, Hom A(M; N)p and Ext 1A(M; N)p have the same length over Ap, and so by Lemma 2.3 we have (Hom A(M; N)) = (Ext 1A(M; N)). So both sides of the equation are zero. 2 We write M* for the graded dual module Hom A(M; A). Corollary 2.5 If A is integrally closed and M is a finitely generated torsion* *-free graded A-module, then (M) + (M*) = 2rank A(M) (A): Proof If M is torsion-free then Ext 1A(M; A) has codimension at least two and * *so (Ext 1A(M; A)) = 0. 2 4 3 The different Suppose that A is integrally closed with field of fractions L, and that L0=L is* * a finite separable field extension. Let B be the integral closure of A in L0, and suppos* *e that B is graded in such a way that the inclusion A ,! B and the trace TrL0=L: B ! A preserve degrees. Since rankA (B) = |L0: L| we have deg(B) = |L0: L| deg(A). P Theorem 3.1 |L0: L| (A) - (B) = 1_2P vP (DB=A ) (B=P), where the sum runs over the homogeneous height one primes. Proof The trace form L0x L0! L given by (x; y) 7! TrL0=L(xy) is non-degenerate, by separability. So the B-module homomorphism : D-1B=A ! B* x 7! (y 7! TrL0=L(xy)) is an isomorphism. By assumption, the restriction of to B is degree preservin* *g. Here, we are using Lemma 2.1 to identify the ungraded homomorphisms with the graded homomorphisms. Writing C for the cokernel of |B , we have (B*) - (B) = (C): Now C is a torsion graded module, so X (C) = lengthBP (CP ) (B=P) P and as an ungraded module C ~=D-1B=A=B so that lengthBP (CP ) = -vP (D-1B=A) = vP (DB=A ): Combining these facts with the equality (B*) + (B) = 2|L0 : L| (A) of Corol- lary 2.5 gives the assertion. * * 2 Now let B be a positively graded integrally closed Noetherian domain with f* *rac- tion field L0. If G is a finite group of automorphisms of B preserving the grad* *ing, then the theorem applies to the inclusion of A = BG in B, with the inherited gr* *ading. If P is a homogeneous height one prime in B then the inertia group is GP = {g 2 G | g(P) = P and gx x (mod P) 8 x 2 B}: The differential exponent of P over A is determined locally, being equal to* * the differential exponent of PBS-1 over Ap, where p = A \ P and S = B \ p. Since Ap 5 and BS-1 are Dedekind domains, the theory of local fields can be used to compute this exponent. In particular, if BP =PBP is a separable extension of Ap=pAp, * *the differential exponent vP (DB=A ) may be computed in terms of the orders of the ramification groups. In our application this is not the case, but the followin* *g well known formula suffices. Lemma 3.2 We have vP (DB=A ) = vP (DB= BGP ). Proof First note that BGP =A is unramified at P = P \ BGP , cf. [2], xI.7. Th* *is implies that D BGP =A6 P , so that BD BGP =A6 P. Now the transitivity of the tr* *ace shows that D-1B=A= (D B=BGPDBGP =A)-1, so that vP DB=A = vP DB= BGP + vP BD BGP =A= vP DB= BGP : 2 Theorem 3.1 has the following corollary, which shows that may be calculat* *ed locally: Corollary 3.3 Suppose that GP \ GP0 = 1 for distinct homogeneous height one prime ideals P and P0. Then X i j |G| (BG ) - (B) = |GP | (BGP ) - (B) P where the sum runs over homogeneous height one primes. Proof By applying Theorem 3.1 to the action of GP on B and using the assumption on the inertia subgroups, we have |GP | (BGP ) - (B) = 1_2vP (DB= BGP ) (B=P) = 1_2vP (DB=A ) (B=P): A second application of the theorem gives the result. * * 2 4 The Carlisle-Kropholler conjecture Let K be a field, and G be a finite subgroup of GL(V ) = GLn(K), acting on the ring B = K[V ] = K[x1; : :;:xn] of polynomials in the obvious way, so that x1; * *: :;:xn form a basis for the linear dual V *of V = Kn, dual to a chosen basis v1; : :;:* *vn of V . We give this its natural grading, so that each xi has degree one, and it follow* *s that (B) = 0. Let A = BG , the ring of invariants. If W is a codimension one subspa* *ce of V , we write PW for the corresponding height one prime ideal of B. It is ge* *nerated by any non-zero linear form vanishing on W . In particular, it is homogeneous. * *We say that W is a reflecting hyperplane if its pointwise stabilizer GW is non-tr* *ivial. 6 Lemma 4.1 If P is a height one homogeneous prime in B and GP 6= 1 then P = PW for some codimension one subspace W of V . Moreover, the inertia group GPW is equal to the hyperplane stabilizer GW , so that W is a reflecting hyperplane. Proof Since B is a unique factorization domain, the ideal P is generated by a homogeneous element f of some degree d. If d 2 then any element of GP fixes the degree one elements of B and is hence the identity element. The rest of the lem* *ma is now clear. 2 We now specialize to the case where K = Fp. In this case, if W is a reflect* *ing hyperplane, we may choose a basis v1; : :;:vn for V in such a way that W is the subspace generated by v2; : :;:vn. Then the matrices for the action of GW have* * the form 0 1 * 0 . . .0 BB* 1 0 C BB . . CCC @ .. ..A * 0 . . .1 So GW is a split extension with normal elementary abelian subgroup (Z=p)aW (aW n - 1) and quotient Z=hW with hW a divisor of p - 1. After a suitable adjustm* *ent of the basis, G is generated by elements g and g2; : :;:gaW +1 acting via g(v1) = v1 gi(v1) = v1 + vi g(vj) = vj gi(vj) = vj (j > 1); where is a primitive hW th root of unity in Fp. The dual action on x1; : :;:x* *n is given by g(x1) = -1x1 gi(x1) = x1 g(xj) = xj gi(xj) = xj - ffiijx1(j > 1) Here, ffiijdenotes the Kronecker delta. Lemma 4.2 The invariants BGW = Fp[x1; : :;:xn]GW form a polynomial ring Fp[xhW1; xp2- x2xp-11; : :;:xpaW +1- xaW +1xp-11; xaW +2; : :;:xn]: In particular, the Poincare series of this ring is given by 1 p(BGW ; t) = ________________________________ (1 - thW )(1 - tp)aW (1 - t)n-aW -1 and so (BGW ) = 1_2((p - 1)aW + hW - 1)=(hW paW ): 7 Proof The given elements are clearly invariant, since they are products of the* * orbits of the degree one generators. Furthermore, it is clear for the same reason that* * B is integral over this subring. At the level of fields of fractions, the extension* * has the right degree, and is hence right by Galois theory. Finally, this polynomial sub* *ring is integrally closed, and therefore equal to the invariants. * * 2 1 X Theorem 4.3 (BG ) = _____ ((p - 1)aW + hW - 1) 2|G| W Proof This follows from Corollary 3.3 and the above lemma. 2 References [1] D. Carlisle and P. Kropholler. Modular invariants of finite symplectic gro* *ups. Preprint. [2] J. P. Serre. Local Fields. Graduate Texts in Mathematics 67, Springer-Verl* *ag, Berlin/New York 1979. 8