A ramification formula for Poincaré 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

Abstract

Let ψ(A) be the coefficient of the second term in the Laurent expansion about t = 1 of the Poincaré series p(A, t) of a graded ring A. For a finite extension of integrally closed domains we prove a formula relating ψ(B), ψ(A) and the differential exponent of B over A. As an application we prove a conjecture of Carlisle and Kropholler which computes ψ for rings of modular 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 A_{0} = K is a field. If M is a finitely generated graded A-module, its Poincaré series p(M, t) = ∑_{i} t^{i} dim_{K}(M_{i}) 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 ψ(M) are defined by the Laurent expansion about t = 1:

p(M, t) = \frac{deg(M)}{(1 − t)^{n}} + \frac{ψ(M)}{(1 − t)^{n−1}} + O (\frac{1}{(1 − t)^{n−2}} ) .

Now suppose that A is integrally closed and has fraction field L. Let L′/L be a finite field extension, and let B be the integral closure of A in L′. Suppose that B is graded in such a way that the inclusion A ↪ B preserves degrees. In this case we have deg(B) = |L′ : L| deg(A). We prove the following formula for ψ in Theorem 3.1:

∗Current address: Department of Mathematics, University of Georgia, Athens GA 30602, USA

†Supported by the SERC of Great Britain

|L′ : L|ψ(A) − ψ(B) = \frac{1}{2} ∑_{𝔓} v_{𝔓}(𝔇_{B/A})ψ(B/𝔓).

Here the sum runs over the homogeneous height one primes 𝔓 of B. The different 𝔇_{B/A} is the inverse of the divisorial fractional ideal

𝔇_{B/A}^{−1} = {x ∈ L′ | Tr_{L′/L}(Bx) ⊆ A},

and v_{𝔓}(𝔇_{B/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 _{p}-vector space, and let G be a subgroup of GL(V) = GL_{n}( _{p}), with its natural action on the ring B = _{p}[V] = _{p}[x_{1}, …, x_{n}] of polynomials. Let B^{G} be the ring of invariants. One can show that deg(B^{G}) = 1/|G|. We use the above formula to prove (Theorem 4.3) the following conjecture of Carlisle and Kropholler [1]:

ψ( _{p}[V]^{G}) = \frac{1}{2|G|} ∑_{W}((p − 1)a_{W} + h_{W} − 1)

Here, the sum runs over all hyperplanes W of V, and the integers h_{W} and a_{W} are defined by |G_{W} | = h_{W} p^{a_{W}} with (h_{W} , p) = 1, where G_{W} is the pointwise stabilizer 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 pointwise 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 A_{0} = K is a field. If M is a finitely generated graded A-module, then M_{i} 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 graded module of homomorphisms, so that in degree i it consists of those homomorphisms which raise degree by i.

Lemma 2.1 If M and N are finitely generated graded A-modules then the inclusion 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 homogeneous 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 from M to N are contained in the graded homomorphisms Hom_{A}(F, N). If ϕ = ∑_{i} ϕ_{i} is an ungraded homomorphism from M to N with ϕ_{i} ∈ Hom_{A}(F, N)_{i}, then each ϕ_{i} vanishes on the kernel of F → M, and so each ϕ_{i} is in Hom_{A}(M, N). □

It follows that the modules Ext_{A}^{r}(M, N) may also be regarded as graded Amodules.

Clearly deg and ψ are additive over short exact sequences of modules. We define 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} = M_{i+d}. The formula p(M[d], t) = t^{−d}p(M, t) and dimension theory prove the following lemma:

Lemma 2.2 (i) deg(M[d]) = deg(M).

(ii) ψ(M[d]) = ψ(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 ψ(M) = 0. □

Lemma 2.3 We have deg(M) = rank_{A}(M) deg(A). If M is torsion then

ψ(M) = ∑_{𝔭} length_{A_{𝔭}}(M_{𝔭})ψ(A/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. □ .

We now prove a formula for ψ which is reminiscent of the Riemann–Roch formula for curves.

Theorem 2.4 Suppose that A is integrally closed. If M and N are finitely generated graded A-modules then

ψ(Hom_{A}(M, N)) − ψ(Ext_{A}^{1}(M, N)) =
rank_{A}(M)rank_{A}(N)ψ(A) + rank_{A}(M)ψ(N) − rank_{A}(N)ψ(M).

Proof The right hand side of the equation is clearly additive on short exact sequences 0 → M → M′ → M′′ → 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 global dimension one. Thus Ext_{A}^{2}(M′′, N) has codimension at least two, and hence so does the image of the last map in the exact sequence

0 → Hom_{A}(M′′, N) → Hom_{A}(M′, N) → Hom_{A}(M, N) →
Ext_{A}^{1}(M′′, N) → Ext_{A}^{1}(M′, A) → Ext_{A}^{1}(M, N) → Ext_{A}^{2}(M′′, N).

It follows that ψ vanishes on the image of this last map, and so ψ(Hom_{A}(M, N)) − ψ(Ext_{A}^{1}(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 equation are additive on short exact sequences 0 → N → N′ → N′′ → 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 forms of the primes involved. First, we note that if either prime has height two or more, all terms in the equation are zero.

In the case where M = A[d] we have Hom_{A}(M, N) ≅ N[−d] and Ext_{A}^{1}(M, N) = 0, and so by Lemma 2.2, both sides of the equation are equal to ψ(N) + d deg(N).

In the case where M = (A/𝔭)[d], N = A[d′], and 𝔭 is a height one homogeneous prime, it follows from the fact that A_{𝔭} is a Dedekind domain that Ext_{A}^{1}(M, N)_{𝔭} and M_{𝔭} have the same length over A_{𝔭}. So by Lemma 2.3 we have ψ(Ext_{A}^{1}(M, N)) = rank_{A}(N)ψ(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/𝔭′)[d′] with 𝔭 and 𝔭′ height one homogeneous primes, again all terms in the equation vanish unless 𝔭 = 𝔭′. In this case, Hom_{A}(M, N)_{𝔭} and Ext_{A}^{1}(M, N)_{𝔭} have the same length over A_{𝔭}, and so by Lemma 2.3 we have ψ(Hom_{A}(M, N)) = ψ(Ext_{A}^{1}(M, N)). So both sides of the equation are zero. □

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

ψ(M) + ψ(M^{∗}) = 2rank_{A}(M)ψ(A).

Proof If M is torsion-free then Ext_{A}^{1}(M, A) has codimension at least two and so ψ(Ext_{A}^{1}(M, A)) = 0. □

3 The different

Suppose that A is integrally closed with field of fractions L, and that L′/L is a finite separable field extension. Let B be the integral closure of A in L′, and suppose that B is graded in such a way that the inclusion A ↪ B preserves degrees. Since B has a basis over A consisting of homogeneous elements, the trace map Tr_{L′/L} : B → A also preserves degrees. Since rank_{A}(B) = |L′ : L| we have deg(B) = |L′ : L| deg(A).

Theorem 3.1 |L′ : L|ψ(A) − ψ(B) = \frac{1}{2} ∑_{𝔓} v_{𝔓}(𝔇_{B/A})ψ(B/𝔓), where the sum runs over the homogeneous height one primes.

Proof The trace form L′ × L′ → L given by (x, y) ↦ Tr_{L′/L}(xy) is non-degenerate, by separability. So the B-module homomorphism

θ : 𝔇_{B/A}^{−1} → B^{∗}
x ↦ (y ↦ Tr_{L′/L}(xy))

is an isomorphism. By assumption, the restriction of θ to B is degree preserving. Here, we are using Lemma 2.1 to identify the ungraded homomorphisms with the graded homomorphisms.

Writing C for the cokernel of θ|_{B}, we have

ψ(B^{∗}) − ψ(B) = ψ(C).

Now C is a torsion graded module, so

ψ(C) = ∑_{𝔓} length_{B_{𝔓}}(C_{𝔓})ψ(B/𝔓)

and as an ungraded module C ≅ 𝔇_{B/A}^{−1}/B so that

length_{B_{𝔓}}(C_{𝔓}) = −v_{𝔓}(𝔇_{B/A}^{−1}) = v_{𝔓}(𝔇_{B/A}).

Combining these facts with the equality ψ(B^{∗}) + ψ(B) = 2|L′ : L|ψ(A) of Corollary 2.5 gives the assertion. □

Now let B be a positively graded integrally closed Noetherian domain with fraction field L′. If G is a finite group of automorphisms of B preserving the grading, then the theorem applies to the inclusion of A = B^{G} in B, with the inherited grading.

If 𝔓 is a homogeneous height one prime in B then the inertia group is

G_{𝔓} = {g ∈ G | g(𝔓) = 𝔓 and gx ≡ x (mod 𝔓) ∀ x ∈ B}.

The differential exponent of 𝔓 over A is determined locally, being equal to the differential exponent of 𝔓BS^{−1} over A_{𝔭}, where 𝔭 = A ∩ 𝔓 and S = B \ 𝔭. Since A_{𝔭} and BS^{−1} are Dedekind domains, the theory of local fields can be used to compute this exponent. In particular, if B_{𝔓}/𝔓B_{𝔓} is a separable extension of A_{𝔭}/𝔭A_{𝔭}, the differential exponent v_{𝔓}(𝔇_{B/A}) may be computed in terms of the orders of the ramification groups. In our application this is not the case, but the following well known formula suffices.

Lemma 3.2 We have v_{𝔓}(𝔇_{B/A}) = v_{𝔓}(𝔇_{B/ B^{G_{𝔓}}}).

Proof First note that B^{G_{𝔓}}/A is unramified at P = 𝔓 ∩ B^{G_{𝔓}}, cf. [2], §I.7. This implies that 𝔇_{B^{G_{𝔓}}/A} ⊈ P, so that B𝔇_{B^{G_{𝔓}}/A} ⊈ 𝔓. Now the transitivity of the trace shows that 𝔇_{B/A}^{−1} = (𝔇_{B/B^{G_{𝔓}}}𝔇_{B^{G_{𝔓}}/A})^{−1}, so that

v_{𝔓}𝔇_{B/A} = v_{𝔓}𝔇_{B/ B^{G_{𝔓}}} + v_{𝔓}B𝔇_{B^{G_{𝔓}} /A} = v_{𝔓}𝔇_{B/ B^{G_{𝔓}}} .
□

Theorem 3.1 has the following corollary, which shows that ψ may be calculated locally:

Corollary 3.3 Suppose that G_{𝔓} ∩ G_{𝔓′} = 1 for distinct homogeneous height one prime ideals 𝔓 and 𝔓′. Then

|G|ψ(B^{G}) − ψ(B) = ∑_{𝔓} (|G_{𝔓}|ψ(B^{G_{𝔓}}) − ψ(B))

where the sum runs over homogeneous height one primes.

Proof By applying Theorem 3.1 to the action of G_{𝔓} on B and using the assumption on the inertia subgroups, we have

|G_{𝔓}|ψ(B^{G_{𝔓}}) − ψ(B) = \frac{1}{2}v_{𝔓}(𝔇_{B/ B^{G_{𝔓}}})ψ(B/𝔓) = \frac{1}{2}v_{𝔓}(𝔇_{B/A})ψ(B/𝔓).

A second application of the theorem gives the result. □

4 The Carlisle–Kropholler conjecture

Let K be a field, and G be a finite subgroup of GL(V) = GL_{n}(K), acting on the ring B = K[V] = K[x_{1}, …, x_{n}] of polynomials in the obvious way, so that x_{1}, …, x_{n} form a basis for the linear dual V^{∗} of V = K^{n}, dual to a chosen basis v_{1}, …, v_{n} of V. We give this its natural grading, so that each x_{i} has degree one, and it follows that ψ(B) = 0. Let A = B^{G}, the ring of invariants. If W is a codimension one subspace of V, we write 𝔓_{W} for the corresponding height one prime ideal of B. It is generated 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 G_{W} is non-trivial.

Lemma 4.1 If 𝔓 is a height one homogeneous prime in B and G_{𝔓} ≠ 1 then 𝔓 = 𝔓_{W} for some codimension one subspace W of V. Moreover, the inertia group G_{𝔓_{W}} is equal to the hyperplane stabilizer G_{W}, 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 G_{𝔓} fixes the degree one elements of B and is hence the identity element. The rest of the lemma is now clear. □

We now specialize to the case where K = _{p}. In this case, if W is a reflecting hyperplane, we may choose a basis v_{1}, …, v_{n} for V in such a way that W is the subspace generated by v_{2}, …, v_{n}. Then the matrices for the action of G_{W} have the form

( \begin{matrix}∗ & 0 & ⋯ & 0 \\ ∗ & 1 & & 0 \\ ⋮ & & & ⋮ \\ ∗ & 0 & ⋯ & 1 \end{matrix} )

So G_{W} is a split extension with normal elementary abelian subgroup ( /p)^{a_{W}} (a_{W} ≤ n − 1) and quotient /h_{W} with h_{W} a divisor of p − 1. After a suitable adjustment of the basis, G is generated by elements g and g_{2}, …, g_{a_{W} +1} acting via

\begin{matrix}g(v_{1}) = λv_{1} & g_{i}(v_{1}) = v_{1} + v_{i} \\ g(v_{j}) = v_{j} & g_{i}(v_{j}) = v_{j} & (j > 1), \end{matrix}

where λ is a primitive h_{W} th root of unity in _{p}. The dual action on x_{1}, …, x_{n} is given by

\begin{matrix}g(x_{1}) = λ^{−1}x_{1} & g_{i}(x_{1}) = x_{1} \\ g(x_{j}) = x_{j} & g_{i}(x_{j}) = x_{j} − δ_{ij}x_{1} & (j > 1) \end{matrix}

Here, δ_{ij} denotes the Kronecker delta.

Lemma 4.2 The invariants B^{G_{W}} = _{p}[x_{1}, …, x_{n}]^{G_{W}} form a polynomial ring

p[x_{1}^{h_{W}} , x_{2}^{p} − x_{2}x_{1}^{p−1}, …, x_{a_{W} +1}^{p} − x_{a_{W} +1}x_{1}^{p−1}, x_{a_{W} +2}, …, x_{n}].

In particular, the Poincaré series of this ring is given by

p(B^{G_{W}} , t) = \frac{1}{(1 − t^{h_{W}})(1 − t^{p})^{a_{W}} (1 − t)^{n−a_{W} −1}}

and so

ψ(B^{G_{W}}) = \frac{1}{2}((p − 1)a_{W} + h_{W} − 1)/(h_{W} p^{a_{W}}).

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 subring is integrally closed, and therefore equal to the invariants. □

Theorem 4.3 ψ(B^{G}) = \frac{1}{2|G|} ∑_{W}((p − 1)a_{W} + h_{W} − 1)

Proof This follows from Corollary 3.3 and the above lemma. □

References

[1] D. Carlisle and P. Kropholler. Modular invariants of finite symplectic groups. Preprint.

[2] J. P. Serre. Local Fields. Graduate Texts in Mathematics 67, Springer-Verlag, Berlin/New York 1979.