A degree formula for equivariant cohomology
Transactions of the American Mathematical Society
The primary theorem of this paper concerns the Poincaré (Hilbert) series for the cohomology ring of a finite group G with coefficients in a prime field of characteristic p. This theorem is proved using the ideas of equivariant cohomology whereby one considers more generally the cohomology ring of the Borel construction H * (EG × G X), where X is a manifold on which G acts. This work results in a formula that computes the "degree" of the Poincaré series in terms of corresponding degrees of
... ng degrees of certain subgroups of the group G. In this paper, we discuss the theorem and the method of proof. Reverts to public domain 28 years from publication 2.2. Poincaré series of graded modules. Now we define and state several properties of the Poincaré series. In addition, we define the number deg(M ). If M ∈ C(R), then M i is a finite dimensional vector space over k for every i; this is a consequence of our hypothesis that R is a finitely generated k-algebra and the fact that M is a finitely generated R-module. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A DEGREE FORMULA FOR EQUIVARIANT COHOMOLOGY 311 the Poincaré series (also called the Hilbert series ) for M . In fact, PS(M, t) is defined in the same way for every nonnegatively graded k-vector space M such that dim k M i < ∞ for all i. Proposition 2.3. For M, N ∈ C(R), the following properties hold: (1) PS(M (−r), t) = t r PS(M, t) for all r ≥ 0. Definition 2.14. The Krull dimension (or just dimension) of a graded ring R, written Dim(R), is the supremum of the lengths of chains of distinct homogeneous prime ideals in R. We define Dim(0)= − ∞. Recall the invariants d(M ) and s(M ) as defined in Section 2.2. The following proposition will prove to be valuable later in this section. Proposition 2.15 (See, for example, , Thm. 5.5 and Prop. 6.2). Let M ∈ C(R), and let d(M ) and s(M ) be as defined in Section 2.2. (1) d(M ) = s(M ) = Dim(M ) < ∞. (2) If d(M ) = s(M ) = Dim(M ) = D, and y 1 , . . . , y D ∈ m are homogeneous elements such that M is finitely generated over k y 1 , . . . , y D , then y 1 , . . . , y D are algebraically independent over k. Definition 2.16. If M ∈ C(R) with M = 0, let s 1 (M ) be the least s such that there exist homogeneous elements y 1 , . . . , y s ∈ m such that M/(y 1 , . . . , y s )M is a finite dimensional graded vector space over k. Note that s 1 (M ) = 0 if and only if M is a finite dimensional graded vector space over k. We define s 1 (0)= − ∞. Theorem 2.17 ([24, Sec. III.B.2, Thm. 1] or ). If M ∈ C(R), then Dim(M ) = s 1 (M ). Corollary 2.18. For M ∈ C(R), we define D(M )=d(M ) = s(M ) = s 1 (M ) = Dim(M ). Definition 2.19. If M ∈ C(R) with M = 0, and D(M ) = D, then a sequence y 1 , . . . , y D of homogeneous elements of m ⊂ R such that M is a finitely generated k y 1 , . . . , y D -module is called a system of parameters for M , as an R-module.