On the homology of associative algebras

We present a new free resolution for k as an G-module, where G is an associative augmented algebra over a field k. The resolution reflects the combinatorial properties of G. Introduction. Let k be a field and let G be an associative augmented k-algebra. For many purposes one wishes to have a projective resolution of k as a G-module. The bar resolution is always easy to define, but it is often too large to use in practice. At the other extreme, minimal resolutions may exist, but they are often hard to write down in a way that is amenable to calculations. The main theorem of this paper presents a compromise resolution. Though rarely minimal, it is small enough to offer some bounds but explicit enough to facilitate calculations. As it relies heavily upon combinatorial constructions, it is best suited for analyzing otherwise tricky algebras given via generators and relations. Since several results we get as consequences of the main theorem have been obtained before through other means, this paper may be viewed as generalizing and unifying several seemingly unrelated ideas. In particular, we are generalizing Priddy's results on Koszul algebras [12], extending homology computations by Govorov [9] and Backelin [3], and complementing Bergman's methods regarding the diamond lemma [6]. Three results may be of interest. The homology of the mod p Steenrod algebra is given in terms of the homology of a new chain complex smaller than the A-algebra in Theorem 3.5. Formula (16) offers an efficient algorithm for the determination of Hilbert series, and Theorem 4.2 asserts the existence of new bounds on the torsion groups of commutative graded rings. 1. Definitions and the main theorem. Throughout this paper, k denotes any field and G is an associative k-algebra with unity. The field k embeds in G via 77: k -*G and we suppose that G has an augmentation, i.e., a k-algebra map s: G -? k for which 77 is a right inverse. S denotes a set of generators for G as a k-algebra and k(S) is the free associative k-algebra with unity on S. There is a canonical surjection f: k(S) -? G once S is chosen, and the augmentation E is determined once we know s(x) for each x E S. In particular, this means that k(S) may be augmented such that f becomes a map of augmented algebras. To S we associate a function e: S -? Z+ called a grading. In the absence of a more compelling choice we often take e to be grading by length, i.e., e(x) = 1 for Received by the editors May 23, 1983 and, in revised form, February 22, 1984. This paper was the subject of an invited one-hour address in Boulder, Colorado, during the week-long AMS summer program on Combinatorics and Algebra, June 1983. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A62; Secondary 13D03, 55S10. (?)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page