Horrocks' question for monomially graded modules.

Abstract

In 1978 Hartshorne reported a question due to Horrocks which essentially asks whether the i-th Betti number of a finite length module M over an n-dimensional regular local ring (R,m,k) is at least as large as the corresponding binomial coefficient. If R is an n-dimensional regular local ring for which this result is known to hold, then we show that the result can be extended to modules graded by monomials over polynomial rings S with coefficients in R. This extension is established as a special case of a very general inequality relating the length of Tor(k,M) to the lengths of Tors of certain associated S-modules where the former Tor is computed over S while the latter is computed over R. The inequalities developed lead quite naturally to similar results on lengths of Koszul homology modules and eventually to a rather amazing inequality relating the length of iterated Koszul homology to appropriately chosen binomial coefficients. Koszul homology provides a much more general setting for the Horrocks' question and, in the absence of monomial grading, iterated Koszul homology yields a much stronger conjecture.