Commutative Algebra

Let (R, m) be a local ring, M a finitely generated module over R. Lech's limit formula states that for a fixed system of parameters f_1, ..., f_d on M, the length of M/(f_1^t_1, ...,f^d/t^d)M divided by the product of the t's approaches the multiplicity of the f's on M as the t_i approach infinity. It is natural to ask whether powers of a fixed sequence of parameters may be replaced by any sequence of parameter ideals I_n contained in m^n. Recalling that the multiplicity may be realized as the alternating sum of the lengths of Koszul homology modules, it is also natural to ask for which i > 0 we have the length of the i^th homology module of the I_n on M divided by the length of M/I_nM approaching 0 as n approaches infinity. In this talk, we will consider the latter question in the case where R is a complete regular local ring containing a field and M is faithful. We will show that under those conditions the M satisfying that condition for all i > 0 are exactly those that are locally Cohen-Macaulay.
Speaker(s): Patricia Klein (University of Michigan)