2008/07/03 by Puthenpurakal, Tony J.
#13A30 #13D45 (Primary) 13H10 #13H15 (Secondary) #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.0807.0471
We give a two step method to study certain questions regarding associated graded module of a Cohen-Macaulay (CM) module M w.r.t an \mathfrakm-primary ideal \mathfraka in a complete Noetherian local ring (A,\mathfrakm). The first step, we call it complete intersection approximation, enables us to reduce to the case when both A, G_\mathfraka(A) = \bigoplusn ≥ 0 \mathfrakan/\mathfrakan+1 are complete intersections and M is a maximal CM A-module. The second step consists of analyzing the classical filtration \HomA(M,\mathfrakan) \ℤ of the dual HomA(M,A). We give many applications of this point of view. For instance let (A,\mathfrakm) be equicharacteristic and CM. Let a(G_\mathfraka(A)) be the a-invariant of G_\mathfraka(A). We prove: 1. a(G_\mathfraka(A)) = -dim A iff \mathfraka is generated by a regular sequence. 2. If \mathfraka is integrally closed and a(G_\mathfraka(A)) = -dim A + 1 then \mathfraka has minimal multiplicity. We extend to modules a result of Ooishi relating symmetry of h-vectors. As another application we prove a conjecture of Itoh, if A is a CM local ring and \mathfraka is a normal ideal with e3^\mathfraka(A) = 0 then G_\mathfraka(A) is CM.