A K-theoretic approach to multiplicities (Q798380)

We examine the relationship between Gersten's conjecture on a regular local ring R, and the structure of the group \(K_ 0(M_ R^{(i)})\), where \(M_ R^{(i)}\) is the category of finitely generated R modules supported in codimension at least i. We show that, for R of the form \(\Lambda [[x_ 1,...,x_ n]]/\sum^{k}_{i=1}x^ 2_ i-p,\) with \(\Lambda\) a DVR with maximal ideal (p), \(K_ 0(M_ R^{(i)})\) is generated by cyclic modules of the form \(R/(f_ 1,...,f_ i)\), where \((f_ 1,...,f_ i)\) is a regular sequence. As an application, this proves the vanishing part of Serre's multiplicity conjecture for rings with completion isomorphic to R.