On a theorem of Knörrer concerning Cohen–Macaulay modules

Abstract

Let K be a field, X={X1,…,Xn} and Y={Y1,…,Yr} sets of indeterminates, and f∈K[[X]],g∈K[[Y]] two non-zero formal power series with f∈(X),g∈(Y). Set Δg=(∂g/∂Yi:1≤i≤r) and suppose that Δg is a (Y)-primary ideal (e.g., if char K=0 and g is an isolated singularity). Write R≔K[[X,Y]]/(f+g),R1≔K[[X]]/(f) and R̃≔R/ΔgR. The main aim of this paper is to relate an arbitrary maximal Cohen–Macaulay (MCM for short) R-module N to the higher order syzygy ΩRr(N/ΔgN), and in this way relate indecomposable MCM R-modules to higher order syzygies of certain indecomposable R̃-modules. The R̃-modules in question are deformations of MCM R1-modules and are weakly liftable. We find resolutions of the higher order syzygy modules in question which are shown to be minimal in certain situations, and express these in terms of matrix factorizations. The theory is shown to be applicable with almost complete success to singularities of Knörrer-type and, in any case, to give detailed information about MCM R-modules.Our techniques generalise and simplify those of Herzog and Popescu [7], and we further use a lifting theorem for maps in Koszul complexes and a technique involving iterated mapping cones which may be of independent interest. [3pt]