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Minimal Resolutions Over Codimension 2 Complete Intersections
David Eisenbud
,
Irena Peeva
Abstract
We construct an explicit free resolution $\mathbf{T}$ for a maximal Cohen-Macaulay module $M$ over a local complete intersection of codimension 2 with infinite residue field. The resolution is minimal when the module $M$ is a sufficiently high syzygy. Our starting point is a layered free resolution $\mathbf{L}$, described in [7], of length 2 over a regular local ring. We provide explicit formulas for the differential in $\mathbf{T}$ in terms of the differential and homotopies on the finite resolution $\mathbf{L}$. One application of our construction is to describe Ulrich modules over a codimension 2 quadratic complete intersection.