Resolutions and Cohomology over Complete Intersections

This chapter contains a new proof and new applications of a theorem of Shamash and Eisenbud, providing a construction of proj ective resolutions of modules over a complete intersection. The duals of these infinite projective resolutions are finitely generated differential graded modules over a graded polynomial ring, so they can be represented in the computer, and can be used to compute Ext modules simultaneously in all homological degrees. It is shown how to write Macaulay 2 code to implement the construction, and how to use the computer to determine invariants of modules over complete intersections that are difficult to obtain otherwise.