Tate resolutions and MCM approximations

Let M M be a finitely generated Cohen-Macaulay module of codimension m m over a Gorenstein Ring R = S / I R = S/I , where S S is a regular ring. We show how to form a quasi-isomorphism ϕ \phi from an R R -free resolution of M M to the dual of an R R -free resolution of M ∨ ≔ E x t R m ( M , R ) M^{\vee }\colonequals {\mathrm {Ext}}_{R}^{m}(M,R) using the S S -free resolutions of R R and M M . The mapping cone of ϕ \phi is then a Tate resolution of M M , allowing us to compute the maximal Cohen-Macaulay approximation of M M .

In the case when R R is 0-dimensional local, and M M is the residue field, the formula for ϕ \phi becomes a formula for the socle of R R generalizing a well-known formula for the socle of a zero-dimensional complete intersection.

When I ⊂ J ⊂ S I\subset J\subset S are ideals generated by regular sequences, the R R -module M = S / J M = S/J is called a quasi-complete intersection, and ϕ \phi was studied in detail by Kustin and Şega. We relate their construction to the sequence of “Eagon-Northcott”-like complexes originally introduced by Buchsbaum and Eisenbud.