Homological Algebra Modulo a Regular Sequence with Special Attention to Codimension Two

Let M be a finite module over a ring R obtained from a commutative ring Q by factoring out an ideal generated by a regular sequence. The homological properties M over R and over Q are intimately related. Their links are analyzed here from the point of view of differential graded homological algebra over a Koszul complex that resolves R over Q. One outcome of this approach is a transparent derivation of some central results of the theory. Another is a new insight into codimension two phenomena, yielding an explicit finitistic construction of the generally infinite minimal R-free resolution of M. It leads to theorems on the structure and classification of finite modules over codimension two local complete intersections that are exact counterparts of Eisenbud’s results for modules over hypersurfaces