Projective covers and minimal free resolutions

Mark A Goddard

doi:10.1155/s0161171296000269

Abstract

Using a generalization of the definition of the projective cover of a module, a special type of surjective free resolution, known as the projective cover of a complex, may be defined. The projective cover is shown to be a direct summand of every surjective free resolution and to be the direct sum of the minimal free resolution and an exact complex. Necessary and sufficient conditions for the projective cover and minimal free resolution to be identical are discussed.

Highlights

Let R be a commutative ring with identity
It can be shown that every bounded above complex of finitely generated modules over a Noetherian local ring has a projective cover and that the projective cover is a unique surjective quasi-isomorphism
Since the existence of projective covers and the hypotheses of lemma both place the same requirements upon our complex C, we shall assume hereafter that C refers to a bounded above complex of finitely generated modules over a Noetherian local ring

Summary

Introduction

It can be shown (see Goddard[1]) that every bounded above complex of finitely generated modules over a Noetherian local ring has a projective cover and that the projective cover is a unique surjective quasi-isomorphism. Since the existence of projective covers and the hypotheses of lemma both place the same requirements upon our complex C, we shall assume hereafter that C refers to a bounded above complex of finitely generated modules over a Noetherian local ring.

Results

Conclusion