Minimal Projective Resolution Of A Class Chain Complex

Projective resolution is a central topic in homological algebra and has important applications in theory of rings and modules,and representation theory of algebra.This paper mainly studies the existence of minimal projective resolution（P,d）of a chain complex（Y,d’）over R and a q-isomorphism chain map f:P→Y,where R is a commutative local ring with identity.The paper also constructs some concrete examples in order to understand the structure and applications of the minimal projective resolutions.Meanwhile,this paper preliminarily investigates the relation between the minimal projective resolution of chain complexes and projective covers.This paper consists of five chapters:Chapter one provides the background and preliminary knowledge,relevant definitions and theorems,which will be used throughout the paper.Chapter two proves the main theorem of this paper-the existence of the minimal projective resolution（P,d）of the chain complex（Y,d’）with Y_n finitely generated for every n∈Z and a q-isomorphism chain map f:P→Y over R,where R is a commutative noetherian local ring with identity.Although it is a special case of a result in[11],our approach is more concise and constructive.Chapter three first introduces some preliminaries to be used in this chapter and then illustrates some simple applications of minimal projective resolutions with some concrete examples,which leads us to a better understanding about minimal projective resolution.Chapter four preliminarily investigates the relation betweenPin the minimal projective resolution（P,d）of a chain complex（Y,d’）and the projective cover of Y_n,and proves that there exists minimal projective resolution（P,d）of chain complex（Y,d’）and q-isomorphism chain map f:P→Y over R,such that（P₀,f₀）is projective cover of Y₀.Chapter five concludes the paper with an outlook for the problem.