The number of projective dimensions and the module depth; Koszul Complex

Abstract

Finite free resolution is a problem presented in quite a few books on Homomorphic Algebra, this problem is systematically summed up by J. Herzog in his article: Finite free resolutions. The lecture by J. Herzog gave very good results on the properties of a module when it has finite free resolution.  In this paper, I present the introduction of the lecture: number of projective dimensions and module depth; Koszul complex. The projective dimension and depth module gives theorem formulas about the projective dimension, depth; The Koszul complex part presents only the most basic concepts and results to solve the problem: On a regular ring, every generated finite module has finite free resolution (later part of the lecture). For the purpose of systematically and clearly restating the knowledge that the author has used and proved the clauses and consequence that the author has stated but not proved, so that readers can approach than his lecture and the content on finite free resolution.