LIFTING MODULES AND A THEOREM ON FINITE FREE RESOLUTIONS

Abstract

This chapter discusses the lifting modules and a theorem on finite free resolutions. It discusses a new theorem about finite free resolutions over commutative noetherian rings that has proved useful in connection with the lifting problem. The result on free resolutions has some other interesting applications. It yields Burch's theorem on the structure of cyclic modules of homological dimension. In particular, it yields a new proof that any regular local ring is a unique factorization domain. A positive solution to the lifting problem would allow one to reduce the general (complete regular local) case of Serre's conjecture to the unramified case. It is found that because of Serre's reduction of the conjecture to the cyclic case, it would be sufficient to be able to lift cyclic modules. The chapter presents the results connected with the lifting problem of Grothendieck. The result on lifting cyclic modules S/I, where S/I has finite homological dimension and I is generated by three elements, follows from a broad generalization of Burch's theorem that gives information about the form of a finite free resolution of any length.