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Hilbert -Samuel polynomials and building indecomposable modules
Let (R, m , k) be a Noetherian local ring and M and N be finitely generated. In this thesis, we give precise formulas for the generalized Hilbert-Samuel polynomials associated to the torsion and contravariant extension functors, that is, polynomials giving the lengths of the modules TorRi M,N/mnN and ExtiR M,N/mnN , respectively. One application of these results is that they can be used to give information about the dimensions of syzygies of finite length modules. ^ We also show this if R is complete and has depth at least 2, then one can build indecomposable modules of arbitrarily prescribed constant rank. Moreover, if R is assumed to be Cohen-Macaulay, then these modules can be chosen to be maximal Cohen-Macaulay when localized on the punctured spectrum. ^
Crabbe, Andrew, "Hilbert -Samuel polynomials and building indecomposable modules" (2008). ETD collection for University of Nebraska - Lincoln. AAI3315330.