Off-campus UNL users: To download campus access dissertations, please use the following link to log into our proxy server with your NU ID and password. When you are done browsing please remember to return to this page and log out.
Non-UNL users: Please talk to your librarian about requesting this dissertation through interlibrary loan.
Vanishing of Ext and Tor over complete intersections
Let (R, m ) be a local complete intersection, that is, a local ring whose m -adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of TorRi (M, N) and ExtiR (M, N).^ In this context, M satisfies Serre's condition ( Sn) if and only if M is an n th syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r − 1 for all sufficiently large n. We use this notion of Serre's condition and complexity to study the vanishing of TorRi (M, N). In particular, building on results of C. Huneke, D. Jorgensen and R. Wiegand , and H. Dao , we obtain new results showing that good depth properties on the R-modules M, N and M ⊗R N force the vanishing of TorRi (M, N) for all i ≥ 1. We give examples showing that our results are sharp. We also show that if R is a one-dimensional domain and M and M ⊗R HomR( M, R) are torsion-free, then M is free if and only if M has complexity at most one.^ If R is a hypersurface and ExtiR (M, N) has finite length for all i » 0, then the Herbrand difference  is defined as length( Ext2R (M, N)) – length( Ext2n-1R (M, N)) for some (equivalently, every) sufficiently large integer n. In joint work with Hailong Dao, we generalize and study the Herbrand difference. Using the Grothendieck group of finitely generated R-modules, we also examined the number of consecutive vanishing of ExtiR (M, N) needed to ensure that ExtiR (M, N) = 0 for all i » 0. Our results recover and improve on most of the known bounds in the literature, especially when R has dimension two.^
Celikbas, Olgur, "Vanishing of Ext and Tor over complete intersections" (2010). ETD collection for University of Nebraska - Lincoln. AAI3411994.