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Vanishing of Ext and Tor over complete intersections

Olgur Celikbas, University of Nebraska - Lincoln


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 [32], and H. Dao [21], we obtain new results showing that good depth properties on the R-modules M, N and MR 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 MR 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 [18] 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.^

Subject Area


Recommended Citation

Celikbas, Olgur, "Vanishing of Ext and Tor over complete intersections" (2010). ETD collection for University of Nebraska - Lincoln. AAI3411994.