UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

December 04, 2007

939 Evans Hall

12:15AM: Linear Algebra Over Commutative Rings

T.Y. Lam

Notes: http://math.berkeley.edu/~lam/amspfaff.pdf

Using the notion of the McCoy rank of a matrix, a considerable amount of linear algebra can be developed over a commutative ring. After a quick introduction to this, we'll offer some new results on the determinantal ideals of symmetric and skew-symmetric matrices over commutative rings. If time permits, we'll also discuss the corresponding results on the Pfaffians of alternating matrices. This is a joint work with Kodiyalam and Swan.

12:17AM: Complete Intersections of Codimension 2 (after Avramov-Buchweitz)

Ed Carter

Over a regular local ring $Q$, a theorem of Auslander, Buchsbaum, and Serre states that any finitely generated $Q$-module has finite projective dimension. However, over a ring $R$ obtained by taking the quotient of $Q$ by a regular sequence, this is not the case, and in general modules have infinite sequences of nonzero Betti numbers. When one considers the subsequences of Betti numbers with even and odd indices, they are eventually given by two different polynomial functions which have the same leading term. When the regular sequence used to obtain $R$ from $Q$ consists of two elements, a theorem of Avramov and Buchweitz allows us to place an upper bound, depending on the $R$-module in question, on how large the indices must be for the even and odd Betti numbers to be given by polynomials.

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