UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

October 30, 2007

939 Evans Hall

3:45PM: Reimagining universal covers and fundamental groups in algebraic

Ravi Vakil (Stanford

In topology, the notions of the fundamental group and the universal cover are inextricably intertwined. In algebraic geometry, the traditional development of the \'etale fundamental group is somewhat different, reflecting the perceived lack of a good universal cover. However, I will describe how the usual notions from topology carry over directly to the algebraic and arithmetic setting without change, rectifying imperfections in the \'etale fundamental group. One key example is the absolute Galois group scheme, which contains more information than the traditional absolute Galois group, in a choice-free manner, and has a rich arithmetic structure. Its geometric fiber is the classical absolute Galois group as a topological group (the profinite topology {\em is} the Zariski topology, and comes from geometry). I will also discuss the example of abelian varieties (and the Tate module). This is joint work with Kirsten Wickelgren.

5:00PM: Resolutions of Monomial Ideals

David Berlekamp

The simplest ideals in a polynomial ring are those generated by monomials. For example, there is a closed form formula for the free resolution of any monomial ideal---a generalization of the Koszul complex called the Taylor complex. But this resolution is not minimal, and it is an open and active problem to produce minimal resolutions, at least in special cases. I will explain the Taylor complex and a simple extension of it that gives a method for combining given resolutions of two monomial ideals to get a resolution of the sum of the monomial ideals.

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