##### Department of Mathematics,

University of California San Diego

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### Algebra

## Allen Knutson

#### UCSD

## The associated graded ring of a homogenized filtration

##### Abstract:

If $R$ is a commutative ring and $I$ an ideal, the associated graded ring ${\rm gr}\, R := R/I + I/I^2 + I^2/I^3 + \ldots$ is a very useful replacement for $R$ for many algebraic and geometric purposes. Frequently, though, it will have nilpotent elements even when $R$ didn't. \vskip .1in \noindent I'll discuss a replacement for it, essentially due to Samuel, Rees, and Nagata, using the ``homogenization" of the $I$-adic filtration. This new filtration is better geometrically motivated, and its associated graded ring never has nilpotents. \vskip .1in \noindent The applications include nilpotent-free versions of intersection theory on smooth varieties, and of Gr\"obner bases.

Host:

### October 24, 2005

### 3:00 PM

### AP&M 7218

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