##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## Dan Rogalski

#### UCSD

## ${\bf Z}$-Graded Simple Rings

##### Abstract:

\indent Let $k$ be of a field of characteristic $0$. The first Weyl algebra $A_1(k) = k/(yx-xy-1)$ is $Z$-graded with deg$(x) = 1$, deg$(y) = -1$, and is a simple ring of $GK$-dimension $2$. Sierra has studied its category of graded modules and shown how to find all $Z$-graded algebras with an equivalent graded module category. Smith has also shown how the geometry of this example is related to a certain stack. Our goal is to study more general classes of $Z$-graded simple rings to find more examples which may have interesting algebraic and geometric properties. Specifically, we study the structure of $Z$-graded simple rings $A$ with graded quotient ring $Q$ such that $Q_0$ is a field with trdeg $Q_0 = GK A - 1$. As a special case, we can classify all $Z$-graded simple rings of $GK$-dimension $2$. This is joint work with Jason Bell.

### October 17, 2011

### 3:00 PM

### AP&M 7218

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