Seminar Calendar
for Commutative Ring Theory events the next 12 months of Saturday, August 1, 2009.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery. 
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Thursday, September 24, 2009

Commutative Ring Theory Seminar
3:00 pm   in 243 Altgeld Hall,  Thursday, September 24, 2009
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Submitted by beder.
Jesse Beder (UIUC Math)
Quillen's Theorem on Chow Groups
Abstract: Claborn and Fossum defined, for a commutative ring A, the Chow groups W_i(A) as a generalization to the usual class group. They proved that for A = k[X_1, ..., X_d] or k[[X_1, ..., X_d]], with k a field or complete DVR, that all the groups W_i(A) = 0, and conjectured that this holds for any regular local ring A. I will present Quillen's theorem, which proves this statement when A is essentially of finite type over a field.

Thursday, October 1, 2009

Commutative Ring Theory Seminar
3:00 pm   in 243 Altgeld Hall,  Thursday, October 1, 2009
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Submitted by beder.
Jesse Beder (Department of Mathematics, University of Illinois)
Quillen's Theorem on Chow Groups
Abstract: Claborn and Fossum defined, for a commutative ring A, the Chow groups W_i(A) as a generalization to the usual class group. They proved that for A = k[X_1, ..., X_d] or k[[X_1, ..., X_d]], with k a field or complete DVR, that all the groups W_i(A) = 0, and conjectured that this holds for any regular local ring A. I will present Quillen's theorem, which proves this statement when A is essentially of finite type over a field.

Thursday, October 8, 2009

Commutative Ring Theory Seminar
3:00 pm   in 243 Altgeld Hall,  Thursday, October 8, 2009
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Submitted by beder.
Kuei-Nuan Lin (Purdue University Math)
Rees Algebras of Diagonal Ideals
Abstract: Given two determinantal rings over a field k. We consider the diagonal ideal D, the kernel of the diagonal map. By the work of Simis-Ulrich, we know the defining equations of special fiber ring of D. When the two determinantal rings are equal, the special fiber ring is known as a homogeneous coordinate ring of secant variety. We aim at a more refined study of the ideal defining Rees algebra of D. By knowing the defining equations, we can show that Rees algebra is Cohen-Macaulay.

Thursday, October 15, 2009

Commutative Ring Theory Seminar
3:00 pm   in 243 Altgeld Hall,  Thursday, October 15, 2009
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Submitted by beder.
Alexandra Seceleanu (Department of Mathematics, University of Illinois)
Weak Lefschetz Property and Powers of Linear Forms
Abstract: Classically, the weak Lefschetz theorems compare the topology of complex projective varieties and of their hyperplane sections. In an algebraic setting, an Artinian graded algebra has the Weak Lefschetz Property (WLP) if multiplication by a general linear form, from any graded component to the next, has maximal rank. I will explain how recent work of Brenner and Kaid expresses the weak Lefschetz property in terms of the cohomology of a certain syzygy bundle. I will prove, using this technique, that an Artinian quotient of K[x; y; z] by an ideal I generated by powers of linear forms has the Weak Lefschetz Property. Our proof works without the semistability hypothesis of Brenner and Kaid, which typically does not hold. The latter part is joint work with H. Schenck.

Thursday, November 12, 2009

Commutative Ring Theory Seminar
3:00 pm   in 243 Altgeld Hall,  Thursday, November 12, 2009
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Submitted by beder.
Alexandra Seceleanu (Department of Mathematics, University of Illinois)
The Syzygy Theorem in Mixed Characteristic
Abstract: The Evans-Griffith Syzygy Theorem states that the rank of a non-free kth syzygy of a module over a equicharacteristic Noetherian local ring is at least k. In the original proof, the height of order ideals of minimal generators for syzygy modules plays a prominent role. I will recall the connection between ranks of syzygies and order ideals and I will introduce a comparison theorem for heights of order ideals of consecutive syzygies modulo a hyperplane section. I will show how one can use this comparison theorem and the Syzygy Theorem in equal characteristic p to prove some relevant cases of the Syzygy Theorem in mixed characteristic.