Algebra & Number Theory Seminar
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Texas Tech University

Algebra & Number Theory Seminar

2017-2018

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Department of Mathematics & Statistics

Math 109

Wednesdays 3:00-4:00 pm

### Peder Thompson

Texas Tech University

#### Wednesday, September 6, 2017

#### Math 109, 3:00 pm

**Title:** Using cyclic Adams operations to prove Serre's Vanishing Conjecture

**Abstract:** Let Q be a regular local ring and let M and N be finitely generated Q-modules whose supports intersect only at the maximal ideal. In 1975, Serre conjectured that the intersection multiplicity \chi(M,N)=\sum_i (-1)^i length(Tor_i^Q(M,N)) vanishes when the codimensions of the supports of M and N are "too big". Using Adams operations, this was proved by Gillet and Soulé in 1987. We give a shorter proof of this conjecture, which will involve developing "cyclic" Adams operations on the Grothendieck group of bounded complexes of finitely generated projective modules over any commutative noetherian ring. This talk will include an introduction to the classical notion of Grothendieck groups and should be accessible by graduate students.