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SUMMARY:A motivic approach to efficient generation of projective modules -
  Morgan Opie (University of California\, Los Angeles)
DTSTART:20250512T130000Z
DTEND:20250512T140000Z
UID:TALK230518@talks.cam.ac.uk
DESCRIPTION:A classical question in commutative algebra is the following: 
 given a finitely generated projective module M over a ring R\, what is the
  minimal number of generators of M as an R-module? A classical theorem of 
 Forster and Swan implies that\, if R is of dimension d over a field k and 
 M is of rank r\, then M can always be generated by r+d elements. Work of M
 urthy shows that\, if k is algebraically closed\, the only obstruction to 
 r+d-1 generation of M is vanishing of the top Segre class of M. I will rep
 ort on an approach to this problem using motivic obstruction theory. This 
 approach recovers and improves these classical bounds: we prove results de
 pending only on the homotopy dimension of R over k\, we remove hypotheses 
 on the base field\, and we study r+d-2 generation in certain cases. We als
 o prove a symplectic Forster&ndash\;Swan theorem. This is joint work with 
 Aravind Asok\, Brian Shin\, and Tariq Syed.
LOCATION:Seminar Room 1\, Newton Institute
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