Deformation of quotients on a product

Details Deformation of quotients on a product Deformation of quotients on a product

2011-03-28

arxiv.org/abs/1103.5482v1

Mathematics

Algebraic Geometry

Scientific paper

We consider the general problem of deforming a surjective map of modules $f : E \to F$ over a coproduct sheaf of rings $B=B_1 \otimes_A B_2$ when the domain module $E = B_1 \otimes_A E_2$ is obtained via extension of scalars from a $B_2$-module $E_2$. Assuming $B_1$ is flat over $A$, we show that the Atiyah class morphism $F \to \LL_{B/B_2} \otimes^{\bL} F[1]$ in the derived category $D(B)$ factors naturally through (the shift of) a morphism $\beta : \Ker f \to \LL_{B/B_2} \otimes^{\bL} F$. We describe the obstruction to lifting $f$ over a (square zero) extension $B_1' \to B_1$ in terms of $\beta$ and the class of the extension. As an application, we use the reduced Atiyah class to construct a perfect obstruction theory on the Quot scheme of a vector bundle on a smooth curve (and more generally).

Affiliated with

Also associated with

No associations

Rating

Deformation of quotients on a product does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Deformation of quotients on a product, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Deformation of quotients on a product will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-162260