Jeroen Hekking: On ideals in derived geometries

Abstract:

At the heart of algebraic geometry lies a correspondence between closed immersions from Z into X, and ideals I on X. To study Z ‘infinitely near’ X, one studies the infinitesimal neighborhoods of Z in X which correspond to the powers of the ideal I. Linearly, this is closely related to the I-adic completion of a give module M over X.

Until recently, there was no good definition of ideals in the derived setting, although there were various definitions of derived I-adic completeness. In joint work with Adeel Khan and David Rydh, we study the derived extended Rees algebra which leads to a natural candidate of infinitesimal neighborhoods. And the theory of Smith ideals has now also lead to a definition of ideals in the derived setting.

In this talk, I will explain work in progress (joint with Zachary Gardner) where we compare and generalize the main definitions surrounding ideals and completeness available in the literature. If time permits, we can look at the following applications: the construction of derived schematic images, and the equivalence between the derived normal bundle and the derived normal cone. The key of the derived story is the correspondence between all affine objects Y over X, and all derived (possibly nonconnective) ideal I on X.