Exceptional complete intersection maps of local rings

TL;DR

This paper characterizes specific surjective local ring maps with kernels generated by regular sequences, using derived category lattice structures and the truncated Atiyah class, to identify exceptional complete intersection maps.

Contribution

It introduces criteria based on derived category lattices and the truncated Atiyah class for detecting exceptional complete intersection maps of local rings.

Findings

Provides criteria for detecting exceptional complete intersection maps.

Characterizes such maps via the truncated Atiyah class.

Connects lattice structures of derived categories to ring map properties.

Abstract

This work concerns surjective maps $\varphi\colon R\to S$ of commutative noetherian local rings with kernel generated by a regular sequence that is part of a minimal generating set for the maximal ideal of $R$. The main result provides criteria for detecting such exceptional complete intersection maps in terms of the lattices of thick subcategories of the derived category of complexes of finite length homology. A key input is a characterization of such maps in terms of the truncated Atiyah class of $\varphi$.