Shiji Lyu

University of Illinois Chicago

Regular local rings over valuation rings

We discuss various properties of “regular local rings” $A$ over a (not necessarily discrete) valuation ring $V$, defined as essentially finitely presented $V$-algebras of finite weak global dimension. We show that many properties/characterizations of Noetherian regular local rings hold for those $A$ as well. For example, such an $A$ is always a splinter, and if $A$ contains a perfect field $k$, then the cotangent complex $L_{A/k}$ is a flat module at degree $0$. We also discuss an application to vanishing theorems over Noetherian rings.