Speaker: Takumi Maruyama
Purdue University
Will give a Algebraic Geometry and Commutative Algebra Seminar entitled:
Boutot’s theorem and vanishing theorems for Zariski–Riemann spaces
Abstract: Let S be a regular ring. By work of Hochster–Roberts, Boutot, Smith, Hochster–Huneke, Schoutens, and Heitmann–Ma, every pure subring of S is pseudo-rational, and in particular, Cohen–Macaulay. This applies for example to rings of invariants of linearly reductive groups. For pure maps R → S of rings of finite type over the complex numbers, Boutot's result is even stronger: If S has rational singularities, then R has rational singularities. In this talk, I will discuss my generalization of Boutot’s theorem which applies to arbitrary Noetherian Q-algebras R and S. The key new ingredient is a new vanishing theorem for Zariski–Riemann spaces of Noetherian schemes in equal characteristic zero. My vanishing theorem has many applications. For example, it implies relative vanishing theorems and the existence (proved jointly with Shiji Lyu) of the relative minimal model program with scaling for algebraic spaces, formal schemes, and both complex and non-Archimedean analytic spaces.
Date: 03-07-2024
Time: 3:30 pm
Location: 258 Hurley Bldg