Cohomologically Full Rings. (28th October 2019)

• Record Type: Journal Article
• Authors: Dao, Hailong; De Stefani, Alessandro; Ma, Linquan
• Is Part Of: International mathematics research notices. Volume 2021:Number 17(2021)
• Page Start: 13508
• Page End: 13545
• Publication Date: 2019-10-28
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnz203
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: Inspired by a question raised by Eisenbud–Mustaţă–Stillman regarding the injectivity of maps from ${\operatorname{Ext}}$ modules to local cohomology modules and the work by the third author with Pham, we introduce a class of rings, which we call cohomologically full rings. This class of rings includes many well-known singularities: Cohen–Macaulay rings, Stanley–Reisner rings, F-pure rings in positive characteristics, and Du Bois singularities in characteristics $0$ . We prove many basic properties of cohomologically full rings, including their behavior under flat base change. We show that ideals defining these rings satisfy many desirable properties, in particular they have small cohomological and projective dimension. When $R$ is a standard graded algebra over a field of characteristic $0$, we show under certain conditions that being cohomologically full is equivalent to the intermediate local cohomology modules being generated in degree $0$ . Furthermore, we obtain Kodaira-type vanishing and strong bounds on the regularity of cohomologically full graded algebras.