(Non) linear behavior of Castelnuovo-Mumford regularity

Abstract: In the first half, we shall discuss the importance of Castelnuovo-Mumford regularity and the main motivation to define this invariant along with examples. This part is aimed for the graduate students and should be elementary. In the second half, we discuss the asymptotic behavior of Castelnuovo-Mumford regularity of powers of ideals, and that of Ext and Tor, with respect to the homological degree, over graded complete intersection rings. We derive from a theorem of Gulliksen, a linearity result for the regularity of Ext modules in high homological degrees. We show a similar result for Tor, under the additional hypothesis that high enough Tor modules are supported in dimension at most one; we then provide examples showing that the behavior could be pretty hectic when the latter condition is not satisfied. This talk is based on our joint papers with Marc Chardin, Navid Nemati and Tony Puthenpurakal.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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