F-splittings of the polynomial ring and compatibly split homogeneous ideals

Abstract: A polynomial ring R in n variables over a field K of positive characteristic is F-split. It has many F-splittings. When K is a perfect field every F-splitting is given by a polynomial g in R with the monomial u^{p-1} in its support (where u is the product of all the variables) occurring with coefficient 1, plus a further condition, which is not needed if g is homogeneous (w.r.t. any positive grading). Fixed an F-splitting s : R -> R, an ideal I of R such that s(I) is contained in I is said compatibly split (w.r.t. the F-splitting s). In this case R/I is F-split. Furthermore, by Fedder’s criterion when I is a homogeneous ideal of R, R/I is F-split if and only if I is compatibly split for some F-splitting s : R -> R. If, moreover, u^{p-1} is the initial monomial of the associated polynomial g of s w.r.t. some monomial order, then in(I) is a square-free monomial ideal… In this talk I will survey these facts (some of them classical, some not so classical), and make some examples, focusing especially on determinantal ideals.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

Series comments: Note: To get meeting ID,fill the Google form on the web-site of the series sites.google.com/view/virtual-comm-algebra-seminar/home