Cartier algebras through the lens of p-families

A Cartier subalgebra of a prime characteristic commutative ring R is an associated non-commutative ring of operators on R that play nicely with the Frobenius map. When R is regular, its Cartier subalgebras correspond exactly with sequences of ideals called F-graded systems. One special subclass of F-graded system is called a p-family; these appear in numerical applications such as the Hilbert-Kunz multiplicity and the F-signature. In this talk, I will discuss how to characterize some properties of a Cartier subalgebra in terms of its F-graded system. I will further present a way to construct, for an arbitrary F-graded system, a closely related p-family with especially nice properties.