F-stable secondary representations and deformation of F-injectivity

TL;DR

This paper demonstrates that F-injectivity, a property in commutative algebra, deforms under certain conditions related to secondary representations and Frobenius stability, with implications for Cohen-Macaulay and graded rings.

Contribution

It establishes new conditions under which F-injectivity deforms, linking secondary representations and Frobenius actions to deformation properties.

Findings

F-injectivity deforms for rings with Frobenius-stable secondary representations.

Deformation holds for sequentially Cohen-Macaulay rings and those with no embedded attached primes.

Additional cases include perfect residue fields and graded rings.

Abstract

We prove that deformation of F-injectivity holds for local rings $(R,\mathfrak{m})$ that admit secondary representations of $H^i_{\mathfrak{m}}(R)$ which are stable under the natural Frobenius action. As a consequence, F-injectivity deforms when $(R,\mathfrak{m})$ is sequentially Cohen-Macaulay (or more generally when all the local cohomology modules $H^i_{\mathfrak{m}}(R)$ have no embedded attached primes). We obtain some additional cases if $R/\mathfrak{m}$ is perfect or if $R$ is $\mathbb{N}$-graded.