On a question of Mehta and Pauly

In this short note we provide explicit examples in characteristic $p$ on certain smooth projective curves where for a given semistable vector bundle $\mathcal{E}$ the length of the Harder-Narasimhan filtration of $F^\ast \mathcal{E}$ is longer than $p$. This answers a question of Mehta and Pauly raised in arXiv:math/0607565.Comment: 3 pages; v1: Changes in exposition, added abstract in french, final versio

Dagger closure in regular rings containing a field

We prove that dagger closure is trivial in regular domains containing a field and that graded dagger closure is trivial in polynomial rings over a field. We also prove that Heitmann's full rank one closure coincides with tight closure in positive characteristic under some mild finiteness conditions. Furthermore, we prove that dagger closure is always contained in solid closure and that the forcing algebra for an element contained in dagger closure is parasolid.Comment: 12 pages, v2: added one corollary and two references, v3: Major simplification in proof of main thm due to the suggestion of a referee, minor changes in expositio

Dagger closure and solid closure in graded dimension two

We introduce a graded version of dagger closure and prove that it coincides with solid closure for homogeneous ideals in two dimensional $\mathbb{N}$-graded domains of finite type over a field.Comment: 26 page

On the behavior of FF-signatures, splitting primes, and test modules under finite covers

We give a comprehensive treatment on how certain fundamental objects in Cartier theory such a $F$-signatures, splitting primes, splitting ratios, and test modules behave under finite covers. We recover previously known results as particular instances. To this end, we expand on the notion of transposability along a section section of the relative canonical module as first introduced by K.~Schwede and K.~Tucker.Comment: 49 pages, comments are more than welcome. v2: we removed normality from our hypothesis, expanded on the concept of transposability, added a new transformation rule for adjoint-like ideals. There are substantial changes from v