0000000000144564
AUTHOR
Manuel Blickle
showing 7 related works from this author
Functorial Test Modules
2016
In this article we introduce a slight modification of the definition of test modules which is an additive functor $\tau$ on the category of coherent Cartier modules. We show that in many situations this modification agrees with the usual definition of test modules. Furthermore, we show that for a smooth morphism $f \colon X \to Y$ of $F$-finite schemes one has a natural isomorphism $f^! \circ \tau \cong \tau \circ f^!$. If $f$ is quasi-finite and of finite type we construct a natural transformation $\tau \circ f_* \to f_* \circ \tau$.
F-signature of pairs and the asymptotic behavior of Frobenius splittings
2012
We generalize $F$-signature to pairs $(R,D)$ where $D$ is a Cartier subalgebra on $R$ as defined by the first two authors. In particular, we show the existence and positivity of the $F$-signature for any strongly $F$-regular pair. In one application, we answer an open question of I. Aberbach and F. Enescu by showing that the $F$-splitting ratio of an arbitrary $F$-pure local ring is strictly positive. Furthermore, we derive effective methods for computing the $F$-signature and the $F$-splitting ratio in the spirit of the work of R. Fedder.
Stabilization of the cohomology of thickenings
2016
For a local complete intersection subvariety $X=V({\mathcal I})$ in ${\mathbb P}^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $X$, the cohomology of vector bundles on the formal completion of ${\mathbb P}^n$ along $X$ can be effectively computed as the cohomology on any sufficiently high thickening $X_t=V({\mathcal I^t})$; the main ingredient here is a positivity result for the normal bundle of $X$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings $X_t$ in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on $X$, and the main new…
F-singularities via alterations
2011
For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair $(X,\Delta)$. Our description is in terms of regular alterations over $X$, and one consequence of it is a common characterization of rational singularities (in characteristic zero) and F-rational singularities (in characteristic $p$) by the surjectivity of the trace map $\pi_* \omega_Y \to \omega_X$ for every such alteration $\pi \: Y \to X$. Furthermore, building on work of B. Bhatt, we establish up-to-finite-map versions of Grauert-Riemenscheneider and Nadel/Kawamata-V…
Test ideals via algebras of 𝑝^{-𝑒}-linear maps
2012
Building on previous work of Schwede, Böckle, and the author, we study test ideals by viewing them as minimal objects in a certain class of modules, called F F -pure modules, over algebras of p − e p^{-e} -linear operators. We develop the basics of a theory of F F -pure modules and show an important structural result, namely that F F -pure modules have finite length. This result is then linked to the existence of test ideals and leads to a simplified and generalized treatment, also allowing us to define test ideals in non-reduced settings. Combining our approach with an observation of Anderson on the contracting property of p − e p^{-e} -linear operators yields an elementary approach to tes…
p −1-Linear Maps in Algebra and Geometry
2012
At least since Habousch’s proof of Kempf’s vanishing theorem, Frobenius splitting techniques have played a crucial role in geometric representation theory and algebraic geometry over a field of positive characteristic. In this article we survey some recent developments which grew out of the confluence of Frobenius splitting techniques and tight closure theory and which provide a framework for higher dimension geometry in positive characteristic. We focus on local properties, i.e. singularities, test ideals, and local cohomology on the one hand and global geometric applicatioms to vanishing theorems and lifting of sections on the other.
F-signature of pairs: Continuity, p-fractals and minimal log discrepancies
2011
This paper contains a number of observations on the {$F$-signature} of triples $(R,\Delta,\ba^t)$ introduced in our previous joint work. We first show that the $F$-signature $s(R,\Delta,\ba^t)$ is continuous as a function of $t$, and for principal ideals $\ba$ even convex. We then further deduce, for fixed $t$, that the $F$-signature is lower semi-continuous as a function on $\Spec R$ when $R$ is regular and $\ba$ is principal. We also point out the close relationship of the signature function in this setting to the works of Monsky and Teixeira on Hilbert-Kunz multiplicity and $p$-fractals. Finally, we conclude by showing that the minimal log discrepancy of an arbitrary triple $(R,\Delta,\b…