arXiv subjects

Rolf Källström

Publications and source records attributed to Rolf Källström.

11 recordsLinked to original sources

arXiv2026

Purity of branch and critical locus

To a dominant morphism $X/S \to Y/S$ of Nœtherian integral $S$-schemes one has the inclusion $C_{X/Y}\subset B_{X/Y}$ of the critical locus in the branch locus of $X/Y$. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of relative differentials $Ω_{X/Y}$, $Ω_{X/S}$, and $Ω_{Y/S}$ that imply bounds on the codimensions of $ C_{X/Y}$ and $ B_{X/Y}$. These bounds generalise to a wider class of morphisms the classical purity results for finite morphisms by Zariski-Nagata-Auslander, and Faltings and Grothendieck, and van der Waerden's purity for birational morphisms.

math.AG↗

arXiv2018

D-modules and finite maps

We study the preservation of semisimplicity for holonomic D-modules with respect to the direct and inverse image of mainly finite maps $π: X \to Y$ of smooth varieties. A natural filtration of the direct image $π_+({\mathcal O}_X)$ is defined by the vanishing of local cohomology along a natural stratification of $π$. The notions are exemplified with the invariant map $X\to X^G$, where $G$ is a complex reflection group. Simply connected varieties are treated algebraically by considering connections instead of fundamental groups. For example, a "Grothendieck-Lefschetz" theorem for connections is proven and also a generalized version of the assertion that rationally connected varieties be simply connected, entirely by algebraic means, using the idea of a "differential covering".

math.AG↗

arXiv2018

The Zariski-Lipman conjecture for complete intersections

The tangential ramification locus $B_{X/Y}^t\subset B_{X/Y}$ is the subset of points in the ramification locus where the sheaf of relative vector fields $T_{X/Y}$ fails to be locally free. It was conjectured by Zariski and Lipman that if $V/k$ is a variety over a field $k$ of characteristic 0 and $B^t_{V/k}= \emptyset$, then $V/k$ is smooth (=regular). We prove this conjecture when $V/k$ is a locally complete intersection. We prove also that $B_{V/k}^t= \emptyset$ implies $\operatorname {codim}_V B_{V/k}\leq 1 $ in positive characteristic, if $V/k $ is the fibre of a flat morphism satisfying generic smoothness.

math.AG↗

arXiv2018

Stratifications of schemes using tangent vector fields

Let $X/k$ be a noetherian scheme over a field $k$ of characteristic 0, such that the residue field at its closed points are algebraic extensions of $k$. Let ${\mathfrak g}_{X/k}\subset T_{X/k}$ be an ${\mathcal O}_{X}$-submodule of the tangent sheaf which is closed under the Lie bracket. We construct a stratification of the subset of closed points in $X$ by locally closed subsets that are preserved by ${\mathfrak g}_{X/k}$, on which ${\mathfrak g}_{X/k}$ acts transitively.

math.AG↗

arXiv2016

Decomposition of modules over invariant differential operators

Let $G$ be a finite subgroup of the linear group of a finite-dimensional complex vector $V$, $B={\operatorname S}(V)$ be the symmetric algebra, ${\mathcal D}=\mathcal D^G_B$ the ring of $G$-invariant differential operators, and ${\mathcal D}^-$ its subring of negative degree operators. We prove that $M\mapsto M^{ann}= {\operatorname Ann}_{\mathcal D^-}(M)$ defines an isomorphism between the category of ${\mathcal D}$-submodules of $B$ and a category of modules formed as lowest weight spaces. This is applied to a construction of simple ${\mathcal D}$-submodules of $B$ when $G$ is a generalized symmetric group, to show that $B^{ann}$ is a so-called Gelfand model. Using differential algebra and lowest weight methods we also prove branching rules, entailing the main results in the representation theory of the symmetric group, such as a differential construction of the Young basis.

math.RT↗

arXiv2015

Hilbert series of modules over Lie algebroids

We consider modules $M$ over Lie algebroids ${\mathfrak g}_A$ which are of finite type over a local noetherian ring $A$. Using ideals $J\subset A$ such that ${\mathfrak g}_A \cdot J\subset J $ and the length $\ell_{{\mathfrak g}_A}(M/JM)< \infty$ we can define in a natural way the Hilbert series of $M$ with respect to the defining ideal $J$. This notion is in particular studied for modules over the Lie algebroid of $k$-linear derivations ${\mathfrak g}_A=T_{A/k}(I)$ that preserve an ideal $I\subset A$, for example when $A={\mathcal O}_n$, the ring of convergent power series. Hilbert series over Stanley-Reisner rings are also considered.

math.AC↗

arXiv2012

Two extensions of Hilbert's finiteness theorem

Let $S^{\cdot}$ be a noetherian graded algebra over a commutative $k$-algebra $A$, where $k$ is a commutative ring, and assume it is a module over a Lie algebroid ${\mathfrak g}_{A/k}$. If $S^\cdot$ is semi-simple over ${\mathfrak g}_{A/k}$ we prove that its ring of invariants $\bar S^{\cdot}$ is notherian. When ${\mathfrak g}_{A/k}$ is a solvable Lie algebra over $A$ we construct noetherian subalgebras of $S^\cdot$ from subsets of characters of ${\mathfrak g}_{A/k}$. We give similar results for noetherian modules over the pair $(S^{\cdot}, {\mathfrak g}_{A/k})$.

math.RA↗

arXiv2012

D-modules with finite support are semi-simple

Let $(R, \mf, k_R)$ be regular local $k$-algebra satisfying the weak Jacobian criterion, such that $k_R/k$ is an algebraic field extension. Let $D_R$ be the ring of $k$-linear differential operators of $R$. We give an explicit decomposition of the $D_R$-module $D_R/D_R \mf_R^{n+1}$ as a direct sum of simple modules, all isomorphic to $D_R/D_R \mf$, where certain "Pochhammer" differential operators are used to describe generators of the simple components.

math.AC↗

arXiv2011

Purity of branch, critical and discriminant locus

To a dominant morphism $π:X/S \to Y/S$ of Nœtherian integral $S$-schemes one has the inclusion $C_π\subset B_π$ of the critical locus in the branch locus of $B_π$. Conditions on the relative differentials $Ω_{X/Y}$, $Ω_{X/S}$, and $Ω_{Y/S}$ are stated that imply bounds on the codimensions of $ C_π$ and $ B_π$. We also give bounds on the codimension of the discriminant locus, proving and generalising a conjecture of I. Dolgachev.

math.AG↗

arXiv2007

Liftable derivations for generically separably algebraic morphisms of schemes

We consider dominant, generically algebraic, and tamely ramified (if the characteristic is positive) morphisms $π: X/S \to Y/S$, where Y,S are Noetherian and integral and X is a Krull scheme (e.g. normal Noetherian), and study the sheaf of tangent vector fields on Y that lift to tangent vector fields on X. We give an easily computable description of these vector fields using valuations along the critical locus. We generalise to positive characteristic Seidenberg's theorem on lifting tangent vector fields to the normalisation. If $π$ is a blow-up of a coherent ideal I we show that tangent vector fields that preserve the Ratliff-Rush hull of I are liftable, and that all liftable tangent vector fields preserve the integral closure of I.

math.AG↗

arXiv2004

Geometric interplay between function subspaces and their rings of differential operators

We study, in the setting of algebraic varieties, finite-dimensional spaces of functions V that are invariant under a ring D^V of differential operators, and give conditions under which D^V acts irreducibly. We show how this problem, originally formulated in physics (Kamran-Milson-Olver), is related to the study of principal parts bundles and Weierstrass points (Laksov-Thorup), including a detailed study of Taylor expansions. Under some conditions it is possible to obtain V and D^V as global sections of a line bundle and its ring of differential operators. We show that several of the published examples of D^V are of this type, and that there are many more -- in particular arising from toric varieties.

math.AG↗