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Niels Lauritzen

Publications and source records attributed to Niels Lauritzen.

16 recordsLinked to original sources
arXiv2026

Hochschild cohomology and lifts of endomorphisms

We study when algebra endomorphisms can be lifted to first-order flat lifts. To a first-order flat lift of an algebra and an endomorphism, we associate a canonical class in Hochschild cohomology with coefficients in a naturally twisted bimodule. The cohomology class vanishes exactly when the endomorphism admits a multiplicative lift. For an Azumaya algebra of constant rank over a formally smooth center, we prove that an endomorphism lifts if and only if the induced endomorphism of the center preserves the Poisson structure given by the lift of the algebra.

math.RA↗
arXiv2026

Lifts of endomorphisms of Weyl algebras modulo $p^2$

Let $\varphi$ denote a $k$-algebra endomorphism of the $n$-th Weyl algebra $A_n(k)$ over a perfect field $k$ of positive characteristic $p$. We prove that $\varphi$ can be lifted to an endomorphism of the Weyl algebra $A_n(W_2(k))$ over the Witt vectors $W_2(k)$ of length two over $k$ if and only if $\varphi$ induces a Poisson morphism of the center of $A_n(k)$. Furthermore, we improve a result of Tsuchimoto, which enables us to conclude that these equivalent statements hold at least when ${\rm deg}(\varphi) < p$. In particular, we conclude that $\varphi$ is injective if ${\rm deg}(\varphi) < p$.

math.RA↗
arXiv2025

A note on a paper by Hashemi and Kapur

Recently Hashemi and Kapur published an algorithm [1] for Groebner basis conversion by truncating polynomials according to a source and a target monomial order. Here we present a counterexample to this algorithm.

cs.SC↗
arXiv2024

Finitely generated bimodules over Weyl algebras

Let $A$ be the $n$-th Weyl algebra over a field of characteristic zero, and $φ:A\rightarrow A$ an endomorphism with $S = φ(A)$. We prove that if $A$ is finitely generated as a left or right $S$-module, then $S = A$. The proof involves reduction to large positive characteristics. By holonomicity, $A$ is always finitely generated as an $S$-bimodule. Moreover, if this bimodule property could be transferred into a similar property in large positive characteristics, then we could again conclude that $A=S$. The latter would imply the Dixmier Conjecture.

math.AG↗
arXiv2019

The graph of a Weyl algebra endomorphism

Endomorphisms of Weyl algebras are studied using bimodules. Initially, for a Weyl algebra over a field of characteristic zero, Bernstein's inequality implies that holonomic bimodules finitely generated from the right or left form a monoidal category. The most important bimodule in this paper is the graph of an endomorphism. We prove that the graph of an endomorphism of a Weyl algebra over a field of characteristic zero is a simple bimodule. The simplicity of the tensor product of the dual graph and the graph is equivalent to the Dixmier conjecture. It is also shown how the graph construction leads to a non-commutative Groebner basis algorithm for detecting invertibility of an endomorphism for Weyl algebras and computing the inverse over arbitrary fields.

math.RA↗
arXiv2015

Two properties of endomorphisms of Weyl algebras

We show that endomorphisms of Weyl algebras over fields of characteristic zero are flat and that birational endomorphisms are automorphisms by reducing to positive characteristic. We also give examples showing that endomorphisms of Weyl algebras are not in general flat over fields of positive characteristic.

math.RA↗
arXiv2012

Exact Algorithms for Solving Stochastic Games

Shapley's discounted stochastic games, Everett's recursive games and Gillette's undiscounted stochastic games are classical models of game theory describing two-player zero-sum games of potentially infinite duration. We describe algorithms for exactly solving these games.

cs.GT↗
arXiv2010

Maximal compatible splitting and diagonals of Kempf varieties

Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal multiplicity vanishing in Frobenius splitting. In this paper we define the algebraic analogue of this concept and construct a Frobenius splitting vanishing with maximal multiplicity on the diagonal of the full flag variety. Our splitting induces a diagonal Frobenius splitting of maximal multiplicity for a special class of smooth Schubert varieties first considered by Kempf. Consequences are Frobenius splitting of tangent bundles, of blow-ups along the diagonal in flag varieties along with the LMP and Wahl conjectures in positive characteristic for the special linear group.

math.AG↗
arXiv2007

Maximal lattice free bodies, test sets and the Frobenius problem

Maximal lattice free bodies are maximal polytopes without interior integral points. Scarf initiated the study of maximal lattice free bodies relative to the facet normals in a fixed matrix. In this paper we give an efficient algorithm for computing the maximal lattice free bodies of an integral matrix A. An important ingredient is a test set for a certain integer program associated with A. This test set may be computed using algebraic methods. As an application we generalize the Scarf-Shallcross algorithm for the three-dimensional Frobenius problem to arbitrary dimension. In this context our method is inspired by the novel algorithm by Einstein, Lichtblau, Strzebonski and Wagon and the Groebner basis approach by Roune.

math.CO↗
arXiv2007

Global F-regularity of Schubert varieties with applications to D-modules

We prove that Schubert varieties are globally F-regular in the sense of Karen Smith. We apply this result to the category of equivariant and holonomic D-modules on flag varieties in positive characteristic. Here recent results of Blickle are shown to imply that the simple D-modules coincide with local cohomology sheaves with support in Schubert varieties. Using a local Grothendieck-Cousin complex we prove that the decomposition of local cohomology sheaves with support in Schubert cells is multiplicity free.

math.AG↗
arXiv2005

Truncated Groebner fans and lattice ideals

We outline a generalization of the Groebner fan of a homogeneous ideal with maximal cells parametrizing truncated Groebner bases. This "truncated" Groebner fan is usually much smaller than the full Groebner fan and offers the natural framework for conversion between truncated Groebner bases. The generic Groebner walk generalizes naturally to this setting by using the Buchberger algorithm with truncation on facets. We specialize to the setting of lattice ideals. Here facets along the generic walk are given by unique (facet) binomials. This along with the representation of binomials as integer vectors give an especially simple version of the generic Groebner walk. Computational experience with the special Aardal-Lenstra integer programming knapsack problems is reported.

math.AC↗
arXiv2002

Local cohomology and D-affinity in positive characteristic

Comparing vanishing of local cohomology in zero and positive characteristic, we give an example of a D-module on a Grassmann variety in positive characteristic with non-vanishing first cohomology group. This is a counterexample to D-affinity and the Beilinson-Bernstein equivalence for flag manifolds in positive characteristic.

math.AG↗
arXiv2001

Twisted Verma modules

Using principal series Harish-Chandra modules, local cohomology with support in Schubert cells and twisting functors we construct certain modules parametrized by the Weyl group and a highest weight in the subcategory O of the category of representations of a complex semisimple Lie algebra. These are in a sense modules between a Verma module and its dual. We prove that the three different approaches lead to the same modules. Moreover, we demonstrate that they possess natural Jantzen type filtrations with corresponding sum formulae.

math.QA↗
arXiv1998

Frobenius splitting of cotangent bundles of flag varieties and geometry of nilpotent cones

We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bundle, which in turn implies good filtrations of cohomology of induced representations.

math.AG↗
arXiv1995

Embeddings of homogeneous spaces in prime characteristics

Let $G$ be a reductive linear algebraic group. The simplest example of a projective homogeneous $G$-variety in characteristic $p$, not isomorphic to a flag variety, is the divisor $x_0 y_0^p+x_1 y_1^p+x_2 y_2^p=0$ in $P^2\times P^2$, which is $SL_3$ modulo a non-reduced stabilizer containing the upper triangular matrices. In this paper embeddings of projective homogeneous spaces viewed as $G/H$, where $H$ is any subgroup scheme containing a Borel subgroup, are studied. We prove that $G/H$ can be identified with the orbit of the highest weight line in the projective space over the simple $G$-representation $L(λ)$ of a certain highest weight $λ$. This leads to some strange embeddings especially in characteristic $2$, where we give an example in the $C_4$-case lying on the boundary of Hartshorne's conjecture on complete intersections. Finally we prove that ample line bundles on $G/H$ are very ample. This gives a counterexample to Kodaira type vanishing with a very ample line bundle, answering an old question of Raynaud.

alg-geom↗