Lyubeznik Numbers Research Articles

Equivariant 𝒟-modules on 2×2×𝓃 hypermatrices

We study D \mathcal {D} -modules and related invariants on the space of 2 × 2 × n 2\times 2\times n hypermatrices for n ≥ 3 n\geq 3 , which has finitely many orbits under the action of G = G L 2 ( C ) × G L 2 ( C ) × G L n ( C ) G=GL_2(\mathbb {C}) \times GL_2(\mathbb {C}) \times GL_n(\mathbb {C}) . We describe the category of coherent G G -equivariant D \mathcal {D} -modules as the category of representations of a quiver with relations. We classify the simple equivariant D \mathcal {D} -modules, determine their characteristic cycles and find special representations that appear in their G G -structures. We determine the explicit D \mathcal {D} -module structure of the local cohomology groups with supports given by orbit closures. As a consequence, we calculate the Lyubeznik numbers and intersection cohomology groups of the orbit closures. All but one of the orbit closures have rational singularities: we use local cohomology to prove that the one exception is neither normal nor Cohen–Macaulay. While our results display special behavior in the cases n = 3 n=3 and n = 4 n=4 , they are completely uniform for n ≥ 5 n\geq 5 .

Certain endomorphism rings of local cohomology modules and Lyubeznik numbers

The goal of this paper is twofold: on the one hand, motivated by questions raised by Schenzel, we explore situations where the Hartshorne–Lichtenbaum Vanishing theorem for local cohomology fails, leading us to simpler expressions of certain local cohomology modules. As application, we give new expressions of the endomorphism ring of these modules. On the other hand, building upon previous work by Àlvarez Montaner, we exhibit the shape of Lyubeznik tables of the so-called partially sequentially Cohen–Macaulay rings as introduced by Sbarra and Strazzanti.

Local cohomology on a subexceptional series of representations

We consider a series of four subexceptional representations coming from the third line of the Freudenthal-Tits magic square; using Bourbaki notation, these are fundamental representations $(G',X)$ corresponding to $(C_3, \omega_3),\, (A_5, \omega_3), \, (D_6, \omega_5)$ and $(E_7, \omega_6)$. In each of these four cases, the group $G=G'\times \mathbb{C}^*$ acts on $X$ with five orbits, and many invariants display a uniform behavior, e.g. dimension of orbits, their defining ideals and the character of their coordinate rings as $G$-modules. In this paper, we determine some more subtle invariants and analyze their uniformity within the series. We describe the category of $G$-equivariant coherent $\mathcal{D}_X$-modules as the category of representations of a quiver with relations. We construct explicitly the simple $G$-equivariant $\mathcal{D}_X$-modules and compute the characters of their underlying $G$-structures. We determine the local cohomology groups with supports given by orbit closures, determining their precise $\mathcal{D}_X$-module structure. As a consequence, we calculate the intersection cohomology groups and Lyubeznik numbers of the orbit closures. While our results for the cases $(A_5, \omega_3), \, (D_6, \omega_5)$ and $(E_7, \omega_6)$ are still completely uniform, the case $(C_3, \omega_3)$ displays a surprisingly different behavior. We give two explanations for this phenomenon: one topological, as the middle orbit of $(C_3, \omega_3)$ is not simply-connected; one geometric, as the closure of the orbit is not Gorenstein.

Lyubeznik numbers, 𝐹-modules and modules of generalized fractions

This paper presents an algorithm for calculation of the Lyubeznik numbers of a local ring which is a homomorphic image of a regular local ring R R of prime characteristic. The methods used employ Lyubeznik’s F F -modules over R R , particularly his F F -finite F F -modules, and also the modules of generalized fractions of Sharp and Zakeri [Mathematika 29 (1982), pp. 32–41]. It is shown that many modules of generalized fractions over R R have natural structures as F F -modules; these lead to F F -module structures on certain local cohomology modules over R R , which are exploited, in conjunction with F F -module structures on injective R R -modules that result from work of Huneke and Sharp [Trans. Amer. Math. Soc. 339 (1993), pp. 765–779], to compute Lyubeznik numbers. The resulting algorithm has been implemented in Macaulay2.

Lyubeznik numbers of almost complete intersection and linked ideals

Lyubeznik numbers of almost complete intersection and linked ideals

Connectedness of square-free Groebner deformations

Let I ⊆ S = K [ x 1 , … , x n ] I\subseteq S=K[x_1,\ldots ,x_n] be a homogeneous ideal equipped with a monomial order > > . We show that if in > ⁡ ( I ) \operatorname {in}_{>}(I) is a square-free monomial ideal, then S / I S/I and S / in > ⁡ ( I ) S/\operatorname {in}_>(I) have the same connectedness dimension. We also show that graphs related to connectedness of these quotient rings have the same number of components. We also provide consequences regarding Lyubeznik numbers. We obtain these results by furthering the study of connectedness modulo a parameter in a local ring.

Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings

We construct complex projective schemes with Lyubeznik numbers of their cones depending on the choices of projective embeddings. This answers a question of G. Lyubeznik in the characteristic 0 case. It contrasts with a theorem of W. Zhang in the positive characteristic case where the Frobenius endomorphism is used. Reducibility of schemes is essential in our argument. B. Wang recently constructed examples of irreducible projective schemes (which are not normal) from our examples of reducible ones. So the question is still open in the normal singular case.

Linear strands of edge ideals of multipartite uniform clutters

We construct the first linear strand of the minimal free resolutions of edge ideals of d -partite d -uniform clutters. We show that the first linear strand of such ideals are supported on relative simplicial complexes . In the case that the edge ideals of such clutters have linear resolutions, we give an explicit and surprisingly simple description of their minimal free resolutions, generalizing known resolutions for edge ideals of Ferrers graphs and hypergraphs and co-letterplace ideals. As an application, we show that the Lyubeznik numbers that appear on the last column of the Lyubeznik table of the cover ideal of such clutters are Betti numbers of certain simplicial complexes. Furthermore, we restate a characterization for edge ideals of d -partite d -uniform clutters which have linear resolutions based on the recent characterization of arithmetically Cohen-Macaulay sets of points in multiprojective spaces.

Lyubeznik and Betti numbers for homogeneous ideals

Let R=K[x1,…,xn] be the standard graded polynomial ring over a field K. We’ll study the Lyubeznik numbers of two homogeneous ideals of R whose initial ideals are linked for some monomial order over R. We’ll also investigate some relations between Lyubeznik and Betti numbers of homogeneous ideals.

Iterated local cohomology groups and Lyubeznik numbers for determinantal rings

We give an explicit recipe for determining iterated local cohomology groups with support in ideals of minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable 𝒟-modules. For nonsquare matrices these indecomposables are simple, but this is no longer true for square matrices where the relevant indecomposables arise from the pole order filtration associated with the determinant hypersurface. Specializing our results to a single iteration, we determine the Lyubeznik numbers for all generic determinantal rings, thus answering a question of Hochster.

PROPERTIES OF $T$-SPREAD PRINCIPAL BOREL IDEALS GENERATED IN DEGREE TWO

In this paper, we have studied the stability of $t$-spread principal Borel ideals in degree two. We have proved that $\Ass^\infty(I) =\Min(I)\cup \{\mathfrak{m}\}$ , where $I=B_t(u)\subset S$ is a $t$-spread Borel ideal generated in degree $2$ with $u=x_ix_n, t+1\leq i\leq n-t.$ Indeed, $I$ has the property that $\Ass(I^m)=\Ass(I)$ for all $m\geq 1$ and $i\leq t,$ in other words, $I$ is normally torsion free. Moreover, we have shown that $I$ is a set theoretic complete intersection if and only if $u=x_{n-t}x_n$. Also, we have derived some results on the vanishing of Lyubeznik numbers of these ideals.

Bass numbers of local cohomology of cover ideals of graphs

We develop splitting techniques to study the Lyubeznik numbers of cover ideals of graphs which allow us to describe them for large families of graphs including forests, cycles, wheels and cactus graphs. More generally, we are able to compute all the Bass numbers and the shape of the injective resolution of local cohomology modules by considering the connected components of all the induced subgraphs. Indeed, our method gives us a very simple criterion for the vanishing of these local cohomology modules in terms of the number of connected components of the induced subgraphs.

Lyubeznik numbers of irreducible projective varieties depend on the embedding

We construct irreducible complex projective varieties such that the Lyubeznik numbers of their affine cones depend on the choices of projective embeddings. The main ingredient is the recent work of Reichelt-Saito-Walther, where the Lyubeznik numbers are reinterpreted using perverse sheaves and reducible projective varieties with dependent Lyubeznik numbers are constructed.

EQUIVARIANT -MODULES ON ALTERNATING SENARY 3-TENSORS

We consider the space $X=\bigwedge ^{3}\mathbb{C}^{6}$ of alternating senary 3-tensors, equipped with the natural action of the group $\operatorname{GL}_{6}$ of invertible linear transformations of $\mathbb{C}^{6}$. We describe explicitly the category of $\operatorname{GL}_{6}$-equivariant coherent ${\mathcal{D}}_{X}$-modules as the category of representations of a quiver with relations, which has finite representation type. We give a construction of the six simple equivariant ${\mathcal{D}}_{X}$-modules and give formulas for the characters of their underlying $\operatorname{GL}_{6}$-structures. We describe the (iterated) local cohomology groups with supports given by orbit closures, determining, in particular, the Lyubeznik numbers associated to the orbit closures.

Lyubeznik numbers for Pfaffian rings

We study the structure of local cohomology with support in Pfaffian varieties as a module over the Weyl algebra DX of differential operators on the space of skew-symmetric matrices X=⋀2Cn. The simple composition factors of these modules are known by the work of Raicu-Weyman, and when n is odd, the general theory implies that the local cohomology modules are semi-simple. When n is even, we show that the local cohomology is a direct sum of indecomposable modules coming from the pole order filtration of the Pfaffian hypersurface. We then determine the Lyubeznik numbers for Pfaffian rings by computing local cohomology with support in the origin of the indecomposable summands referred to above.

Finiteness properties and numerical behavior of local cohomology

This manuscript discusses many problems connected with finiteness properties of local cohomology modules, e.g. the associated primes, the minimal primes, Bass numbers, as well as the behavior of injective dimension, Lyubeznik numbers, and content.This article is dedicated to Gennady Lyubeznik on the occasion of his 60th birthday, and celebrates his many contributions to commutative algebra.

Lyubeznik numbers and injective dimension in mixed characteristic

We investigate the Lyubeznik numbers, and the injective dimension of local cohomology modules, of finitely generated $\mathbb{Z}$-algebras. We prove that the mixed characteristic Lyubeznik numbers and the standard ones agree locally for almost all reductions to positive characteristic. Additionally, we address an open question of Lyubeznik that asks whether the injective dimension of a local cohomology module over a regular ring is bounded above by the dimension of its support. Although we show that the answer is affirmative for several families of $\mathbb{Z}$-algebras, we also exhibit an example where this bound fails to hold. This example settles Lyubeznik's question, and illustrates one way that the behavior of local cohomology modules of regular rings of equal characteristic and of mixed characteristic can differ.

Cohomological dimension, Lyubeznik numbers, and connectedness in mixed characteristic

We establish a “second vanishing theorem” for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings.

Local cohomology and Lyubeznik numbers of F-pure rings

In this article, we study certain local cohomology modules over F-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula for computing these invariants when the F-pure ring is standard graded and, by its means, we provide some new examples of Lyubeznik tables. We study associated primes of certain Ext-modules, showing that they are all compatible ideals. Finally, we focus on properties that Lyubeznik numbers detect over a globally F-split projective variety.

Connectedness and Lyubeznik Numbers

AbstractWe investigate the relationship between connectedness properties of spectra and the Lyubeznik numbers, numerical invariants defined via local cohomology. We prove that for complete equidimensional local rings, the Lyubeznik numbers characterize when connectedness dimension equals 1. More generally, these invariants determine a bound on connectedness dimension. Additionally, our methods imply that the Lyubeznik number $\lambda _{1,2}(A)$ of the local ring $A$ at the vertex of the affine cone over a projective variety is independent of the choice of its embedding into projective space.