Let $A$ be a $d$-dimensional local ring containing a field. We will prove that the highest Lyubeznik number $λ_{d,d}(A)$ (defined in \cite{l1}) is equal to the number of connected components of the Hochster-Huneke graph (defined in \cite{hh}) associated to $B$, where $B=\hat{\hat{A}^{sh}}$ is the completion of the strict Henselization of the completion of $A$. This was proven by Lyubeznik in characteristic $p>0$. Our statement and proof are characteristic-free.
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.
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 non-square 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.
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.
We 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 one. More generally, these invariants determine a bound on connectedness dimension. Additionally, our methods imply that the Lyubeznik number with indices (1,2) of the local ring at the vertex of the affine cone over a projective variety is independent of the choice of its embedding into projective space.
We prove that sequentially Cohen-Macaulay rings in positive characteristic, as well as sequentially Cohen-Macaulay Stanley-Reisner rings in any characteristic, have trivial Lyubeznik table. Some other configurations of Lyubeznik tables are also provided depending on the deficiency modules of the ring.
The Lyubeznik numbers are invariants of a local ring containing a field that capture ring-theoretic properties, but also have numerous connections to geometry and topology. We discuss basic properties of these integer-valued invariants, as well as describe some significant results and recent developments (including certain generalizations) in the area.
We study the structure of local cohomology with support in Pfaffian varieties as a module over the Weyl algebra D_X of differential operators on the space X of n x n complex skew-symmetric matrices. The simple composition factors of these modules are known by the work of Raicu-Weyman in 2016, 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 Pfaff…
In this work we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a $\mathbb{Z}$-graded ideal $I\subseteq R=\Bbbk[x_1, \ldots, x_n]$. We also prove that these invariants satisfy some properties analogous to those of Lyubeznik numbers of local rings. In particular, they satisfy a consecutiveness property that we prove first for Lyubeznik numbers. For the case of squarefree monomial ideals we get more insight on the relation between Lyubeznik numbers and the linear strands of their associated Alexander dual ideals. Finally, we prove that Lyubeznik…
We exhibit a global bound for the Lyubeznik numbers of a ring of prime characteristic. In addition, we show that for a monomial ideal, the Lyubeznik numbers of the quotient rings of its radical and its polarization are the same. Furthermore, we present examples that show striking behavior of the Lyubeznik numbers under localization. We also show related results for generalizations of the Lyubeznik numbers.
Let $R\rightarrow S$ be an arbitrary ring extension of Noetherian rings. In this article we study the behaviour of Zariski closedness of the support of Lyubeznik functors $\mathrm{T}$, when the ring extension $R\rightarrow S$ is namely `flat', `faithfully flat', `pure' and lastly `cyclically pure'. We show that the Zariski closedness of the support comes down from extended ring to the base ring for faithfully flat, pure and finally for cyclically pure ring extensions. Lastly, we focus on a special case of pure extension i.e. when $R$ is a direct summand of $S$ and we compare the sets $\mathrm{…
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 exa…
We prove that the top mixed characteristic Lyubeznik number of a ring $S$ that is a quotient of a complete unramified regular local ring of mixed characteristc with algebraically closed residue field is $1$ provided that depth $S \geq 2$ and dim $S \geq 3$ using a second vanishing theorem in mixed characteristic proved in arXiv:1609.05846
In this paper, we determine completely the Lyubeznik numbers $λ_{i,j}(A)$ of the local ring $A$ at the vertex of the affine cone over a nonsingular projective variety $V$, where $V$ is defined over a field of characteristic zero, in terms of the dimensions of the algebraic de Rham cohomology spaces of $V$. In particular, we prove that these numbers are intrinsic numerical invariants of $V$, even though a priori their definition depends on an embedding into projective space. This provides supporting evidence for a positive answer to the question of embedding-independence for arbitrary varieties…
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 $R$ of prime characteristic. The methods used employ Lyubeznik's $F$-modules over $R$, particularly his $F$-finite $F$-modules, and also the modules of generalized fractions of Sharp and Zakeri. It is shown that many modules of generalized fractions over $R$ have natural structures as $F$-modules; these lead to $F$-module structures on certain local cohomology modules over $R$, which are exploited, in conjunction with $F$-module structures on injective …
Let R = K[x_1; : : : ; x_n] be a polynomial ring over a field K, and I := I_C_n be an edge ideal of n-cycle graph C_n. In the present paper, we compute the last column of the Lyubeznik table of R/I.
Let $(R,\fm)$ be a local ring and $I$ an ideal. The aim of the present paper is twofold. At first we continue the investigation to compare $\fgrade(I,R)$ with $\depth R/I$ and further we derive some results on the vanishing of Lyubeznik numbers.
We study the behaviour of the finiteness of the set of associated primes of local cohomology modules, more generally of Lyubeznik functors, under various ring extensions. At first, we review the results for flat and faithfully flat extensions and we present new applications of them. Then, we focus how the finiteness property of the set of associated primes of local cohomology modules and Lyubeznik functors is transferred from extended ring to the base ring of pure and cyclically pure ring extensions. We show that finiteness property can be transferred from a ring to its pure local subring and …
The Frobenius depth denoted by F-depth defined by Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a different way, for rings of positive characteristic. The first aim of the present paper is to compare the F-depth with formal grade and reprove some results of Lyubeznik using formal local cohomology. Then the endomorphism rings of local cohomology modules will be considered. As an application, we reprove the results due to Huneke-Koh in positive characteristic and Lyubeznik in characteristic zero on the annihilators of local cohomology modules.
School of Mathematics, Institute for Advanced Study（高等研究院数学学院）；Department of Mathematics, Princeton University（普林斯顿大学数学系）；Department of Mathematics, Purdue University（普渡大学数学系）
The Frobenius depth denoted by $\Fdepth$ defined by Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a different way, for rings of positive characteristic. The aim of the present paper is to compare the $\Fdepth$ with formal grade, and $\depth$ to shed more light on the notion of Frobenius depth from a different point of view.
The aim of this paper is to extend the main result of C. Huneke and G. Lyubeznik in [Adv. Math. 210 (2007), 498--504] to the class of rings that are images of Cohen-Macaulay local rings. Namely, let $R$ be a local Noetherian domain of positive characteristic that is an image of a Cohen-Macaulay local ring. We prove that all local cohomology of $R$ (below the dimension) maps to zero in a finite extension of the ring. As a direct consequence we obtain that the absolute integral closure of $R$ is a big Cohen-Macaulay algebra. Since every excellent local ring is an image of a Cohen-Macaulay local …
Lyubeznik's conjecture, (\cite{Ly1}, Remark 3.7) asserts the finiteness of the set ssociated primes of local cohomology modules for regular rings. But, in the case of ramified regular local ring, it is open. Recently, in Theorem 1.2 of \cite{Nu1}, it is proved that in any Noetherian regular local ring $S$ and for a fixed ideal $J\subset S$, associated primes of local cohomology $H^i_J(S)$ for $i\geq 0$ is finite, if it does not contain $p$. In this paper, we use this result to construct examples of local cohomology modules for ramified regular local ring so that they have finitely many associa…
One classical topic in the study of local cohomology is whether the non-vanishing of a specific local cohomology module is equivalent to the vanishing of its annihilator; this has been studied by several authors, including Huneke, Koh, Lyubeznik and Lynch. Motivated by questions raised by Lynch and Zhang, the goal of this paper is to provide some new results about this topic, which provide some partial positive answers to these questions. The main technical tool we exploit is the structure of local cohomology as module over rings of differential operators.
The Lyubeznik size of a monomial ideal $I$ of a polynomial ring $S$ is a lower bound for the Stanley depth of $I$ decreased by $1$. A proof given by Herzog-Popescu-Vladoiu had a gap which is solved here.
Let $R=k[x,y]$ be a polynomial ring over a field $k$ of prime characteristic $p$ and let $E$ denote the injective hull of $k$ (which is isomorphic to $H^2_{(x,y)}(R)$). We prove that $E$ is not an injective object in the category of graded $F$-modules over $R$. This answers in the negative a question raised by Lyubeznik-Singh-Walther.
Let $I\subseteq S=K[x_1,\ldots,x_n]$ be a homogeneous ideal equipped with a monomial order $<$. We show that if $\operatorname{in}_<(I)$ is a square-free monomial ideal, then $S/I$ and $S/\operatorname{in}_<(I)$ 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.
In this paper we show that the Frobenius test exponent for parameter ideals of a local ring of prime characteristic is always bigger than or equal to its Hartshorne-Speiser-Lyubeznik number. Our argument is based on an isomorphism of Nagel and Schenzel on local cohomology that we will provide an elementary proof.
Let $A$ be a complete local ring with a coefficient field $k$ of characteristic zero, and let $Y$ be its spectrum. The de Rham homology and cohomology of $Y$ have been defined by R. Hartshorne using a choice of surjection $R \rightarrow A$ where $R$ is a complete regular local $k$-algebra: the resulting objects are independent of the chosen surjection. We prove that the Hodge-de Rham spectral sequences abutting to the de Rham homology and cohomology of $Y$, beginning with their $E_2$-terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of fini…
We establish a criterion for the strong $F$-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least $2$, containing a perfect field of prime characteristic $p$. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an $n\times (n-1)$ matrix $X$ of indeterminates. For $p\geq 5$, these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring defined by the maximal minors of $X$ is strongly $F$-regular.
Let $(A,\mathfrak{m})$ be a complete equicharacteristic Noetherian domain of dimension $d + 1 \geq 2$. Assume $k = A/\mathfrak{m}$ has characteristic zero and that $A$ is not a regular local ring. Let $Sing(A)$ the singular locus of $A$ be defined by an ideal $J$ in $A$. Note $J \neq 0$. Let $ f \in J$ with $f \neq 0$. Set $R = A_f$. Then $R$ is a regular domain of dimension $d$. We show $R$ contains naturally a field $\ell \cong k((X))$. Let $\mathfrak{g}$ be the set of $\ell$-linear derivations of $R$ and let $D(R)$ be the subring of $Hom_\ell(R,R)$ generated by $\mathfrak{g}$ and the multip…
Let $R$ be a standard graded polynomial ring that is finitely generated over a field, and let $I$ be a homogenous prime ideal of $R$. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of $R/I^t$, as $t$ grows arbitrarily large. Such rings are known as thickenings of $R/I$. We consider $R = \mathbb{F}[X]$ where $\mathbb{F}$ is a field of characteristic 0, $X$ is a $2 \times m$ matrix, and $I$ is the ideal generated by size two minors. We give concrete constructions for the local cohomology modules of thickenings of $R/I$. Bizarrely, these local cohomology modules can be …
Let $R=k[x_1,\dots,x_n]$ be a ring of polynomials over a field $k$ of characteristic $p>0$. There is an algorithm due to Lyubeznik for deciding the vanishing of local cohomology modules $H^i_I(R)$ where $I\subset R$ is an ideal. This algorithm has not been implemented because its complexity grows very rapidly with the growth of $p$ which makes it impractical. In this paper we produce a modification of this algorithm that consumes a modest amount of memory.
Broadening existing results in the literature to much wider classes of rings, we prove among other things: 1. Reduced quotients of excellent regular rings of characteristic $p$ admit big test elements, 2. The set of F-jumping numbers of a principal ideal in a locally excellent regular ring is a discrete subset of $\mathbb Q$, and 3. If $R$ is a quotient of a locally excellent regular ring of prime characteristic, then there is a uniform upper bound on the Hartshorne-Speiser-Lyubeznik numbers of the injective hulls of the residue fields of $R$. To do so, we develop the parallel theories of Ohm-…
In this article, we study the behaviour of smooth algebra $R$ over local Noetherian local ring $A$. At first, we observe that for every $f\in R$, $R_f$ has finite length in the category of $D(R,A)$-module if dimension of $A$ is zero. This extends the result of Theorem 2 of \cite{Ly3}. We use this fact to generalize the result of Theorem 4.1 of \cite{BBLSZ}, from the finiteness of the set of associated primes of local cohomology module to that of Lyubeznik functor. Finally, we introduce the definition of $Σ$-finite $D$-modulue for smooth algebra and we extend the result of Theorem 1.3 of \cite{…
