Two interesting examples of $\mathcal{D}$-modules in characteristic $p>0$
We provide two examples of $\mathcal{D}$-modules in prime characteristic $p$ which answer two open problems in \cite{Lyubeznik} in the negative.
math.AC↗
arXiv subjects
Mordechai Katzman
Publications and source records attributed to Mordechai Katzman.
An upper bound on the number of F-jumping coefficients of a principal ideal
We prove a result relating the Jacobian ideal and the generalized test ideal associated to a principal ideal in $R=k[x_1,...,x_n]$ with $[k:k^p]<\infty$ or in $R=k[[x_1,...,x_n]]$ with an arbitrary field $k$ of characteristic $p>0$. As a consequence of this result, we establish an upper bound on the number of $F$-jumping coefficients of a principal ideal with an isolated singularity.
math.AC↗
A non-finitely generated algebra of Frobenius maps
The purpose of this paper is to answer a question raised by Gennady Lyubeznik and Karen Smith. This question involves the finite generation of the following non-commutative algebra. Let $S$ be any commutative algebra of prime characteristic $p$. For any $S$-module $M$ and all $e\geq 0$ we let $\mathcal{F}^e(M)$ denote the set of all additive functions $ϕ: M \to M$ with the property that $ϕ(s m)=s^{p^e} ϕ(m)$ for all $s\in S$ and $m\in M$. For all $e_1, e_2 \geq 0$, and $ϕ_1\in \mathcal{F}^{e_1}(M)$, $ϕ_2\in \mathcal{F}^{e_2}(M)$ the composition $ϕ_2 \circ ϕ_1$ is in $\mathcal{F}^{e_1+e_2}(M)$. Also, each $\mathcal{F}^{e}(M)$ is a module over $\mathcal{F}^{0}(M)=\Hom_{S}(M,M)$ via $ϕ_0 ϕ=ϕ_0 \circ ϕ$. We now define $\mathcal{F}(M)=\oplus_{e\geq 0} \mathcal{F}^e(M)$ and endow it with the structure of a $\Hom_{S}(M,M)$-algebra with multiplication given by composition. We construct an example of an Artinian module over a complete local ring $S$ for which $\mathcal{F}(M)$ is not a finitely generated $\Hom_{S}(M,M)$-algebra, thus giving a negative answer to the question raised by Lyubeznik and Smith.
math.AC↗
Frobenius maps on injective hulls and their applications to tight closure
This paper studies Frobenius maps on injective hulls of residue fields of complete local rings with a view toward providing constructive descriptions of objects originating from the theory of tight closure. Specifically, the paper describes algorithms for computing parameter test ideals, and tight closure of certain submodules of the injective hull of residue fields of a class of well-behaved rings which includes all quasi-Gorenstein complete local rings.
math.AC↗
On the discreteness and rationality of F-jumping coefficients
This paper studies the jumping coefficients of principal ideals of regular local rings. Recently M. Blickle, M. Mustata and K. Smith showed that, when $R$ is of essentially finite type over a field and $F$-finite, bounded intervals contain finitely many jumping coefficients and that those are rational. In a later paper they extended these results to principal ideals of $F$-finite complete regular local rings. The aim of this paper is to extend these results on the discreteness and rationality of jumping coefficients to principal ideals of arbitrary (i.e. not necessarily $F$-finite) excellent regular local rings containing fields of positive characteristic. Our proof uses a very different method: we do not use $D$-modules and instead we analyze the modules of nilpotents elements in the injective hull or $R$ under some non-standard Frobenius actions. This new method undoubtedly holds a potential for more applications.
math.AC↗
Parameter test ideals of Cohen Macaulay rings
We describe an algorithm for computing parameter-test-ideals in certain local Cohen-Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Rodney Sharp's notion of ``special ideals''. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the fact that for a power series ring $R$ of prime characteristic, for all nonzero $f\in R$, $1/f$ generates $R_f$ as a $D_R$-module.
math.AC↗
Frobenius test exponents for parameter ideals in generalized Cohen-Macaulay local rings
This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring $R$ of prime characteristic $p$. For a given ideal $\fa$ of $R$, there is a power $Q$ of $p$, depending on $\fa$, such that the $Q$-th Frobenius power of the Frobenius closure of $\fa$ is equal to the $Q$-th Frobenius power of $\fa$. The paper addresses the question as to whether there exists a {\em uniform} $Q_0$ which `works' in this context for all parameter ideals of $R$ simultaneously. In a recent paper, Katzman and Sharp proved that there does exists such a uniform $Q_0$ when $R$ is Cohen--Macaulay. The purpose of this paper is to show that such a uniform $Q_0$ exists when $R$ is a generalized Cohen--Macaulay local ring. A variety of concepts and techniques from commutative algebra are used, including unconditioned strong $d$-sequences, cohomological annihilators, modules of generalized fractions, and the Hartshorne--Speiser--Lyubeznik Theorem employed by Katzman and Sharp in the Cohen--Macaulay case.
math.AC↗
$F$-stable submodules of top local cohomology modules of Gorenstein rings
This paper applies G. Lyubeznik's notion of $F$-finite modules to describe in a very down-to-earth manner certain annihilator submodules of some top local cohomology modules over Gorenstein rings. As a consequence we obtain an explicit description of the test ideal of Gorenstein rings in terms of ideals in a regular ring.
math.AC↗
On ideals of minors of matrices with indeterminate entries
This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The main result in the first part of this paper is a theorem which gives sufficient conditions for the ideal of minors of a matrix to be prime. This theorem is general enough to include interesting examples, such as the ideal of maximal minors of catalecticant matrices and their generalisations discussed in the second part of the paper. The second aim of this paper is to settle a specific problem raised by David Eisenbud and Frank-Olaf Schreyer on the primary decomposition of an ideal of maximal minors. We solve this problem by applying the theorem above together with some ad-hoc techniques.
math.AC↗
Characteristic-independence of Betti numbers of graph ideals
In this paper we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first six Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables $n$ is at most 10, all Betti numbers are independent of the ground field. For $n=11$, there exists precisely 4 examples in which the Betti numbers depend on the ground field. This is equivalent to the statement that the homology of flag complexes with at most 10 vertices is torsion free and that there exists precisely 4 non-isomorphic flag complexes with 11 vertices whose homology has torsion. In each of the 4 examples mentioned above the 8th Betti numbers depend on the ground field and so we conclude that the highest Betti number which is always independent of the ground field is either 6 or 7; if the former is true then we show that there must exist a graph with 12 vertices whose 7th Betti number depends on the ground field.
math.AC↗
Counting monomials
This paper presents two enumeration techniques based on Hilbert functions. The paper illustrates these techniques by solving two chessboard problems.
math.AC↗
The Betti numbers of forests
This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs.
math.AC↗
Uniform Behaviour of the Frobenius closures of ideals generated by regular sequences
This paper is concerned with ideals in a commutative Noetherian ring $R$ of prime characteristic. The main purpose is to show that the Frobenius closures of certain ideals of $R$ generated by regular sequences exhibit a desirable type of `uniform' behaviour. The principal technical tool used is a result, proved by R. Hartshorne and R. Speiser in the case where $R$ is local and contains its residue field which is perfect, and subsequently extended to all local rings of prime characteristic by G. Lyubeznik, about a left module over the skew polynomial ring $R[x,f]$ (associated to $R$ and the Frobenius homomorphism $f$, in the indeterminate $x$) that is both $x$-torsion and Artinian over $R$.
math.AC↗
On the arithmetic of tight closure
We provide a negative answer to an old question in tight closure theory by showing that the containment x^3y^3 \in (x^4,y^4,z^4)^* in K[x,y,z]/(x^7+y^7-z^7) holds for infinitely many but not for almost all prime characteristics of the field K. This proves that tight closure exhibits a strong dependence on the arithmetic of the prime characteristic. The ideal (x,y,z) \subset K[x,y,z,u,v,w]/(x^7+y^7-z^7, ux^4+vy^4+wz^4+x^3y^3) has then the property that the cohomological dimension fluctuates arithmetically between 0 and 1.
math.AC↗
The Hilbert series of algebras of Veronese type
This paper gives a fairly explicit formula for the Hilbert series of algebras of Veronese type.
math.AC↗
The support of top graded local cohomology modules
Let $R_0$ be any domain, let $R=R_0[U_1, ..., U_s]/I$, where $U_1, ..., U_s$ are indeterminates of some positive degrees, and $I\subset R_0[U_1, ..., U_s]$ is a homogeneous ideal. The main theorem in this paper is states that all the associated primes of $H:=H^s_{R_+}(R)$ contain a certain non-zero ideal $c(I)$ of $R_0$ called the ``content'' of $I$. It follows that the support of $H$ is simply $V(\content(I)R + R_+)$ (Corollary 1.8) and, in particular, $H$ vanishes if and only if $c(I)$ is the unit ideal. These results raise the question of whether local cohomology modules have finitely many minimal associated primes-- this paper provides further evidence in favour of such a result. Finally, we give a very short proof of a weak version of the monomial conjecture based on these results.
math.AC↗
Finiteness of $\bold{\bigcup_e \Ass F^e(M)}$ and its connections to tight closure
The paper shows that if the set of associated primes of Frobenius powers of ideals or a closely related set of primes is finite then if tight closure does not commute with localisation one can find a counter-example where $R$ is complete local and we are localizing at a prime ideal $P \subset R$ with $\dim (R/P)=1$. If one assumes further that for any local ring $(R,m)$ of prime characteristic $p$ and every finitely generated $R$-module $\bar M$ the set $ \bigcup_e \Ass G^e (\bar M) $ has finitely many maximal elements and, in addition, for every $R$-module $\bar M$ there exists a positive integer $B>0$ such that $m^{qB}$ kills $\H_m^0(F^e(\bar M))$ (or $\H_m^0(G^e(\bar M))$) then it is shown tight closure commutes with localization. The author then produces an example of an ideal in an hypersurface whose union of sets associated primes of all its Frobenius powers form an infinite set.
math.AC↗
Bipartite graphs whose edge algebras are complete intersections
Let R be monomial sub-algebra of $k[x_1,...,x_N]$ generated by square free monomials of degree two. This paper addresses the following question: when is R a complete intersection? For such a k-algebra we can associate a graph G whose vertices are $x_1,...,x_N$ and whose edges are $\{(x_i, x_j) | x_i x_j \in R \}$. Conversely, for any graph G with vertices $\{x_1,...,x_N\}$ we define the {\it edge algebra associated with G} as the sub-algebra of $k[x_1,...,x_N]$ generated by the monomials ${x_i x_j | (x_i,x_j) \text{is an edge of} G}$. We denote this monomial algebra by k[G]. This paper describes all bipartite graphs whose edge algebras are complete intersections.
math.AC↗