We study the behavior of various properties of commutative Noetherian rings under Segre products, with a special focus on properties in positive prime characteristic defined using the Frobenius endomorphism. Specifically, we construct normal graded rings of finite Frobenius representation type that are not Cohen-Macaulay.
In this paper we give a nice formula for the Castelnuovo-Mumford regularity of the Segre product of modules, under some suitable hypotheses. This extends recent results of David A. Cox, and Evgeny Materov (2009).
In a recent preprint, Ilse Fischer and Martina Kubitzke, proved the bilinearity of the Segre transform under some restricted hypothesis, motivated by their results we show in this paper the bilinearity of the Segre transform in general. We apply these results to compute the postulation number of a series. Our second application is motivated by the paper of David A. Cox, and Evgeny Materov (2009), where is computed the Castelnuovo-Mumford regularity of the Segre Veronese embedding, we can extend partially their result and compute the Castelnuovo-Mumford regularity of the Segre product of Cohen-…
This paper studies the regularity of certain coherent sheaves that arise naturally from Segre-Veronese embeddings of a product of projective spaces. We give an explicit formula for the regularity of these sheaves and show that their regularity is subadditive. We then apply our results to study the Tate resolutions of these sheaves.
Let $R=k[x_1,\dots,x_n]/I$ be a standard graded $k$-algebra where $k$ is a field of prime characteristic and let $J$ be a homogeneous ideal in $R$. Denote $(x_1,\dots,x_n)$ by $\mathfrak{m}$. We prove that there is a constant $C$ (independent of $e$) such that the regularity of $H^s_{\mathfrak{m}}(R/J^{[p^e]})$ is bounded above by $Cp^e$ for all $e\geq 1$ and all integers $s$ such that $s+1$ is at least the dimension of the locus where $R/J$ doesn't have finite projective dimension.
The behavior of the Frobenius map is investigated for valuation rings of prime characteristic. We show that valuation rings are always F-pure. We introduce a generalization of the notion of strong F-regularity, which we call F-pure regularity, and show that a valuation ring is F-pure regular if and only if it is Noetherian. For valuations on function fields, we show that the Frobenius map is finite if and only if the valuation is divisorial; in this case the valuation ring is Frobenius split. For Noetherian valuation rings in function fields, we show that the valuation ring is Frobenius split …
Let $R$ be a standard graded algebra over an $F$-finite field of characteristic $p > 0$. Let $ϕ:R\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let ${}^ϕ\!M$ be the abelian group $M$ with the $R$-module structure induced by the Frobenius endomorphism. The $R$-module ${}^ϕ\!M$ has a natural grading given by $\text{deg} x=j$ if $x\in M_{jp+i}$ for some $0\le i \le p-1$. In this paper, we prove that $R$ is Koszul if and only if there exists a non-zero finitely generated graded $R$-module $M$ such that $\text{reg}_R\,{}^ϕ\!M <\infty$. This result supplies …
For a scheme of fat points $Z$ defined by the saturated ideal $\mathcal{I}_Z$, the regularity index computes the Castelnuovo-Mumford regularity of the Cohen-Macaulay ring $R/\mathcal{I}_Z.$ For points in " general position" we improve the bound for the regularity index computed by Segre for $\mathbb {P}^2$ and generalised by Catalisano, Trung and Valla for $\mathbb {P}^n$. Moreover, we prove that the generalised Segre's bound conjectured by Fatabbi and Lorenzini holds for $n+3$ arbitrary points in $\mathbb {P}^n$. We propose a modification of Segre's conjecture for arbitrary points and we disc…
Recently, the regular local rings of prime characteristic were characterized in terms of the finiteness of injective dimension of the Frobenius map. We obtain relative versions of this result.
We prove the Trung's conjecture about Segre's upper bound for s equimultiple fat points not on a linear (r-1)-space, s\le r+3, by algebraic method used in [3]. This method also may used to research other cases of fat points.
Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely one where finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence. Connections are made with Ohm-Rush content theory, intersection-flatness of the Frobenius map, and various flatness criteria.
Let $R=\mathbb{K}[X_1, \ldots , X_n ]$ be a polynomial ring over a field $\mathbb{K}$. We introduce an endomorphism $\mathcal{F}^{[m]}: R \rightarrow R $ and denote the image of an ideal $I$ of $R$ via this endomorphism as $I^{[m]}$ and call it to be the $m$ \textit{-th square power} of $I$. In this article, we study some homological invariants of $I^{[m]}$ such as regularity, projective dimension, associated primes and depth for some families of ideals e.g. monomial ideals.
This is a survey on the relation between homological properties of the Frobenius endomorphism and finiteness of various homological dimensions of the ring or of modules over it, such as global dimension and projective dimension. We begin with Kunz's surprising result in 1969 that the regularity of a Noetherian local ring is equivalent to the flatness of its Frobenius endomorphism, as well as the subsequent generalizations to the module setting by Peskine and Szpiro and continue up through the recent flurry of results in the last five years. An attempt is made to include proofs whenever feasibl…
We study commutative algebra arising from generalised Frobenius numbers. The $k$-th (generalised) Frobenius number of natural numbers $(a_1,\dots,a_n)$ is the largest natural number that cannot be written as a non-negative integral combination of $(a_1,\dots,a_n)$ in $k$ distinct ways. Suppose that $L$ is the lattice of integers points of $(a_1,\dots,a_n)^{\perp}$. Taking cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modules $M_L^{(k)}$ whose Castelnuovo-Mumford regularity captures the $k$-th Frobenius number of $(a_1,\dots,a_n)$. We study th…
The goal of this short note is to study the secant varieties of the triple Segre product of type (1,a,b) by means of the standard tools of combinatorial commutative algebra. We reprove and extend to arbitrary characteristic results of Landsberg and Weyman regarding the defining ideal and the Cohen-Macaulay property of the secant varieties. Furthermore for these varieties we compute the degree and give a bound for their Castelnuovo-Mumford regularity which is sharp in many cases.
We demonstrate that the ring of invariants for the natural action of a subgroup G of GL_n(F_q) on a polynomial ring R=K[X_1,...,X_n] need not be F-pure. In these examples G is the symplectic group over a finite field, and the invariant subrings are always complete intersections by the work of Carlisle and Kropholler. These examples are of special interest from the point of view of studying the Frobenius closures and tight closures of ideals as contractions from certain extension rings: they provide instances when the socle element modulo an ideal generated by a system of parameters is forced i…
On the test properties of the Frobenius endomorphism
West Virginia University（西弗吉尼亚大学）；School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所数学学院）；Georgia State University（佐治亚州立大学）
We investigate the behavior of Castelnuovo-Mumford regularity with respect to some classical functors : Tor, the Frobenius functor in positive characteristic, taking a power or a product (on ideals). These generalizes and refines previous results on these issues by several authors. As an application we provide results on the regularity of an intersection of subschemes of a projective scheme, under appropriate geometric hypotheses. Results on the rigidity of multiple Tor modules and on the characterization of their vanishing are given, motivated by geometric applications.
It is proved that when R is a local ring of positive characteristic, $ϕ$ is its Frobenius endomorphism, and some non-zero finite R-module has finite flat dimension or finite injective dimension for the R-module structure induced through $ϕ$, then R is regular. This broad generalization of Kunz's characterization of regularity in positive characteristic is deduced from a theorem concerning a local ring R with residue field of k of arbitrary characteristic: If $ϕ$ is a contracting endomorphism of R, then the Betti numbers and the Bass numbers over $ϕ$ of any non-zero finitely generated R-module …
We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of I but not the degree. We relate the Hilbert series and the generators of I and I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the regularity of certain lattice ideals is additive in a certain sense. The…
Regularity, complete intersection and Gorenstein properties of a local ring can be characterized by homological conditions on the canonical homomorphism into its residue field (Serre, Avramov, Auslander). It is also known that in positive characteristic, the Frobenius endomorphism can also be used for these characterizations (Kunz, ...), and more generally any contracting endomorphism. We introduce here a class of local homomorphisms, in some sense larger than all above, for which these characterizations still hold, providing an unified treatment for this class of homomorphisms.
In this paper, we investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to "F-singularities of pairs." The notions of F-regular and F-pure rings in characteristic $p>0$ are characterized by a splitting of the Frobenius map, and define some classes of rings having "mild" singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolutions of singularities in characteristic zero. These are defined also for pairs of a normal…
The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism f: R --> S. Various techniques are developed to study the new invariants and to establish their basic properties. In several cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite R-modules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may…
