arXiv subjects
Publications and source records attributed to Holger Brenner.
We study how forcing algebras give rise to ${\mathbb A}^1$-bundles and ${\mathbb A}^1$-torsors and how they are related to ${\mathbb A}^1$-patches. In particular we discuss the affineness of torsors and how algebraic properties of ${\mathbb A}^1$-patches can be deduced from this property. This is in part a survey article.
We introduce a graded version of dagger closure and prove that it coincides with solid closure for homogeneous ideals in two dimensional $\mathbb{N}$-graded domains of finite type over a field.
Let K be an algebraically closed field of characteristic p > 0. We apply a theorem of C. Han to give an explicit description for the weak Lefschetz property of the monomial Artinian complete intersection A = K[X,Y,Z]/(X^d,Y^d,Z^d) in terms of d and p. This answers a question of J. Migliore, R. M. Miro-Roig and U. Nagel and, equivalently, characterizes for which characteristics the rank-2 syzygy bundle Syz(X^d,Y^d,Z^d) on PP^2 satisfies the Grauert-Muelich theorem. As a corollary we obtain that for p=2 the algebra A has the weak Lefschetz property if and only if d=(2^t+1)/3 or d=(2^t-1)/3 for some positive integer t. This was recently conjectured by J. Li and F. Zanello.
We give an example showing that tight closure does not commute with localization.
We give a survey about some recent work on tight closure and Hilbert-Kunz theory from the viewpoint of vector bundles
Let R be an integral domain of finite type over Z and let f:X --> Spec R be a smooth projective morphism of relative dimension d >= 1. We investigate, for a vector bundle E on the total space X, under what arithmetical properties of a sequence (p_n, e_n)_{n \in \NN}, consisting of closed points p_n in Spec R and Frobenius descent data E_{p_n} \cong F^{e_n}^*(F) on the closed fibers X_{p_n}, the bundle E_0 on the generic fiber X_0 is semistable.
We give a class of examples of vector bundles on a relative smooth projective curve over Spec Z such that for infinitely many prime reductions the bundle has a Frobenius descent, but the restriction to the generic fiber in characteristic zero is not semistable. In the third section of the paper we prove for a large class of varieties (including abelian varieties) that any vector bundle with this Frobenius descent property is generically semistable.
We study (slope-)stability properties of syzygy bundles on a projective space P^N given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring.
Generalizing homogeneous spectra for rings graded by natural numbers, we introduce multihomogeneous spectra for rings graded by abelian groups. Such homogeneous spectra have the same completeness properties as their classical counterparts, but are possibly nonseparated. We relate them to ample families of invertible sheaves and simplicial toric varieties. As an application, we generalize Grothendieck's Algebraization Theorem and show that formal schemes with certain ample families are algebraizable.
We relate closure operations for ideals and for submodules to non-flat Grothendieck topologies. We show how a Grothendieck topology on an affine scheme induces a closure operation in a natural way, and how to construct for a given closure operation fulfilling certain properties a Grothendieck topology which induces this operation. In this way we relate the radical to the surjective topology and the constructible topology, the integral closure to the submersive topology, to the proper topology and to Voevodsky's h-topology, the Frobenius closure to the Frobenius topology and the plus closure to the finite topology. The topologies which are induced by a Zariski filter yield the closure operations which are studied under the name of hereditary torsion theories. The Grothendieck topologies enrich the corresponding closure operation by providing cohomology theories, rings of global sections, concepts of exactness and of stalks.
Let K be an algebraically closed field of characteristic zero and let I=(f_1,...,f_n) be a homogeneous R_+-primary ideal in R:=K[X,Y,Z]. If the corresponding syzygy bundle Syz(f_1,...,f_n) on the projective plane is semistable, we show that the Artinian algebra R/I has the Weak Lefschetz property if and only if the syzygy bundle has a special generic splitting type. As a corollary we get the result of Harima et alt., that every Artinian complete intersection (n=3) has the Weak Lefschetz property. Furthermore, we show that an almost complete intersection (n=4) does not necessarily have the Weak Lefschetz property, answering negatively a question of Migliore and Miro-Roig. We prove that an almost complete intersection has the Weak Lefschetz property if the corresponding syzygy bundle is not semistable.
We introduce the universal unitarily graded A-algebra for a commutative ring A and an arbitrary abelian extension U of the group of units of A, and use this concept to give simplified proofs of the main theorems of co-Galois theory in the sense of T. Albu. The main tool is a generalisation of a theorem by M. Kneser which, in our language, is a criterion for the universal algebra to be a field when the base ring A is itself a field. This theorem implies also the theorem of A. Schinzel on linearly independent roots. We discuss examples involving the injective hull of the multiplicative group of a field and we develop criteria for Galois extensions which allow a co-Galois grading, in particular for the cyclic case.
We ask for a given system of polynomials f_1,...,f_n and f over the complex numbers when there exist continuous functions q_1,...,q_n such that q_1 f_1+...+q_n f_n = f. This condition defines the continuous closure of an ideal. We give inclusion criteria and exclusion results for this closure in terms of the algebraically defined axes closure. Conjecturally, continuous and axes closure are the same, and we prove this in the monomial case.
We show that the Hilbert-Kunz multiplicity is a rational number for an R_+-primary homogeneous ideal I=(f_1, ..., f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic. Moreover we give a formula for the Hilbert-Kunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of the syzygy bundle Syz(f_1, . . ., f_n) on the projective curve Y = Proj R.
Suppose that R is a two-dimensional normal standard-graded domain over a finite field. We prove that there exists a uniform Frobenius test exponent b for the class of homogeneous ideals in R generated by at most n elements. This means that for every ideal I in this class we have that f^(p^b) belongs to I^([p^b]) if and only if f belongs to the Frobenius closure I^F. This gives in particular a finite test for the Frobenius closure. On the other hand we show that there is no uniform bound for Frobenius test exponent for all homogeneous ideals independent of the number of generators. Under similar assumptions we prove also the existence of a bound for tight closure test ideal exponents for ideals generated by at most n elements.
Let I denote an R_+ -primary homogeneous ideal in a normal standard-graded Cohen-Macaulay domain over a field of positive characteristic p. We give a linear degree bound for the Frobenius powers I^[q] of I, q=p^e, in terms of the minimal slope of the top-dimensional syzygy bundle on the projective variety Proj R. This provides an inclusion bound for tight closure. In the same manner we give a linear bound for the Castelnuovo-Mumford regularity of the Frobenius powers I^[q].
We characterize the tight closure of graded primary ideals in a homogeneous coordinate ring over an elliptic curve by numerical conditions and we show that it is in positive characteristic the same as the plus closure.
We study computational aspects of the tight closure of a homogeneous primary ideal in a two-dimensional normal standard-graded domain. We show how to use slope criteria for the sheaf of syzygies for generators of the ideal to compute the tight closure. Our method gives in particular an algorithm to compute the tight closure of three elements under the condition that we are able to compute the Harder-Narasimhan filtration. We apply this to the computation of (x^a,y^a,z^a)^* in K[x,y,z]/(F), where F is a homogeneous polynomial.