Homological dimensions and semidualizing complexes

For finite modules over a local ring and complexes with finitely generated homology, we consider several homological invariants sharing some basic properties with projective dimension. In the second section, we introduce the notion of a semidualizing complex, which is a generalization of both a dualizing complex and a suitable module. Our goal is to establish some common properties of such complexes and the homological dimension with respect to them. Basic properties are investigated in Sec. 2.1. In Sec. 2.2, we study the structure of the set of semidualizing complexes over a local ring, which is closely related to the conjecture of Avramov-Foxby on the transitivity of the G-dimension. In particular, we prove that, for a pair of semidualizing complexes X ₁ and X ₂ such that G _X2, we have X ₂ ≃ X ₁ ⊗ _R ^L RHom _R (X ₁, X ₂). Specializing to the case of semidualizing modules over Artinian rings, we obtain a number of quantitative results for the rings possessing a configuration of semidualizing modules of special form. For the rings with m ³=0, this condition reduces to the existence of a nontrivial semidualizing module, and we prove a number of structural results in this case. In the third section, we consider the class of modules that contains the modules of finite CI-dimension and enjoys some nice additional properties, in particular, good behavior in short exact sequences. In the fourth section, we introduce a new homological invariant, CM-dimension, which provides a characterization for Cohen-Macaulay rings in precisely the same way as projective dimension does for regular rings, CI-dimension for locally complete intersections, and G-dimension for Gorenstein rings. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 30, Algebra, 2005.     

Vanishing and nilpotence of locally trivial symmetric spaces over regular schemes

We prove two results about Witt rings W(−) of regular schemes. First, given a semi-local regular ring R of Krull dimension d, if U is the punctured spectrum, obtained from Spec(R) by removing the maximal ideals of height d, then the natural map is injective. Secondly, given a regular integral scheme X of finite Krull dimension, consider Q its function field and the natural map . We prove that there is an integer N, depending only on the Krull dimension of X, such that the product of any choice of N elements in is zero. That is, this kernel is nilpotent. We give upper and lower bounds for the exponent N. Received: December 4, 2001     

On s-Semipermutable Maximal and Minimal Subgroups of Sylow p-Subgroups of Finite Groups

Let R be a Noetherian algebra over a field k. A formula is given for the Krull dimension of the ring R?_k k(X) in terms of the heights of simple modules with large endomorphism rings.     

Gaussian Property of the Rings R(X) and R〈X〉

The content of a polynomial f over a commutative ring R is the ideal c(f) of R generated by the coefficients of f. A commutative ring R is said to be Gaussian if c(fg) = c(f)c(g) for every polynomials f and g in R[X]. A number of authors have formulated necessary and sufficient conditions for R(X) (respectively, R?X?) to be semihereditary, have weak global dimension at most one, be arithmetical, or be Prüfer. An open question raised by Glaz is to formulate necessary and sufficient conditions that R(X) (respectively, R?X?) have the Gaussian property. We give a necessary and sufficient condition for the rings R(X) and R?X? in terms of the ring R in case the square of the nilradical of R is zero.     

A Gersten-Witt spectral sequence for regular schemes

A spectral sequence is constructed whose non-zero E₁-terms are the Witt groups of the residue fields of a regular scheme X, arranged in Gersten-Witt complexes, and whose limit is the four global Witt groups of X. This has several immediate consequences concerning purity for Witt groups of low-dimensional schemes. We also obtain an easy proof of the Gersten Conjecture in dimension smaller than 5. The Witt groups of punctured spectra of regular local rings are also computed.     

Cohomological approach to asymptotic dimension

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdim _Z X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe’s coarse cohomology is strictly less than its asymptotic dimension.      

Lazy bases: a minimalist constructive theory of Noetherian rings

We give a constructive treatment of the theory of Noetherian rings. We avoid the usual restriction to coherent rings; we can even deal with non‐discrete rings. We introduce the concept of rings with certifiable equality which covers discrete rings and much more. A ring R with certifiable equality can be fitted with a partial ideal membership test for ideals of R. Lazy bases of ideals of R [X ] are introduced in order to derive a partial ideal membership test for ideals of R [X ]. It is then proved that if R is Noetherian, then so is R [X ]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rings Whose Modules Have Grade Zero

In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of RR is not a radical module for some right coherent rings. We call a ring a right X ring if Homa(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith‘s conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.     

Injective homogeneity and homological homogeneity of the ore extensions

In this paper we prove that under some natural conditions, the Ore extensions and skew Laurent polynomial rings are injectively homogeneous or homologically homogeneous if so are their coefficient rings. Specifically, we prove that ifR is a commutative Noetherian ring of positive characteristic, thenA _n (R), then ^th Weyl algebra overR, is injectively homogeneous (resp. homologically homogeneous) ifR has finite injective dimension (resp. global dimension).     

The Higher K-Theory of a Complex Surface

Let X be a smooth complex variety of dimension at most two, and let F be its function field. We prove that the K-groups of F are divisible above the dimension of X, and that the K-groups of X are divisible-by-finite. We also describe the torsion in the K-groups of F and X.     

Algebras with small homological dimensions

We consider the algebras Λ which satisfy the property that for each indecomposable module X, either its projective dimension pd_Λ X is at most one or its injective dimension id_Λ X is at most one. This clearly generalizes the so-called quasitilted algebras introduced by Happel–Reiten–Smal?. We show that some of the niciest features for this latter class of algebras can be generalized to the case we are considering, in particular the existence of a trisection in its module category. Received: 26 August 1998     

Hilbert-Kunz function of monomial ideals and binomial hypersurfaces

The aim of this note is to determine the Hilbert-Kunz functions of rings defined by monomial ideals and of rings defined by a single binomial equationX ^a−X^b with gcd(X ^a, X^b)=1.     

Semigroups with I-Quasi Length and Syntactic Semigroups

A module M is said to be extending (𝒢-extending) if for each submodule X of M there exists a direct summand D of M such that X is essential in D (X ∩ D is essential in both X and D). It is known that for a nonsingular module the concepts of 𝒢-extending and extending coincide. However, in the not nonsingular case, they are distinct. In this article, we obtain a characterization of the right 𝒢-extending generalized triangular matrix rings. This result and its corollaries improve and generalize the existing results on right extending generalized triangular matrix rings. It is well known that the ring of n-by-n triangular matrices over a right selfinjective ring is not, in general, right extending. One application of our characterization shows that such rings are right 𝒢-extending. Connections to Operator Theory and a characterization of the class of right extending right SI-rings are also obtained. Examples are given to illustrate and delimit the theory.     

Dimension pure globale des anneaux commutatifs

For every (n,m) ? IN^?× IN ^?, verifying l We study direct systems of rings and give a bound for the pure - global - dimension of their limits, not depending on the ring cardinality. Using this result and a theorem of C0UCH0T we give examples of pure -heriditary group- rings (ie. of pure -global - dimension one) of large cardinality.

Finaly we prove - using a result of VASC0NCEL0S and a theorem of JENSEN - that some countably products of noetherian rings are coherent rings and that their pure - global - dimension is exactly two.     

The local structure of length spaces with curvature bounded above

We show that a number of different notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT(0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are determined by the maximal dimension of a flat in X. Received April 21, 1998     

Cox rings of surfaces and the anticanonical Iitaka dimension

In this paper we investigate the relation between the finite generation of the Cox ring R(X) of a smooth projective surface X and its anticanonical Iitaka dimension κ(−KX).     

Multiplicative properties of projectively dual varieties

We present a general formula for the dimension of the projectively dual of the product of two projective varietiesX ₁ andX ₂, in terms of dimensions ofX ₁,X ₂ and their projective duals (Theorem 0.1). The proof is based on the formula due to N. Katz expressing the dimension of the dual variety in terms of the rank of certain Hessian matrix. Some consequences and related results are given, including the “Cayley trick” from [3] and its dual version. Partially supported by the NSF (DMS-9102432) Partially supported by the NSF (DMS-9104867) This article was processed by the author using the Springer-Verlag T_EE mamath macro package 1990.     

On nef reductions of projective irreducible symplectic manifolds

Let X be a projective irreducible symplectic manifold and L be a non trivial nef divisor on X. Assume that the nef dimension of L is strictly less than the dimension of X. We prove that L is semiample. Partially supported by Grant-in-Aid no. 15740002 (Japan Society for Promotion of Sciences)     

Linear ultrafilters

Let X be a k-vector space, and U a maximal proper filter of subspaces of X. Then the ring of endomorphisms of X that are “continuous” with respect to U modulo the ideal of those that are “trivial” with respect to U forms a division ring E(U). (These division rings can also be described as the endomorphism rings of the simple left End(X)-modules.) We study this and the dual construction, based on maximal cofiIters of subspaces of X, in particular, the relation between the constructed division rings and the original field or division ring k. We end by examining a more general construction in which X is a module over a general ring, given with both a filter and a cofilter of submodules.