<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript> and rings over which the class of finitely generated strongly gorenstein projective modules is closed under extensions

In this paper, we consider the rings over which the class of finitely generated strongly Gorenstein projective modules is closed under extensions (called fs-closed rings). We give a characterization about the Grothendieck groups of the category of the finitely generated strongly Gorenstein projective R-modules and the category of the finitely generated R-modules with finite strongly Gorenstein projective dimensions for any left Noetherian fs-closed ring R.     

Module valuations and representation theory of orders I: Algebraic aspects

ABSTRACT

In this article, we study finitely generated reflexive modules over coherent GCD-domains and finitely generated projective modules over polynomial rings. In particular, we give a sufficient condition for a finitely generated reflexive module over a coherent GCD-domain to be a free module. By use of this result, we prove that every finitely generated projective R + [X]-module can be extended from R if R is a commutative ring with gl.dim(R) ≤ 2.     

Translation invariant topologies on commutative *-algebras

Let A be a commutative *-algebra. Under certain conditions on the involution, we construct a topology makingA a HausdorffQ-ring with continuous involution and inversion; this topology is induced by anA-valued norm.Applying the previous considerations to the algebraC(X) of continuous complex-valued functions over a topological spaceX, we obtain a new characterization of Weierstrass spaces.Furthermore, we provide every projective finitely generated moduleM over a topological ring R with a unique topology, under whichM is a topologicalR-module and every R-linear mapf :M N is continuous, for any topologicalR-moduleN. In caseR =A, we prove that this topology is induced by anA-norm. Mathematics subject classification numbers, 1980/85. Primary 46K05, Secondary 16A80.     

Sufficient conditions for the projective freeness of Banach algebras

Let R be a unital semi-simple commutative complex Banach algebra, and let M(R) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M(R) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notably the Hardy algebra H^∞(X) of bounded holomorphic functions on a Riemann surface of finite type, and also some algebras of stable transfer functions arising in control theory.     

Answer to a conjecture of J.M. Cohen

The origin of Gelfand rings comes from [9] where the Jacobson topology and the weak topology are compared. The equivalence of these topologies defines a regular Banach algebra. One of the interests of these rings resides in the fact that we have an equivalence of categories between vector bundles over a compact manifold and finitely generated projective modules over C(M), the ring of continuous real functions on M [17].These rings have been studied by R. Bkouche (soft rings [3]) C.J. Mulvey (Gelfand rings [15]) and S. Teleman (harmonic rings [19]).Firstly we study these rings geometrically (by sheaves of modules (Theorem 2.5)) and then introduce the ?ech covering dimension of their maximal spectrums. This allows us to study the stable rank of such a ring A (Theorem 6.1), the nilpotence of the nilideal of K₀(A) - The Grothendieck group of the category of finitely generated projective A-modules - (Theorem 9.3), and an upper limit on the maximal number of generators of a finitely generated A-module as a function of the afore-mentioned dimension (Theorem 4.4).Moreover theorems of stability are established for the group K₀(A), depending on the stable rank (Theorems 8.1 and 8.2). They can be compared to those for vector bundles over a finite dimensional paracompact space [18].Thus there is an analogy between finitely generated projective modules over Gelfand rings and ?ech dimension, and finitely generated projective modules over noetherian rings and Krull dimension.     

Finitely generated projective modules over exchange rings

This paper studies finitely generated projective modules over exchange rings. We prove that cancellation holds inp(R), andK _o (R) is completely determined by the continuous maps from the spectrum ofR toZ ifR is an exchange ring andR/J(R) is a ring with central idempotent elements.     

Galois Objects of Finitely Generated Projective Hopf Algebras

Let C be the category of cocommutative coalgebras over a commutative ring R and let H be a group object in C, i.e., let H be a cocommutative Hopf algebra. Assume that H is a finitely generated, projective R-module and that the integrals (of [4]) in H* ≡ Hom_R(H, R) are cocommutative elements. We will show that any Galois H-object (as defined in [3, Def. 1.2, p. 8]) is a finitely generated, projective R-module.     

Projectivized Rank Two Toric Vector Bundles are Mori Dream Spaces

We prove that the Cox ring of the projectivization P(?) of a rank two toric vector bundle ?, over a toric variety X, is a finitely generated k-algebra. As a consequence, P(?) is a Mori dream space if the toric variety X is projective and simplicial.     

Hermite and ps-rings of hurwitz series ∗

Let R be a commutative ring and H R the ring of Hurwitz series over R. In this note, we consider some properties of rings, which are shared by R and HR. In particular, we show that for the rings R and H R, if either ring is (i) a Hermite ring, or (ii) a PF-ring in the sense that every finitely generated projective R-module is free, then so is another. We also show that if R is a PS-ring in the sense that the socle Soc( _RR) is projective and char(R) = 0, then H R is also a PS-ring.     

On finiteness of multiplication modules

Our main aim in this note, is a further generalization of a result due to D. D. Anderson, i.e., it is shown that if R is a commutative ring, and M a multiplication R-module, such that every prime ideal minimal over Ann (M) is finitely generated, then M contains only a finite number of minimal prime submodules. This immediately yields that if P is a projective ideal of R, such that every prime ideal minimal over Ann (P) is finitely generated, then P is finitely generated. Furthermore, it is established that if M is a multiplication R-module in which every minimal prime submodule is finitely generated, then R contains only a finite number of prime ideals minimal over Ann (M).      

The subring generated by the units in a compact ring

ABSTRACT

In this paper, the authors introduce the concept of integrally closed modules and characterize Dedekind modules and Dedekind domains. They also show that a given domain R is integrally closed if and only if a finitely generated torsion-free projective R-module is integrally closed. In addition, it is proved that any invertible submodule of a finitely generated projective module over a domain is finitely generated and projective. Also they give the equivalent conditions for Dedekind modules and Dedekind domains.

     

Cartesian Closed Topological Categories and Tensor Products

The projective tensor product in a category of topological R-modules (where R is a topological ring) can be defined in Top, the category of topological spaces, by the same universal property used to define the tensor product of R-modules in Set. In this article, we extend this definition to an arbitrary topological category X and study how the Cartesian closedness of X is related to the monoidal closedness of the category of R-module objects in X. Mathematics Subject Classifications (2000) 18D15, 18D35, 18A40.     

Almost split sequences for dedekind-like rings. II

Let R be a two-pullback ring, that is, the pullback of two Dedekind domains over a common residue field. This paper contains explicit constructions of all the almost split sequences in the category of finitely generated R-modules, under the assumption that the Dedekind domains are local.     

Representations of étale Lie groupoids and modules over Hopf algebroids

The classical Serre-Swan’s theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita category of étale Lie groupoids and we show that the given correspondence represents a natural equivalence between them.     

First,Second, and Third Change of Rings Theorems for Gorenstein Homological dimensions

In this article, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this article for the Gorenstein projective dimension is a generalization of a G-dimension of a finitely generated module M over a noetherian ring R.     

Gorenstein Modules and Dualizing Modules

In this article, we study the characterizations of Gorenstein injective left S-modules and finitely generated Gorenstein projective left R-modules when there is a dualizing S-R-bimodule associated with a right noetherian ring R and a left noetherian ring S.     

On the Existence of Certain Modules of Finite Gorenstein Homological Dimensions

Let (R, 𝔪) be a commutative Noetherian local ring. It is known that R is Cohen–Macaulay if there exists either a nonzero finitely generated R-module of finite injective dimension or a nonzero Cohen–Macaulay R-module of finite projective dimension. In this article, we investigate the Gorenstein analogues of these facts.     

The Beilinson equivalence for differential operators

Let D be the ring of differential operators on a smooth irreducible affine variety X over C, or, more generally, the enveloping algebra of any locally free Lie algebroid on X. The category of finitely generated graded modules of the Rees algebra has a natural quotient category P_D which imitates the category of modules on Proj of a graded commutative ring. We show that the derived category D^b(P_D) is equivalent to the derived category of finitely generated modules of a sheaf of algebras E on X which is coherent over X. This generalizes the usual Beilinson equivalence for projective space, and also the Beilinson equivalence for differential operators on a smooth curve used by Ben-Zvi and Nevins in [6] to describe the moduli space of left ideals in D.     

Algebraic geometry of topological spaces I

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case . We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C ^*-algebras, and for a homology theory of commutative algebras to vanish on C ^*-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil K-theory implies that commutative C ^*-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.