scholarly journals Local dimension theory of tensor products of algebras over a ring

2018 ◽  
Vol 17 (06) ◽  
pp. 1850106
Author(s):  
Samir Bouchiba
Keyword(s):  
General Setting ◽  
Tensor Products ◽  
Krull Dimension ◽  
Dimension Theory ◽  
Boolean Ring ◽  
Local Dimension ◽  
Arbitrary Ring ◽  
Commutative Algebras ◽  
Local Properties ◽  
Nonzero Characteristic

Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring [Formula: see text]. Actually, we translate the theory initiated by Grothendieck and Sharp and subsequently developed by Wadsworth on Krull dimension of tensor products of algebras over a field [Formula: see text] into the general setting of algebras over an arbitrary ring [Formula: see text]. For this sake, we introduce and study the notion of a fibered AF-ring over a ring [Formula: see text]. This concept extends naturally the notion of AF-ring over a field introduced by Wadsworth in [The Krull dimension of tensor products of commutative algebras over a field, J. London Math. Soc. 19 (1979) 391–401.] to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fiber rings of tensor products of algebras over a ring. Also, given a triplet of rings [Formula: see text] consisting of two [Formula: see text]-algebras [Formula: see text] and [Formula: see text] such that [Formula: see text], we introduce the inherent notion to [Formula: see text] of a [Formula: see text]-fibered AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product [Formula: see text]. As an application, we provide a formula for the Krull dimension of [Formula: see text] when either [Formula: see text] or [Formula: see text] is zero-dimensional as well as for the Krull dimension of [Formula: see text] when [Formula: see text] is a fibered AF-ring over the ring of integers [Formula: see text] with nonzero characteristic and [Formula: see text] is an arbitrary ring. This enables us to answer a question of Jorge Martinez on evaluating the Krull dimension of [Formula: see text] when [Formula: see text] is a Boolean ring. Actually, we prove that if [Formula: see text] and [Formula: see text] are rings such that [Formula: see text] is not trivial and [Formula: see text] is a Boolean ring, then dim[Formula: see text].

scholarly journals KRULL DIMENSION OF TENSOR PRODUCTS OF PULLBACKS

2009 ◽  
Vol 08 (03) ◽  
pp. 319-338
Author(s):  
SAMIR BOUCHIBA
Keyword(s):  
Open Problem ◽  
Tensor Products ◽  
Krull Dimension ◽  
The Other ◽  
Dimension Theory ◽  
Other Hand

This paper is concerned with the study of the dimension theory of tensor products of algebras over a field k. We answer an open problem set in [6] and compute dim (A ⊗k B) when A is a k-algebra arising from a specific pullback construction involving AF-domains and B is an arbitrary k-algebra. On the other hand, we deal with the question (Q) set in [5] and show, in particular, that such a pullback A is in fact a generalized AF-domain.

scholarly journals Props in model categories and homotopy invariance of structures

2010 ◽  
Vol 17 (1) ◽  
pp. 79-160
Author(s):  
Benoit Fresse
Keyword(s):  
General Setting ◽  
Model Category ◽  
Topological Spaces ◽  
Algebra Structure ◽  
Model Structure ◽  
Homotopy Invariance ◽  
Model Categories ◽  
Arbitrary Ring ◽  
General Argument ◽  
Simplicial Sets

Abstract We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object 𝑋 homotopy equivalent to an algebra 𝐴 over a cofibrant prop P inherits a P-algebra structure so that 𝑋 defines a model of 𝐴 in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of 𝐴∞-algebras.

scholarly journals Nonnoetherian geometry

2016 ◽  
Vol 15 (09) ◽  
pp. 1650176 ◽  
Author(s):  
Charlie Beil
Keyword(s):  
Commutative Algebra ◽  
Projective Dimension ◽  
Krull Dimension ◽  
Maximal Dimension ◽  
Algebraic Varieties ◽  
Finitely Generated ◽  
One Dimensional ◽  
Simple Modules ◽  
Commutative Algebras

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive-dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of [Formula: see text] are all isomorphic if and only if [Formula: see text] is noetherian, if and only if the center [Formula: see text] of [Formula: see text] is noetherian, if and only if [Formula: see text] is a finitely generated [Formula: see text]-module. Furthermore, we show that [Formula: see text] is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.

Primary rings and tensor products of algebras

1982 ◽  
Vol 92 (1) ◽  
pp. 41-48 ◽  
Author(s):  
L. O'Carroll ◽  
M. A. Qureshi
Keyword(s):  
Tensor Products ◽  
Commutative Algebras

All rings considered in this paper are non-trivial commutative algebras over a field k; unless indicated otherwise, all tensor products are understood to be taken over k. (Some of the results and concepts extend to general rings, but it is not worth while noting such generalizations.)

scholarly journals A non-abelian tensor product of Lie algebras

1991 ◽  
Vol 33 (1) ◽  
pp. 101-120 ◽  
Author(s):  
Graham J. Ellis
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Algebraic Theory ◽  
Tensor Products ◽  
Algebraic Structures ◽  
Commutative Algebras ◽  
Group Case ◽  
Do So ◽  
Product Of Groups

A generalized tensor product of groups was introduced by R. Brown and J.-L. Loday [6], and has led to a substantial algebraic theory contained essentially in the following papers: [6, 7, 1, 5, 11, 12, 13, 14, 18, 19, 20, 23, 24] ([9, 27, 28] also contain results related to the theory, but are independent of Brown and Loday's work). It is clear that one should be able to develop an analogous theory of tensor products for other algebraic structures such as Lie algebras or commutative algebras. However to do so, many non-obvious algebraic identities need to be verified, and various topological proofs (which exist only in the group case) have to be replaced by purely algebraic ones. The work involved is sufficiently non-trivial to make it interesting.

Sign in / Sign up

Export Citation Format

Share Document