Arbitrary Commutative Ring Research Articles

When quotient of a polynomial or power series ring by a monomial ideal is half-factorial

Suppose that K is a field, S = K [ x 1 , … , x n ] or S = K [ [ x 1 , … , x n ] ] . We present a characterization of those monomial ideals I of S, for which S/I is a half-factorial ring. This characterization is related to the structure of the base field K and also depends on the combinatorics of a graph constructed from the monomial ideal I. We also show that if R [ x ] or R [ [ x ] ] is half-factorial for an arbitrary commutative ring R, then R is an integral domain.

Rigid integral representations of quivers over arbitrary commutative rings

Rigid integral representations of quivers over arbitrary commutative rings

Weil cohomology theories and their motivic Hopf algebroids

In this paper we discuss a general notion of Weil cohomology theories, both in algebraic geometry and in rigid analytic geometry. We allow our Weil cohomology theories to have coefficients in arbitrary commutative ring spectra. Using the theory of motives, we give three equivalent viewpoints on Weil cohomology theories: as a cohomology theory on smooth varieties, as a motivic spectrum and as a realization functor. We also associate to every Weil cohomology theory a motivic Hopf algebroid generalizing the construction we gave in Ayoub (2014) for the Betti cohomology. Exploiting results and constructions from Ayoub (2020), we are able to prove that the motivic Hopf algebroids of all the classical Weil cohomology theories are connective. In particular, they give rise to motivic Galois groupoids which are spectral affine groupoid schemes.

On the subalgebra of invariant elements: Finiteness and immersions

Given the n-dimensional affine space over an arbitrary commutative ring k, G a group scheme flat and finitely presented over k and X⊂Akn a G-invariant affine and closed subscheme, we prove that the GIT quotient ▪ is of finite type over k, even if k is not noetherian, provided G is linearly reductive. This is well-known if ▪ is faithfully flat, which does not hold in general. We also explore the infinite-dimensional case. Concretely, we consider a G−k finitely presented projective module M and an arbitrary k-module N. We prove, under certain conditions on k and G, that the degrees of the generators of (S•(M∨⊗N))G and the degrees of the generators of the ideal of relations are bounded. We encode this property into the notion of partially generated graded (pgg) algebra and we give their main properties. In particular, we prove the existence of canonical equivariant immersions of spectra of pgg algebras in certain projective spaces.

The Diophantine problem in Chevalley groups

In this paper we study the Diophantine problem in Chevalley groups Gπ(Φ,R), where Φ is a reduced irreducible root system of rank >1, R is an arbitrary commutative ring with 1.We establish a variant of double centralizer theorem for elementary unipotents xα(1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup Xα, α∈Φ, is Diophantine in G. Then we prove that the Diophantine problem in Gπ(Φ,R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.

Equivariant resolutions over Veronese rings

AbstractWorking in a polynomial ring , where is an arbitrary commutative ring with 1, we consider the th Veronese subalgebras , as well as natural ‐submodules inside . We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple ‐equivariant minimal free ‐resolutions for the quotient ring and for these modules . These also lead to elegant descriptions of for all and for any pair of these modules .

Cellular Noetherian algebras with finite global dimension are split quasi-hereditary

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers.

Local Nullstellensatz over Commutative Ground Rings

It is shown that a local Nullstellensatz holds over an arbitrary commutative ring A (with identity 1 ≠ 0); specifically, if B = A[x1, . . . , xn] is a finitely generated extension ring of A and N is a maximal ideal in B, then NBN = (N ∩ A, x1 − c1, . . . , xn − cn)BN for some c1, . . . , cn ∈ BN .

Strongly regular matrices revisited

We prove a necessary condition and a sufficient condition for an n×n-matrix A with determinantal rank ρ(A)=t over an arbitrary commutative ring to be (von Neumann) strongly regular in terms of the trace of its tth compound matrix Ct(A). In particular, a non-zero n×n-matrix A with ρ(A)=t over a local commutative ring R is strongly regular if and only if Tr(Ct(A)) is a unit in R, and in this case we construct a strong inner inverse of A. We derive applications to products of local commutative rings and group algebras. Finally, we count strongly regular matrices over some finite rings of residue classes and group algebras.

Dihedral modules with ∞-simplicial faces and dihedral homology for involutive 𝐴_{∞}-algebras over rings

On the basis of combinatorial techniques of dihedral modules with ∞ \infty -simplicial faces, dihedral homology is constructed for involutive A ∞ A_{\infty } -algebras over arbitrary commutative unital rings.

An integral second fundamental theorem of invariant theory for partition algebras

We prove that the kernel of the action of the group algebra of the Weyl group acting on tensor space (via restriction of the action from the general linear group) is a cell ideal with respect to the alternating Murphy basis. This provides an analogue of the second fundamental theory of invariant theory for the partition algebra over an arbitrary commutative ring and proves that the centraliser algebras of the partition algebra are cellular. We also prove similar results for the half partition algebras.

Which homotopy algebras come from transfer?

We characterize A ∞ A_\infty -structures that are equivalent to a given transferred structure over a chain homotopy equivalence or a quasi-isomorphism, answering a question posed by D. Sullivan. Along the way, we present an obstruction theory for weak \text{ A ∞ A_\infty -morphisms} over an arbitrary commutative ring. We then generalize our results to P ∞ \mathcal {P}_\infty -structures over a field of characteristic zero, for any quadratic Koszul operad P \mathcal {P} .

Ideals of étale groupoid algebras and Exel’s Effros–Hahn conjecture

We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author’s Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal) and if the base ring is Artinian (e.g., a field), then we can show that every primitive ideal is the kernel of an irreducible representation induced from isotropy.

A classification of ideals in Steinberg and Leavitt path algebras over arbitrary rings

We give a one-to-one correspondence between ideals in the Steinberg algebra of a Hausdorff ample groupoid G, and certain families of ideals in the group algebras of isotropy groups in G. This generalises a known ideal correspondence theorem for Steinberg algebras of strongly effective groupoids. We use this to give a complete graph-theoretic description of the ideal lattice of Leavitt path algebras over arbitrary commutative rings, generalising the classification of ideals in Leavitt path algebras over fields.

Implicit Linear Differential-Difference Equations in the Module of Formal Generalized Functions over a Commutative Ring

We study an implicit inhomogeneous linear differential-difference equation in the module of formal generalized functions over a commutative ring. We prove the well-posedness of the equation and find the fundamental solution. We obtain a representation of a unique solution to the equation in the form of the convolution of the fundamental solution and a given formal generalized function. We also consider the inhomogeneous second order differential equation over an arbitrary commutative ring.

Cohomological supports over derived complete intersections and local rings

A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over an exterior algebra. The main objects of study are certain DG modules over a polynomial ring; these determine the aforementioned cohomological supports and are shown to encode (co)homological information about pairs of DG modules over a Koszul complex. The perspective in this article leads to new proofs of well-known results for pairs of complexes over a complete intersection. Furthermore, these cohomological supports are used to define a support theory for pairs of objects in the derived category of an arbitrary commutative noetherian local ring. Finally, we calculate several examples; one of which answers a question of D. Jorgensen in the negative.

Toward the Reverse Decomposition of Unipotents. II. The Relative Case

Recently, Raimund Preusser displayed very short polynomial expressions of elementary generators in classical groups over an arbitrary commutative ring as products of conjugates of an arbitrary matrix and its inverse by absolute elementary matrices. In particular, this provides very short proofs for description of normal subgroups. In 2018, the author discussed various generalizations of these results to exceptional groups, specifically those of types E6 and E7. Here, a further variation of Preusser’s wonderful idea is presented. Namely, in the case of GL(n, R), n ≥ 4, similar expressions of elementary transvections as conjugates of g ∈ GL(n, R) and g−1 by relative elementary matrices x ∈ E(n, J) and then x ∈ E(n, R, J), for an ideal J ⊴ R, are obtained. Again, in particular, this allows to give very short proofs for the description of subgroups normalized by E(n, J) or E(n, R, J), and thus also of subnormal subgroups in GL(n, R).

Iterated Laurent series over rings and the Contou-Carrère symbol

This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrère symbol. In addition to the survey, the article also contains new results related to this symbol. The higher-dimensional Contou-Carrère symbol arises naturally when one considers deformation of a flag of algebraic subvarieties of an algebraic variety. The non-triviality of the problem is due to the fact that, in the case , for the group of invertible elements of the algebra of -iterated Laurent series over a ring, no representation is known in the form of an ind-flat scheme over this ring. Therefore, essentially new algebro-geometric constructions, notions, and methods are required. As an application of the new methods used, a description of continuous homomorphisms between algebras of iterated Laurent series over a ring is given, and an invertibility criterion for such endomorphisms is found. It is shown that the higher- dimensional Contou-Carrère symbol, restricted to algebras over the field of rational numbers, is given by a natural explicit formula, and this symbol extends uniquely to all rings. An explicit formula is also given for the higher-dimensional Contou-Carrère symbol in the case of all rings. The connection with higher-dimensional class field theory is described. As a new result, it is shown that the higher-dimensional Contou-Carrère symbol has a universal property. Namely, if one fixes a torsion-free ring and considers a flat group scheme over this ring such that any two points of the scheme are contained in an affine open subset, then after restricting to algebras over the fixed ring, all morphisms from the -iterated algebraic loop group of the Milnor -group of degree to the above group scheme factor through the higher-dimensional Contou-Carrère symbol. Bibliography: 67 titles.

Гомоморфiзми з умовою (*), якщо 2 – оборотний елемент

Вивчення гомоморфiзмiв матричних груп над асоцiативними кiльцями розпочалося майже 100 рокiв тому роботами Шраєра i Ван-дер-Вардена i в подальшому розвивалися в працях Дьєдоне, Хуа Ло-гена, Райнера, О’Мiри, Хана, Ю.I. Мерзлякова, Уотерхауса, О.В. Мiхальова, Ю.I. Зельманова, I.З. Голубчика, В.М. Петечука та iнших авторiв. В основi вивчення знаходяться груповi властивостi повної лiнiйної групи GL(n, R) – множини всiх оборотних матриць над асоцiативним кiльцем R з 1. При n ≥ 3 у всiх вiдомих випадках, незважаючи на вiдмiннiсть методiв, якi застосовувалися, автоморфiзми повної лiнiйної групи виявлялись добутком стандартних автоморфiзмiв. Саме оборотнiсть елемента 2 давала можливiсть розглядати все бiльш широкi класи кiлець над якими можливий стандартний опис гомоморфiзмiв матричних груп. Якщо 2 – необоротний елемент, то при n ≥ 3 В.М. Петечук зробив опис автоморфiзмiв групи GL(n, R) у випадку, коли R – комутативне локальне кiльце. Виявилося, що при n ≥ 4 всi автоморфiзми таких груп є добутком стандартних автоморфiзмiв, а при n = 3 їх можна виразити через стандартнi i деякий нестандартний автоморфiзми. Спираючись на цей результат, В.М. Петечук [2] отримав опис iзоморфiзмiв групи GL(n, R), n ≥ 3, якщо R – довiльне комутативне кiльце. Зокрема, вiн здiйснив опис гомоморфiзмiв Λ : P E (n, R) → P GL(m, K), m ≥ 3, n ≥ 3 таких, що ΛP E (n, R) = P H i H ⊇ E (m, K) над довiльними комутативними кiльцями R i K. I.З. Голубчик i О.В. Мiхальов [3], використовуючи системи iдемпотентiв, i незалежно Ю.I. Зельманов [4], використовуючи методи йорданових алгебр, отримали опис iзоморфiзмiв групи E (n, R), n ≥ 3, 2 ∈ R∗ на групу E (m, K), 2 ∈ K∗ над довiльними асоцiативними кiльцями R i K з 1. В.М. Петечук [5] зробив опис гомоморфiзмiв групи P E (n, R), n ≥ 3 в групу GL(m, K), m ≥ 2, 2 ∈ K∗ у випадку, коли нерухомi пiдмодулi деяких елементiв четвертого порядку збiгаються з нерухомими пiдмодулями їх квадратiв. З нього випливають результати I.З. Голубчика, О.В. Мiхальова i Ю.I. Зельманова. Розвиваючи технiку, пов’язану з iдемпотентами, I.З. Голубчик [6] здiйснив опис iзоморфiзмiв груп GL(n, R) i GL(m, K) при n, m ≥ 4 над асоцiативними кiльцями R i K. Виявилося, що вони допускають стандартний опис на групi E (n, R). Авторами В.М. Петечук, Ю.В. Петечук [7, 8] описанi гомоморфiзми з умовою (*) з чого зокрема випливає i опис iзоморфiзмiв повних лiнiйних груп над асоцiативними кiльцями. У данiй роботi удосконалюються i розширюються методи опису гомоморфiзмiв з умовою (*), якщо елемент 2 є оборотним в кiльцi K i n ≥ 3. Основним результатом роботи є наступна теорема. Нехай R i K – асоцiативнi кiльця з 1, 2 ∈ K∗ , E (n, R) ⊆ G ⊆ GL(n, R), n ≥ 3, W – лiвий K-модуль, гомоморфiзм Λ : G → GL(W) задовольняє умову (*). Тодi Λ має стандартний опис на групi E (n, R).

A strictly commutative model for the cochain algebra of a space

AbstractThe commutative differential graded algebra$A_{\mathrm {PL}}(X)$of polynomial forms on a simplicial set$X$is a crucial tool in rational homotopy theory. In this note, we construct an integral version$A^{\mathcal {I}}(X)$of$A_{\mathrm {PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections$\mathcal {I}$to model$E_{\infty }$differential graded algebras (dga) by strictly commutative objects, called commutative$\mathcal {I}$-dgas. We define a functor$A^{\mathcal {I}}$from simplicial sets to commutative$\mathcal {I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the$E_{\infty }$dga of cochains. The functor$A^{\mathcal {I}}$shares many properties of$A_{\mathrm {PL}}$, and can be viewed as a generalization of$A_{\mathrm {PL}}$that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that$A^{\mathcal {I}}(X)$determines the homotopy type of$X$when$X$is a nilpotent space of finite type.