The main theorem (Theorem 4.1) of this paper claims that any ring morphism from an Azumaya algebra of constant rank over a commutative ring to another one of the same constant rank and over a reduced commutative ring induces a ring morphism between the centers of these algebras. The second main theorem (Theorem 5.3) implies (through its Corollary 5.4) that a ring morphism between two Azumaya algebras of the same constant rank is an isomorphism if and only if it induces an isomorphism between their centers. As preliminary to these theorems we also prove a theorem (Theorem 2.8) describing the Brauer group of a commutative Artin ring and give an explicit proof of the converse of the Artin-Procesi theorem on Azumaya algebras (Theorem 2.6).
We define a nonassociative generalization of cyclic Azumaya algebras employing skew polynomial rings $D[t;σ]$, where $D$ is an Azumaya algebra of constant rank with center $C$ and $σ$ an automorphism of $D$, such that $σ|_{C}$ has finite order. The automorphisms of these algebras are canonically induced by ring automorphisms of the skew polynomial ring $D[t;σ]$ used in their construction. We achieve a description of their inner automorphisms. Results on the automorphisms of classical Azumaya algebras and central simple algebras of this type are obtained as special cases.
We provide easily-verified necessary and sufficient conditions for a skew group ring, or more generally, a crossed product ring, to be an Azumaya algebra. We use our results to show that (suitable localisations of) skew group rings associated to the quantum Kleinian singularities introduced by Chan, Kirkman, Walton, and Zhang in [CKWZ16a] are Azumaya, and use this to show that these algebras are maximal orders. We also give a new proof of Auslander's Theorem for quantum Kleinian singularities.
Let A be an Azumaya algebra of constant rank n^2 over a Hensel pair (R,I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK(A) coincides with its reduction SK(A/IA).
We attach to any commutative ring R a subgroup of the Brauer group of R, called the Brauer-Galois group of R. Its elements are the classes of the Azumaya R-algebras which can be represented, via Brauer equivalence, by a Galois extension of R. We compute this group for some particular commutative fields.
Let $R$ be a commutative ring. An Azumaya coring consists of a couple $(S,\Cc)$, with $S$ a faithfully flat commutative $R$-algebra, and an $S$-coring $\Cc$ satisfying certain properties. If $S$ is faithfully projective, then the dual of $\Cc$ is an Azumaya algebra. Equivalence classes of Azumaya corings form an abelian group, called the Brauer group of Azumaya corings. This group is canonically isomorphic to the second flat cohomology group. We also give algebraic interpretations of the second Amitsur cohomology group and the first Villamayor-Zelinsky cohomology group in terms of corings.
We introduce the notion of Azumaya object in general homotopy-theoretic settings. We give a self-contained account of Azumaya objects and Brauer groups in bicategorical contexts, generalizing the Brauer group of a commutative ring. We go on to describe triangulated bicategories and prove a characterization theorem for Azumaya objects therein. This theory applies to give a homotopical Brauer group for derived categories of rings and ring spectra. We show that the homotopical Brauer group of an Eilenberg-Mac Lane spectrum is isomorphic to the homotopical Brauer group of its underlying commutativ…
We use the definition of differential operators on noncommutative rings given by V.Lunts and A.Rosenberg to find the differential operators on Azumaya algebras and the Heisenberg algebras.
Let $A$ be an algebra over a commutative ring $k$. We prove that braidings on the category of $A$-bimodules are in bijective correspondence to canonical R-matrices, these are elements in $A\ot A\ot A$ satisfying certain axioms. We show that all braidings are symmetries. If $A$ is commutative, then there exists a braiding on ${}_A\Mm_A$ if and only if $k\to A$ is an epimorphism in the category of rings, and then the corresponding $R$-matrix is trivial. If the invariants functor $G = (-)^A:\{}_A\Mm_A\to \Mm_k$ is separable, then $A$ admits a canonical R-matrix; in particular, any Azumaya algebra…
The main purpose of this paper is to investigate epsilon-strongly graded rings that are partial crossed products. Let $G$ be a group, $A=\oplus_{g\in G}\,A_g$ an epsilon-strongly graded ring and ${\bf pic}{R}$ the Picard semigroup of $R:=A_1$. We prove that the isomorphism class $[A_g]$ is an element of ${\bf pic}{R}$, for all $g\in G$. Thus, the association $g\mapsto [A_g]$ determines a partial representation of $G$ on ${\bf pic}{R}$ which induces a partial action $γ$ of $G$ on the center $Z(R)$ of $R$. Sufficient conditions for $A$ to be an Azumaya $R^γ$-algebra are presented in the case tha…
The definition of Azumaya algebras over commutative rings $R$ require the tensor product of modules over $R$ and the twist map for the tensor product of any two $R$-modules. Similar constructions are available in braided monoidal categories and Azumaya algebras were defined in these settings. Here we introduce Azumaya monads on any category $\A$ by considering a monad $\bF$ on $\A$ endowed with a distributive law $λ: FF\to FF$ satisfying the Yang-Baxter equation (BD-law). This allows to introduce an {\em opposite monad} $\bF^\la$ and a monad structure on $FF^\la$. For an {\em Azumaya monad} we…
We consider the general circumstance of an Azumaya algebra $A$ of degree $n$ over a locally ringed topos $(\mathbf{X}, {\mathcal{O}}_{\mathbf{ X}})$ where the latter carries a (possibly trivial) involution, denoted $λ$. This generalizes the usual notion of involutions of Azumaya algebras over schemes with involution, which in turn generalizes the notion of involutions of central simple algebras. We provide a criterion to determine whether two Azumaya algebras with involutions extending $λ$ are locally isomorphic, describe the equivalence classes obtained by this relation, and settle the questi…
A paper of U. First & Z. Reichstein proves that if $R$ is a commutative ring of dimension $d$, then any Azumaya algebra $A$ over $R$ can be generated as an algebra by $d+2$ elements, by constructing such a generating set, but they do not prove that this number of generators is required, or even that for an arbitrarily large $r$ that there exists an Azumaya algebra requiring $r$ generators. In this paper, for any given fixed $n\ge 2$, we produce examples of a base ring $R$ of dimension $d$ and an Azumaya algebra of degree $n$ over $R$ that requires $r(d,n) = \lfloor \frac{d}{2n-2} \rfloor + 2$ …
Let $F$ be an algebraically closed field of characteristic zero and let $G$ be a finite group. Consider $G$-graded simple algebras $A$ which are finite dimensional and $e$-central over $F$, i.e. $Z(A)_{e} := Z(A)\cap A_{e} = F$. For any such algebra we construct a \textit{generic} $G$-graded algebra $\mathcal{U}$ which is \textit{Azumaya} in the following sense. $(1)$ \textit{$($Correspondence of ideals$)$}: There is one to one correspondence between the $G$-graded ideals of $\mathcal{U}$ and the ideals of the ring $R$, the $e$-center of $\mathcal{U}$. $(2)$ \textit{Artin-Procesi condition}: $…
The differential Brauer monoid of a differential commutative ring R s defined. Its elements are the isomorphism classes of differential Azumaya R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them are differentially isomorphic. The Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the Brauer monoids of R and its ring of constants and the submonoid whose underlying Azumaya algebras are matrix rings.
We consider the following problem: Under what assumptions do one or more of the following are equivalent for a ring $R$: (A) $R$ is Morita equivalent to a ring with involution, (B) $R$ is Morita equivalent to a ring with an anti-automorphism, (C) $R$ is Morita equivalent to its opposite ring. The problem is motivated by a theorem of Saltman which roughly states that all conditions are equivalent for Azumaya algebras. Basing on the recent "general bilinear forms", we present a general machinery to attack the problem, and use it to show that (C)$\iff$(B) when $R$ is semilocal or $\mathbb{Q}$-fin…
Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra $A$ over a ring represents a $2$-torsion class in the Brauer group if and only if there is an algebra $A'$ in the Brauer class of $A$ admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose $A'$ such that $\mathrm{deg}\, A'=2\mathrm{deg}\, A$. We show that $2\mathrm{deg}\, A$ is the lowest degree one can expect in general. Specifically, we construct an Azumaya algebra $A$ of degree $4$ and period $2$ such that the degree of any algebra $A'$ in the Brauer class of $A$ admitt…
In this paper, we study triple derivations and triple homomorphisms of perfect Lie superalgebras over a commutative ring $R$. It is proved that, if the base ring contains $\frac{1}{2}$, $L$ is a perfect Lie superalgebra with zero center, then every triple derivation of $L$ is a derivation, and every triple derivation of the derivation algebra $ Der (L)$ is an inner derivation. Let $L,~L^{'}$ be Lie superalgebras over a commutative ring $R$, the notion of triple homomorphism from $L$ to $L^{'}$ is introduced. We proved that, under certain assumptions, homomorphisms, anti-homomorphisms, and sums…
We examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano's ring extension of the real numbers $^{\bullet}\mathbb{R}$ are smaller than any infinitesimal hyperreal number from Abraham Robinson's nonstandard analysis $^\ast\mathbb{R}$. Our approach consists in the study of two canonical order-preserving homomorphisms taking values in ${^{\bullet}\mathbb{R}}$ and in ${^\ast\mathbb{R}}$, respectively, and whose domain is Henle's extension of the real numbers in the framework of "non-nonstandard" analysis. In particular, we will show that there exists a nonzero…
The Artin-Schreier polynomial $Z^p - Z - a$ is very well known. Polynomials of this type describe all degree $p$ (cyclic) Galois extensions over any commutative ring of characteristic $p$. Equally attractive is the associated Galois action. If $θ$ is a root then $σ(θ) = θ+ 1$ generates the Galois group. Less well known, but equally general, is the so called "differential crossed product" Azumaya algebra generated by $x,y$ subject to the relation $xy - yx = 1$. In characteristic $p$ these algebras are always Azumaya and algebras of this sort generate the $p$ torsion subgroup of the Brauer group…
Let k be a base commutative ring, R a commutative ring of coefficients, X a quasi-compact quasi-separated k-scheme, A a sheaf of Azumaya algebras over X of rank r, and Hmo(R) the category of noncommutative motives with R-coefficients. Assume that 1/r belongs to R. Under this assumption, we prove that the noncommutative motives with R-coefficients of X and A are isomorphic. As an application, we show that all the R-linear additive invariants of X and A are exactly the same. Examples include (nonconnective) algebraic K-theory, cyclic homology (and all its variants), topological Hochschild homolo…
We investigate a notion of Azumaya algebras in the context of structured ring spectra and give a definition of Brauer groups. We investigate their Galois theoretic properties, and discuss examples of Azumaya algebras arising from Galois descent and cyclic algebras. We construct examples that are related to topological Hochschild cohomology of group ring spectra and we present a K(n)-local variant of the notion of Brauer groups.
We prove that Toen's secondary Grothendieck ring is isomorphic to the Grothendieck ring of smooth proper pretriangulated dg categories previously introduced by Bondal, Larsen and Lunts. Along the way, we show that those short exact sequences of dg categories in which the first term is smooth proper and the second term is proper are necessarily split. As an application, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injective properties: in the case of a commutative ring of characteristic zero, it distinguishes between dg Azuma…
Let $\mathcal{C}$ be a pocategory, $FI(\mathcal{C})$ the finitary incidence ring of $\mathcal{C}$ and $φ$ a Jordan isomorphism of $FI(\mathcal{C})$ onto an associative ring $A$. We study the problem of decomposition of $φ$ into the (near-)sum of a homomorphism and an anti-homomorphism.
Let $A$ and $B$ be unital rings. An additive map $T:A\to B$ is called a weighted Jordan homomorphism if $c=T(1)$ is an invertible central element and $cT(x^2) = T(x)^2$ for all $x\in A$. We provide assumptions, which are in particular fulfilled when $A=B=M_n(R)$ with $n\ge 2$ and $R$ any unital ring with $\frac{1}{2}$, under which every surjective additive map $T:A\to B$ with the property that $T(x)T(y)+T(y)T(x)=0$ whenever $xy=yx=0$ is a weighted Jordan homomorphism. Further, we show that if $A$ is a prime ring with char$(A)\ne 2,3,5$, then a bijective additive map $T:A\to A$ is a weighted Jo…
Let $R$ be a ring, a right ideal $I$ of $R$ is called small if for every proper right ideal $K$ of $R$, $I+K\neq R$. A ring $R$ is called right small injective if every homomorphism from a small right ideal to $R_{R}$ can be extended to an $R$-homomorphism from $R_{R}$ to $R_{R}$. Properties of small injective rings are explored and several new characterizations are given for $QF$ rings and $PF$ rings, respectively.
We study continuous homomorphisms between algebras of iterated Laurent series over a commutative ring. We give a full description of such homomorphisms in terms of a discrete data determined by the images of parameters. In similar terms, we give a criterion of invertibility of an endomorphism and provide an explicit formula for the inverse endomorphism. We study the behavior of the higher-dimensional residue under continuous homomorphisms.
A ring $R$ is called right $\aleph_{0}$-injective if every homomorphism from a countably generated right ideal of $R$ to $R_{R}$ can be extended to a homomorphism from $R_{R}$ to $R_{R}$. In this note, some characterizations of $\aleph_{0}$-injective rings are given. It is proved that if $R$ is semilocal, then $R$ is right $\aleph_{0}$-injective if and only if every homomorphism from a countably generated small right ideal of $R$ to $R_{R}$ can be extended to one from $R_{R}$ to $R_{R}$. It is also shown that if $R$ is right noetherian and left $\aleph_{0}$-injective, then $R$ is \emph{QF}. Th…
An associative ring with 1 is said to be semilocal provided it is semisimple artinian modulo its Jacobson radical, that is, modulo its Jacobson radical it is isomorphic to a finite product of matrices over division rings. Modules with a semilocal endomorphism ring inherit some properties of semisimple modules. For example, they can be decomposed into a finite direct sum of indecomposable submodules, such decomposition is not unique but there are only a finite number of different ones, they cancel from direct sums and they satisfy the n-th root uniqueness property. A ring homomorphism is said t…
In this paper we present a systematic study of the reflexivity properties of homologically finite complexes with respect to semidualizing complexes in the setting of nonlocal rings. One primary focus is the descent of these properties over ring homomorphisms of finite flat dimension, presented in terms of inequalities between generalized G-dimensions. Most of these results are new even when the ring homomorphism is local. The main tool for these analyses is a nonlocal version of the amplitude inequality of Iversen, Foxby, and Iyengar. We provide numerous examples demonstrating the need for cer…
By Rickard's work, two rings are derived equivalent if there is a tilting complex, constructed from projective modules over the first ring such that the second ring is the endomorphism ring of this tilting complex. In this work I describe, under some conditions, the homomorphism space in the derived category between two complexes as iterated pull-back of homomorphism spaces between the homologies of the complexes.
Thanks to recent results on ring homomorphisms of Azumaya algebras and to the following ones about endomorphisms of canonical Poisson algebras and Dirac quantum algebras, and about the reformulation in positive characteristic of these conjectures in characteristic zero, we prove the equivalence of these three conjectures in any characteristic, giving also by this way a new proof of the equivalence of the complex version of the two first conjectures recently proved by Y. Tsuchimoto.
This is an English translation of the author's Ph.D. thesis, accumulating his results on a construction of Cohen-Macaulay modules over a polynomial ring that appeared in the study of Cauchy-Fueter equations. This construction is generalized from quaternions to arbitrary finite-dimensional associative algebras. We show that for maximally central algebras (as introduced by Azumaya) this construction produces Cohen-Macaulay modules and is an exact functor (tensoring with a bimodule, actually) and this class of algebras cannot be enlarged. For this class several invariants of the resulting modules…
The purpose of this short note is to prove that if $A$ and $B$ are unital C*-algebras and $ϕ: A \to B$ is a unital *-preserving ring homomorphism, then $ϕ$ is contractive; i.e., $\| ϕ(a) \| \leq \| a \|$ for all $a \in A$. (Note that we do not assume $ϕ$ is linear.) We use this result to deduce a number of corollaries as well as characterize the form of such unital *-preserving ring homomorphisms. (This note may be of interest to C*-algebraists as well as algebraists who study noncommutative rings and algebras. It is meant to be accessible to a general mathematician and does not require any pr…
We study a ring containing a complete set of orthogonal idempotents as a generalized matrix ring via its Peirce decomposition. We focus on the case where some of the underlying bimodule homomorphisms are zero. Upper and lower triangular generalized matrix rings are pertinent examples of the class of rings which we study. The triviality of the particular bimodule homomorphisms motivates the introduction of three new types of idempotents, namely inner Peirce trivial idempotents, outer Peirce trivial idempotents and Peirce trivial idempotents. These idempotents provide the main tools in our inves…
In \cite{rupel3},the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category $\mathcal{A}$ to an appropriate $q$-polynomial algebra. In the case that $\mathcal{A}$ is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in \cite{DX}. Moreover, if the underlying graph of $Q$ is bipartite and the matrix $B$ associated to the quiver $Q$ is of full rank, we show that the image of the algebra homomorphisms is in the corresponding quantum cluster algebra.
We study properties of a group, abelian group, ring, or monoid $B$ which (a) guarantee that every homomorphism from an infinite direct product $\prod_I A_i$ of objects of the same sort onto $B$ factors through the direct product of finitely many ultraproducts of the $A_i$ (possibly after composition with the natural map $B\to B/Z(B)$ or some variant), and/or (b) guarantee that when a map does so factor (and the index set has reasonable cardinality), the ultrafilters involved must be principal. A number of open questions, and topics for further investigation, are noted.
Braman [B08] described a construction where third-order tensors are exactly the set of linear transformations acting on the set of matrices with vectors as scalars. This extends the familiar notion that matrices form the set of all linear transformations over vectors with real-valued scalars. This result is based upon a circulant-based tensor multiplication due to Kilmer et al. [KMP08]. In this work, we generalize these observations further by viewing this construction in its natural framework of group rings.The circulant-based products arise as convolutions in these algebraic structures. Our …
We define a functor which gives the "global rank of a quiver representation" and prove that it has nice properties which make it a generalization of the rank of a linear map. We demonstrate how to construct other "rank functors" for a quiver Q, which induce ring homomorphisms (called "rank functions") from the representation ring of Q to Z. These rank functions give discrete numerical invariants of quiver representations, useful for computing tensor product multiplicities of representations and determining some structure of the representation ring. We also show that in characteristic 0, rank f…
Given a surjective ring homomorphism, we study when the induced group homomorphism on unit groups is surjective. To this end, we introduce notions of generalized inverses and units, as well as a class of rings such that the set of closed points in the spectrum is a closed set. It is shown that any surjection out of such a ring induces a surjection on unit groups.
Under semi-weak and weak compatibility of bimodules, we establish sufficient and necessary conditions of Gorenstein-projective modules over rings of Morita contexts with one bimodule homomorphism zero. This generalises and extends results on triangular matrix Artin algebras and on Artin algebras of Morita contexts with two bimodule homomorphisms zero in the literature, where only sufficient conditions are given under a strong assumption of compatibility of bimodules. An application is provided to describe Gorenstein-projective modules over noncommutative tensor products arising from Morita con…
We continue the first and second authors' study of $q$-commutative power series rings $R=k_q[[x_1,\ldots,x_n]]$ and Laurent series rings $L=k_q[[x^{\pm 1}_1,\ldots,x^{\pm 1}_n]]$, specializing to the case in which the commutation parameters $q_{ij}$ are all roots of unity. In this setting, $R$ is a PI algebra, and we can apply results of De Concini, Kac, and Procesi to show that $L$ is an Azumaya algebra whose degree can be inferred from the $q_{ij}$. Our main result establishes an exact criterion (dependent on the $q_{ij}$) for determining when the centers of $L$ and $R$ are commutative Laure…
For any countable graph $E$, we investigate the relationship between the Leavitt path algebra $L_{\C}(E)$ and the graph C*-algebra $C^*(E)$. For graphs $E$ and $F$, we examine ring homomorphisms, ring *-homomorphisms, algebra homomorphisms, and algebra *-homomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$. We prove that in certain situations isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ yield *-isomorphisms between the corresponding C*-algebras $C^*(E)$ and $C^*(F)$. Conversely, we show that *-isomorphisms between $C^*(E)$ and $C^*(F)$ produce isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ …
