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3
votes
1answer
73 views

Motivation for the Preprojective Algebra

Let $Q=(Q_0,Q_1)$ be a quiver and $k$ a field. We construct a new quiver $\bar{Q}$ in the following way. Let the vertices of $\bar{Q}$ be the same as the vertices of $Q$, and let the arrows of ...
1
vote
1answer
77 views

Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?
2
votes
0answers
53 views

Poincaré Duality of a quasi-free algebra

I'm completely stumped on this one (yet I feel it is obviously true or obviously false) If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality? All i need to find is a protective ...
1
vote
0answers
43 views

For Finite Dual when is $(A \otimes A)^o = A^o \otimes A^0$?

Let $A$ be any $k$-algebra. The finite dual or restricted dual of $A$ is $$ A^o = \{f \in A^* ~ | ~ f(I)= 0, \text{ for some ideal } I \subseteq A, \text{ such that } \text{dim}_k(A/I) < \infty\}. ...
0
votes
0answers
65 views

Explicit calculation of module of derivations on noncommutative polynomial ring

Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$. Explicitly how would one go about computing ...
0
votes
1answer
85 views

Jacobi-Zariski exact sequence question

Denote by $HC(A,M)$ the Hochschild homological complex of an algebra $A$ with coefficients in an $A$-bimodule $M$, and let $B\rightarrow A$ be an $R$-flat extension of $R$-algebras, for some $CRing$ ...
2
votes
1answer
95 views

Endomorphism Ring of Indecomposable MCM Modules

Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...
3
votes
1answer
192 views

Hochschild homology of quiver algebras

Let $K$ be a field and $\Gamma$ a quiver (=multidigraph) and $K[\Gamma]$ its quiver algebra (free $K$-module on the set of all paths of length $\geq0$ where multiplication is concatenation if ...
8
votes
1answer
223 views

Explicit isomorphism for quaternion algebras over $\mathbb{Q}$?

It is known that the isomorphism class of a quaternion algebra $A=\binom{a,b}{K}$ over a number field $K$ is determined by the finite set of places $v$ of $K$ where $A\otimes_K K_v$ is a division ...
5
votes
2answers
220 views

Hochschild homology of upper triangular matrix algebra?

Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim_K$ of its hochschild homology $HH_k(A;A)$? Is there any ...
1
vote
1answer
39 views

Integral elements of quaternion algebras with predescribed properties

In the course of doing some calculations I have found myself wanting to answer the following question: Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let ...
13
votes
2answers
491 views

Why is “naive” definition of non-commutative spectrum bad?

It is well-known that the category of affine schemes is equivalent to the opposit category of commutative unital rings. So naively, one would think that the same should hold in non-commutative ...
2
votes
0answers
138 views

Discriminants of Clifford algebras

I have a Clifford algebra defined over a field of characteristic not equal to 2. Is there a formula for its discriminant in terms of the corresponding symmetric bilinear form (or in terms of its ...
1
vote
1answer
210 views

Computing noncommutative geometries

I find myself needing to construct some noncommutative geometries. I want to take various (algeba-) geometric objects and look at their noncommutative analogs. Is there a constructive way to do this? ...
0
votes
0answers
60 views

Problem with Smoothness and quasi-freeness

Let A be a unital associative algebra over a field k. Then A is smooth if and only if X:=Spec(A) is smooth. That is $\Omega_{X|Spec(k)}$ is locally-free. The later module is isomorphic to ...
0
votes
0answers
133 views

Is the universal enveloping algebra of the free Poisson algebra generated by finite set (left)-noetherian?

Let $P$ be the free Poisson algebra over $k$ (a field) generated by a finite set $x_1,\dots,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. A Poisson ...
3
votes
0answers
154 views

Computing the Abelianization of an Automorphism Group

Setup: We are working in a Henselian local ring $(R, \mathfrak m, k)$ that way may assume is Cohen-Macaulay, admits a canonical module and is of finite type (so is an isolated singularity). Let ...
1
vote
0answers
40 views

When is the dual module isomorphic to conjugate module of a *-algebra

Let $A$ be a $\ast$-algebra, and let ${\cal M}$ be a module over $A$. As usual, we denote the dual module of $M$ by $M^*$. Moreover, we denote the conjugate module of $M$ by $\overline{M}$, where as ...
1
vote
1answer
113 views

Variety of factorizations of differential operator

Take differential operator as polynomial of letter $d$ with coefficients in some function field, where $d$ act by derivation in this function field. Call it a differential field. For simplicity let ...
1
vote
0answers
81 views

Chinese Remainder Theorem

On a non-commutative ring we have this generalization of CRT. I wonder if there is a statement involving only left or right ideals. do you know any?
2
votes
1answer
135 views

$Aut(\mathbb{Z}G)=?$ for $G=\mathbb{Z}^2\rtimes_n\mathbb{Z}$

I am interested in the automorphism group of the group ring $\mathbb{Z}G$ for some noncommutative group $G$ of the form $\mathbb{Z}^2\rtimes_n\mathbb{Z}$, say ...
3
votes
2answers
307 views

noncommutative polynomials equality

Suppose $x$, $y$, $z$ are three variables satisfying $yz=zy$, $zx=xz$, $xy=yzx$. Could anyone give me two (non-commutative) polynomials $f$ and $g$ in the above three variables such that the ...
4
votes
3answers
406 views

Units in a group algebra

Let k be a field and let G be a finite group. I would like to know if there is any nice description of the group of units in the group algebra kG. (If there is no nice answer in this generality, ...
2
votes
0answers
52 views

The norm of a polynomial f in a skew polynomial ring must be in the center

This is proved in Prop 1.7.1 in Jacobson's book ``Finite dimensional division algebras over fields". But I am not clear why the norm n(f), defined as the norm of the matrix representation of f by ...
0
votes
1answer
83 views

$\mathrm{rk}_R M$ vs $\mathrm{rk}_S M$ - how nice need $R,S$ be?

Let $R\hookrightarrow S$ be Noetherian (noncommutative) rings without zero divisors with $\mathrm{rk}_{R} S < \infty$ (e.g. $S=R*G$ the crossed product of $R$ with a finite group $G$). Let $M$ be a ...
8
votes
1answer
326 views

For $G=\mathbb{Z}^2\rtimes \mathbb{Z}$, $Spec(\mathbb{Z}G)$=?

Let $G$ be the group $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}=\langle y,z\rangle\rtimes_{\sigma}\langle x\rangle$, where $\sigma(x)=\begin{pmatrix}a, b\\c,d\end{pmatrix}\in SL_2(\mathbb{Z})$, which ...
4
votes
0answers
119 views

Injectivity criterion for surjective coalgebra maps: does it hold in full generality?

Let $k$ be a commutative ring. Let $C$ be a filtered $k$-coalgebra. This means a $k$-coalgebra equipped with an increasing $k$-module filtration $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ ...
5
votes
2answers
189 views

How big can a commutative subalgebra of Weyl algebra be?

Consider the smallest Weyl algebra $A_1=\{q,p; qp-pq=1\}$. It is known that there exist pairs of commuting elements, say $L$ and $M$, that obey various polynomial relations, e.g. elliptic curves. I ...
5
votes
2answers
128 views

Dimension of commutative subalgebras of a central simple algebra

let $k$ be a field, and let $A$ be a central simple $k$-algebra over $k$. What is the maximal dimension of a commutative $k$-subalgebra of $A$? If $A=M_r(D)$, where $D$ is a central division ...
1
vote
1answer
96 views

Conjugate linear maps between $*$-algebra modules

Let $A$ be a $*$-algebra, $E,$ and $F$ two $A$-modules, and a map $f:E \to F$ such that $$ f(ae) = a^*f(e), ~~~~~~~ a \in A. $$ This seems to me to be the natural generalisation of a conjugate linear ...
6
votes
2answers
436 views

Structure theorem for finite dimensional $C^*$-algebras and their representations

I would like a source for some Artin-Wedderburn type facts about these algebras which seem to have easy proofs, and are probably written somewhere. Let $\mathcal{A} \subset M_n(\mathbb{C})$ be an ...
11
votes
1answer
318 views

$\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group. Denote the ...
14
votes
1answer
355 views

Geometry of numbers for three by three matrices?

While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem: What is the volume of the largest symmetric convex subset ...
2
votes
2answers
160 views

Normal regular sequence in noncommutative algebras

Does anyone know anything about the normal regular sequences in the quantum plane? Here are the definitions: Normal regular sequence: Let $R$ be a ring (not necessarily commutative). A sequence ...
0
votes
1answer
143 views

Non-simple and non-unital rings with trivial centres

Let $R$ be an associative and non-unital ring. (Suppose that $R$ is $s$-unital, i.e. for each $x\in R$ there is $u,v\in R$ such that $ux=xv=x$.) It is not difficult to show that if $R$ is a simple ...
1
vote
2answers
265 views

Gerstenhaber bracket out of $L_\infty$ algebras

Given a Lie algebra g, with $Ug$ being its universal enveloping algebra, one can construct a cochain complex $d: Ug^n \rightarrow Ug^{n+1}$, and a Gerstenhaber bracket on $\oplus_n Ug^n$ so that ...
5
votes
0answers
121 views

Central Element in Sklyanin Algebras?

I'm interested in Sklyanin Algebras or Artin-Shelter regular algebras of type A. These are generated in degree 1 by three variables x,y,z, and have three defining relations in degree 2, which you can ...
10
votes
4answers
333 views

Applications of Govorov-Lazard Theorem?

I asked this question on SE a long time ago, but never received an answer: The Govorov-Lazard Theorem states that a (left) module over an unital ring is flat iff it is a direct limit of finitely ...
9
votes
0answers
221 views

Does there exist a Noetherian ring of finite injective dimension but higher Krull dimension?

Definition: a (not necessarily commutative) left and right Noetherian ring $R$ is said to be Auslander-Gorenstein if (i) $R$ has finite left and right injective dimension (in which case it turns out ...
12
votes
0answers
318 views

Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
2
votes
1answer
139 views

Algorithmically finite-dimensional (noncommutative) algebras.

Can anyone help to find some information about these structures?
9
votes
0answers
441 views

Is “being a full ring of quotients” a Morita invariant property?

Definition and context: An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
5
votes
2answers
297 views

Properties of ring epimorphisms that are true only over commutative rings

I'm interested in knowing/collecting some properties of epimorphisms of rings (with identity) that are true over commutative rings but are false in the non-commutative case. Example: I learned from ...
4
votes
2answers
228 views

Projectives in the category of discrete G-modules

If $G$ is a profinite group, then the category $Mod(G)$ of discrete $G$-modules has sufficiently many injectives (Neukirch, Schmidt, Wingberg: Cohomology of Number Fields, 2.6.5). Since the cited ...
2
votes
0answers
156 views

What are the enforceable models of local artinian rings?

I was reading Hodges' "Model Theory" Chapter 8 a propos existentially closed models of $\forall_2$ theories in a countable first order language $L$. He extends the proof of the omitting type theorem ...
1
vote
0answers
194 views

Annihilator ideals

For an ideal $I$ of a ring $R$ with identity, let $r(I)=\{r\in R: Ir=0\}$ and $l(I)=\{r\in R: rI=0\}$. Question: If for any two ideals (two-sided ideal) $I, J$ of $R$, we have $l(I)+l(J)=l(I\cap J)$, ...
4
votes
1answer
79 views

Example of computation of moduli space of $n$-pointe modules?

I am looking for an example of computation of the isomorphism classes of $n$-point modules over a non-commutative generated graded algebra (assuming all good properties such as Noetherian property). ...
5
votes
2answers
680 views

Epimorphisms and free submodules

By inspecting the accepted answer to this question Are epimorphisms from a division ring isomorphisms ? one obtains the following necessary condition for epimorphisms: Let $R \le S$ be rings ...
6
votes
3answers
673 views

Are epimorphisms from a division ring isomorphisms ?

According to Corollary 1.2(3) of the paper Silver: Noncommutative Localizations and Applications. J. of Alg. 7(1964), 44-67: If $R$ is a (commutative) field and $\alpha: R \to S$ an epimorphism in ...
3
votes
0answers
132 views

AS Cohen Macaulay algebras and dualizing complexes

Let $A$ be an $\mathbb N$-graded algebra such that $A_0 = k$ is a field. This are usually called graded connected algebras. One can define a torsion functor with respect to the ideal $\mathfrak m = ...