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 ...
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 ...
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 ...
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
1answer
524 views

Classification of long exact sequences

Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish. The category $\mathcal C$ is naturally additive as a subcategory of ...
7
votes
2answers
960 views

Global dimensions of non-commutative rings

This is related to my previous question: When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau? I now would like to know the global dimension of the ring $R=\mathbb{C}\langle ...
2
votes
0answers
338 views

Noncommutative analog of Koszul complex

Let $R=R_0 \oplus R_1 \oplus R_2 ...$ be a graded not necessarily commutative algebra over field $k$ and $R$ is generated by $R_1$, $R_0=k$. In commutative situation if one wants free resolution of ...
4
votes
1answer
369 views

A Question on Koszul duality and $B(\infty)$ structures on $HH^*$

The following theorem is known from a paper "Duality in Gerstenhaber Algebras" by Felix, Menichi, Thomas. Given a simply connected space X of finite type. There is an equivalence of Gerstenhaber ...
3
votes
2answers
371 views

An example where finitistic dimension does not equal right global dimension?

The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup ...
6
votes
1answer
266 views

A potential resolution of $R/r$

The DGA For $k$ some field, let $R$ be a $k$-algebra, and let $r\in R$. Define a differential graded algebra $\mathbf{R}_r$ as follows. As a graded algebra, it is isomorphic to $R\langle t\rangle$, ...
2
votes
1answer
297 views

Finite Homological Dimension of R/P for all P for module finite non-commutative rings

I have a reasonably precise question which I hope is clear enough to get a nice answer. Let R be a Noetherian non-commutative ring which is finite as a module (and flat/free if it helps) over it's ...
4
votes
1answer
241 views

Classifying Algebra Extensions over a fixed extension?

There are lots of "Ext groups" in homological algebra which measure extensions of various things. I'm sure there must be a homological algebra machine for computing the following, and I'm hoping that ...
4
votes
7answers
1k views

Hochschild/Cyclic Homology of von Neumann Algebras: Useless?

Hochschild homology gives invariants of (unital) $k$-algebras for $k$ a unital, commutative ring. If we let our algebra $A$ be the group ring $k[G]$ for $G$ a finite group, we get group homology. ...