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2 votes
1 answer
306 views

Serre functors and global dimensions

Let $k$ be a field. Let $\mathcal{C}$ be an abelian category (over $k$). We say that $\mathcal{C}$ has a finite global dimension if there exists integer $n > 0$ such that $$ \operatorname{Ext}^i(M,...
YkMz's user avatar
  • 889
5 votes
1 answer
197 views

Examples of cyclic A-infinity algebra

I am wondering about (references to) examples of cyclic A-infinity algebras- especially including explicit descriptions of the structure maps and pairing. Thanks a lot!
Jak's user avatar
  • 51
9 votes
1 answer
236 views

Formal smoothness of path algebras and connections

Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if $$ \Omega^1_kA = \operatorname{Ker}(\...
Qwert Otto's user avatar
7 votes
0 answers
574 views

Why is Hochschild homology interesting if its cohomology groups are infinite-dimensional?

I am trying to understand Hochschild homology, in particular the Hochschild–Kostant–Rosenberg theorem. As far as I understand this result gives an isomorphism between the algebraic (Kähler) ...
Jake Wetlock's user avatar
  • 1,144
13 votes
1 answer
669 views

Is a "smooth" finite-dimensional algebra separable modulo its radical?

Let $k$ be a field, and let us write the "unadorned" tensor $\otimes$ in place of $\otimes_k$. For a unital finite-dimensional $k$-algebra $A$, let $A^e = A \otimes A^{op}$ denote the enveloping ...
Manny Reyes's user avatar
  • 5,407
9 votes
2 answers
2k 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 x_1,\dots,x_n\rangle/...
user2013's user avatar
  • 1,663