All Questions
Tagged with noncommutative-geometry ct.category-theory
22 questions
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,...
- 889
7
votes
0
answers
369
views
Bondal-Orlov' theorem for noncommutative projective schemes
My question is very simple.
Is Bondal-Orlov's theorem known for noncommutative projective schemes in the sense of Artin and Zhang?
The commutative version is the following :
Let $X, Y$ be smooth ...
- 889
-3
votes
1
answer
325
views
Is the category of $Z^* $ algebra equivalent to the category of $C^*$ algebras
A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor.
They form a category with usual structures.
Question. Is this category equivalent to the category of $C^*$ algebras?
...
- 356
3
votes
0
answers
184
views
Hochschild homology of stable categories as topological chiral homology
Sorry for repost from Math Stack Exchange:
Let $\mathscr{C}_0$ be a small idempotent complete stable category tensored over some symmetric monoidal category $\mathcal{E}$.
Its Ind-completion $\mathscr{...
- 525
15
votes
1
answer
672
views
Reference for the Swan-Serre theorem as a monoidal equivalence
Let $X$ be a compact Hausdorff The well-known Swan--Serre theorem gives an equivalence between the continuous vector bundles over a compact Hausdorff space $X$, and finitely-generated projective $C(X)$...
- 393
3
votes
1
answer
446
views
A set of objects classically generates the full subcategory of compact objects iff it generates the whole category
Sorry in advance if my question doesn't have the level of this community.
I am studying this paper of Bondal and Van Den Bergh and in particular section 2. Generators and resolutions in triangulated ...
- 224
8
votes
1
answer
538
views
Vanishing of Hochschild homology of a category
Let $A$ be a dg- or $A_{\infty}$-category (with $\mathbb{Z}$-graded Hom sets, over a field of characteristic $0$). Let $HH_*(A)$ be the Hochschild homology of $A$.
Suppose that $HH_n(A)=0$ for all $n ...
- 937
12
votes
0
answers
688
views
Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'
It appears to me (though I may be wrong) that the common opinion is that the main difference between derived noncommutative geometry and Rosenberg's noncommutative 'spaces' is that Rosenberg's version ...
- 439
3
votes
0
answers
227
views
Categorical features of Hilbert spaces
Does the category of Hilbert spaces and bounded maps have any particular categorical feature which can be studied systematically?
I mean, I know that it's a $*$-category, but it seems to have much ...
user95283
10
votes
1
answer
477
views
Equivalence of rational Voevodsky motives: partial Converse to Conjecture of Orlov
There is a conjecture of Orlov stating that if $X$ and $Y$ are smooth projective complex varieties that are derived equivalent (equivalent bounded derived categories of coherent sheaves), then their ...
- 143
7
votes
1
answer
263
views
Most natural equivalence between $C^*$-algebras in NCG
I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism.
Can someone explain this sentence or ...
- 561
13
votes
1
answer
500
views
Is there something like "Noncommutative geometry internal to a category"?
I have heard that one can do algebraic geometry internal to symmetric monoidal categories. Topological quantum field theories also exist internal to symmetric monoidal categories, and the usual ...
- 5,565
16
votes
0
answers
591
views
Lifting DG-categories to characteristic zero
The question of lifting (smooth projective) varieties from an algebraically closed field $k$ of characteristic $p$ to characteristic zero (i.e., to the Witt vectors $W(k)$) is a classical one. It's ...
- 25.6k
1
vote
1
answer
416
views
Are these two "FUNCTORS" adjoint?
I am considering the following correspondence:
Let $X$ be quasi compact quasi separated schemes.Consider a pseudo functor \begin{equation}Sch\rightarrow CAT :U\mapsto Qcoh(U),f:U\rightarrow V\mapsto f^...
- 1,982
16
votes
2
answers
3k
views
If the direct image of f preserves coherent sheaves on noetherian schemes, how to show f is proper?
The other direction is well known.
I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove. I am also wondering ...
- 1,982
1
vote
3
answers
450
views
Smooth affine algebras are Calabi-Yau
Are all smooth affine algebras over a field Calabi-Yau?
I'm thinking yes since they satisfy Van den Bergh duality with dualizing module themselves (have I made a mistake in this reasoning)/
- 27
5
votes
0
answers
547
views
Open question: non-commutative site following Grothendieck, Quillen, Connes and Crane for quantum gravity.
This is an open question and it's to find out who is interested in this kind of thing, who can benefit from thinking about this. It is very brief but hopefully will only be unclear to people who are ...
- 171
15
votes
1
answer
1k
views
Convolution algebras for double groupoids?
There is a lot of work of course on convolution algebras of measured groupoids, and this gives "Noncommutative geometry". However there is a lot of interest in algebraically structured groupoids, for ...
- 12.3k
9
votes
2
answers
667
views
Is a groupoid determined by its Hopfish algebra?
This is a follow up to my question What is the precise relationship between groupoid language and noncommutative algebra language?. I will briefly review some definitions; for details, a good place ...
- 54.6k
21
votes
3
answers
2k
views
What is the precise relationship between groupoid language and noncommutative algebra language?
I have sitting in front of me two 2-categories. On the left, I have the 2-category GPOID, whose:
objects are groupoids;
1-morphisms are (left-principal?) bibundles;
2-morphisms are bibundle ...
- 54.6k
9
votes
4
answers
2k
views
is localization of category of categories equivalent to |Cat|
It might be a stupid question.
Suppose There is a category of categories,denoted by CAT,where objects are categories, morpshims are functors between categories
Take multiplicative system S={category ...
- 5,505
14
votes
2
answers
982
views
Recovering a monoidal category from its category of monoids
What kind of additional properties and/or structures one needs to impose on the category
of (commutative or noncommutative) monoids of some monoidal category
so that one can recover the original ...
- 37.8k