All Questions
Tagged with derived-algebraic-geometry algebraic-stacks
14 questions
13
votes
1
answer
961
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Several simple questions on the geometry of higher stacks
I'm trying to understand definition/work out some examples. So, there are some simple questions about higher stacks.
For the simplicity assume that we are working with higher DM (Deligne-Mumford) ...
- 525
9
votes
0
answers
287
views
derived schemes and perfect obstruction theories
In a survey article of Toen's it is claimed that that there is forgetful $\infty$-functor between the $\infty$-category of derived schemes locally of finite presentation over a field $k$ and the $\...
- 131
8
votes
1
answer
721
views
Milnor excision for algebraic stacks
Recall that a commutative square of commutative rings
$$\begin{matrix}
A&\to&B\\
\downarrow &&\downarrow\\
A^\prime&\to&B^\prime\end{matrix}$$
is called a Milnor square if the ...
- 19.6k
8
votes
1
answer
1k
views
Derived noncommutative geometry includes derived, or spectral algebraic geometry?
Let $k$ be a commutative ring. In derived noncommutative (algebraic) geometry a "noncommutative space over $k$" is a $k$-linear $\mathrm{DG}$-category.
This is motivated by the fact that homological ...
- 439
7
votes
1
answer
629
views
Canonical comparison between $\infty$ and ordinary derived categories
This question is a follow-up to a previous question I asked.
If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...
- 1,349
4
votes
0
answers
310
views
Dimension of derived Artin stacks and perfect complexes
I am interested in the concept of dimension of derived and $n$-Artin stacks. Take for example the definition 4.10 of From HAG to DAG: derived moduli stacks. in which they define the dimension of a ...
- 787
4
votes
0
answers
242
views
Topological invariance of periodic cyclic homology of stacks
Goodwillie proved (in Cyclic homology, derivations, and the free loopspace) that the periodic cyclic homology of a connective dg algebra is that of its reduced classical ring. Preygel proved (in Ind-...
- 1,423
3
votes
0
answers
317
views
Reference request: Derived structure on the moduli stack of Higgs bundles
I am reading arXiv:1708.08124. When talking about the moduli stack of Higgs bundles on a projective curve $X$. It is said on page 59, first paragraph that
It is often better to put
derived ...
- 385
3
votes
0
answers
286
views
Exterior tensor of derived categories of coherent sheaves
Let $X, Y$ be Noetherian perfect derived stacks over $S$ a regular perfect derived stack. Consider the exterior tensor functor
$$\text{DCoh}(X) \otimes_{\text{DCoh}(S)} \text{DCoh}(Y) \rightarrow \...
- 1,423
2
votes
0
answers
181
views
Dualizing sheaf for classifying stack and duality
For an algebraic group $G$ there should be an equivalence $\operatorname{Rep}(G) \simeq \operatorname{IndCoh}(BG)$. I'm trying to understand what the dualizing sheaf (or complex) of $BG$ is. Here's ...
- 834
2
votes
0
answers
327
views
Linear $\infty$-categories $\mathrm{QC(X)}$ and $\mathrm{Perf(X)}$ of a "derived" stack $\mathrm{X}$
For each scheme or algebraic stack their $\infty$-category of quasicoherent sheaves (resp., perfect complexes) on it is $k$-linear for a commutative ring $k$. That is (by a recent result of L.Cohn), ...
- 21
1
vote
0
answers
224
views
Two definitions of cotangent complex
I have reading a paper by Professor Pridham(https://arxiv.org/abs/0905.4044v4). Page 47-48 contains a comparison of the two definitions of the cotangent complex, but there is a part I don't understand....
- 479
1
vote
0
answers
185
views
Is there a stacky definition of irreducible symplectic manifold?
I am now interested in studying symplectic structures in the field of stacks.
In particular, is there a stacky definition of irreducible symplectic manifold ?
I'm also interested in similar things in ...
- 479
1
vote
0
answers
148
views
Perfect complexes on affine schemes
I'm reading a paper on algebraic stacks and in some part is stated the following:
Let $X$ be an algebraic stack and let
$P\in D_{qc}(X)$ be a perfect complex. Then, for every $x\in |X|$, there ...
- 41