All Questions
Tagged with ct.category-theory derived-algebraic-geometry
17 questions
6
votes
1
answer
274
views
Reference: the category of derived affine schemes is extensive
The category (that is, $(\infty, 1)$-category) of derived affine schemes is the opposite category of the localization of simplicial commutative rings in weak equivalences.
See extensive category. Does ...
- 6,962
2
votes
0
answers
354
views
Higher-order HKR theorems?
Recall that Hochschild-Kostant-Rosenberg -type theorems identify certain smoothness conditions guaranteeing an isomorphism between the cotangent complex and (a shift of) the Hochschild homology of an ...
- 64k
3
votes
1
answer
176
views
Left adjoint for nested admissible categories
This question is motivated by the construction of the Kuznetsov component on a prime Fano threefold $X$ of index 1 (say genus $g \geq 6$, $g \neq 7, 9$):
$$
D^b(X) = \langle Ku(X), E, \mathcal{O}_X \...
- 317
5
votes
1
answer
465
views
Questions about $\text{Perf}(A)$ of dg algebra $A$
[ALEXEY ELAGIN AND VALERY A. LUNTS, p.4.] Recall that triangulated category $\text{Perf}(A)$ is defined as the full thick triangulated subcategory of $D(A)$ generated by the dg $A$-module $A$.
[...
- 603
9
votes
2
answers
380
views
How to characterize $E_n$ morphisms from $\mathrm{Mod}(A)$ to $\mathrm{Mod}(B)$?
Suppose $A$ and $B$ are $E_{\infty}$ rings, then $\mathrm{Mod}(A)$ and $\mathrm{Mod}(B)$ are $E_{\infty}$ monoidal categories (left modules over those rings). We can ask about $E_n$ colimit-preserving ...
- 2,356
3
votes
0
answers
162
views
Homotopy Kan extensions, formally coherent functors and derived Schlessinger criterion
Let $k$ be a finite field. Denote by $discArt_k$ the category of Artinian rings with residue field $k$ and $Art_k$ the category of Artinian simplicial rings. Consider a functor $\mathcal{F}:disArt_k\...
- 4,418
4
votes
0
answers
246
views
Tannaka duality for $DG$ indschemes
In Lurie's paper on Tannaka duality for geometric stacks he proves that there is a natural isomorphism
$$\operatorname{Hom}(X,Y) \cong \operatorname{Hom}(\mathbf{QC}(Y), \mathbf{QC}(X)$$
where $X$ and ...
- 3,019
2
votes
0
answers
181
views
Why does the following construction describe the Serre functor?
In the book "Spectral Agebraic Geometry" that Jacob Lurie is currently writing, he gives a construction (11.1.5.1), which describes the Serre functor: it has already been shown that any proper $A$-...
- 21
8
votes
3
answers
471
views
Cofiber of the inclusion of an $E_0$-algebra $M$ into the free $E_k$-algebra generated by it
Let $\mathcal{C}$ be the $E_k$-monoidal $\infty$-category of left modules over a fixed connective $E_{k+1}$-ring spectrum $A$. Suppose that $M$ is an object of $\mathcal{C}$ which is an $E_0$-algebra, ...
- 10.5k
5
votes
0
answers
447
views
Infinity categories with an action of a simplicial group
Recent papers in derived algebraic geometry use a notion of $S^1$-actions on infinity categories. I think I understand what this "should" be and how to calculate with it; however, I can't find a much ...
- 1,423
2
votes
0
answers
157
views
Why is the stabilization of augmented $\mathbb{E}_\infty$-algebras equivalent to $k$-module spectra?
(I have already asked this on Math.SE, but it didn't draw much attention there, so I am reposting it here.)
Example 1.1.4 of Jacob Lurie's DAGX says that the stabilization $\operatorname{Stab}((\...
- 95
19
votes
3
answers
2k
views
Is it always possible to write a scheme as a colimit of affine schemes?
My question is: Is it possible to write any scheme as a (1-categorical) colimit of a diagram of affines? If no, what are some examples?
I ask this question because I have read that one can write any ...
- 595
10
votes
1
answer
851
views
Bar/Cobar Adjunction Between Modules and Comodules
There is a pretty well known, and widely written about, adjunction between augmented algebras and coaugmented coalgebras given by taking the bar construction on algebras and the cobar construction on ...
- 10.5k
7
votes
2
answers
672
views
Direct proof that the model category of cdgas is left proper
Let $k$ be a field of characteristic $0$. The projective model structure on the category $cdga$ of commutative differential graded $k$-algebras is proper. Since this model structure is transferred ...
- 15.5k
3
votes
1
answer
465
views
Monoidal Forgetful/Free Adjunction for $E_2$-algebras
Suppose I am given two $E_2$-ring spectra $A$ and $B$ and a morphism of $E_2$-rings $\phi:A\to B$. Then I have $E_1$-monoidal categories of modules $LMod_A$ and $LMod_B$. Moreover I have morphisms $F:...
- 10.5k
2
votes
1
answer
246
views
Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?
Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: \...
- 54.6k
6
votes
2
answers
944
views
Small objects in categories
I would like to pick out small objects from a category. I would like to find such a notion which
Dream 1. Picks out the schemes of finite type over $k$ from the category of $k$-schemes. Or at least ...
- 213