All Questions
Tagged with derived-algebraic-geometry homotopy-theory
46 questions
4
votes
1
answer
376
views
Is the geometric realization of simplicial functors interesting?
While studying a completely unrelated problem, I have proved something on the following line: given a diagram of simplicial sets $X: C \to \textrm{sSet}$, some deformations of the geometric ...
- 2,152
4
votes
1
answer
329
views
Infinite suspension is cotangent complex
In Higher Algebra by Lurie, we define the absolute cotangent complex $L_{A}$ through the composition $C\stackrel{\triangle}{\longrightarrow} Fun(\triangle^{1},C)\stackrel{F}{\longrightarrow}T_{C}$ ...
- 618
3
votes
0
answers
225
views
Derived $\infty$-category of quasi-coherent sheaves on schemes
Let $X$ be a scheme. On the one hand, we have the derived $\infty$-category constructed from the abelian category of quasi-coherent sheaves on $X$. On the other hand, we can define the stable $\infty$-...
- 151
6
votes
0
answers
170
views
New investigations on Homotopical Algebraic Contexts
Homotopical algebraic context are models that allows Toën and Vezzosi to do derived geometry. It have been defined in their seminal paper Homotopical Algebraic Geometry II.
These are general abstract ...
- 177
4
votes
2
answers
413
views
“Geometric” vs Homotopical completion
There are two notions of completions of slightly different nature, and I am wondering if there is a precise statement relating them.
The first one is the “homotopical” (or maybe it should be called ...
- 1,374
2
votes
0
answers
116
views
Quasicompact quasiaffine classical schemes are nonconnectively-affine
In this answer to What is the relationship between connective and nonconnective derived algebraic geometry? I learned that any quasicompact open subscheme of an affine scheme is affine in the sense of ...
- 448
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
2
votes
0
answers
189
views
Is the homotopy limit of derived schemes along affine maps a derived scheme?
The title question is true in the setting of ordinary limits and ordinary schemes; that is, given an inverse limit of schemes along affine maps, the limit still lives in the category of schemes.
I'd ...
- 301
9
votes
1
answer
748
views
In the not necessarily abelian cat setting, is there a Grothendieck spectral sequence for computing the homotopy of a composition of derived functors?
Recall the Grothendieck spectral sequence which computes the homology groups of a composition of left derived functors $F$ and $G$ on abelian categories:
\begin{align*}
E_{p,q}^{2}(A)=L_{p}G\circ L_{q}...
- 301
5
votes
1
answer
298
views
Interpolating between the flat and smooth affine lines in spectral algebraic geometry
Consider the following construction (which came up recently in a question about "spectral exterior algebras"):
Pick a ring spectrum $R$ and consider the $\infty$-category $\mathsf{Mod}_R$ ...
- 11.8k
21
votes
1
answer
2k
views
Why does elliptic cohomology fail to be unique up to contractible choice?
It is often stated that the derived moduli stack of oriented elliptic curves $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some ...
- 423
27
votes
0
answers
1k
views
Spectral sequences as deformation theory
I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
- 64k
4
votes
0
answers
153
views
Preorientation of additive formal group
In "A Survey of Elliptic Cohomology", Section 3.2, Lurie asserts that the preorientations of the additive formal group $\widehat{\mathbf G}_a$ over $\mathbf Z$ are classified by the $\mathbb ...
- 2,011
44
votes
5
answers
6k
views
What is the cotangent complex good for?
The cotangent complex seems to be a pretty fundamental object in algebraic geometry, but if it's treated in Hartshorne then I missed it. It seems to be even more important in derived algebraic ...
- 64k
9
votes
0
answers
288
views
Every Spectral Deligne-Mumford stack satsifies fpqc descent?
In SAG Remark 6.3.3.8, Lurie asserts that if we have a representable (by Spectral Deligne-Mumford stacks) natural transformation $X\to Y$ where $Y$ is a functor satisfying fpqc descent, then so too ...
- 19.6k
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
1
vote
0
answers
281
views
Étale homotopy type of (derived) loop space
A feature of derived algebraic geometry is that we have internal homs. Furthermore, we can think of $B\mathbb{Z}$ as the derived algebraic geometric analogue of $S^1$. Thus we have an analogue of the ...
- 4,418
15
votes
1
answer
805
views
When does QCoh have 'enough perfect complexes'?
Let $X$ be a derived fpqc stack on the $\infty$-category of connective spectral affine schemes $\mathbf{Aff}^{\mathrm{cn}}=(\mathbf{Ring}^{\mathrm{cn}}_{E_\infty})^{\mathrm{op}}$, that is to say, a ...
- 19.6k
2
votes
0
answers
104
views
Linearity of a dg category $C$ over $HH^0(C)$
Let $C$ be a pre-triangulated dg-category over a field $k$ whose Hochschild cohomology groups $\operatorname{HH}^*(C)$ are concentrated in non-negative degree (cohomologically). Is $C$ Morita ...
- 3,758
4
votes
0
answers
118
views
Does the ∞-category of Derived/Spectral schemes admit all colimits over constant diagrams?
In the case of ordinary schemes, all coproducts exist, so given any constant diagram $D_S:C\to \operatorname{Sch}$, the colimit over $D_S$ is isomorphic to the coproduct of $S$ over the connected ...
- 493
10
votes
1
answer
605
views
Descent properties of topological Hochschild homology
Question: What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which THH (Topological Hochschild Homology) satisfies descent?
Adaptations of the arguments appearing in ...
- 532
11
votes
1
answer
2k
views
Derived base change in étale cohomology
Given a commutative square of ringed topoi
$$\begin{array}{ccc}X'\!\! & \overset{f'}\to & Y'\!\! \\ \!\!\!\!\!{\small g'}\downarrow & & \downarrow{\small g}\!\!\!\! \\ X & \...
- 28.8k
7
votes
0
answers
161
views
Is it always possible to write a derived manifold (in the sense of Spivak) as a homotopy colimit of principal derived manifolds?
Is it always possible to write a derived manifold as a homotopy colimit of principal derived manifolds (i.e. zero sets of smooth functions)? This is true for schemes and derived schemes, so it seems ...
- 31
11
votes
0
answers
887
views
How far can Spectral Algebraic Geometry be developed over $\mathbb{E}_2$-rings (instead of $\mathbb{E}_\infty$-rings)?
Jacob Lurie has extensively developed derived algebraic geometry in the setting of $\mathbb{E}_\infty$-ring spectra [SAG]. The resulting theory of Spectral Algebraic Geometry (SAG) gives (in ...
- 11.8k
3
votes
1
answer
304
views
Local quotient covers for derived Deligne-Mumford geometric stacks of Toen-Vezzosi
Let $\mathcal{X}$ be a separated Deligne-Mumford stack, and $X$ its coarse moduli space. A well-known lemma establishes an etale covering $X_{\alpha} \rightarrow X$, such that for each $\alpha$, there ...
- 31
4
votes
0
answers
477
views
DAG applied to homotopy theory: how to reach research level?
It is my dream to do research on applications of spectral algebraic geometry in homotopy theory one day. Specifically, giving a more uniform treatment for the results proved via scary computations (of ...
10
votes
1
answer
883
views
$\infty$-categorical understanding of Bridgeland stability?
On triangulated categories we have a notion of Bridgeland stability conditions.
Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become ...
- 103
12
votes
1
answer
782
views
Power operations from a Tate construction
In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on $\mathbb{E}_{\infty}$-algebras, one finds a construction of the power ...
- 858
6
votes
0
answers
226
views
Applications of spectral Artin representability?
The spectral Artin representability theorem says that a functor $X:CAlg^{cn}\rightarrow S$ from the $\infty$-category of connective $E_{\infty}$-rings to the $\infty$-category of small topological ...
user74900
16
votes
1
answer
1k
views
$\infty$-operads and $E_\infty$-algebras
I work in algebraic geometry. Lately, the answer to most of my questions seems to be "you should read Lurie's Higher Algebra." I took this advice seriously, however it turned out not to be an easy ...
- 16.2k
5
votes
1
answer
569
views
Derived completion of complexes
Suppose $K$ is a bounded above complex of free abelian groups, and take its derived $\ell$-adic completion $K^{\wedge,\ell} = R\lim (K/\ell^n)$ in the derived category, for $\ell$ a prime.
If $K\to L$...
user120812
14
votes
2
answers
781
views
Interpretation of the cohomology of compact lie groups and their classifying spaces in DAG?
I'll be using homological grading throughout this question.
Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods:
$$H^{\bullet}(...
- 7,799
11
votes
1
answer
348
views
(Pre)orientation vs. formal completion
Let $\mathbb G$ be an abelian vatiety over an $\mathbb E_\infty$-ring $A$. That is to say, it consists of an abelian group object in the $\infty$-category of relative schemes $\mathbb G\to \...
- 2,011
5
votes
0
answers
395
views
Derived tensor products and Tor of commutative monoids
Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)...
- 889
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
35
votes
2
answers
3k
views
What is the relationship between connective and nonconnective derived algebraic geometry?
"Derived algebraic geometry" usually means the study of geometry locally modeled on "$Spec R$" where $R$ is a connective $E_\infty$ ring spectrum (perhaps with further restrictions). Why "connective", ...
- 64k
2
votes
0
answers
268
views
Interesting examples of large, accessible, non-presentable $\infty$-categories?
What are some interesting examples of accessible $\infty$-categories
which are not presentable and not small?
By interesting I mean a category which comes up naturally in a certain context and in a ...
- 7,799
2
votes
0
answers
277
views
classifying space of algebraic groups
Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a Borel pair $(B,T)$.
Let $BG$ be the classifying space of $G$.
Can we say that $BG$ is the homotopy colimit of all $BP$ for $P$ a ...
- 3,472
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
5
votes
0
answers
225
views
Does the Amitsur complex have a universal property?
The question is essentially the title. In other words, is there some universal property that the Amitsur complex for a morphism of rings $\phi:A\to B$ satisfies as a cosimplicial ring, or cosimplicial ...
- 10.5k
23
votes
2
answers
2k
views
Why do people say DG-algebras behave badly in positive characteristic?
It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
- 231
21
votes
1
answer
3k
views
Motivation and potential applications of spectral algebraic geometry
Nowadays there is a lot of talk about derived algebraic geometry, but not so much about the related subject of spectral algebraic geometry.
Now I'm curious what future is there for spectral algebraic ...
- 433
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
2
votes
0
answers
166
views
How to understand $\mathcal{L}BG \simeq G/^{\text{ad}}G$ in term of simplicial sets?
First let $G$ be a topological group and $BG$ its classifying space. Let $\mathcal{L}BG=\text{Map}(S^1, BG)$ be the free loop space of $BG$.
We can see that $\mathcal{L}BG$ has the homotopy type of $...
- 9,019
11
votes
1
answer
650
views
Thom Spectra and Hopf-Galois Extensions of Ring Spectra
So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it ...
- 10.5k
18
votes
1
answer
2k
views
can a common mortal understand why the affine line is not smooth in brave new algebraic geometry?
In the introduction to HAGII Toen and Vezzosi write that in brave new algebraic geometry (that is, algebraic geometry over the category of symmetric spectra) Z[T] is not smooth over Z.
I am told that ...
- 1,889