All Questions
Tagged with derived-algebraic-geometry ag.algebraic-geometry
81 questions with no upvoted or accepted answers
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
336
views
DAG vs Classical algebraic geometry
I have a very vague question, but also a fairly specific wish. Namely, I'm wondering what the similarities and differences are between the theory of ordinary schemes on the one hand, and the theory of ...
- 213
3
votes
0
answers
180
views
vanishing of higher homotopy sheaves of cotangent complex
Let $X$ be a $\mathbb{C}$-scheme and suppose that there is an isomorphism (in the derived category of qc-sheaves on $X$) between the cotangent complex, $\mathbb{L}_{X}$, and its 0th homotopy sheaf, ...
user108998
3
votes
0
answers
188
views
What are projective morphisms in derived algebraic geometry
I was curious if someone could point me to a reference or tell me if there is any notion of projective morphism between derived schemes? What about a notion of Proj? Also I am interested in what blow-...
- 431
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
3
votes
0
answers
502
views
Analysis of Eilenberg-MacLane Stacks
In a series of three papers from the fifties, Eilenberg and MacLane did a pretty exhaustive study of what we now call "Eilenberg-MacLane spaces" and used a lot of machinery to do it, e.g. Whitehead's $...
- 10.5k
2
votes
0
answers
157
views
Resolutions of semi free (or almost free) commutative dg algebras with finitely generated cohomology
Let $A^{\bullet}:=\{ \cdots \rightarrow A^i \overset{d^i}{\rightarrow} A^{i+1} \rightarrow \cdots \rightarrow A^{-1} \rightarrow A^0 \rightarrow 0 \rightarrow \cdots \}$ be a non-positively graded ...
- 889
2
votes
0
answers
123
views
Moduli stack of doubly periodic complexes?
Let $\mathcal{A}$ be an abelian category.
In HAG II Toen and Vessozi built a higher derived stack $X$ whose category of perfect complexes is $\text{Perf}(X)\simeq D^b(\mathcal{A})$. So $X$ is a good ...
- 5,711
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
2
votes
0
answers
235
views
Formally étale maps of animated $k$-algebras
In Lurie's DAG, he defines what it means for a natural transformation $T:\mathcal{F}\to\mathcal{F}'$ of functors $\mathcal{F},\mathcal{F}':\mathcal{SCR}\to\mathcal{S}$ to be formally étale. Namely, it ...
- 301
2
votes
0
answers
482
views
About derived divided power envelope
Assume $A$ is a $\mathbb{Z}_{(p)}$-algebra with ideal $I$ and $A,A/I$ are $p$-torsionfree.
In this survey, Akhil Mathew defines the derived divided power envelope $LD_I(A)$ in Construction 7.15, after ...
- 121
2
votes
0
answers
230
views
Lie bracket on the unshifted tangent complex?
My problem is as follows: if $X$ is a derived scheme, or derived stack, or any kind of a space where tangent complex makes sense, I guess there should be a lie bracket on its tangent complex, ...
- 121
2
votes
0
answers
213
views
Do dg schemes have derived points?
Working over a base field $k$ of characteristic $0$, say $K$ is a field (over $k$) and $X$ is a ("nice" if necessary) dg scheme in the sense of Toen-Vezzosi and others, and say $X^0$ is the reduced ...
- 7,972
2
votes
0
answers
132
views
Compact generation of quasicoherent sheaves on mapping stack
Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(...
- 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
2
votes
0
answers
113
views
quotient a scheme by a stratified vector bundle
Let $k$ be a field.
Let $X$ be a $k$-scheme of finite type, normal and integral. We consider $f,g:R\rightarrow X$ an equivalence relation, surjective and such that it is a stratified vector bundle, i....
- 3,472
1
vote
0
answers
62
views
Geometric stability conditions on calabi-yau's fibred over Fano always identical to geometric stability conditions on Fano
I apologize in advance for the long title. This question is motivated primarily by [2], with the explicit example of $\mathbb{P}^2$ and $\omega_{\mathbb{P}^2}$ computed in [3] and [1], respectively.
...
- 317
1
vote
0
answers
109
views
Computing Grothendieck group of (unnodal) Enriques surface
Let $X$ be an unnodal Enriques surface together with an isotropic 10-sequence $\{ f_1, \dots, f_{10}\} \subset \operatorname{Num}(X)$, and let $F_i^\pm \in \operatorname{NS}(X)$ denote the two ...
- 317
1
vote
0
answers
275
views
Fourier-Mukai transform is the derived functor
In Mukai's paper Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math Journal, 1981, there is one sentence that puzzles me.
Let $X$ be an abelian variety over an ...
- 615
1
vote
0
answers
202
views
Is the cotangent complex sensitive to truncation?
$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Spec{Spec}$If $V$ is a dg vector space (in positive degrees), we can view it as a derived scheme $V = \Sym V^*$. It has (co)tangent complex $V$ (and $...
- 5,711
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
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
1
vote
0
answers
202
views
Schlessinger criterion and finiteness of tangent space
Schlessinger's criterion allows us to study whether or not a functor $\mathcal{F}:C_{\Lambda}\rightarrow \text{Set}$ from the category of local Artin $\Lambda$-algebras to sets is representable. One ...
- 4,418
1
vote
0
answers
137
views
Open problems for shifted symplectic structures
I am now interested in shifted symplectic structures.
What are the open problems of shifted symplectic structures regarding the moduli space of sheaves ?
Especially now I am interested in moduli ...
- 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
1
vote
0
answers
204
views
How can we construct a derived scheme as a gluing of derived schemes?
More precisely, consider a Segal groupoid $X_*$ in an infinity category of derived schemes : dSch
In Toen's note, 'Derived Algebraic Geometry', he defines a 1-Artin stacks as a homotopy colimits of ...
- 421
1
vote
0
answers
86
views
derived invariants, perversity and modular coefficients
Let $\pi:X\rightarrow Y$ a Galois cover of finite type schemes over $\mathbb{C}$ of group $\Gamma$.
Let $n$ an integer such that it is not prime with the order of $\Gamma$.
Then $\pi_{*}\mathbb{Z}/n\...
- 3,472
1
vote
0
answers
300
views
Classifying Spaces and Eilenberg-Maclane objects in the category of simplicial rings
[Skip down to the bottom for a correction] Let's work over a field k, assume it is as nice as you need it to be.. Suppose I have an ordinary (edit: commutative) affine group scheme G = Spec(A) over k, ...
- 11
0
votes
0
answers
170
views
Cone of morphism induced by Serre duality
For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category :
$$
S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X]
$$
...
- 317