# Questions tagged [derived-algebraic-geometry]

The derived-algebraic-geometry tag has no usage guidance.

192
questions

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votes

**1**answer

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### 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 $\mathsf{Mod}_R$ ...

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97 views

### Base-change theorems for stable $\infty$-categories

Omitting some technicalities, the base-change theorem for quasicoherent sheaves says that if we have the following diagram of (derived) schemes
$\require{AMScd}$
\begin{CD}
X \times_S Y @>\pi_2>&...

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168 views

### What's the definition of a microlocal sheaf?

I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general.
In this paper of ...

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244 views

### Coherent objects in a hypercomplete $\infty$-topos

In Lurie's "Spectral Algebraic Geometry", Proposition A.6.6.1 (2) shows that for $\mathcal{X}$ an $\infty$-topos that is both locally coherent and hypercomplete, the full subcategory $\...

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134 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 ...

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695 views

### Connes's Absolute Geometry and Lurie's Spectral Algebraic Geometry

Alain Connes and Caterina Consani seem to be currently working on "absolute algebraic geometry", which is a kind of "algebraic geometry over the sphere spectrum" (https://arxiv.org/...

**3**

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**1**answer

199 views

### Intuition for points of the moduli of objects for a dg-category

Problem summary: I'm trying to get some intuition for what the moduli space of objects for a dg-category (as in this paper by Brav and Dyckerhoff) actually looks like/how to give an alternative ...

**21**

votes

**1**answer

1k 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 ...

**4**

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**1**answer

345 views

### Categorical-geometric Langlands for tori

Fix a "nice" curve $X$ (smooth, projective, proper, geometrically connected, what-have-you) and an algebraic torus $G$, both over a field of characteristic $0$ (possibly algebraically closed?...

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73 views

### A question about relative deformations of the structure sheaf of the diagonal

Let $X$ be a smooth proper algebraic variety over $\mathbb{C}$. Let us consider an associative
algebra $${p_2}_*{\mathcal{H}{om}}_{X\times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta) \in \text{Alg}_\...

**25**

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904 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 ...

**2**

votes

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158 views

### Derived quot schemes and the derived linearity locus

I am studying derived quot schemes in the paper Ciocan-Fontanine and Kapranov, “Derived Quot schemes” .
On page 36 ~ 37, the derived linearity locus is defined.
Let $S$ be a $\mathbb{Z}_-$-graded dg-...

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210 views

### Relationship between different definitions of the Hochschild homology

Throughout the literature, one can find many definitions of the Hochschild homology of various objects. However, the precise relationship between these definitions is not always so clear, at least to ...

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134 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 ...

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315 views

### Construction of derived Quot schemes

I am studying the construction of derived Quot schemes in the paper Borisov, Katzarkov, and Sheshmani - “Shifted symplectic structures on derived Quot-stacks”.
Derived quot stacks are constructed from ...

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134 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, ...

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131 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 $...

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122 views

### Structure sheaf of derived intersection

Everything is over a field $k$ of characteristic $0$.
Let $X$ and $Y$ two closed dg subschemes over a dg scheme $Z$. I am trying to understand the structure sheaf of the derived intersection of $X$ ...

**4**

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159 views

### Example of pseudocoherent complex which is not locally quasi-isomorphic to a strict pseudocoherent one

I ask this question here even if I posted it also on math.stackexchange (recieving no answer so far) because I have read some analogous question but for perfect complexes on this site, even though ...

**4**

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**1**answer

201 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 $A$.
[...

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188 views

### The dual abelian scheme in derived algebraic geometry

$\def\Pic{\mathcal{Pic}}\def\Gm{\mathbb{G}_m}\def\Hom{\mathop{Hom}}\def\HOM{\mathcal{Hom}}$
If $A/S$ is an abelian scheme, the fppf sheaf $\Pic^0_{A/S}$ is representable by an abelian scheme $\hat{A}$....

**21**

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563 views

### What is the status of a result of Kontsevich and Rosenberg?

In their influential paper Noncommutative Smooth Spaces (https://arxiv.org/abs/math/9812158), Kontsevich and Rosenberg define the notion of a noncommutative projective space. In Section 3.3 they ...

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184 views

### Solution spaces of algebraic differential equations and derived geometry

We consider potentially non-linear differential equations on the formal one dimensional disc $\Delta$. Such equations are given by expressions $$P(z,f,f',f'',...)=0,$$ where $P$ is an element of the ...

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**1**answer

232 views

### Is there a definition of reduced $E_\infty$ ring?

[Edit: I have completely changed the question in response to the replies given]
I am curious if there is well defined notion of reduced $E_\infty$-ring.
Let $CAlg$ denote the $\infty$-category of $E_\...

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175 views

### Derived category and L-function

For abelian varieties over $\mathbb{Q}$ $\mathscr{A}$ and $\mathscr{A}'$, if derived categories $D(\mathscr{A})$ and $D(\mathscr{A}')$ are equivalent then L-functions are same $L(s,\mathscr{A})=L(s,\...

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306 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 ...

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194 views

### Derived geometry and theoretical physics

Is there any link between derived geometry and theoretical physics?
for example with particle physics or quantum mechanics?
Specifically something that included the obstruction bundle.
If possible I ...

**5**

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145 views

### Smoothness of a variety implies homological smoothness of DbCoh

I have been told that $D^bCoh(X)$ is homologically smooth if $X$ is a smooth variety, and I am trying to construct a proof. My background is not in algebra, so I apologize for elementary questions.
It ...

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245 views

### When is the cotangent complex perfect?

Let $X\rightarrow S$ be a proper flat morphism of schemes.
When is the cotangent complex $L_{X/S}$ perfect ?
It is well known, that for local complete intersections the cotangent complex is perfect, ...

**34**

votes

**5**answers

3k 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 ...

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221 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 ...

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153 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 ...

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**1**answer

427 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 ...

**4**

votes

**1**answer

204 views

### Does formation of the derived $\infty$-category preserve pushouts?

Let $B\leftarrow A\to C$ be a diagram of commutative rings, and let $\mathcal{D}(A)$ be the derived $\infty$-category of $A$-modules (as in Lurie's "Higher Algebra"). Then is there an ...

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188 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....

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79 views

### Can an $\infty$-action on a derived affine scheme by an affine group scheme always be strictified?

Let $X$ be an affine derived scheme, say $X = \operatorname{Spec} A$, for $A$ a simplicial commutative ring. Let $G$ be an affine group scheme (classical), say $G = \operatorname{Spec}B$, and let an $\...

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131 views

### Relation between $\mathbb{A}^1$-homotopy theory and derived algebraic geometry [duplicate]

I've often heard that one of the benefits of derived algebraic geometry, next to a cleaner intersection theory, is that "provides natural settings " for the $\mathbb{A}^1$-homotopy theory (...

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224 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 ...

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134 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\...

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160 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 ...

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229 views

### The correct homotopically relevant notion of ideals of dg-algebras (or $\mathbb E_1$-rings)

I'm trying to figure out what an ideal of a, say, dg-algebra (or, if you prefer, $\mathbb E_1$-ring) $R$ is in a homotopically relevant fashion, but I can't actually figure it out. I can assume that $...

**14**

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**1**answer

504 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 ...

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106 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 ...

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161 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 ...

**2**

votes

**1**answer

278 views

### Two results about (shifted) symplectic structures

I am now interested in shifted symplectic structures.
I found Zhang's results about symplectic structures (2011, p.3-4, arXiv link,
Comm. Anal. Geom. 2017) and Pantev–Toen–Vaquié–Vezzosi's results ...

**2**

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78 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 ...

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174 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 ...

**9**

votes

**1**answer

576 views

### Connectedness, loops and formal moduli problems

Let $k$ be an algebraically closed field of characteristic zero. Formalizing a classical folk concept, Pridham and (in a different way,) Lurie defined a formal moduli problem (over $k$) to be a ...

**3**

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**1**answer

303 views

### Is the category of spectra on $\mathbb{P}^1$ a module category?

I cannot really state my question in an incredibly precise way as I'm very new to this area, but I hope what I'm asking will be clear. Let $\mathcal{C}$ be the infinity category of sheaves of quasi-...

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71 views

### Derived manifold and real virtual dimension

In https://arxiv.org/pdf/1504.00690.pdf, it seems like the "derived manifold structure" given on a certain complex analytic space seems to have the real virtual dimension the same as the complex ...