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Questions tagged [cotangent-complex]

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What classifies deformations of group schemes (or Hopf algebras)?

The cotangent complex of a scheme classifies its deformations. That is, if $X$ is a scheme over a field $k$ (with conditions?) and $\mathbf{T}^*_X\in D^b(\text{QCoh}(X))$ its cotangent complex, the ...
Pulcinella's user avatar
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Cotangent complex of a blowup

Let $X$ be a nonsingular variety over an algebraically closed field $k$, and let $Y \subset X$ be a nonsingular subvariety. Consider the blowup $p: \tilde{X} \to X$ of $X$ along $Y$, with exceptional ...
John Nolan's user avatar
3 votes
0 answers
162 views

Linear deformations of a morphism between stacks

Given smooth algebraic stacks $\mathcal{X}$, $\mathcal{Y}$ what are the linear deformations $\operatorname{Def}^1(f: \mathcal{X} \to \mathcal{Y})$ of a morphism $f:\mathcal{X} \to \mathcal{Y}$? In ...
Robert Hanson's user avatar
3 votes
2 answers
403 views

Pushout along weak equivalence gives weakly equivalent object

This question arose through reading "Interactions between homotopy theory and algebra" (the first chapter by Goerss and Schemmerhorn). In particular, I am struggling with the proof of ...
Sofía Marlasca Aparicio's user avatar
6 votes
2 answers
464 views

Distinguished triangle of dualizing complexes and/or determinants?

Q1 : If $X \to Y \to Z$ are maps of schemes, is there a relation such as $$\omega_{X/Z} \overset{?}{=} \omega_{Y/Z}|_X \overset{L}{\otimes} \omega_{X/Y}$$ between their dualizing complexes? Or maybe ...
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What does the cotangent complex tell you when it takes animated inputs?

These two links: What is the cotangent complex good for? and Intuition about the cotangent complex? are quite helpful in giving intution for the cotangent complex in terms of deformations but I don't ...
Eric's user avatar
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Cotangent complex of a formal thickening

Let $R$ be an (animated) commutative ring, with cotangent complex $L_R$ and let $\mathcal{C}(R) = \mathcal{D}(R)_{\Sigma^{-1}L_R/}$ be the category of nice square zero extensions of $R$. A typical ...
pupshaw's user avatar
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$\Omega^1_{B_\bullet/A_\bullet}$ is acyclic if $A_\bullet \to B_\bullet$ is quasi-isomorphism

Let $A_\bullet \to B_\bullet$ be a quasi-isomorphism of simplicial rings in the sense of (P.62, I.3.1.7, Complexe Cotangent et Déformations I, Luc Illusie). Then, we define the simplicial $B_\bullet$-...
Kenny Lau's user avatar
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1 answer
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Proposition 5.13 (ii) in Scholze's Perfectoid Spaces

In Proposition 5.13 (ii) in Scholze's Perfectoid Spaces, we have $R \to S$ a morphism of $\Bbb F_p$-algebras and the assumption that the relative Frobenius $\Phi_{S/R}$ induces an isomorphism $R_{(\...
Kenny Lau's user avatar
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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 $...
Pulcinella's user avatar
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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, ...
Can Yaylali's user avatar
43 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 ...
Tim Campion's user avatar
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formal smoothness and cotangent complex

If $k$ is a field and $A$ is a formally smooth $k$-algebra, then we know that $\Omega^{1}_{A/k}$ is projective. What about its cotangent complex $L_{A/k}$? When is it quasi-isomorphic to $\Omega^{1}_{...
prochet's user avatar
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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 ...
Martin Hurtado's user avatar
28 votes
2 answers
3k views

What (or how) are the new spaces of derived algebraic geometry?

I am a beginner in derived algebraic geometry and I am trying to develop some visual and geometrical intuition about derived schemes (and stacks), or more precisely about the new geometrical phenomena ...
Martin Hurtado's user avatar
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Comparing definitions of cotangent complex

Consider the following two ways of defining the cotangent complex of a ring map $R \rightarrow A$ (Let $P^{\bullet} \rightarrow A$ be a polynomial resolution): As the complex $\Omega^1_{P^{\bullet}/R}...
safety stegosaurus's user avatar
13 votes
1 answer
730 views

Cotangent Complex in Analytic Category

I am looking for a reference which develops the theory of the cotangent complex for complex analytic spaces. I need this to justify some computations I did assuming some formal properties which hold ...
Mohan Swaminathan's user avatar
7 votes
0 answers
228 views

Why does the cotangent complex really have a distinguished triangle?

Associated to any ring maps $A\to B\to C$ there is the distinguished triangle $$\mathbf{L}_{B/A}\otimes^L_BC\ \longrightarrow \ \mathbf{L}_{C/A} \ \longrightarrow \ \mathbf{L}_{C/B} \ \stackrel{+1}{\...
Pulcinella's user avatar
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8 votes
1 answer
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Elementary (English) reference for the cotangent complex?

I'm trying to understand cotangent complexes and their role in deformation theory, and later the statement that they're somehow natural in a derived scheme/stack. I understand that the standard ...
Joseph's user avatar
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1 answer
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Vanishing of the cotangent complex for non-perfect algebras

Let $k$ be a perfect field of characteristic $p>0$. If $R$ is a perfect $k$-algebra, then the cotangent complex $L_{R/k}$ vanishes (the Frobenius is zero and induces an isomorphism on $L_{R/k}$ ...
skd's user avatar
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11 votes
2 answers
1k views

Cotangent complex of perfect algebra over a perfect field

Let $A$ be a perfect $\kappa$-algebra over a perfect field $\kappa$ of positive characteristic $p$. Then the algebraic (= classical) cotangent complex $L_{A/\kappa}^{\operatorname{alg}}$ is known to ...
A Rock and a Hard Place's user avatar
3 votes
1 answer
503 views

characterisation of regular morphisms

Recall that a morphism of schemes $X\rightarrow Y$ is regular if it's flat with geometric fibres that are regular schemes. Fix a field $k$ and consider a morphism $f:X\rightarrow Y$ of noetherian $k$-...
prochet's user avatar
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6 votes
0 answers
456 views

Explicit computation of the cotangent complex in a non-lci case

Is there an example of a non-lci morphism $X\rightarrow Y$ for which the entire cotangent complex (or just Andre-Quillen cohomology) can be explicitly computed? I believe it is a theorem of Avramov ...
dhy's user avatar
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6 votes
1 answer
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Cotangent complex of certain dg-scheme

This is a somewhat embarrassing question, but still I will ask it. Let $V$ be a vector space over $\mathbb C$ of dimension $d$. Let $X$ be the dg-preimage of $0$ under the natural map $V\to Sym^2(V)$ (...
Alexander Braverman's user avatar
3 votes
1 answer
232 views

Is the cotangent complexes of groupoids bounded above by degree $1$?

Let $\mathcal{X}$ be a stack given by a groupoid $X_1\rightrightarrows X_0$, where $X_0$ and $X_1$ are smooth $k$-varieties. Let $\mathbb{L}_{\mathcal{X}/k}$ be the cotangent complex of $\mathcal{X}$....
Zhaoting Wei's user avatar
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5 votes
0 answers
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Flat Connections on the Cotangent Complex

I'm trying to find a reference which defines and discusses some properties of connections and flat connections on the cotangent complex in a homotopical setting. That is to say, a connection or flat ...
Jonathan Beardsley's user avatar
17 votes
1 answer
840 views

Are deformations of a scheme some kind of a "derived gerbe" under the cotangent complex?

(This is probably a very naive question. My understanding of the cotangent complex is quite vague.) Let me first recall the picture for deformations of a smooth morphism: If $f:X_0\to S_0$ is a ...
Piotr Achinger's user avatar
6 votes
1 answer
664 views

A construction of Kähler differentials and Illusie cotangent complex as colimit over embeddings

Let $\Bbbk$ be a field, $X$ affine scheme of finite type over $\Bbbk$. Let $\mathcal C_X$ be the category of closed embeddings of $X$ into (say affine) smooth $Y$'s of finite type over $\Bbbk$, ...
rrrrrttttttt's user avatar
1 vote
1 answer
201 views

Obstruction map for local singularities via tangent (Andre-Quillen) cohomology

Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent (Andre-...
Sasha Pavlov's user avatar
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3 votes
0 answers
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Does the cotangent complex commute with coequalisers?

I would like to know if the cotangent complex (say of rings) commutes with coequalisers. More precisely, let $B_1\rightrightarrows B_2\rightarrow C$ be a coequaliser of $A$-algebras. Is then the ...
user36504's user avatar
6 votes
2 answers
2k views

What is a good reference (preferably thorough) for the Derived Category of a scheme/orbifold/stack?

I've sort of circled around the idea of derived categories a few times, read a few introductory papers ("Derived Categories for the working mathematician", e.g.), and feel now that this is something ...
Simon Rose's user avatar
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1 vote
0 answers
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Question about $\Delta(n)_{U}$ notaion in illusie's cotangent compelexe et deformations

In illusie's book cotangent complexe et deformations, 38page, the notation $\Delta(n)_{U}$ appears, and I cannot find the direct explanation or hint about meaning of this notation in this book. I ...
keaton's user avatar
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4 votes
2 answers
908 views

resolution by simplicial objects versus resolution by chain complex

I'm reading Illusie's book "complexe de cotangent et Deformations I". And I'm puzzled on the definition of cotangent complex. I formulate my question as follows: Suppose $C$ and $D$ are abelian ...
Xiaobo Zhuang's user avatar
7 votes
0 answers
323 views

Computer Algebra solution for simplicial resolutions for André-Quillen cohomology

Hello, I would like to experiment with André-Quillen (co)homology. Especially for singular rings. A key problem is that the construction of a simplicial resolution of a ring seems to require a rather ...
Ojen's user avatar
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3 votes
1 answer
714 views

cotangent complex of a trivial extension

Let $k$ be a field of characteristic zero, $A$ a simplicial commutative k-algebra, and $M$ a simplicial $A$-module. Consider the trivial square-zero extension $A\oplus M$ as an $A$-algebra. Is it true ...
Martin Lagenbach's user avatar
10 votes
1 answer
2k views

Is Illusie's generalization of the cotangent complex to arbitrary ringed toposes necessary in algebraic geometry?

André and Quillen both gave constructions of the relative cotangent complex for commutative rings, so pretty immediately that gives us that we understand the cotangent complex for affine schemes. ...
Harry Gindi's user avatar
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9 votes
1 answer
2k views

Formally smooth morphisms, the cotangent complex, André-Quillen cohomology, and representability of nilpotent extensions as trivial extensions over a cofibrant replacement

Recall that an $R$-algebra $R\to S$ is called formally smooth (resp. formally unramified resp. formally étale) if given any lifting problem of the form $$\begin{matrix} R&\to &T\\ \downarrow&...
Harry Gindi's user avatar
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6 votes
1 answer
599 views

Detecting etale maps on reduced points

Suppose I have a morphism of schemes for which I know the relative cotangent complex is trivial, and the map on reduced subschemes is an isomorphism. Is the map an isomorphism? More generally, given ...
David Ben-Zvi's user avatar
63 votes
5 answers
9k views

Intuition about the cotangent complex?

Does anyone have an answer to the question "What does the cotangent complex measure?" Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...
Peter Arndt's user avatar
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