# Questions tagged [cotangent-complex]

The cotangent-complex tag has no usage guidance.

37
questions

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### 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 ...

3
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2
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### 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 ...

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### 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 ...

4
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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 ...

4
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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 ...

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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 $B_\bullet$-...

6
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1
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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_{(\...

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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 $...

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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, ...

39
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### 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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### 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}_{...

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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 ...

27
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2
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### 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 ...

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191
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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}...

13
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1
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641
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### 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 ...

7
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### 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}{\...

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1
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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 ...

5
votes

1
answer

260
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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 $L_{R/k}$ ...

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2
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### 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 ...

3
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1
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443
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### 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 $k$-...

6
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### 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 ...

6
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1
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351
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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 $V\to Sym^2(V)$ (...

3
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1
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### 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 $\mathcal{X}$....

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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 ...

15
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1
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790
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### 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 ...

6
votes

1
answer

638
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### 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 $\Bbbk$, ...

1
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1
answer

198
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### 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-...

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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 ...

7
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2
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### 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 ...

1
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0
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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 ...

4
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2
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856
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### 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 ...

7
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### 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 ...

3
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1
answer

648
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### 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 ...

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1
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### 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. ...

9
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1
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### 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&...

6
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1
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### 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 ...

63
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5
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### 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 ...