# Questions tagged [cotangent-complex]

The cotangent-complex tag has no usage guidance.

24
questions

**50**

votes

**5**answers

7k 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 ...

**14**

votes

**1**answer

553 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 ...

**12**

votes

**1**answer

328 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 ...

**11**

votes

**2**answers

551 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 ...

**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&...

**7**

votes

**1**answer

1k 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. ...

**7**

votes

**2**answers

818 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 ...

**7**

votes

**0**answers

153 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}{\...

**7**

votes

**0**answers

290 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 ...

**6**

votes

**1**answer

528 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 ...

**6**

votes

**1**answer

830 views

### 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 ...

**6**

votes

**1**answer

299 views

### 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 $V\to Sym^2(V)$ (...

**6**

votes

**0**answers

331 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 ...

**5**

votes

**1**answer

204 views

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

**5**

votes

**1**answer

503 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 $\Bbbk$, ...

**5**

votes

**0**answers

230 views

### 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 ...

**3**

votes

**2**answers

668 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 ...

**3**

votes

**1**answer

199 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 $\mathcal{X}$....

**3**

votes

**1**answer

243 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 $k$-...

**3**

votes

**0**answers

122 views

### 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 ...

**2**

votes

**1**answer

401 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 ...

**2**

votes

**0**answers

142 views

### 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}...

**1**

vote

**1**answer

187 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-...

**1**

vote

**0**answers

71 views

### 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 ...