Questions tagged [cotangent-complex]
The cotangent-complex tag has no summary.
47 questions
9
votes
1
answer
761
views
Quillen's conjecture
I was reading about cotangent complexes in derived algebraic geometry. I saw the following two surprising conjectures by Quillen (from "Cohomology of commutative rings").
My question is: are ...
- 1,463
3
votes
0
answers
190
views
Relation between Du Bois singularity and cotangent complex
I am currently going through the notes Hodge theory and singularities by M. Popa. By reading the definition of Deligne Du Bois complex, I am wondering if there exists any relation between this with ...
- 1,463
7
votes
0
answers
602
views
Defining the cotangent complex via equivalence classes of derived étale curves
In the classical setting, the tangent space $T_p X$ at a point $p$ of a smooth scheme $X$ can be defined using equivalence classes of morphisms from the affine line $\mathbb{A}^1$. To avoid the issue ...
- 35
3
votes
1
answer
185
views
How does the tangent complex of a deformation functor represented by a smooth complete k-algebra relate to the the Jacobi matrix of that algebra?
I am quite confused at the moment how to show that two deformation moduli problems which are represented by smooth complete k-algebras (in mixed characteristic) are equivalent, as I've come across ...
- 3,579
1
vote
0
answers
102
views
Stack of sections for exact triangle
Given a derived Artin stack $X$, suppose we have an exact triangle,
$E_1\to E_2 \to E_3\to .$
of objects in $\mathrm{Perf}(X)$.
I wonder is there any relation between the stacks of section of $E_i$?
...
- 563
2
votes
2
answers
377
views
Constructing a quasi-coherent sheaf on a derived scheme/stack as a compatible family
Set up: Suppose that $\mathcal{X}$ is a derived stack (I'm using the conventions of Yaylali's "Notes on derived algebraic geometry," but I'd be happy to work with other conventions of the ...
- 1,612
4
votes
0
answers
235
views
'Naive cotangent complex' as 1-truncation of cotangent complex
In the stacks project, there is a 'naive version' of cotangent complex $NL_{S/R}$ for a ring morphism $R\rightarrow S$, given by the chan complex $(I/I^{2}\rightarrow \Omega_{R[S]/R}\otimes_{R[S]}S)$ ...
- 1,010
2
votes
0
answers
304
views
Square zero extension in the derived setting
Here we take the infinity category of simplicial ring $SCRing=Fun^{\prod}(Poly^{op},Spc)$ and follow the construction 25.3.1.1 in DAG by Lurie, where we extend the construction of square zero ...
- 1,010
5
votes
0
answers
186
views
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 ...
- 6,211
3
votes
0
answers
300
views
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 ...
- 373
3
votes
0
answers
200
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 ...
- 323
3
votes
2
answers
642
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 ...
6
votes
2
answers
786
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 ...
- 1,134
4
votes
0
answers
416
views
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 ...
- 301
4
votes
0
answers
318
views
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 ...
- 868