All Questions
Tagged with derived-algebraic-geometry homological-algebra
18 questions
23
votes
3
answers
2k
views
Where does one go to learn about DG-algebras?
The theory of differential graded algebras (in char 0) and their modules has numerous applications in rational homotopy theory as well as algebraic geometry.
I'm looking for a reasonably complete ...
- 7,799
15
votes
1
answer
805
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 ...
- 19.6k
9
votes
1
answer
748
views
In the not necessarily abelian cat setting, is there a Grothendieck spectral sequence for computing the homotopy of a composition of derived functors?
Recall the Grothendieck spectral sequence which computes the homology groups of a composition of left derived functors $F$ and $G$ on abelian categories:
\begin{align*}
E_{p,q}^{2}(A)=L_{p}G\circ L_{q}...
- 301
9
votes
0
answers
507
views
Categorification of definitions in the context of the derived category of quasi-coherent sheaves
Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned ...
- 7,799
8
votes
1
answer
979
views
Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$
All varieties appearing below are assumed smooth projective over $\mathbb C$ and all vector bundles, sections etc are assumed to be algebraic/holomorphic. We use the word resolution to mean quasi-...
- 1,761
6
votes
2
answers
686
views
Somewhat general question that includes: "Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?"
Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree?
Example: Given a dg ...
- 54.6k
6
votes
1
answer
394
views
2-shifted Poisson bracket on Lie algebra cohomology
Let $\frak{g}$ be a semisimple Lie algebra, and let $({-},{-})$ be an invariant inner product on $\frak{g}$. The Chevalley–Eilenberg complex $C^*(\frak{g})$ has a natural Poisson bracket of degree $-2$...
- 819
6
votes
0
answers
201
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 ...
5
votes
1
answer
262
views
Derived Morita equivalence of associative algebras
An associative algebra $A$ is said to be Morita equivalent to another one $B$ if there is an equivalence $$\mathsf{Mod}_A\simeq \mathsf{Mod}_B$$ between its corresponding abelian categories of modules....
user95283
5
votes
0
answers
395
views
Derived tensor products and Tor of commutative monoids
Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)...
- 889
4
votes
0
answers
195
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 ...
- 351
4
votes
0
answers
181
views
Flat resolutions of DG-schemes
Recall that a DG-scheme is a pair $(X,\mathcal{O}_X)$,
where $(X,\mathcal{O}^0_X)$ is a scheme, $\mathcal{O}_X$ is a sheaf of commutative DG-algebras over $(X,\mathcal{O}^0_X)$, and each $\mathcal{O}^...
- 153
3
votes
1
answer
354
views
Who are the compact generators in the derived category of $\mathcal{D}_X$-modules?
Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators.
Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category ...
- 7,799
3
votes
0
answers
225
views
Derived $\infty$-category of quasi-coherent sheaves on schemes
Let $X$ be a scheme. On the one hand, we have the derived $\infty$-category constructed from the abelian category of quasi-coherent sheaves on $X$. On the other hand, we can define the stable $\infty$-...
- 151
3
votes
0
answers
111
views
Is there a relation between Projection formula and Verdier duality
For suitable settings, $f\colon X\to Y$, $F,G$ we have projection formula and Verdier duality:
Projection formula: $Rf_!(F\otimes^\mathbb{L}f^{-1}G)\cong Rf_!F\otimes^{\mathbb{L}}G$
Verdier Duality:...
user39380
2
votes
0
answers
213
views
Do dg schemes have derived points?
Working over a base field $k$ of characteristic $0$, say $K$ is a field (over $k$) and $X$ is a ("nice" if necessary) dg scheme in the sense of Toen-Vezzosi and others, and say $X^0$ is the reduced ...
- 7,972
2
votes
0
answers
166
views
How to understand $\mathcal{L}BG \simeq G/^{\text{ad}}G$ in term of simplicial sets?
First let $G$ be a topological group and $BG$ its classifying space. Let $\mathcal{L}BG=\text{Map}(S^1, BG)$ be the free loop space of $BG$.
We can see that $\mathcal{L}BG$ has the homotopy type of $...
- 9,019
1
vote
0
answers
128
views
full strong exceptional collection
I am wondering whether, if a triangulated category $\mathcal{D}$ has a full strong exceptional collection (infinite), it is triangle-equivalent to the bounded derived category of finitely generated ...
- 359