5
votes
1answer
163 views

Equivariant Formality

Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$. Assume further there exists a ...
3
votes
0answers
93 views

Formal DG-algebras

Sorry for this question but I really have difficulties with model categories. Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...
5
votes
2answers
235 views

Equivariant derived category and invariant divisor

I'm looking for a reference of the following (folklore?) result. Let $X$ be a smooth projective variety equipped with a $G=\mathbb{Z}/2\mathbb{Z}$ action (we consider the simplest case, everything ...
3
votes
1answer
158 views

Formality of classifying spaces (for not necessarily connected groups)

As should be evident from the title this question has a similar flavor to: Formality of classifying spaces However, unlike Geordie's question, I will be working with torsion free coefficients (say ...
4
votes
1answer
242 views

Additive functors and Derived Categories

I have been learning about derived categories from Hartshorne's "Residues and Duality". One of the main theorems, as it is presented there seems to be the canonical isomorphism $RF \circ RG \cong R(F ...
7
votes
2answers
338 views

Recovering an abelian category out of its derived category

I'm trying to learn more about derived category stuff and my curiosity has made me to ask these questions. Sorry if I'm being sloppy, I'm a new learner. In Wikipedia it has been stated that since ...
4
votes
2answers
358 views

Examples of tilting objects that don't come from exceptional sequences

This is a question on geometric tilting theory. On smooth projective variety it is possible to define in general tilting object as perfect complex that satisfy some properties, but are there examples ...
5
votes
0answers
181 views

Not isomorphic varieties with isomorphic tilting algebras

Let $X$ be a smooth projective variety over a field, than tilting object $T$ on $X$ is a perfect complex that is a compact generator of the derived category $\operatorname{D}(QCoh(X))$ and satisfies ...
0
votes
1answer
173 views

Recovering torsion in singular homology from cplx of singular chains

For a simply connected simplicial complex, a theorem of Whitehead (Derived categories for the working mathematician, bottom of page 2) explains that the associated chain complexes with coefficients in ...
3
votes
1answer
318 views

Semiorthogonal decompositions for Fano 3-folds and 4folds

Let $X$ be a projective Fano 3-fold or 4-fold and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. For what $X$ is it known a semi orthogonal decomposition into indecomposable ...
10
votes
0answers
350 views

The derived category of integral representations of a Dynkin quiver.

Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write ...
3
votes
2answers
260 views

Earliest/most standard reference for derived categories of hereditary algebras

Let $A$ be a hereditary algebra and let $\mathcal{D}$ be the derived category of bounded complex of finitely generated $A$-modules. Then, for any complex $C_{\bullet}$ in $\mathcal{D}$, we have ...
25
votes
7answers
3k views

A down-to-earth introduction to the uses of derived categories

When I was learning about spectral sequences, one of the most helpful sources I found was Ravi Vakil's notes here. These notes are very down-to-earth and give a kind of minimum knowledge needed about ...
7
votes
1answer
221 views

Classification of t-structures in derived category of R-mod?

I am looking for a reference talking about the complete(or not)description of t-structures in bounded derived category of $R-mod$, i.e. $D^b(R-mod)$.where $R$ is commutative ring, in particular, ...
12
votes
2answers
876 views

What is a flop (and when are they conjectured to give derived equivalences)?

(1) Is the definition of flop given by Wikipedia the industry standard? (2) Regardless of the answer to (1), when is it expected that a birational transformation gives rise to a derived ...
3
votes
1answer
2k views

Is there a general projection formula for morphisms of ringed topoi?

What's the general projection formula in algebraic geometry, for instance on the level of derived categories of ringed topoi? And what's the reference? I guess it might be in SGA 4, but couldn't find ...
12
votes
0answers
486 views

References for a certain generalization of Hochschild cohomology?

Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C ...