For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.

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2
votes
1answer
284 views

dual modules in the derived category

Hello, This question is a follow up to Sasha's comment in Duals and Tensor products. In the comment there, it is claimed that the given a ring $A$ and modules $M$, $N$, there is an isomorphism $$ ...
2
votes
1answer
263 views

For an l-adic sheaf (F_n), why is the complex F_n of finite Tor dimension?

Let $X$ be a variety and let $\mathcal{F}=(\mathcal{F}_n)_{n\geq 0}$ be a (constructible) $\ell$-adic sheaf on $X$. Let $K_n$ be the object in the derived category $D(X,\mathbf{Z}/\ell^{n+1})$ of ...
3
votes
1answer
552 views

Perfect complexes and RGamma(X,F) without mentioning derived categories

Let $A$ be a commutative noetherian ring. Let $K_{parf}(A)$ be the full subcategory of the homotopy category $K(A)$ of $A$-modules whose objects are bounded complexes of finitely generated ...
9
votes
3answers
1k views

Derived categories of (coherent) sheaves of modules: exceptional images, gluing, and proper descent?

I am interested in the properties of (the derived categories) of various categories of (coherent) sheaves of modules (over varieties). I would like to understand in what extent these properties are ...
6
votes
1answer
615 views

Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?

I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R ...
12
votes
1answer
659 views

Analytic Torsion in the Derived Category

I recently learned about analytic torsion and about the amazing Cheeger-Muller theorem identifying analytic and Reidemeister torsion for compact Riemannian manifolds. Now analytic torsion is defined ...
2
votes
3answers
736 views

Exceptional collections with many Exts

Background definitions: Let $D$ be a triangulated category arising in nature (for instance as the cohomology category of a dg category). An object $E$ in $D$ is called exceptional if $RHom(E,E)$ is ...
6
votes
2answers
584 views

t-structures on the derived category of finitely generated abelian groups

is it possible to explicitly parametrise all the t-structures on the derived category of finitely generated abelian groups?
8
votes
1answer
978 views

What is the Hochschild cohomology of the dg category of perfect complexes on a variety?

Let $X$ be a quasi-projective variety over a field $k$. Let $D_{qcoh}$ be a dg enhancement of the unbounded derived category of quasi-coherent sheaves over $X$, and $D_{perf}$ its full subcategory of ...
3
votes
1answer
270 views

Inverse of a tilting module

Let $k$ be a field, $A$ an associative unital $k$-algebra, $\operatorname{\mathsf{Mod}} A$ the category of left $A$-modules and $D^b(\operatorname{\mathsf{Mod}} A)$ the bounded derived category. Let ...
18
votes
0answers
909 views

Is there a Grothendieck-Riemann-Roch type of theorem generalizing Grothendieck's Lefschetz trace formula

Grothendieck deduced that the L-function of a (constructible) $\ell$-adic sheaf on a variety over $\mathbf{F}_p$ is rational from the generalized trace formula. My first question is based on the ...
10
votes
5answers
2k views

Sheaves without global sections

The line bundle $O(-1)$ on a projective space or $O(-\rho)$ on a flag variety has a property that all its cohomology vanish. Is there a story behind such sheaves? Here are more precise questions. Let ...
7
votes
4answers
771 views

Intuition about the triangulation of a homotopy category K(A)

Let $\cal{A}$ be an additive category. Given a morphism of (cochain) complexes $f:X\rightarrow Y$ we can form the mapping cone $C_f$, which is the complex $X[1]\oplus Y$ with differential given by ...
6
votes
4answers
2k views

group of Yoneda extensions and the EXT groups defined via derived category

Given an abelian category C, we can form the Yoneda extensions $YExt^i(X,Y)$ to the equivalent classes of $i$-extensions of X by Y. Given any abelian category C, we can always formulate the derived ...
27
votes
7answers
4k 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 ...
9
votes
0answers
657 views

Generating the derived category with line bundles

The following lemma is useful and well-known: LEMMA If $L^{\pm 1}$ is ample on proper scheme over a field $k$, then some number of powers $\mathcal{O},L,...,L^{m}$ generate the unbounded derived ...
7
votes
1answer
227 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
1k 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 ...
8
votes
2answers
1k views

Derived Physics

Hello to all, This question will probably be closed down as being off-topic faster than one can say "string theory", but here it goes: I've noticed that the problems I'm working on -the structure of ...
5
votes
3answers
1k views

Is this a definition of equivariant derived category?

Let $X$ be a topological space and $G$ be a topological group acting on $X$, both locally compact Hausdorff. Denote by $D^b(X)$ the derived category of sheaves (say of abelian groups) on $X$. We ...
7
votes
3answers
777 views

Are the underlying undirected graphs of two mutation-equivalent acylic quivers isomorphic?

Quiver mutation, defined by Fomin and Zelevinsky, is a combinatorial process. It is important in the representation theory of quivers, in the theory of cluster algebras, and in physics. We consider ...
5
votes
2answers
681 views

Topological homotopy category as derived category

In the Introduction of his Derived Categories for the working mathematician Richard Thomas mentions the following theorem of Whitehead. Suppose that $X,Y$ are simplicial complexes, then the ...
2
votes
1answer
574 views

Number of sheaves in a full exceptional collection

Suppose we have a full exceptional collection (F1,...,Fn) of coherent sheaves on a smooth projective variety X. The number n of sheaves in this collection is equal to the rank of the Grothendieck ...
9
votes
1answer
690 views

How to write down the determinant of a quasi-isomorphism?

This question about the determinant of a perfect complex reminded me of an old question that I had. The construction of the determinant (as in MR1914072 or MR0437541) is a difficult piece of ...
13
votes
0answers
492 views

When is every “solid” perfect complex faithful?

Let $R$ be a noetherian commutative ring. Consider $D^{perf}(R)=K^b(R-proj)$ the category of bounded complexes of finitely generated projective $R$-modules, with maps of complexes up to homotopy. ...
12
votes
1answer
945 views

Why does the naive definition of compactly supported étale cohomology give the wrong answer?

Illusie's article about étale cohomology available here (in French) mentions that the standard definition of compactly supported cohomology (and higher direct images with compact support) does not ...
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 ...
4
votes
2answers
453 views

Closed monoidal structure on the derived category of sheaves

Given a topological space X, i'd like to find Der X - the derived category of sheaves of abelian groups on X - to be a closed monoidal category. Hom should be cohomological and the internal-hom should ...
7
votes
1answer
620 views

Is an irreducible holomorphic symplectic manifold a simple Lie algebra?

The tangent bundle of a hyper-Kahler manifold gives a quadratic Lie algebra in the derived category. Can this be regarded as a simple Lie algebra according to Vogel's definition? A point of view that ...
3
votes
2answers
305 views

Exceptional collections and cohomological criteria for isomorphism

Suppose that we are given a smooth projective variety $X$ with a full exceptional collection of vector bundles $(F_1, F_2, \ldots, F_k)$ in $D^b(X)$ and two vector bundles $E_1$, $E_2$ on $X$. ...
3
votes
1answer
404 views

reference for a result on thick subcategories and t-structures

A thick subcategory of a triangulated category $C$ is essentially one that one can get away with declaring to be zero, i.e. it is the subcategory which sent to 0 when declares that all maps whose ...
22
votes
2answers
3k views

How do I know the derived category is NOT abelian?

I have heard the claim that the derived category of an abelian category is in general additive but not abelian. If this is true there should be some toy example of a (co)kernel that should be there ...
7
votes
1answer
780 views

Verdier duality via Brown representability?

Hello, I wonder if the techniques introduced in Neemans paper: "The Grothendieck duality theorem via Bousfield's techniques and Brown representability " can be used to establish Verdier duality. More ...
2
votes
1answer
227 views

How is this action of monoidal derived category induced?

I am reading a paper concerning the action of monoidal category to another category. Let $k$ be a commutative ring, $R$ is a k-algebra. $A=R-mod$, $B=R^{e}-mod=R\bigotimes _{k}R^{o}-mod$. Consider ...
19
votes
6answers
3k views

Heuristic behind the Fourier-Mukai transform

What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform? Moreover, could someone recommend a concise introduction to the subject?
3
votes
0answers
469 views

Sebastiani-Thom isomorphism for D-modules

Considering $f:X\to \mathbb{C}$, $g:X\to \mathbb{C}$ and $f\oplus g:(x,y)\mapsto f(x)+g(y)$. The Sebastiani-Thom isomorphism is an isomorphism $\Phi_{f\oplus g}(M\boxtimes N) = \Phi_{f}(M) \otimes ...
12
votes
0answers
500 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 ...
5
votes
2answers
616 views

What is the geometric meaning of reconstruction of quantum group via Ringel Hall algebra

If I remembered correctly. There are some work done by C.M.Ringel,he defined so called Ringel-Hall algebra on abelian category and then show that Ringel-hall algebra is isomorphic to positive part of ...
16
votes
5answers
2k views

determinant of a perfect complex

Say $K_\bullet$ is a bounded complex of vector bundles. I seem to want the determinant of $K_\bullet$ to be the alternating tensor product of the terms of the complex: $\det(K) = \bigotimes_n ...
4
votes
3answers
381 views

Derived category with total cohomology finite dimensional: is there a better name for it?

One of the annoying things about derived categories is that they come with a host of different finiteness conditions, which are all subtlely different, and for each situation you want a particular ...
23
votes
9answers
2k views

What is a deformation of a category?

I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references. What is a deformation of a (linear, dg, ...
7
votes
3answers
710 views

Compact generation for modular representations

Are the derived categories of modular representations of algebraic groups compactly generated? (e.g. consider SL_2 in characteristic 2). Note modular reps of finite groups are compactly generated (by ...
8
votes
4answers
1k views

Categories which are not compactly generated

Do you know natural examples of triangulated categories (or [presentable] stable $\infty$-categories) which are not compactly generated? (ideally they'd be defined algebraically, but curious to hear ...
5
votes
2answers
805 views

Singular K3 — mathematical meaning?

There's a very interesting text by Cumrun Vafa called Geometric Physics. Here I'm particularly interested in Chapter 4, where we take a Calabi-Yau manifold presented as a degenerating fibration: ...
0
votes
1answer
366 views

Understanding a lemma in “Loop Spaces and Langlands Parameters” article

First, some background. I was trying to read the article Loop Spaces and Langlands Parameters but I get immediately stuck at Theorem 2.1 in the introduction. This was actually forward-referring to ...
11
votes
1answer
916 views

When do six operations work?

This question comes (heavily edited) from my notes, thus slightly unusual structure. We know that algebraic maps have very strict structure, and in many settings the operations ...
14
votes
3answers
1k views

distinguished triangles and cohomology

Start with A an abelian category and form the derived category D(A). Take a triangle (not necessarily distinguished) and take it's cohomology. We obtain a long sequence (not necessarily exact). If the ...
5
votes
1answer
378 views

Equivariant Derived Categories via their properties.

There are some ways to define equivariant derived categories of all sorts. But all the ways i know of involve giving a concrete construction. Is the other way around possible? Is there some universal ...
8
votes
2answers
937 views

Derived functors vs universal delta functors

I would like to understand the relationship between the derived category definition of a right derived functor Rf (which involves an initial natural transformation n: Qf → (Rf)Q, where Q is the ...
15
votes
4answers
3k views

Is the Fukaya category “defined”?

Sometimes people say that the Fukaya category is "not yet defined" in general. What is meant by such a statement? (If it simplifies things, let's just stick with Fukaya categories of compact ...