6
votes
1answer
226 views

Idea and intuition behind Penrose transform

I would like to know what a Penrose transform is, or more precisely, what is it intended to be - I'm interested in ideas, intuition and some examples of application. My knowledge of differential ...
2
votes
0answers
101 views

Unbounded derived category that is not left-complete

Let me first recall some definition: Let $A$ be a Grothendieck Abelian category. Then, then category $\mathrm{Ch}(A)$ (I am using homological indexing) admits a combinatorial model structure (see for ...
2
votes
1answer
57 views

dg-flat complexes and their characters

Let $\otimes$ denotes the usual tensor products of complexes and symbols live in the category of chain complexes of $R$-modules. Let $X$ be a dg-flat complex (i.e. $X_n$ is flat for each n and ...
0
votes
2answers
191 views

Injective resolution for right derived functor

This question is base on my previous question, and I repeat it here: Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of ...
4
votes
2answers
475 views

Why is the derived tensor product only defined for bounded above derived categories?

In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has $$\otimes: D^{-}(X) \times D^{-}(X) \to ...
0
votes
0answers
245 views

generators for derived category

Let $G$ be a algabraic group $G$ over a field $k$. We denote by $D^b(\mathrm{Repr}(G))$ the derived category of finite dimensional representations. Under what kind of assmumptions one has a generating ...
1
vote
0answers
49 views

injective dimension of dualizing complex

Let $R$ be a balanced dualizing complex of a Noetherian connected graded algebras $A$. Dose one always have $\text{id}_A R = \text{id}_{A^{op}} R$? Thanks a lot.
1
vote
1answer
175 views

Could we extend the Atiyah class to the sheaf of poly-vector fields to get a Poisson bracket?

Let $X$ be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
9
votes
4answers
595 views

What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
0
votes
2answers
245 views

Balanced dualizing complexes according to A. Yekutieli

I am reading A. Yekutieli's original article on dualizing complexes for noncommutative algebras and I found a problem I cannot solve. First, some background. We start with a field $k$ and a ...
3
votes
1answer
168 views

The Hochschild cohomology of a variety “with coefficient” in a vector bundle

This question is related to one of my previous question Do we have the following isomorphism for $\mathcal{Ext}$? Let $X$ be a smooth variety (over $\mathbb{C}$) and $\Delta: X \rightarrow X \times ...
5
votes
1answer
112 views

Is this a description of the $\aleph_1$-localizing subcategory generated by a compact generator?

This should be obvious but I'm not seeing it: The $\mathfrak T$ be a triangulated category with coproducts and with a compact generator $A$ (that is, the functor $\mathfrak T(A,\_)$ preserves ...
3
votes
3answers
267 views

When an exact embedding of abelian categories induces a full embedding of their derived categories?

Let $F:A\to A'$ be a (full) exact embedding of abelian categories. When $D(F):D(A)\to D(A')$ (or its bounded version) is a full embedding also? I would be interested in any necessary or sufficient ...
1
vote
0answers
146 views

Model structure on the category of chain complexes in an abelian category gives rise to the derived category

Let $\mathcal A$ be an abelian category with enough projectives. Consider a morphism $f \colon X^\bullet \longrightarrow Y^\bullet$ in the category $C(\cal A)$ of chain complexes in $\mathcal A$. Is ...
3
votes
1answer
254 views

Commutativity of Tor

Let $A$ be a commutative ring with $1$ and $M,N$ be $A$-modules. Can you give a quick proof that $\textrm{Tor}_i(M,N) \cong \textrm{Tor}_i(N, M)$ using derived categories? In his Homological algebra ...
4
votes
1answer
254 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 ...
3
votes
1answer
141 views

morphism of injective objects

Let $A,B$ be two bounded below complexes in module category, and $A \longrightarrow I$ (resp. $B \longrightarrow J$) a injective resolution. If $f: A \longrightarrow B$ is a morphism of complexes. My ...
1
vote
1answer
168 views

Compact generator of $D(\mathbb{P}^1)$

I suppose that Beilinson's compact generator (and, in fact, tilting object) $\mathcal{O} \oplus \mathcal{O}(1)$ in $D(\mathbb{P}^1)$ is the most well known example. I have the following simple ...
7
votes
2answers
354 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
384 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
194 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 ...
6
votes
0answers
165 views

Evens norm as a derived functor

In the theory of cohomology of groups there are several important maps between cohomologies: restriction, inflation, transfer and Evens norm. The first three of them can be clearly seen on the level ...
1
vote
2answers
464 views

Theorem on composition of derived functors, question about proof

I got a question about a proof I found in Gelfand-Manin's "Methods of homological algebra" (Page 200): Theorem 1. Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be three abelian categories, $F: ...
1
vote
0answers
296 views

Additive functors preserving quasi-isomorphism

Let $F: \mathcal{A} \rightarrow \mathcal{B}$ be an additive functor between abelian categories (with enough injectives and projectives) and $K^\cdot, L^\cdot$ objects of $\textrm{Ch}(\mathcal{A})$. ...
3
votes
1answer
373 views

Unbounded complexes, resolutions and computation of derived functors

Hey guys, let $F: \mathcal{A} \rightarrow \mathcal{B}$ be a left exact functor between abelian categories with enough injectives, let $K \in Kom(\mathcal{A})$ be an unbounded complex, I've heard that ...
2
votes
2answers
275 views

Obtaining derived functors from derived functors of similar complexes or “bluntly truncated” unbounded complexes (without adding 0's to the left)

I don't know if I'm actually using the right terminology here, to be clear I'm going to state explicitly what I'm trying to figure out to see if I can be pointed in the right direction: Let $F: ...
3
votes
2answers
280 views

Cone of a morphism in an abelian category when considered as a morphism in derived category. Connection between 4-term exact sequences and distinguished triangles.

Let $\mathcal{A}$ be an abelian category and let $$ 0 \rightarrow E \rightarrow F \rightarrow G \rightarrow 0 $$ be a short exact sequence. Then in $D(\mathcal{A})$, the derived category of ...
1
vote
1answer
194 views

When do dg-lifts exist?

Let $\mathcal A$ and $\mathcal B$ be abelian categories (with enough injectives and countable products) and $$F:D^+(\cal A) \rightarrow D^+(\cal B)$$ be a triangulated functor. I am interested ...
5
votes
1answer
386 views

Generators of the derived category

For a ring $R$, which is a finite-dimensional algebra over a field, the category of finite-dimensional, projective, right $R$-modules, $\mathcal{P}_R$ is generated by the indecomposable projective ...
3
votes
2answers
272 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 ...
7
votes
2answers
744 views

The composition of derived functors - commutation fails hazardly?

Hello, When we have left exact functors $F: A \to B , G: B \to C$ (between abelian categories), we would like sometimes to state that $D(GF)=D(G)D(F)$ (functors between bounded below derived ...
1
vote
1answer
609 views

Adjunctions between derived functors

Given an adjunction $F\dashv G$ between functors between Abelian categories, we know that $F$ is right exact and $G$ is left exact so there are derived functors $LF$ and $RG$ between (bounded above, ...
0
votes
1answer
361 views

Recollements and global dimension

Let $A, B, C$ be algebras. Suppose that $D^b(A)$ (the bounded derived category of $A$) admits a recollement relative to $D^b(B)$ and $D^b(C)$. Then, by a result of Alfred Wiedemann's paper "On ...
0
votes
1answer
530 views

derived functors and triangulated categories

If you derive a right exact functor $F$ you get a functor normally denoted by $RF$ on the derived category. Similarly, if you start with a left exact functor $G$ you get a functor normally denoted by ...
3
votes
0answers
170 views

Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor

Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see ...
6
votes
2answers
569 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?
7
votes
4answers
706 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
1k 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 ...
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 ...
9
votes
1answer
663 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
468 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. ...
4
votes
2answers
437 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 ...
3
votes
1answer
394 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 ...
21
votes
2answers
2k 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 ...
12
votes
0answers
493 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 ...
4
votes
3answers
377 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 ...
11
votes
1answer
859 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 ...
22
votes
6answers
2k views

How to think about model categories?

I've read about model categories from an Appendix to one of Lurie's papers. What are the examples of model categories? What should be my intuition about them? E.g. I understand the typical examples ...
11
votes
3answers
678 views

Freyd-Mitchell for triangulated categories?

Is there a nice analog of the Freyd-Mitchell theorem for triangulated categories (potentially with some requirements)? Freyd-Mitchell is the theorem which says that any small abelian category is a ...