In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. (Def: http://en.m.wikipedia.org/wiki/Derived_functor)

learn more… | top users | synonyms

6
votes
1answer
200 views

Description of Tor via the derived category?

If $A,B$ are objects of an abelian category $\mathcal{A}$ and $n \in \mathbb{N}$, there is a very nice and useful description of $\mathrm{Ext}^n(A,B)$. Namely, it is just the set of morphisms $A \to ...
4
votes
1answer
42 views

What is higher kernel explicitly?

Let $\mathcal{A}$ be an abelian category (for simplicity you can think that $\mathcal{A}$ is the category of modules over ring $R$). Let $[1]$ be the category with two objects and one arrow between ...
3
votes
1answer
63 views

Physical interpretation of categorical structures related to Dirichlet Branes

In Dirichlet Branes and Mirror Symmetry by Aspinwall et al, section 5.9 discusses various questions that remain open. In particular they say: "There are many constructions from homological ...
0
votes
1answer
64 views

Tor functors: a basic explanation?

Could anyone give a basic explanation about Tor functors and, particularly, any idea about how they might be useful for the description of natural language?
14
votes
0answers
342 views

Why do universal $\delta$-functors annihilate injectives?

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to ...
7
votes
0answers
294 views

Composition of derived functors and comparison between hypercohomology and sheaf cohomology

I had a few questions about compositions of derived functors, the comparison between hypercohomology, and sheaf cohomology and the following theorem from the Gelfand, Manin homological algebra book: ...
6
votes
0answers
151 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
4
votes
0answers
76 views

Spectral sequence $\bigoplus_{k-j=q}\mathrm{Ext}^p(\mathcal{H}^j,\mathcal{H}^k)\Rightarrow \mathrm{Hom}^{p+q}(P,P)$

Reading the proof in Bondal-Orlov reconstruction theorem (http://arxiv.org/pdf/alg-geom/9712029v1.pdf), I found the spectral sequence in the title ...
4
votes
0answers
283 views

Long exact sequence in cohomology associated to a short exact sequence of *functors*

In homological algebra, when you have a left exact functor $F$ From an abelian category $\mathcal{A}$ to an abelian category $\mathcal{B}$ and you have enough injectives in $\mathcal{A}$, then you ...
4
votes
0answers
61 views

The functor $\underline{\mathbf{R}}^if_*$

Let $f: X \to Y$ be a proper morhpism of varieties, and $\mathcal{F}$ be a sheaf on $X$. Then we have $f_* \mathcal{F}$ as a sheaf on Y and we also have a higher derived functor $\mathbf{R}^i ...
4
votes
0answers
153 views

Cartan-Eilenberg resolutions, adapted classes and acyclic resolutions

I may get grilled for this but here I go: Let $\mathcal{A}$ be an abelian category with enough injectives. What I want to know is VERY VERY specific. Let's say I have a complex in $\mathcal{A}$ $0 ...
3
votes
0answers
81 views

A certain natural map between Tor functors

Consider the following Here $A$ is a flat (commutative, unital) $k$-algebra ($k$ a commutative ring) and $\mu:A\otimes_k A\rightarrow A$ is by $\mu(a\otimes b)=ab$, $\mathcal{M}$ denotes a maximal ...
3
votes
0answers
144 views

Homotopy limits

Let $\mathfrak C$ be a Grothendieck category and let ${\bf D}=\mathrm{D}(\frak C)$ be its derived category, that is, consider the injective model structure on the category $\mathrm{Ch}(\frak C)$ of ...
3
votes
0answers
182 views

Adapted classes of objects and left (right) exact functors

I had a question about adapted classes of objects, I was confused by the definition and how it relates to left exact functors. Let $\mathcal{A}$ be an abelian category with enough injectives, let $F: ...
3
votes
0answers
175 views

Left-derived functors

Let $F:\mathcal{A}\to\mathcal{B}$ be a covariant right-exact functor between two abelian categories. Suppose $\mathcal{A}$ has enough projectives. Then we define the left derived functors of $F$ by ...
2
votes
0answers
36 views

Computation of $Ext^*_R(k,k)$ as an algebra using a dga-resolution

There is a theorem (VIII.2.3) in Mac Lane's Homology that reads: Let $k$ be a commutative ring. Let $R,S$ be $k$-algebras, and let $U$ be a $k$-differential graded algebra. Suppose there is a ...
2
votes
0answers
33 views

Derived categories of filtered modules

For a ${\mathbb Z}$-filtered ring ${\mathbb k}$ one can consider the category ${\mathbb k}\text{-filt}$ of ${\mathbb Z}$-filtered ${\mathbb k}$-modules, equipped with the exact structure which ...
2
votes
0answers
64 views

Derived version of projection formula

Let $f \colon X \to Y$ be a continuous map of locally compact spaces. Denote by $Sh(X)$, $Sh(Y)$ the categories of sheaves of $k$-vector spaces for some field $k$ and by $D^b(X)$, $D^b(Y)$ their ...
2
votes
0answers
84 views

Derived functors and coboundary operator

I understand that one can define the cohomology of an object $A$ in terms of a complex (non-zero in positive degrees) in some Abelian category, together with differentials, such that the composition ...
2
votes
0answers
53 views

Hypercohomology and acyclicity of direct images

Let $f: X\rightarrow Y$, $g:Y \rightarrow Z$ be morphisms of topological spaces and let $K^{.}$ be an injective object in the category of complexes of abelian sheaves on $X$. Write $\mathbb R^0f_* ...
2
votes
0answers
153 views

Computing Derived Pullback on the Complement

Let $X$ be a scheme and $\iota: Z\hookrightarrow X$ the embedding of a closed subscheme $Z$; let $j: U\hookrightarrow X$ be the open complement. Suppose $\mathcal{F}$ is a coherent sheaf on $X$. ...
1
vote
0answers
23 views

Derived direct image of emdedding of projective varieties

Let $i:Y\to X$ be an embedding map between projective varieties. What is the example of $X$ and $Y$ such that the functor $Ri_*:D^b(\text{Coh}\,Y)\to D^b(\text{Coh}\,X)$ is not a fully faithful ...
1
vote
0answers
18 views

Naturality of connecting homomorphisms

Let $\mathcal{F}$ be a right-exact additive functor on the category of R-modules (R a fixed ring). Proposition A3.17(d.) in Eisenbud's Commutative algebra with a view towards algebraic geometry states ...
1
vote
0answers
26 views

Reference for derived functor

I'm following a course in algebraic geometry and in 2-3 month we will see the cohomology of schemes using derived functors. I don't know anything about it, (and about category theory in general), ...
1
vote
0answers
40 views

derived versions of natural isomorphisms

I have just recently started approaching the topic of derived categories in algebraic geometry, and I'm doing so reading Huybrechts "Fourier-Mukai transforms in algebraic geometry". I have a doubt ...
1
vote
0answers
42 views

Equivalences between categories $\mathcal{K}^b(\text{Injectives})$ and $\mathcal{D}^b(\mathcal A)$ if $\mathcal{A}$ has enough injectives

I have the following question: Let $\mathcal{A}$ be a abelian category and $\mathcal{I}$ be the full subcategory of injective objexts of $\mathcal{A}$. Assume that $\mathcal{A}$ has enough ...
1
vote
0answers
128 views

Derived push-forward of projective sheaf

Let S,X be schemes and $s \in S$ be a closed point. Let $D(X)$ be the derived category of complexes of sheaves. Let $$i_s: X \cong {s} \times X \hookrightarrow S \times X$$ be the natural embedding. ...
1
vote
0answers
50 views

Where is the mistake? (derived functors )

Assume $pd(M) =n \leq \infty$ for a left $R$-module. I then have to show there exists a free module $F$ such that $Ext_{R}^{n}(M,F) \neq 0 $. I have tried these steps and obtained a contradiction: ...
1
vote
0answers
27 views

Why is dual functor continuous?

Recall that a functor F is continuous is the map from Hom(V,W)to Hom(F(V),F(W)) is always continuous. I have already know how to prove the functor V** is continuous, but don't know why the functor ...
0
votes
0answers
47 views

$\mathrm{Ext}^i(-,A/\mathfrak{m})$ in $(A,\mathfrak{m})$ noetherian regular local ring

Dealing with $\mathrm{Ext}^i(\mathcal{F},k(x))$ on a smooth variety over a field $k$, with $\mathcal{F}$ coherent and $k(x)$ skyscraper sheaf of a closed point I foundin a proof that for $i=2,3$ (and ...
0
votes
0answers
21 views

Closure properties for classes of modules that form a cotorsion pair

A torsion theory is a pair of classes of $R$-modules (where $R$ is an associative ring with identity) $({\mathbb T},{\mathbb F})$, such that $r({\mathbb T})={\mathbb F}$ and $l({\mathbb F})={\mathbb ...
0
votes
0answers
41 views

How do you take divergence of this field?

I forgot how to do divergences 3 years ago, this one is very confusing. We used to take them with respect to $x,y,z$ but this one doesn't have them. Help me with this question Prove that $∇.E = 0$ ...
0
votes
0answers
209 views

how to derive the canonical form of a transfer second order equation?

How to derive the canonical form of the second order transfer function?? $$\frac{(\omega_n)^2}{s^2+2\zeta\omega_ns + (\omega_n)^2}$$