5
votes
1answer
53 views

Ext functor commutes with connecting homomorphisms?

Suppose we have an exact sequence $0 \to L \to M \to N \to 0$ and a morphism $f \colon A \to B$ of $R$-modules. If $\delta \colon \text{Ext}^{i}_{R}(B,N) \to \text{Ext}^{i+1}_{R}(B,L)$ and $\delta' ...
2
votes
0answers
119 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 ...
2
votes
2answers
85 views

Having trouble understanding the Tor functor

I am having trouble understanding the Tor functor as presented in Dummit and Foote. Given $\dotsb\to P_n\to P_{n-1}\to\dotsb\to P_0\to B\to 0$ as a projective resolution with homomorphisms ...
4
votes
0answers
86 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 ...
6
votes
2answers
102 views

Why is the definition of $\lim^1$ via a cokernel the first derived functor of $\lim$?

Let $A_*=\ldots\to A_n\to A_{n-1}\to\ldots\to A_0$ be a linear system of abelian groups. The limit of this system may be defined as the kernel of the map $$ \prod A_n\xrightarrow{g-1}\prod A_n $$ ...
0
votes
0answers
26 views

Exactness of derived functor [duplicate]

Is the right derived functor of a left exact functor left exact also? If now, can anything be said about its exactness in general?
4
votes
0answers
48 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 ...
2
votes
0answers
30 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 ...
15
votes
1answer
416 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
2
votes
0answers
61 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 ...
3
votes
1answer
63 views

$\mathrm{Ann}_RA+\mathrm{Ann}_RB\,\subseteq\mathrm{Ann}_R\,\mathrm{Ext}^n_R\!(A,B)$?

Let $R$ be a commutative unital ring and $r\!\in\!R$. Let $A$ and $B$ be $R$-modules. Does $rA\!=\!0$ or $rB\!=\!0$ imply $r\mathrm{Ext}^n_R(A,B)=0$ for all $n\in\mathbb{N}$? For $n=0$ it holds, but ...
3
votes
2answers
92 views

Derived functors definition

I´m searching for a reference that defines $n^{th}$derived functors in an analogous way to the definition given in Mitchell´s "Theory of Categories" for the $0^{th}$ derived functor of $T$ covariant ...
0
votes
1answer
117 views

$\mathrm{Tor}_1(R/a,M)$ and $\mathrm{Ext}^1_R(R/a,M)$, $a\in R$ a non-zero divisor

In Lecture Notes in Algebraic Topology, Davis & Kirk, it is written: Proposition $\mathbf{2.4.}\,\,$ Let $R$ be a commutative ring and $a\in R$ a non-zero divisor (i.e. $ab=0$ implies $b=0$). ...
4
votes
1answer
75 views

Does $\varprojlim\ ^1$ vanish whenever it doesn't have to account for non-right exactness of $\varprojlim$?

The projective limit functor is not right-exact: if $G_\bullet\rightarrowtail H_\bullet\twoheadrightarrow K_\bullet$ is a projective system of extensions, then there is a long exact sequence $$ ...
2
votes
1answer
69 views

On the element $2\in\mathrm{Ext}(\mathbb Z/4,\mathbb Z)$.

There is a canonical isomorphism $\mathrm{Ext}(\mathbb Z/4,\mathbb Z)\cong\mathbb Z/4$ based on the fact that an extension $$0\to\mathbb Z\xrightarrow i G\xrightarrow{p}\mathbb Z/4\to 0$$ is ...
6
votes
1answer
165 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 ...
5
votes
1answer
128 views

Equivalence between Ext and Hom

This is a question from Homology by Saunders Mac Lane. This is problem 5 page 76. I've been struggling to solve this problem for like more than a day, but still nothing valuable comes across my mind ...
2
votes
2answers
140 views

Equivalence of categories and derived functors.

Don't know if this kind of a dumb question but let $A$ and $B$ be abelian categories and suppose they're equivalent: there are two functors $P: A \rightarrow B$ and $Q: B \rightarrow A$ satisfying the ...
9
votes
1answer
165 views

Ext between two coherent sheaves

Let $X$ be a smooth projective variety over a field $k = \overline k$. From Hartshorne we know, that $\textrm{dim} \, H^i (X,F)<\infty$ for any coherent sheaf $F$. How to show, that all $Ext^i ...
2
votes
0answers
48 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_* ...
5
votes
1answer
135 views

Sheaves on $\mathbb{P}^n \times \mathbb{P}^m$, and a commutation relation for derived functors of global sections and tensor products on it.

I'll state my questions first and then provide some background. Question 3 is by far my most important one. We work over $k=\mathbb{C}$ whenever necessary. Is it true that $\text{Pic}(\mathbb{P}^n ...
3
votes
1answer
133 views

Computing left derived functors from acyclic complexes (not resolutions!)

I am reading a paper where the following trick is used: To compute the left derived functors $L_{i}FM$ of a right-exact functor $F$ on an object $M$ in a certain abelian category, the authors ...
4
votes
1answer
195 views

Why are these two functors isomorphic?

Let $A$ be a local noetherian ring, $M$ an $A$-module finitely generated. Let $f$ be an $A$-regular and $M$-regular element (i.e. $f$ is not a zero divisors on $A$ nor on $M$). Then inside the ...
9
votes
2answers
151 views

What does $Tor_{R}^n(M,N)$ represent?

Let $R$ be a commutative ring and $M$ and $N$ be $R$-modules (I am not sure if one really needs commutativity in the following). It is well-known that $Ext_{R}^n(M,N)$ for $n>1$ parametrizes ...
4
votes
0answers
124 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
149 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: ...
7
votes
0answers
244 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: ...
2
votes
1answer
161 views

Question about the $\mathrm{Tor}$ functor

Assume we want to define $\mathrm{Tor}_n (M,N)$ where $M,N$ are $R$-modules and $R$ is a commutative unital ring. We take a projective resolution of $M$: $$ \dots \to P_1 \to P_0 \to M \to 0$$ Now ...
7
votes
1answer
83 views

Is there an easy formula for $\operatorname{Tor}_i^{\mathbb{Z}/(p^n)}(\mathbb{Z}/(p),\mathbb{Z}/(p))$?

I've been following some old slides from the Spring 2010 Algebra Seminar at UWaterloo. I now know that $$ \operatorname{Ext}_{\mathbb{Z}/(p^n)}^i(\mathbb{Z}/(p),\mathbb{Z}/(p))\cong\mathbb{Z}/(p) $$ ...
3
votes
1answer
90 views

Is there a general formula for $\operatorname{Ext}_{\mathbb{Z}/(p^n)}^i(\mathbb{Z}/(p),\mathbb{Z}/(p))$?

The other day I was reading through some slides I found online about Ext and Tor. One of the examples gave a cursory derivation for a general formula $$ ...
11
votes
0answers
295 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 ...
3
votes
1answer
186 views

$\mathrm{Tor}$ functor not left exact

Is there an example which shows that the functor $B\otimes_R(-)$ is not left-exact, given a ring $R$ and a right $R$-module $B$?
5
votes
1answer
246 views

Some questions about the Tor functor as a two-variable functor related to the arbitrary character of the choice of projective resolutions

Given a ring $R$, we can consider the following functors: any $A\in Mod-R$ and choice of projective resolutions $P_\bullet(B)$ for every $B\in R-Mod$ defines a functor $Tor_n^R(A,-):R-Mod\to Ab$, ...
3
votes
0answers
159 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 ...
0
votes
1answer
89 views

Right derived functor of diagonal morphism equals direct image on line bundles?

Let $X$ be a smooth projective variety. The map $i:X\to X\times_k X$ induced by the identity is a closed immersion. Denote its image by $\bigtriangleup$. We have ...
2
votes
2answers
320 views

Derived functors are Kan extensions

In this short paper by G. Maltsiniotis derived functors are presented as Kan extensions along the localization functor. I began studying derived categories only a couple of months ago, so I'm not at ...
1
vote
1answer
118 views

Question about derived functors

Let $F,G, H: Mod \to Mod$ be three left exact functors such that $R^iF(-)\cong R^iG(-)$ for all $i\in\mathbb{N}$. We consider the exact sequence $$\cdots\to R^iF(M)\to R^iG(M)\to R^iH(M)\to ...
9
votes
2answers
463 views

Derived functors of torsion functor

Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a ...
0
votes
1answer
241 views

derived functors and acyclics

I'm not sure how I can show the following: If F is a left exact functor from an abelian category A to an abelian category B, whose derived functor RF in the sense of derived categories exists, then ...
9
votes
1answer
356 views

Derived functor of a derived functor

Given $F$ is a covariant additive functor from left R-module to a left S-module, show that $\mathscr{L}_n(\mathscr{L_m}(F))=0$ if $m>0$ (where $\mathscr{L}$ refers to the derived functor). I am ...
4
votes
5answers
313 views

Derived category and so on

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...
35
votes
3answers
4k views

What is the Tor functor?

I'm doing the exercises in "Introduction to commutive algebra" by Atiyah&MacDonald. In chapter two, exercises 24-26 assume knowledge of the Tor functor. I have tried Googling the term, but I ...