Skip to main content

Questions tagged [derived-functors]

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones.

Filter by
Sorted by
Tagged with
0 votes
0 answers
53 views

derived version of Picard functor as a derived stack

The notation originates from the paper virtual Cartier divisor and derived blow up I read recently, in its proof of proposition 3.2.6, there is a derived stack: $\\$$\underline{Pic} ^{\simeq}$: $(...
Yang's user avatar
  • 85
3 votes
1 answer
111 views

Understanding how group cohomology classifies extensions using the derived functor point of view

I am rereading some material about group extensions, in particular because I needed to recall the formula $$H^2(G;A)\cong \mathcal{E}(G;A).$$ We have that $G$ is some group acting on an abelian group $...
DevVorb's user avatar
  • 1,495
1 vote
0 answers
37 views

Behaviour of graded Betti number of a module

Let $M$ be a graded finitely generated $R=\mathbb{K}[x_1,\dots,x_n]$-module and, the graded Betti number of $M$ is defined by $\beta_{i,p}^{R}(M)=\mathrm{dim}_{k}(\mathrm{Tor}_i^{R}(M,k)_p)$. Suppose $...
Raman's user avatar
  • 199
0 votes
1 answer
53 views

Equivalent conditions of $\textrm{Ext}^2(A,B) = 0$ for all $A$ and $B$

I am studying homological algebra and I am having difficulty proving the equivalences of the following: (i) If $0 \to A \xrightarrow{f} B$ is exact and $B$ is projective, then $A$ is projective (ii) ...
Squirrel-Power's user avatar
1 vote
1 answer
68 views

Prove that for $F$ an additive functor of abelian categories, $R^0F$ is exact iff $R^1F = 0$ iff $R^iF = 0$ for all $i > 0$

I am beginning to study homological algebra. Let $F: \mathcal{A} \to \mathcal{B}$ be an additive functor of abelian categories. Prove that the following are equivalent: The functor $R^0F$ is exact $R^...
Squirrel-Power's user avatar
1 vote
0 answers
34 views

How can Lie algebra cohomology be nontrivial for a semisimple algebra?

Let $\mathfrak{g}$ be a semisimple Lie algebra over an (algebraically closed) field $k$ of characteristic zero. I am going by the definitions in Weibel, chapter 7. Here's my logic: A finite-...
smitke6's user avatar
  • 699
5 votes
0 answers
55 views

On a possible isomorphism from a spectral sequence coming from derived tensor-hom adjunction

Let $M,N,X$ be modules over a commutative ring $R$. We have the derived tensor-hom adjunction $$\mathbf R\text{Hom}_R(M\otimes_R^{\mathbf L} N,X)\cong \mathbf R\text{Hom}_R(M,\mathbf R\text{Hom}_R(N,...
Alex's user avatar
  • 433
1 vote
0 answers
17 views

Rank of a D-module and the solution complex

For a $D$-module $M$ it is common to talk about its rank defined as the $\mathbb{C}(x_i)$-dimension of $\mathbb{C}(x_i)\otimes_{\mathbb{C}[x_i]} M$. Kashiwara’s Cauchy–Kovalevskaya theorem then ...
A.H's user avatar
  • 41
0 votes
1 answer
64 views

Is a finitely generated module over a hereditary ring always finitely presented? When does $Ext^{n}( M, -)$, for $n \geq 0$, commute with direct sum?

In the §6 Appendix II (2) of the article Gorenstein projective modules says that: Lemma 1 : Let $R$ be a ring. If $M$ is a finitely generated $R$-module, then $$Ext^{n}( M, \oplus_{i\in I}\ N_{i}) \...
Liang Chen's user avatar
0 votes
0 answers
34 views

Homology defining quasi-isomorphisms vs sheaf cohomology

I don’t understand how the homology groups in regards to the derived category of sheafs on a space X is connected to the cohomology of a sheaf which is calculated with the images/kernels after ...
Tom Gatward's user avatar
0 votes
0 answers
46 views

Functoriality of derived tensor product ( Gortz's Algebraic Geometry, Vol.2. )

I am reading Gortz's Algebraic geometry book, Vol.2 , p.207 and some qestion arises. I think that I am begginer of cohomology theory of schemes so please understand. Let $f:X\to Y$ be a morphism of ...
Plantation's user avatar
  • 2,656
5 votes
0 answers
101 views

On the two definitions of derived functor in general triangulated category.

I'm learning homological algebra using several references books. But I find two definitions of derived functor in general triangulated category. I wonder to know which definition is more generally ...
Z. He's user avatar
  • 502
9 votes
2 answers
683 views

What is the "goal" of derived functors?

I've been learning about derived functors recently, and I had conceptualized them as fulfilling the following goal: Suppose that we had a left-exact functor $F:\mathcal{A} \to \mathcal{B}$ between two ...
Irving Rabin's user avatar
  • 2,673
1 vote
0 answers
71 views

Basic questions of triangulated functors

I am not familiar with triangulated categories so these questions might be too basic (but I did not find any answers by google). Also, the question can be formulated in purely triangulated category ...
Cyrist's user avatar
  • 39
1 vote
1 answer
57 views

A faulty "proof" regarding exactness of derived functors

I've been trying to wrap my head around derived functors and have come upon the following chain of arguments, which seems to yield the unreasonabel conclusion that all left derived functors are exact. ...
Jeppe Obel's user avatar
1 vote
0 answers
45 views

Can the inflation map be seen as a map between two $\delta$-functors?

Let $G$ be a group and $H$ be a normal subgroup. For each $r \geq 0$, the inflation map from $H^r(G/H, M^H) \to H^r(G,M)$ is defined by the composition of two maps $H^r(G/H, M^H) \to H^r(G, M^H) \to H^...
Kyaw Shin Thant's user avatar
0 votes
1 answer
59 views

Show $E \cong E'$ as R-modules given $ h : N \to N'$ isomorphism, $\mathscr E : 0 \to N \to E \to M \to 0$ and its pushout

Let $R$ be a ring, and consider an extension $\mathscr E : 0 \to N \to E \to M \to 0$ of $R$-modules. If $h : N \to N'$ is a homomorphism, we can form the pushout $h_*(\mathscr E ) : 0 \to N' \to E' \...
darkside's user avatar
  • 589
3 votes
0 answers
198 views

$\mathrm{Ext}$ and direct limit

Let $R$ be a commutative Noetherian ring. Then, for $R$-modules $\{X_i\}_i$ and $Y$, do we always have $$\mathrm{Ext}^n_R(\varinjlim X_i,Y) \cong \varprojlim \mathrm{Ext}^n_R( X_i,Y)$$ for all $n\geq ...
Alex's user avatar
  • 433
1 vote
1 answer
71 views

Topological proof that $Ext(A, \mathbb{Q})=0$

In Hatcher's algebraic topology book, section 3.F, there is this exercise which seems really neat and I wanted to try. The exercise is to show that $Ext^1_\mathbb{Z}(A, \mathbb{Q})=0$ using the ...
DevVorb's user avatar
  • 1,495
5 votes
1 answer
333 views

Suppose $0\to A\to B\to C\to D\to 0$ is exact. Let $0\to A[2]\to B[2]\to C[2]\to X\to 0$ be an exact sequence. Then, $X [2]$ is finite.

Let $A,B,C,D$ be abelian groups and $A$ is finite. Suppose $D[2]$ is finite. Is the following true ? Suppose $0\to A\to B\to C\to D\to 0$ is exact. Let $0\to A[2]\to B[2]\to C[2]\to X\to 0$ be an ...
Poitou-Tate's user avatar
  • 6,351
0 votes
1 answer
38 views

$2$ part of $Ext(\Bbb{Z}/2\Bbb{Z}, A)$ when $A$ is finite

Let $A$ be a finite group. Let denote $Ext^1$ by $Ext$ functor in homological algebra. Is $(Ext^1(\Bbb{Z}/2\Bbb{Z}, A))[2]$ finite? When $A$ is finite cyclic group $\Bbb{Z}/m\Bbb{Z}$, $\#Ext^1(\Bbb{Z}/...
Poitou-Tate's user avatar
  • 6,351
0 votes
0 answers
77 views

Can we explicitly describe the derived pullback $\mathbf L\pi^* \widetilde M$ for a closed immersion $\pi$ of affine schemes?

Let $I$ be an ideal of a commutative Noetherian ring $R$. Let $M$ be a finitely generated $R$-module and $\widetilde M$ be its associated sheaf on $\text{Spec} (R)$. We have the closed immersion $\pi:...
uno's user avatar
  • 1,560
0 votes
1 answer
31 views

Evaluating a derived functor on an object where the original functor is 0

I feel like this question has surely already been asked, but I wasn't able to find a formulation which made my search fruitful, so here goes. Let $C, D$ be two abelian categories, and F a (left/right ...
DevVorb's user avatar
  • 1,495
1 vote
0 answers
41 views

Example of a factorisation of functors $F = HK$ for which the Kan extension of $F$ along $K$ is not $H$.

I was reading Emily Riehl's book: Categorical Homotopy theory, and I encountered exercise 1.1.3: Exercise 1.1.3: Construct a toy example to illustrate that if $F$ factors through $K$ along some ...
julio_es_sui_glace's user avatar
2 votes
0 answers
70 views

Non-abelian cohomology

In this paper on page 8 there is the following passage: Let $A$ be an algebra, $M$ an $A$-module. There are the following approaches to the "cohomology of $A$ with coefficients in $M$". ...
Margaret's user avatar
  • 1,769
0 votes
0 answers
32 views

Help studying the derived functor of the limit functor

I am currently working through the second half of Hatcher's algebraic topology and through Weibel's Homological algbera. In the latter the next chapter I am going to read is the derived functor of the ...
DevVorb's user avatar
  • 1,495
0 votes
1 answer
66 views

Why is $f_{\ast}=f_{!}$ when $f$ is proper?

Let $f$ be a morphism of schemes. I have seen that $f_{\ast}=f_{!}$ when $f$ is proper. Why is it true (if it is not trivial where can I find a proof)? On which generality is it true? (For example, is ...
Marsault Chabat's user avatar
2 votes
0 answers
55 views

$\pi:\Bbb P^1_k\to \text{Spec}(k)$ then is $\Bbb L\pi^*\tilde{k}\cong \mathcal{O}_{\Bbb P^1_k}[0]$?

Let $\pi:\Bbb P^1_k\to \text{Spec}(k)$. Am I correct that $\Bbb{L}\pi^*\tilde{k}$ is just $\mathcal{O}_{\Bbb{P}^1_k}[0]\in D_{\text{qc}}(\Bbb P^1_k)$? I saw $\Bbb{L}\pi^*\tilde{k}$ being talked about ...
F.White's user avatar
  • 501
2 votes
0 answers
44 views

Webeil's Intro to Homological Algebra: does theorem 10.6.3 implicitly reindex cochain complexes to chain complexes?

Let $R$ be a ring, let $\mathbf{D^-(R-mod)}$ denote the derived category of bounded above cochain complexes of $R$-modules, and consider the total tensor product functor $$ \otimes_R^\mathbf{L}: \...
xion3582's user avatar
  • 470
1 vote
1 answer
85 views

$\operatorname{Tor}^{\mathbb{Z}}_1(-,-)$ on finite abelian groups is not right exact?

In his answer here Martin Brandenburg claims that the Tor functor $\operatorname{Tor}^{\mathbb{Z}}_1(-,-)$ in the category of finite abelian groups is not right exact in neither argument. Since Tor is ...
Peter's user avatar
  • 881
0 votes
0 answers
31 views

How to construct $\delta^i$ morphism for right derived functors?

Suppose $F: \mathcal{A} \to \mathcal{B}$ is a left-exact functor between abelian categories. Assume $\mathcal{A}$ has enough injectives. I want to prove the following: For every short exact sequence $...
Anthony Lee's user avatar
0 votes
0 answers
37 views

Proof regarding when left exact functors are $\delta$-functors

Here is the theorem I am trying to prove: $\mathcal{A}$ and $\mathcal{B}$ are abelian categories. Suppose $F: \mathcal{A} \to \mathcal{B}$ is a left exact functor. If $\mathcal{A}$ has enough ...
Anthony Lee's user avatar
1 vote
1 answer
97 views

Right derived functors are additive

I am trying to prove the following statement: Let $F: \mathcal{A} \to \mathcal{B}$ be a left-exact functor between abelian categories. Suppose $\mathcal{A}$ has enough injectives. Then the right ...
Anthony Lee's user avatar
2 votes
0 answers
197 views

Why is the Yoneda product sometimes called cup product?

In algebraic topology there is the cup product, which endows the direct sum of (singular) cohomology groups with the structure of an associative, graded-commutative, unital ring. Given an associative, ...
Margaret's user avatar
  • 1,769
1 vote
2 answers
186 views

When is the induced representation functor exact?

I only have superficial familiarity with concepts of homological algebra, and couldn't find this written down explicitly anywhere so I wanted to make sure. Here is my basic argument: Let $G$ be a ...
smitke6's user avatar
  • 699
0 votes
0 answers
94 views

Compute the stalk and costalk of a pushforward of the constant sheaf

I am reading Achar's book about perverse sheaves. Now I am trying to solve the exercise 1.10.5 in this book (all varieties are assumed over $\mathbb{C}$ and sheaves are over a field $k$): Define $$ \...
Runner's user avatar
  • 145
1 vote
1 answer
48 views

Reference for result and proof that $R(g_*\circ f_*)\cong Rg_*\circ Rf_*$ for morphisms of ringed spaces $X\xrightarrow{f}Y\xrightarrow{g}Z$

I am trying to find a reference that states and proves the following Lemma: Lemma. Let $(X,\mathcal{O}_X)\xrightarrow{f}(Y,\mathcal{O}_Y)\xrightarrow{g}(Z,\mathcal{O}_Z)$ be morphisms of ringed ...
Elías Guisado Villalgordo's user avatar
0 votes
0 answers
71 views

Derived $\mathrm{Hom}$-functors commute with finite direct products?

Let $\mathcal{R}_X$ be a sheaf of (commutative) rings on a topological space $X$. Let $\mathcal{M}_X, \mathcal{N}_X, \mathcal{P}_X$ be $\mathcal{R}_X$-modules. Do the derived $\mathrm{Hom}$-functors ...
Flavius Aetius's user avatar
2 votes
1 answer
157 views

If a resolution $f:Y\to X$ satisfies $R^if_*\omega_Y=0$ and $R^if_*\mathcal{O}_Y=0$ for all $i>0$, then do we have $f_*\omega_Y=\omega_X$?

Let $X$ be a normal projective variety over an algebraically closed field of arbitrary characteristic (but I'm mainly interested in positive characteristic). Assume that $X$ has rational singularities,...
imtrying46's user avatar
  • 2,789
1 vote
1 answer
120 views

Cohomology of an elliptic surface with section

Let $\pi:S \to B$ be an elliptic surface with a section $\sigma$. Let $f$ be the linear equivalence class of any fiber. Then for all integers $a$ $h^0(\mathcal O_S(-\sigma + af)) = 0 $. $R^0\pi_*\...
Conjecture's user avatar
  • 3,270
1 vote
1 answer
48 views

Poisson-like distribution for partially or fully observed events with "duration"

Consider a type of event that has fixed duration $\delta$. Let $\lambda$ be the rate of events starting / ending (assuming steady state). I want to derive the expected number of events observed (...
jessexknight's user avatar
0 votes
0 answers
89 views

Nearby cycle complex of a scheme over a DVR with finite number of singularities

Let $R$ be a DVR and $S = \mathrm{Spec}(R)$ with closed point $s$ and generic point $\eta$. Let $X\to S$ be a scheme over $S$, and denote by $X_s$ and $X_{\eta}$ the special and generic fibers. We ...
Suzet's user avatar
  • 5,571
0 votes
0 answers
40 views

Conceptualizing Derived Functors

I am working through Weibel's "An Introduction to Homological Algebra". After reading the first 3 chapters, to me it seems that there is a much easier way to think about derived functors as ...
user1104937's user avatar
1 vote
1 answer
76 views

Detecting projective dimension of finitely generated modules over Noetherian semi-perfect rings via vanishing of Ext on simple modules

Let $R$ be a Noetherian semi-perfect ring. Let $M$ be a finitely generated left $R$-module. Let $n \geq 0$ be an integer. If $\text{Ext}^{n+1}_R(M,S)=0$ for every simple left $R$-module $S$, then is ...
Alex's user avatar
  • 433
3 votes
1 answer
133 views

Quillen pair with derived equivalence $\implies$ Quillen equivalence

Let $L\dashv R$ a Quillen pairs an adjunction between model categories $M,M'$ with $L$ preserving cofibrations and $R$ fibrations. From then we can construct an adjunction $\mathbb LL\dashv \mathbb RR$...
raisinsec's user avatar
  • 463
0 votes
1 answer
51 views

reference request support of derived tensor product

I currently am working on trying to compute some Hochschild cohomology of some scheme. However, I should be able to do it as soon as I know/have a reference for the following natural statement: Let $X$...
Felix's user avatar
  • 2,339
1 vote
0 answers
143 views

Vanishing of $\text{Ext}^i_{X}(E, F)$ vs. $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ for $E,F \in \mathcal D^b(\text{Coh } X)$

Let $X$ be a Noetherian scheme and $E,F \in \mathcal D^b(\text{Coh } X)$. Let $x\in X$ be a closed point. Then, is there any connection between $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ and $\text{Ext}^...
Alex's user avatar
  • 433
2 votes
0 answers
161 views

Compare two definition of $Rf_!$ (derived pushforward with proper support)

I'm talking about etale sheaves. For a morphism $f:X\to Y$ of schemes, there are two definition of $Rf_!:D(X)\to D(Y)$. The usual derived functor: $\forall\mathcal{F}\in Sh(X)$, let $f_!\mathcal{F}$ ...
Xiong Jiangnan's user avatar
2 votes
0 answers
104 views

Categories and Sheaves Kashiwara Schapira Theorem 14.4.5

In the proof of the theorem I have problems understanding why the functor $\operatorname{Hom}_\mathcal{C}$ has a right derived functor, as C is not necessarily a Grothendieck category. Moreover, but ...
Joscha's user avatar
  • 21
0 votes
0 answers
37 views

left bounded chain complex $Z$ such that $Z \otimes_R^{\mathbf L} (R_P/PR_P)$ is uniformly right bounded as $P$ varies over prime ideals of $R$

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $Z$ be a left bounded chain complex of finitely generated $R$-modules. If there exists an integer $n$ such that for every prime ideal $P$ of $R$, ...
Snake Eyes's user avatar

1
2 3 4 5
10