Questions tagged [derived-functors]
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones.
334
questions
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Mittag-Leffler Condition and fiber products
For a general inverse system of short exact sequences $ 0\rightarrow A_i\rightarrow B_i \rightarrow C_i \rightarrow 0$ for some index $I$, we only get a left exact sequence
$$0\rightarrow \lim A_i \...
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17 views
When is the derived functor of a representable functor representable?
Let $\mathcal{C}$ be an abelian category and $\mathcal{F}:\mathcal{C}\rightarrow \text{Set}$ a representable sheaf. Consider the $i$-th derived functor $R^i\mathcal{F}$ of $\mathcal{F}$. My question ...
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39 views
shriek-star base change for cohomology of quasi-coherent sheaves on schemes
Suppose we have a cartesian diagram of schemes as follows
$\require{AMScd}$
\begin{CD}
V @>{j}>> Y\\
@V\pi VV @VV pV\\
U @>{i}>> X
\end{CD}
Suppose $\mathcal{F}$ is a coherent sheaf ...
1
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1answer
48 views
Steps for computing Tor$(\mathbb{Z}, \mathbb{Z}\times\mathbb{Z})$
I'm reviewing algebraic topology, in particular the Kunneth Formula. I can't find online or in my book (by Hatcher) an explanation for how to calculate $\mbox{Tor}(G,H)$ for any two groups. My ...
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25 views
What is the model structure on simiplicial $B$-modules?
In the following notes Lemma 4.12
Let $A$ be a commutative ring. $sAlg_A$ the category of simplicial$A$-algebras. Let $B\in sAlg_A$, then we can consider the category of $B$-module objects in the ...
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47 views
The Decomposition Theorem for a resolution of singularities
In learning about the celebrated Decomposition Theorem I have found that I am having trouble applying it in even the simplest situations. In particular, I'm considering an example where $f:X\to Y$ is ...
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25 views
Derived functor of additive functor
Let $\mathscr{C}$ be a category which admits enough injectives and let $\mathscr{I}$ be the full subcategory of injective objects.
Let $F:\mathscr{C}\rightarrow\mathscr{C}'$ be an additive functor of ...
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80 views
Why is the nearby cycle of a family of elliptic curves the derived pushforward?
Consider a family of elliptic curves over the open disc $D$ in $\mathbb{C}$, which degenerate to the nodal elliptic curve over $0$, and let $f$ be the map to $D$. I'm interested in the sheaf on the ...
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21 views
References for Completed Tensor Product and Completed Tor
I'm reading Serre's Local Algebra. And there is a topic in there I would like to read more references about before I proceed. In Chapter 5, he defines the Completed Tor as
$$\hat{Tor}_i^k(M,N):= \lim_{...
2
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1answer
43 views
Can two elements of an Ext group come from the same middle object of an SES?
Let $X$ be an object of an abelian category. Is it possible for there to be an object $B$ that is a subobject of $X$ in two distinct ways that yield isomorphic cokernels but is not off by an ...
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1answer
25 views
Transverse intersection and conditions on Tor
Consider $X$ and $Y$ varieties inside a smooth variety $M$.
I say that $X$ and $Y$ intersect transversally at $m\in M$ if the tangent spaces of $X$ and $Y$ span the whole tangent space of $M$ at $m$.
...
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78 views
Restriction-extension identities using six functors formalism
Recently fellow user Thorgott pointed out to me that flat restrictions of flat modules remain flat. That is, let $f : A \to B$ be a flat morphism of rings and $M$ be a flat $B$-module. Then ...
8
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1answer
104 views
Functors which are non-isomorphic but whose derived functors are isomorphic
I was wondering if there is a good/interesting examples of functors that are non-isomorphic as functors but whose derived functors are isomorphic? The example I have encountered so far, the derived ...
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103 views
Grothendieck group of local affine- surfaces with rational singularities
Let $(R, \mathfrak m)$ be an excellent, normal, local domain of dimension $2$ containing an algebraically closed field $k=R/\mathfrak m$. Let $ \pi: Y \to X=\operatorname {Spec}(R)$ be a resolution ...
2
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1answer
48 views
Bijection between $\mathrm{Ext}^1$ and equivalence classes of extensions
I'm reading Weibel's book on homological algebra right now and he's proving that for two $R$-modules $A$ and $B$, the equivalence classes of extensions of $A$ by $B$ (i.e. equivalence classes of short ...
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0answers
41 views
Tor is a covariant functor
Various books and many online notes and other posts such as Proving that $\operatorname{Tor}_n^R$ is a bifunctor give a rather touch-and-go treatment to Tor being a covariant functor. For example, in ...
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25 views
Laurent series ring as ${\lim}^1$
Let $R_*$ be a graded ring concentrated in even degrees. I was presented a construction of $R_*((x))$, the ring of Laurent series in the variable $x$ with degree $-d$, as follows. For every $i \in \...
3
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1answer
105 views
$\operatorname {Ext}$ vanishing and finitely generated reflexive modules over regular local rings
Let $M$ be a finitely generated reflexive module over a regular local ring $(R,\mathfrak m,k)$ such that $\operatorname {Ext}^1_R( \operatorname {Hom}_R(M,M),R)=0$. Then how to show that $M$ is a free ...
2
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29 views
Derived functors via derived categories
I got stuck in the proof of III.6.8 of Gelfand, Manin: Methods of Homological Algebra which essentially asserts the existence of a right derived functor of a left exact functor in a category with ...
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49 views
Kan extensions and satellite functors
As is described for example in Cisinski's book on Higher Category theory, the left (right) derived functor of a functor between model categories $F: \mathcal{C} \rightarrow \mathcal{D} $ can be ...
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Pushforward and Pullback along Square Diagrams of Affine Schemes.
Given an inclusion of rings $A\subseteq B$ and element $m\in A$ we get the following commutative diagram of affine schemes
$$\begin{array}{ccc}\text{Spec}(B_m)&\xrightarrow{j}& \text{Spec}(B)\\...
5
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1answer
87 views
Simple applications of higher derived functors
I was surprised to not have found this question discussed before on this or any other forum, hence I am posting it myself.
From an abstract point of view of homological algebra derived functors are ...
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74 views
Dualizing complex of Cohen-Macaulay variety
I have a question on a proposition from Shihoko Ishii's book "introduction to Singularities":
Preliminaries: A Noetherian local ring $R$ with the maximal ideal $m$ is called Cohen-Macaulay ring if $\...
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Right-deriving a component of bifunctor through F-exact resolutions
This is an exercise in Aluffi's Chapter 0 (Exercise IX.8.14).
We are given a bifunctor $\mathcal{F}: \text{A}^{op}\times \text{B} \to \text{C}$. Denote $\mathcal{F}^A(-)=\mathcal{F}(A,-),\ \mathcal{F}...
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1answer
47 views
Sheafification in the derived category.
Let ${\cal F}$ be a presheaf on a scheme $X$ which associates each abelian group ${\cal F}(U) \in {\mathrm{Ab}}$ for each open $U \hookrightarrow X$. Then we can always associate its associated sheaf $...
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1answer
35 views
$f(x)=-8e^{-0.6x}+e^{-0.4x}$. Determine maximum concentration of the drug, and the value of it assuming the experiment runs for 12 hours
A person is being tested against a certain drug, and has been injected with it. The drug blood concentration, measured in mg/l, is modeled by the function
$f(x)=-8e^{-0.6x}+e^{-0.4x}$
Here, the ...
2
votes
1answer
77 views
some clearly fact about δ-functor in an abelian category
I am reading Hartshorne's AG and confused about the uniqueness of the universal $\delta$-functor of a functor $F$.
$\delta$-functor" /> and $\delta$-functor of $F$" />
p206, Remark 1.2.1: If $F$:$\...
3
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1answer
59 views
Pushforward as integral functor for Azumaya varieties
Consider $(X,\mathcal{A}_X)$ and $(Y,\mathcal{A}_Y)$ two Azumaya varieties over a field $k$. Recall that an Azumaya variety is the data of a variety and a sheaf of semisimple $\mathcal{O}_X$-algebra $\...
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27 views
Derived functors of exact (contravariant) functor are trivial
I'm new to projective resolutions, derived functors, etc, so I'd like a proof check of my proof that the derived functors of an exact contravariant functor are trivial.
Suppose that $F$ is an exact ...
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49 views
Is the sheaf cohomology only a sheaf for flabby sheaves?
Let $X$ be a topological space and let $\mathcal{F}$ be an abelian sheaf on $X$. Choose an arbitary open cover $\{U_i\}_i$ of $X$. Then the Mayer-Vietoris long exact sequence is
$$0\rightarrow \...
1
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1answer
48 views
Understanding a Line Bundle as a Spherical Twist
I am reading Segal's paper "All Autoequivalences are Spherical Twists", https://arxiv.org/abs/1603.06717, and I am trying to understand a detail in example 2.4. The situation is this:
Let $X$...
4
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65 views
Wedge product and right derived functors
let $f:X\to Y$ be a flat map between complex algebraic varieties and let $\mathcal F$ be a locally free sheaf on $X$. Very often I've seen the following map:
$$\wedge: \bigotimes^n (R^1f_{\ast}\...
0
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1answer
132 views
How to construct a short exact sequence of complexes
Suppose that a hsort exact sequence
$$
0 \longrightarrow A \overset{f}{\longrightarrow B} \overset{g}{\longrightarrow} C \longrightarrow 0
$$
of objects in some (Abelian) category is given. Also, ...
1
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0answers
31 views
A complex to compute Čech-Cohomology
Let $\newcommand{\F}{\mathcal{F}} \F$ be a sheaf of abelian groups on a paracompact (Hausdorff) space $X$. The Čech-cohomology of $\F$ is defined as
$$
\check H(X, \F) := \varinjlim_{\mathcal{U}} \...
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35 views
Definition of the Wedge Product of Chain Complex or Derived Object
Let $A$ be a commutative, unitary ring. The derived category of $A$-modules has a tensor product, namely the derived tensor product. To define the wedge product, one only need a tensor product and ...
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45 views
Derived functor and natural transformation
Let $A$ and $B$ be two categories.
Let $F_1, F_2$ be functors from $A$ to $B$ and there is a natural transformation $\eta$ from $F_1$ to $F_2$.
Now let $G$ be another functor from $B$ to $B$.
Now, ...
1
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1answer
110 views
Doubt over a proof about higher direct image functors in Hartshorne
For reference, this is Chapter III Proposition 8.5 in Hartshorne. The claim is this
Let $X$ be a noetherian scheme and let $f: X \rightarrow Y$ be a morphism of $X$ to an affine scheme $Y = \text{...
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29 views
Projective vs Injective Resolution
Let $A$ be an abelian category with enough projectives and take an object C. Then one can construct a projective resolution of C, which is also functorial if we consider the complex of resolution in ...
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54 views
Finding Tor of k[x]-module
I am asked to find $\operatorname{Tor}_{*}^{k[x]}(M,M)$, with $M=k[x,x^{-1}]/xk[x]$.
I start with finding a projective resolution for $M$. An arbitrary element of $M$ is $\sum_{n \leq 0}a_nX^{-n}$, ...
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0answers
118 views
Does $\text{Tor}_1^R(M, M) = M$?
Let $k$ be a field and let $R = k[x]$. Consider the $R$-module $M := \frac{k[x, x^{-1}]}{x \cdot k[x]}$ (i.e. so typical elements are Laurent polynomials with no positive powers).
I have computed $\...
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0answers
30 views
section functor with support in a point of $\mathbb{R}^n$
Let $Z$ be a locally closed subset of a topological space, one denote classically by $\Gamma_{Z}$ the functor of section with support.
Let $x$ be a point of the affine space $\mathbb{R}^n$, we denote ...
0
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1answer
51 views
Tensoring locally free sheaves preserves acyclicity
This is a statement in Huybrechts, Fourier Mukai Transform, pg78, where he defines the derived tensor.
If $E^*$ is an acyclic complex bounded with all $E^i$ locally free. $F$ a coherent sheaf , ...
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Why are $\operatorname{Tor}_2$ and $\operatorname{Ext}^2$ often $0$?
I'm doing a bunch of exercises involving the $\operatorname{Tor}$ and $\operatorname{Ext}$ functors and often the functors are non-trivial for $i = 0,1$ and become $0$ for $i \geq 2$. Why is this?
My ...
1
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1answer
64 views
Opposite and product of derived categories
Let $\mathscr{A, B}$ be abelian categories.
Then $D(\mathscr{A}) \times D(\mathscr{B}) = D(\mathscr{A \times B})$?
And $D(\mathscr{A})^{\text{op}} = D(\mathscr{A}^{\text{op}})$?
I can't find any ...
2
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1answer
154 views
How to compute a derived tensor product?
Let $A_*$ and $B_*$ be simplicial algebras over a simplicial commutative ring $R_*$. I would like to understand how one explicitly computes the derived tensor product $A_* \otimes^L_{R_*} B_*$. More ...
3
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1answer
152 views
On relating $\operatorname {Tor}_i^R (k, M)$ and $ \operatorname {Ext}_R^{d-i} (k, M)$
Let $M$ be a finitely generated module of finite projective dimension over a local Gorenstein ring $(R, \mathfrak m,k)$ of dimension ($=$depth) $d$. Then since $R$ is Gorenstein, so $\operatorname {...
1
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1answer
49 views
Long exact sequence of derived functors from a finite exact sequence (resolution) ending with zeros
Let $\mathcal A, \mathcal B$ be abelian categories such that $\mathcal A$ have enough injectives and projectives.
Let $F: \mathcal A \to \mathcal B$ be an additive left exact functor, and let $R^i F$ ...
5
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1answer
129 views
Betti numbers and dimension of socle of an ideal
Let $I$ be a non-zero ideal in a regular local ring $(R, \mathfrak m,k) $ (where $k:=R/\mathfrak m$) . The socle of $R/I$ is Hom$_R(k, R/I) \cong (I:\mathfrak m)/I$ , which as evident has a natural $k$...
1
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1answer
40 views
Characterisation of trivial $n$-extensions
$\newcommand{\Ext}{\operatorname{Ext}}$Let $\mathcal C$ be an abelian category. For $M, N\in\mathcal C$, we know that the set $\Ext^1(M, N)$ of all equivalence classes of short exact sequences $0\to N\...
0
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1answer
56 views
First Isomorphism Theorem/use
How do I apply $T\varepsilon$ to obtain an isomorphism $\sigma_A:\text{coker}T(d_1)\to TA$ by the First Isomorphism Theorem in the snippet below?
In fact, I cannot see how FIT applies.
