Questions tagged [derived-functors]
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156 questions
3
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Left exact functor $F$ preserves quasi-isomorphism between $F$-acyclics
In this math overflow page, the poster gives a proof of the statement "an additive left exact functor $F$ preserves quasi-isomorphisms between $F$-acyclic objects." I'm having trouble ...
4
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2
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284
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Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?
Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
- 412
11
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1
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506
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When does derived tensor product commute with arbitrary products?
Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
- 412
3
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0
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120
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Derived tensor by perfect complex preserves exact triangle in singularity category?
Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
- 413
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121
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Singular cohomology as a sheaf of $\infty$-categories
In several expositions of $\infty$-categories, I read that singular cohomology of a topological space with integral coefficients is a sheaf valued in $D(\mathbb{Z})$, if we consider Top and $D(\mathbb{...
3
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199
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When a fully faithful functor from an abelian category to itself will be an equivalence
Let $A$ be an abelian category. Suppose $i:A\to A$ is a fully faithful functor from $A$ to itself. I wonder when the functor will be an equivalence.
If $A$ is a "nice" category, I think $i$ ...
- 253
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62
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Base change for finding fibers of the pushforward of a line bundle along a proper non-flat morphism
Let $f: X \to Y$ be a proper morphism whose fibers have different dimensions, in particular $f$ is not flat. Let $L$ be a line bundle on $X$. What conditions would be sufficient to be able to conclude ...
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1
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61
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Compatibility condition with the adjunct pair of derived functors
$\def\C{\mathsf{C}}
\def\D{\mathsf{D}}
\def\hoc{\mathsf{HoC}}
\def\hod{\mathsf{HoD}}
\def\L{\mathbf{L}}
\def\R{\mathbf{R}}$I'm a little confused by [R, Remark 6.4.14]. Before stating the remark, I'll ...
5
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361
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On a simple alternative correction to Roos' theorem on $\varprojlim^1$
Here is a discussion about an incorrect theorem of Roos, later corrected, some counterexamples and so on. Reading over this, I was a bit shocked because it contradicted something from Weibel's ...
- 1,021
2
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0
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93
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Minimal injective resolution and change of rings
Let $R$ be a commutative Noetherian ring. For an $R$-module $M$, let $0\to E^0_R(M)\to E^1_R(M)\to \ldots $ denote the minimal injective resolution of $M$. I have two questions:
(1) If $I$ is an ...
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265
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Is there a correction to the failure of geometric morphisms to preserve internal homs?
Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...
- 171
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497
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Does anyone have a good example of an injective resolution?
I'm learning about injective resolutions and derived functor sheaf cohomology, and it seems that every source on injective resolutions gives no examples. I feel like just one good example would make ...
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3
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149
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(Derived category of) sheaves over an infinite union
The short version of my question is:
Suppose $X$ is a (reasonably nice) topological space such that $X = \bigcup_{n \ge 1} X_n$ for an increasing sequence of (closed) subspaces $X_1 \subset X_2 \...
- 197
3
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1
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195
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Image, upto direct summands, of derived push-forward of resolution of singularities
Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all ...
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3
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1
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273
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Higher direct images along proper morphisms in the non-Noetherian setting
Let $f : X \to Y$ be a finitely presented proper morphism. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Do the functors $R^i f_* \mathcal{F}$ preserve any of the following properties:
(1) ...
- 3,625
2
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1
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98
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A perfect complex over a local Cohen--Macaulay ring whose canonical dual is concentrated in a single degree
Let $R$ be a complete local Cohen--Macaulay ring with dualizing module $\omega$. Let $M$ be a perfect complex over $R$. If the homology of $\mathbf R\text{Hom}_R(M,\omega)$ is concentrated in a ...
- 413
3
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1
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239
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Vanishing of $\operatorname{Ext}_R^{1}(M,R)$ when $R$ is a Gorenstein local ring of dimension $1$ and $M$ is not finitely generated
Let $(R,\mathfrak m)$ be a Gorenstein local ring of dimension $1$. Let $M$ be an $R$-module (not finitely generated) such that $M\neq \mathfrak m M$ and there exists a non-zero-divisor $x\in \mathfrak ...
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131
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derived completion and flat base change
Let $f:A \to B$ be a flat morphism of commutative $p$-adic completely rings.
We denote by $D_{\text{comp}}(A)$ the derived category of complexes over $A$, which is derived $p$-adic complete.
For a ...
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205
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left integration of functor in the category of groups
Assume that a functor on the category of groups vanishes on all projective objects. Is it necessarily the left derived functor of a half exact functor on this category?
- 356
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2
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406
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Proper birational morphism from a Gorenstein normal scheme to a normal local domain, with trivial higher direct images, implies Cohen-Macaulay?
Let $k$ be a field of characteristic $0$. Let $R$ be a Noetherian local normal domain containing $k$. Also assume that $R$ is the homomorphic image of a Gorenstein ring of finite dimension, hence $R$ ...
- 413
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Does a functor preserving injectives also preserve K-injective complexes?
Let $F:A\to B$ be an exact functor of Grothendieck abelian categories. If $F$ preserves injective objects, then does the exact functor $F:K(A)\to K(B)$ preserves K-injective complexes?
For example, ...
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180
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Non-cofiltered derived limits
As far as I know, the inverse limit and its derived functors can be defined even in case we are dealing with a functor $F: I \to A$ from a category $I$ that is not cofiltered. I would content myself ...
- 505
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274
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Fourier-Mukai transform is the derived functor
In Mukai's paper Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math Journal, 1981, there is one sentence that puzzles me.
Let $X$ be an abelian variety over an ...
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380
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What properties do the categories $\mathbf{GrpMod}$ and $\mathbf{GrpMod}^*$ of compatible pairs have? Can we do homological algebra with them?
Consider the following category $\mathbf{GrpMod}^*$ of compatible pairs, that is:
an object is a pair $(G,M)$, where $G$ is a group and $M$ is a left $\Bbb Z[G]$-module. A morphism $(G,M) \to (H,N)$ ...
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281
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What can be said about the derived functor of a composition between unbounded derived categories?
Let $\mathcal A, \mathcal B,\mathcal C$ be abelian categories and let $F:\mathcal A \to \mathcal B,G: \mathcal B \to \mathcal C$ be left exact functors such that $RF:D(\mathcal A) \to D(\mathcal B), ...
- 804
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Restricting perverse intermediate extension to closed complement
Consider a scheme $X$ over athe complex numbers, $j:U\to X$ an open subscheme, $i:Z\to X$ its closed complement, and a perverse sheaf $F$ over $U$ with complex coefficients.
The intermediate extension ...
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If Serre's intersection multiplicity $\chi(R/I, R/J)$ equals $\operatorname{length}_R (R/(I+J))$, then are $R/I, R/J$ Cohen-Macaulay?
Let $(R,\mathfrak m)$ be a regular local ring. Let $I,J$ be proper ideals of $R$ such that $R/(I+J)$ has finite length i.e. $\sqrt{I+J}=\mathfrak m.$ Since $I+J$ annihilates $\text{Tor}_n^R(R/I, R/J)$ ...
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Vanishing of higher limits
Let $I$ be a directed set and let $X_I$ be a corresponding inverse system of, say, (complex) vector spaces or abelian groups (in my case in general not finite-dimensional, resp. not finitely generated)...
- 2,998
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280
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Is the composite of absolute derived functors a derived functor?
Let me recall the following definition. Let $F: C \to D$ be a functor between homotopical categories. Denote by $\gamma_C: C \to \mathrm{Ho} C$ the localization and similary for $D$. A total left ...
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A functor admitting a total, but not point-set derived functor
Mike Shulman, in his article Homotopy limits and colimits and enriched homotopy theory observes (towards the end of page 8) that it may be the case that a functor $F: C \to D$ between homotopical ...
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Can we define derived functors in model categories without functorial factorisations?
Let $F: \mathcal{C} \to \mathcal{D}$ be a left Quillen functor between model categories. In Definition 2.16 of Goerss–Schemmerhorn - Model Categories and Simplicial Methods, the left derived functor $...
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How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?
Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
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Do we have a left adjoint of $i^*$ for a closed immersion $i$?
Let $i: X\hookrightarrow Y$ be a closed immersion of varieties. We have the derived pullback functor $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$.
My questions is: can we construct a left adjoint of $i^*$ in ...
- 9,009
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1
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What is the cone of $\mathcal{F}\to i_*i^*\mathcal{F}$ for a divisor $i: D\hookrightarrow X$?
Let $X$ be a smooth projective variety of dimension $n$ and $i: D\hookrightarrow X$ be a smooth, anti-canonical divisor in $X$. In other words,$[D]+[\omega_X]=[0]$ where $\omega_X$ is the canonical ...
- 9,009
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406
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Relative and absolute Ext groups
Given a homomorphism of rings $S \rightarrow R$, for a pair of $R$-modules $M, N$ the machinery of relative homological algebra defines relative $Ext$-groups
$Ext_{R, S}(M, N)$.
These can be defined, ...
- 2,197
4
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Higher direct image of coherent sheaf and rigid analytification
Let $k$ be a non-archimedean field of characteristic zero. Then let $$f:X \rightarrow Y$$
be a (proper) morphism of smooth projective varieties over $k$. The GAGA functor (for rigid analytic spaces) ...
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Derived functor of functor tensor product
Suppose $\mathcal{A}$ is a Grothendieck abelian category with enough projectives, then $\mathcal{A}$ is tensored and cotensored over $\mathrm{Ab}$ with $\mathbb{Z}^{\oplus S}\otimes X\cong \bigoplus_S ...
5
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Derived functors of inverse limit in abelian categories?
I have a finite poset $I$ and an inverse system $A: I^{op}\longrightarrow \mathscr C$ taking values in an abelian category $\mathscr C$.
I suppose that $\mathscr C$ has direct sums. Given that my ...
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$\operatorname{Ext}$-group in the category of modules versus in the subcategory of finitely generated ones
I am trying to refine my understanding of derived categories.
Let $\text{Mod}_R$ and $\text{Mod}^f_R$ be respectively the categories of modules and finitely generated modules over a Notherian ring $R$ ...
- 1,479
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302
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nPOV: Cohomology and derived functors
In the nPOV, cohomology is realized as the connected component of the derived hom space [1]. Namely,
$$H^n(X,Y) = \pi_0 \mathbb{H}(X,B^nY),$$
where $\mathbb{H}$ is an $(\infty,1)$-topos and $B$ is the ...
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189
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Relation between push forward by diagonal morphism and higher direct image functors
Let $f : X \to Y$ be a morphism between two Noetherian schemes. Then $f_*$(respect to $R^1f_*$) sends coherent sheaves to coherent sheaves if and only if $f$ is universally closed (respect to ...
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If a natural transform is an equivalence, under which circumstances is the induced derived natural transform also an equivalence?
More specifically, let A and B be two abelian categories. Suppose $F:A\to B$ is a left exact functor, $G:A\to A$ and $H:B\to B$ are two right exact functors such that $F\circ G=H\circ F$.
With which ...
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The cohomology groups corresponding to a modified global sections functor
Let $\mathcal{F}$ be a sheaf on the big etale site of $Sm_k$. I am looking for a way to calculate a modified version of sheaf cohomology. Let $X$ be a smooth scheme and $Z$ a closed sub-scheme. After ...
- 5,901
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Why are derived functors triangulated?
I am following Verdier's notion of derived functors as Kan extensions along the localization $K(\mathcal{A}) \to D(\mathcal{A})$ of the homotopy category of complexes to the derived category.
In the ...
- 3,625
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879
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Nonabelian variants of the Breen-Deligne resolution
The Breen-Deligne resolution is an unusual functorial resolution of an abelian group A by finite direct sums of free abelian groups of the form $\Bbb Z[A^n] = Free_{Ab}(A^n)$. It makes several ...
- 52.6k
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334
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Passing to torsion of an exact sequence
If
$$
\Theta\colon\quad 0\to A\to B\to C\to 0
$$
is an exact sequence of abelian groups, and $n$ is an integer, then one obtains an exact sequence $$
0\to A[n] \to B[n] \to C[n] \stackrel{\delta_n(\...
- 13k
3
votes
1
answer
115
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Derived functor and bi-module
If A and B are finite dimensional k-algebras, k is a field. $_{A}G\in A-mod$ is a Gorenstein projective module, then we have $RHom_{A}(G,A)\simeq Hom(G,A)$ since $Ext_{A}^{i}(G,A)=0$ for any $i\in \...
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1
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1k
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Conflicting definitions of RHom
I am trying to understand the bifunctor $R\operatorname{Hom} : D(\mathcal{A}) ^{op} \times D(\mathcal{A}) \to D(\operatorname{Ab})$ (I am also interested in the total right derived functor of the ...
5
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1
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Questions about $\text{Perf}(A)$ of dg algebra $A$
[ALEXEY ELAGIN AND VALERY A. LUNTS, p.4.] Recall that triangulated category $\text{Perf}(A)$ is defined as the full thick triangulated subcategory of $D(A)$ generated by the dg $A$-module $A$.
[...
- 603
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1
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Is every middle exact functor a derived functor?
Assume for the sake of simplicity we are working with categories of modules over some ring. Call a functor $F$ middle exact if for an exact sequence $ 0 \to A \to B \to C \to 0 $, we have that $FA \to ...
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