Questions tagged [derived-functors]
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145
questions
2
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94
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Normality and integrality of schemes and splitting of map from structure sheaf to (derived)pushforward of structure sheaf along proper birational map
Let $R, S$ be commutative Noetherian rings such that $R$ is a subring of $S$. If $S$ is a normal domain, and there exists an $R$-linear map $\phi: S\to R$ whose restriction on $R$ is the identity map, ...
3
votes
1
answer
141
views
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 ...
2
votes
1
answer
209
views
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) ...
2
votes
1
answer
83
views
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 ...
3
votes
1
answer
217
views
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 ...
2
votes
1
answer
79
views
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 ...
0
votes
0
answers
100
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Basis of Lambda algebra for a programmer
First of all, I'm not a specialist in alg. top., but I try to apply computational math to it, so if I'm wrong in something you'd be doing a better thing explaining it to me instead of blaming me :) ...
1
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0
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197
views
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?
2
votes
2
answers
338
views
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 $R$ ...
3
votes
0
answers
112
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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, ...
2
votes
1
answer
160
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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 ...
1
vote
0
answers
261
views
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 ...
7
votes
1
answer
354
views
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 $(G,M) \to (H,N)$ ...
0
votes
0
answers
249
views
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), ...
2
votes
0
answers
75
views
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 ...
6
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2
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253
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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 $\text{Tor}_n^R(R/I, R/J)$ ...
6
votes
1
answer
329
views
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)...
6
votes
1
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246
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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 ...
4
votes
0
answers
64
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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 ...
4
votes
1
answer
389
views
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 $...
2
votes
1
answer
220
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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 ...
1
vote
0
answers
202
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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 ...
0
votes
1
answer
146
views
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 ...
7
votes
2
answers
349
views
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, ...
4
votes
1
answer
186
views
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) ...
7
votes
1
answer
400
views
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 ...
4
votes
1
answer
379
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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 ...
0
votes
0
answers
85
views
$\operatorname{Ext}$-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 $R$ ...
0
votes
0
answers
282
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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 ...
2
votes
0
answers
171
views
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 ...
2
votes
0
answers
33
views
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 ...
2
votes
0
answers
112
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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 ...
2
votes
0
answers
199
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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 ...
8
votes
0
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843
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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 ...
8
votes
0
answers
275
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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(\...
3
votes
1
answer
112
views
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 \...
2
votes
1
answer
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
votes
1
answer
368
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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 $A$.
[...
3
votes
1
answer
300
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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 ...
3
votes
2
answers
335
views
Reference request: excess normal bundle and derived pullback
Consider a fiber square
$\require{AMScd}$
\begin{CD}
X' @>i'>> Y'\\
@V g V V @VV f V\\
X @>>i> Y,
\end{CD}
where $i$ and $i'$ are regular immersions, and consider the ...
1
vote
2
answers
400
views
Action on group $\operatorname{Ext}^i(\mathcal{L}, \mathcal{M})$ by scalar multiplication
Let $X$ be a proper scheme over field $k$ and $\mathcal{L}, \mathcal{M}$ two invertible $\mathcal{O}_X$-modules. Then $Hom_{\mathcal{O}_X}(\mathcal{L}, \mathcal{M}) \cong Hom_{\mathcal{O}_X}(\mathcal{...
2
votes
0
answers
67
views
Does direct image via proper map preserve coherence of unbounded complexes?
As for the title, I'm considering a proper map $f : X \rightarrow Y$ of Noetherian schemes and I'm trying to understand whether the direct image $Rf_{\ast} : D_{qc}(X) \rightarrow D_{qc}(Y)$ sends the ...
3
votes
1
answer
197
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Subspace inclusion with non-vanishing higher direct images
I'm looking for concrete topological intuition for the derived pushforward.
Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf_\ast$ takes a sheaf $F$ to the sheafification of ...
16
votes
1
answer
871
views
Abelian category with enough injectives but not functorially
Let $\mathcal{A}$ be an Abelian category with enough injectives. Is it always possible to make the injective embedding functorial? By this I mean that there should exist a functor $I \colon \mathcal{A}...
3
votes
0
answers
165
views
Conceptual definition of derived functors allowing for quick proof of comparison theorems for sheaf cohomology
There are several approaches of increasing sophistication and simplicity to defining derived functors. I know of universal $\delta$-functors and Kan extensions along localizations. More definitions ...
2
votes
0
answers
159
views
Dualizing complex description in Stacks project
The question is closely related to this one this one (more precisely the reference the comment by AGl earner) and is aimed to understand the proof of Lemma 20.2 from notes from Stacks notes from ...
3
votes
0
answers
102
views
Fourier Mukai kernel which gives an equivalence only in one direction
If $X$ and $Y$ are two schemes and $F \in Perf(X \times Y)$, then we can define a functor from $Perf(X)$ to $Perf(Y)$ as the Fourier Mukai transform $\Phi^{X \rightarrow Y} = q_{\ast}(F \otimes p^{\...
6
votes
3
answers
407
views
multiplicative structure of Ext
Basically, I am trying to compute something with the Adams spectral sequence (as a toy example). The $E^2$ page reduced to computing $Ext^{s,t}_{\Gamma} (\mathbb{F}_2, \mathbb{F}_2)$, where $\Gamma = \...
44
votes
1
answer
2k
views
Existence and uniqueness of Haar measure on compacta; a cohomological approach
I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group.
I think the best way of introducing the idea I am pursuing is ...
1
vote
1
answer
120
views
Relative version of the cohomology product
Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\...