All Questions
Tagged with homological-algebra ag.algebraic-geometry
154 questions with no upvoted or accepted answers
31
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0
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On the definition of regular (non-noetherian, commutative) rings
All rings are commutative with unit. A ring $R$ is called regular if it satisfies
(Reg) Every finitely generated ideal of $R$ has finite projective dimension.
Clearly this gives the usual ...
- 19.6k
20
votes
0
answers
3k
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Idea of presheaf cohomology vs. sheaf cohomology
Let $X$ be a topological space and $U$ an open cover of $X$.
In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology:
The zeroth Cech ...
- 319
18
votes
0
answers
697
views
Do $\infty$-categories make Grothendieck duality simpler?
I've heard multiple times that the main difficulty of Grothendieck duality is that triangulated categories don't 'glue well'.
In my view, there are 3 parts in understanding Grothendieck duality:
We ...
- 711
16
votes
0
answers
591
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Lifting DG-categories to characteristic zero
The question of lifting (smooth projective) varieties from an algebraically closed field $k$ of characteristic $p$ to characteristic zero (i.e., to the Witt vectors $W(k)$) is a classical one. It's ...
- 25.6k
15
votes
0
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720
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If a polynomial ring is a finite flat module over some subring, is that subring itself a polynomial ring?
A question motivated by If a polynomial ring is a free module over some subring, is that subring itself a polynomial ring? and If a polynomial ring is finite free over a subring, is the subring ...
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14
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0
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1k
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Is there a slick proof of the fundamental theorem of dimension theory?
The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of ...
- 1,514
14
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0
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891
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Local proof of Grothendieck-Riemann-Roch theorem
There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem.
Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of ...
- 7,377
13
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0
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474
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Refinement of concept of support of a module
My rings are commutative and noetherian.
The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...
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12
votes
0
answers
688
views
Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'
It appears to me (though I may be wrong) that the common opinion is that the main difference between derived noncommutative geometry and Rosenberg's noncommutative 'spaces' is that Rosenberg's version ...
- 439
11
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0
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552
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The intrinsic meaning of abelian sheaf cohomology of a category
Basically my question is:
Suppose I meet an alien mathematician which understands everything through category theory and category theory alone. How would I convince said mathematician that certain ...
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11
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0
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877
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Who proved the exactness of Amitsur's complex ?
A foundational result in Grothendieck's descent theory and in his étale cohomology is the exactness of Amitsur's complex. More precisely, suppose we have an $A$-algebra
$A\to B$; then there is a ...
- 47.5k
10
votes
0
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813
views
On functoriality of the Leray spectral sequence
The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
- 40.2k
9
votes
0
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300
views
How did Jouanolou define the cup product with no finiteness hypotheses in SGA 5?
In SGA 5 Exposé VII, at the beginning of §2, Jouanolou lets $X$ and $Y$ denote two schemes, $f:X\rightarrow Y$ a morphism, and $A$ the ring $\mathbf{Z}/\nu\mathbf{Z}$ where $\nu$ is an integer prime ...
- 1,217
9
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198
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Does a morphism which induces an isomorphism between Hochschild homology also induce an isomorphism between cyclic homology?
In a 1998 paper by B. Keller, the author consider the following problem in Section 1.4: Let $k$ be a commutative ring and $X$ a scheme over $k$. We can consider the cyclic homology as well as the ...
- 9,019
9
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424
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Are dualizable objects in the derived category of a ringed topos perfect?
Recall that an object $a$ in a symmetric monoidal category $(\mathcal{C}, \otimes, e)$
is dualizable if there exists an object $b$ and morphisms $\varepsilon\colon b \otimes a \to e$
and $\eta\colon e ...
- 1,538
9
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0
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506
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Categorification of definitions in the context of the derived category of quasi-coherent sheaves
Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned ...
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8
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268
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What exactly goes wrong with $f_!$ outside of locally compact spaces?
Let $f:X\to Y$ be a morphism of ringed spaces. We define a functor $f_!:\mathcal{O}_X\mathsf{-Mod}\to\mathcal{O}_Y\mathsf{-Mod}$ as
$$\Gamma(U,f_!\mathscr{F}):=\{s\in \Gamma(f^{-1}(U),\mathscr{F})\:|\:...
- 711
8
votes
0
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394
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Cohomology of constructible sheaves via exit paths
Let $X$ be a stratified space, with stratification $S$ (we will ignore technicalities).
The category of exit paths $Ex(X,S)$ is a directed refinement of the path groupoid of $X$ accounting for the ...
- 1,635
8
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0
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173
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On constructible Hall algebra and instantons
I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-...
- 661
8
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0
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286
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Functorial classes in Brauer group
For a smooth variety $X$ over a perfect field of characteristics $p$ the sheaf of differential operators is an Azumaya algebra(etale locally is isomorphic to endomorphisms of its center, which is ...
- 7,377
7
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0
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352
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What is a morphism of Tannakian categories?
I feel that this question is interesting but has not received enough attention; possibly because it's in MSE. So, the present question is mainly a repost, in the hopes of getting a good answer. (If ...
- 711
7
votes
0
answers
225
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Phantom category with trivial Hochschild cohomology
An admissible subcategory $C\subset D$ of a triangulated category is called phantom if $K_0(C)=0$. Such categories may be detected by their Hochschild cohomology (but usually have trivial Hochschild ...
- 1,980
7
votes
0
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555
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Background on Kontsevich's Work on Quantization
Where can I find background reading material necessary to be able to read about Maxim Kontsevich's work on quantization? I would like to able to follow the ongoing seminar of IHES, "Resurgence and ...
- 3,309
7
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0
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268
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Identifying and reconstructing the derived category from its auto-equivalences
Background: Given a smooth irreducible algebraic variety $Y$ with $\omega_Y$ or $\omega_Y^{-1}$ ample. Then Bondal-Orlov theorem states that if there exists any other smooth algebraic variety $Y'$ ...
- 1,981
7
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0
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374
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Arbitrarily non-degenerate Hodge to de Rham spectral sequence
It is true that for any $n$ there exists a compact complex manifold which Frolicher spectral sequence does not degenerate at the $n$-th page(https://arxiv.org/pdf/0709.0481.pdf).
Does the analogous ...
- 7,377
7
votes
0
answers
218
views
Coherent sheaves and Mitchell's embedding theorem
Let $S$ be a scheme. By Freyd-Mitchell's theorem the category of (quasi)coherent sheaves of $\mathcal{O}_S$-modules is equivalent to a full subcategory of the category of left $R$-modules for some ...
- 1,001
7
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0
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275
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Not isomorphic varieties with isomorphic tilting algebras
Let $X$ be a smooth projective variety over a field, than tilting object $T$ on $X$ is a perfect complex that is a compact generator of the derived category $\operatorname{D}(QCoh(X))$ and satisfies ...
- 1,545
7
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0
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460
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Quantum polynomial rings and singularities
Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with ...
- 3,758
7
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0
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809
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A Question about a theorem in Toën's notes "Lectures on dg-categories"
So I am trying to learn a bit about dg categories from Toën's notes, "Lectures on dg-categories" http://www.math.univ-toulouse.fr/~toen/swisk.pdf and in particular I would like to understand ...
- 3,758
6
votes
0
answers
201
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Smoothness of a variety implies homological smoothness of DbCoh
I have been told that $D^bCoh(X)$ is homologically smooth if $X$ is a smooth variety, and I am trying to construct a proof. My background is not in algebra, so I apologize for elementary questions.
It ...
6
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0
answers
656
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The derived version of the Grothendieck spectral sequence
Consider the (very well known) Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$ and $\mathcal G: \mathcal B \to \mathcal C$ between abelian categories ...
- 61
6
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0
answers
366
views
Transgression map spectral sequence of Ext
Let $X$ be a scheme over $k$ and $p:\ X \to Spec(k)$ the structure morphism. If $M$ is an étale sheaf of abelian groups over $Spec(k)$ I have a Grothendieck spectral sequence $$E^{p,q}_2=Ext^p_k(M,R^...
- 111
6
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0
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337
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Fourier-Mukai bimodule
Let $X$ and $Y$ be two smooth varieties over some field, and let $E$ be a perfect complex on $X \times Y$. It looks like it is not possible to define a DG-functor
$F_E : Perf(X) \to Perf(Y)$ such ...
- 1,545
5
votes
0
answers
288
views
Picard group of almost module category
I am very new to the world of almost mathematics and I am curious about the following:
Fix an almost mathematics situation $(R,I)$ throughout. Very generally, the almost module category comes with a ...
- 51
5
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0
answers
290
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About the left adjoint of $f^*$
In lots of different cases (Verdier duality, Grothendieck duality, étale cohomology, ...) the very existence of a (right) adjoint to the sheaf functor $f_!$ gives useful information. (I'm going to ...
- 711
5
votes
0
answers
689
views
Does there always exists a locally free resolution of quasi-coherent sheaves on quasi-projective noetherian scheme?
We consider a quasi-projective noetherian scheme. It is well known that for a coherent sheaf we can construct a sheaf resolution of locally free of finite rank. It is introduced in Hartshorne chapter ...
- 61
5
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0
answers
214
views
Universal property for derived category of coherent sheaves
Let $X$ be a scheme, and let $D^{*}(X)$ be the unbounded (resp. unbounded, resp. bounded below/above, etc) derived category of coherent sheaves on $X$.
The work of Robalo establishes a universal ...
- 1,635
5
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0
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195
views
To what extent is the derived category of coherent sheaves on a scheme a "homotopy type" of the scheme?
It is well known that the derived category of coherent sheaves (unbounded, bounded, and all cousins) on a scheme $X$ contain most - if not all (depending on specifics) - of the cohomological ...
- 1,635
5
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0
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530
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What is the algebro-geometric or measure-theoretic "content" of Dhillon and Mináč's motivic Artin symbols over an arbitrary ground field?
1. Short version. In this text, Dhillon and Mináč define motivic Artin symbols. Having fixed a ground field $k$ and a smooth projective curve $Y$ over $k$ equipped with the action of a finite group $G$...
- 944
5
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0
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396
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Description of connecting maps of Derived functors
Let $C$ be an abelian category with enough injectives and $F$ be a left exact additive functor. Consider the short exact sequence $0 \to A' \to A \to A ''\to 0$. Therefore, we have connecting maps $\...
- 493
5
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0
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279
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Exactness is often an open condition. How often?
Let $\mathcal C$ denote an abelian category. Consider a formal $\mathbb Z$-graded object $V_\bullet$ in $\mathcal C$, by which I simply mean a $\mathbb Z$-indexed list $n\mapsto V_n$ of objects. A ...
- 54.6k
5
votes
0
answers
232
views
Coherence of the monoid algebra of a non-finitely generated monoid
Let $P$ be an integral, sharp, finitely generated commutative monoid (say even torsion-free and saturated if you like), and consider the "rational cone" $P_\mathbb{Q}\subseteq P^{gp}\otimes_\mathbb{Z}...
- 1,030
5
votes
0
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331
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Extensions of maps between graded modules
Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\...
- 25.5k
5
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0
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1k
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Question about unbounded derived categories of quasicoherent sheaves
This is a bit of a strange question since I more or less want to ask the MO crowd whether I've understood the situation correctly. If you have an unbounded complex of quasicoherent injective sheaves $...
- 3,758
4
votes
0
answers
723
views
$\mathbb{Z}[T]$-Solidification in light condensed setting
In the lectures to "Analytic Stacks" Scholze and Clausen introduced a new concept of "light" condensed mathematics. In Lecture 7 Clausen introduces the derived $T$-solidification ...
4
votes
0
answers
107
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Perfect dg-modules under faithfully flat extension
Let $k \subseteq \bar{k}$ be an extension of fields (Orlov in the reference below seems to indicate the same thing will hold for faithfully flat maps but the case of fields is enough for me).
On page ...
- 959
4
votes
0
answers
324
views
Edge morphisms on the Grothendieck spectral sequence in the case that one of the functors is exact
Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be abelian categories and let $G: \mathcal{A} \to \mathcal{B}, F: \mathcal{B} \to \mathcal{C}$ be left exact functors, with the hypotheses needed to apply ...
4
votes
0
answers
213
views
Computing homology class of curve in product of elliptic curves
I have a smooth, projective curve $X/\mathbb{C}$ of genus $g$, embedded in a product of elliptic curves $A = \prod_{i=1}^g E_i$. Since $H_*(A; \mathbb{Z})$ with the Pontryagin product is isomorphic to ...
- 1,856
4
votes
0
answers
195
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Example of pseudocoherent complex which is not locally quasi-isomorphic to a strict pseudocoherent one
I ask this question here even if I posted it also on math.stackexchange (recieving no answer so far) because I have read some analogous question but for perfect complexes on this site, even though ...
- 351
4
votes
0
answers
168
views
detecting a semi-free module from its bar-resolution
Let $A$ be a DG-algebra over a field (say $k$). A DG-module $M$ over $A$ is said to be semi-free if it admits an exhaustive filtration $0 = M_0 \subset M_1 \subset \ldots \subset M_p = M$ such that ...
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