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3 votes
1 answer
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How to check that exceptional sequence of vector bundles on Fano variety is helix foundation

Let $X$ be smooth Fano variety with $\operatorname{Pic}(X) = \mathbb{Z}$ of dimension $m$ with canonical class $K$, and $E_0,...,E_n$ is exceptional sequence of $(n+1)$ vector bundles in $D^b(Coh(X))$....
Mykola Pochekai's user avatar
3 votes
1 answer
267 views

Morphism of sites and abelian sheaf cohomology

Let $f : \mathcal{C}\to\mathcal{D}$ be a morphism of sites (see the Stacks Project) with induced morphism of topoi $$(f^{-1}, f_*) : Sh(\mathcal{D})\to Sh(\mathcal{C}).$$ By assumption, $f^{-1}$ is an ...
Ari's user avatar
  • 181
12 votes
1 answer
596 views

An inverse problem for Grothendieck rings of varieties

Suppose $A$ is a given commutative ring, and suppose that one knows that $A$ is isomorphic to the Grothendieck ring of $k$-varieties for some unknown field $k$. Can $k$ be recovered from $A$ ? If ...
THC's user avatar
  • 4,547
2 votes
1 answer
276 views

Can one always find a bigger global resolution

Let $X$ be a scheme. Let $E$ be a perfect complex of coherent sheaves on $X$ and suppose it admits two global resolutions $ F$ and $F'$. By global resolution I mean that both $F$ and $F'$ are quasi-...
Anette's user avatar
  • 595
8 votes
1 answer
979 views

Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$

All varieties appearing below are assumed smooth projective over $\mathbb C$ and all vector bundles, sections etc are assumed to be algebraic/holomorphic. We use the word resolution to mean quasi-...
Mohan Swaminathan's user avatar
3 votes
1 answer
238 views

commutative ring satisfying descending chain condition on radical ideals

Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $...
user521337's user avatar
  • 1,209
9 votes
0 answers
424 views

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 ...
Daniel Bergh's user avatar
  • 1,538
3 votes
0 answers
189 views

Pontryagin square on spin and non-spin manifold

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
annie marie cœur's user avatar
0 votes
1 answer
269 views

Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals

If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...
user521337's user avatar
  • 1,209
1 vote
0 answers
32 views

A regular sequence in a quotient by a "half lattice" defined by a toric manifold

I am interested in some properties of polynomial algebras associated with smooth compact toric varieties. Recall that a toric manifold can be obtained as a quotient $$P^{-1}(p) / \mathbb{K}$$ by the ...
BrianT's user avatar
  • 1,227
1 vote
0 answers
140 views

On finite dimensional commutative algebras and regular sequences

Let $\mathbb{C}[u]:= \mathbb{C}[u_1,...,u_n]$ the algebra of polynomial functions on $\mathbb{C}^n$. I consider two ideals $I$, and $J$, where $I$ is the ideal generated by the polynomials vanishing ...
BrianT's user avatar
  • 1,227
4 votes
0 answers
218 views

derived symmetric powers of an ideal

Let $R$ be the polynomial algebra in $n$ variables over a field $F$ of characteristic $0$. Let $m$ be the ideal of the origin: $m=(x_1,...,x_n)$. We have a canonical map $Lsym^k(m)\to m^k$ from the ...
S. carmeli's user avatar
  • 4,189
0 votes
0 answers
86 views

On the dimension of the cohomology of toric manifolds

Let $M$ be a toric manifold. I'm not sure what conditions on $M$ are required, but one can assume, if needed, that it is compact, smooth, etc. We consider $M$ as a quotient given by the momentum map $...
BrianT's user avatar
  • 1,227
1 vote
0 answers
74 views

On some finiteness properties of cohomological algebras of complex tori

Denote by $A := \mathbb{C}[u,u^{-1}]$, $u = (u_1,...,u_n)$, the algebra of polynomial functions on the complex torus $(\mathbb{C} \setminus \{ 0 \})^n$, which we consider as a $\mathbb{C}[u]$-module. ...
BrianT's user avatar
  • 1,227
15 votes
0 answers
720 views

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 ...
Zhiyu's user avatar
  • 6,622
1 vote
0 answers
83 views

Direct limit of complexes from cocycle property up to homotopy

Suppose that we have a directed set $(I, \leq)$, and a set of maps \begin{equation} f_{i,j} : C^*(A_i) \to C^*(A_j), \quad i \leq j, \end{equation} of singular cochain complexes of topological spaces $...
BrianT's user avatar
  • 1,227
4 votes
0 answers
258 views

Generators of unbounded derived categories of (quasi-)coherent sheaves

An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...
Andrea's user avatar
  • 263
8 votes
0 answers
173 views

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-...
Gorbz's user avatar
  • 661
5 votes
1 answer
331 views

Is the sheaf associated to a differential structure of a specific type?

On a set $X$, let us define a set $\mathcal{D}$ of functions from $X$ to $\mathbb{R}$. Consider first the initial topology $\tau_\mathcal{D}$ on $X$ with respect to $\mathcal{D}$, i.e. the coarsest ...
Christophe Wacheux's user avatar
0 votes
0 answers
118 views

Extending Beauville-Bogomolov orthogonal decomposition from variety to scheme

I'm seeking to understand the de Rham cohomology of a Hilbert scheme $K3^{[4]}$ of the K3 surface. By Beauville, this 8-dimensional compact manifold is Kaehler, irreducible, holomorphically symplectic ...
zenith90's user avatar
3 votes
1 answer
247 views

Support of cohomology of a dualizing complex

Let $A$ be a commutative noetherian local ring, and let $D$ be a dualizing complex over $A$. Let $i$ be the minimal integer such that $H^i(D) \ne 0$ (I am assuming cohomological grading, so the ...
Dualizing's user avatar
14 votes
1 answer
2k views

Why Use Hypercohomology When Defining the de Rham Cohomology of a Smooth Scheme over $k$?

Hopefully this question is of an appropriate level for this site: I'm reading some notes by Claire Voisin titled Géométrie Algébrique et Géométrie Complexe. Let $X$ be a smooth $k-$scheme. In these ...
Alekos Robotis's user avatar
4 votes
1 answer
694 views

Proper mapping theorem

Let $Z\to X$ be a closed immersion of schemes. Assume $\mathcal{O}_Z$ and $\mathcal{O}_X$ both are coherent sheaves of $\mathcal{O}_Z$, resp. $\mathcal{O}_X$-modules. In particular, the coherent ...
user avatar
2 votes
0 answers
325 views

A question on direct limits of rings, and descent of ideals

Motivated by an étale cohomology calculation I am going to do, here is a question that should have positive and not too hard answer. Let $A$ be a ring, $A_0$ a ring such that $A_0$ is equipped with ...
user avatar
3 votes
0 answers
175 views

Geometric interpretation of homological quantities in Artinian local Gorenstein algebras

By corollary 3.5. of http://www.ams.org/journals/tran/2012-364-09/S0002-9947-2012-05430-4/S0002-9947-2012-05430-4.pdf the classification of local artinian Gorenstein algebras (all algebras here are ...
Mare's user avatar
  • 26.5k
2 votes
0 answers
20 views

Cohomology of $\mathbf{G}_a\otimes_{\mathbf{Z}}\mathbf{G}_a$

Let $X$ be a smooth projective variety over a field. Is $$H^p(X_{Zar},\mathbf{G}_a\otimes_{\mathbf{Z}}\mathbf{G}_a)$$ at all related to $H^p(X_{Zar},\mathbf{G}_a) = H^p(X,\mathcal{O}_X)$ via tensor ...
user avatar
10 votes
1 answer
342 views

Vanishing natural transformation exact triangle

This question is a follow-up to this question I asked some time ago. Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb{C}$. Let $\omega \in H^{n}(X,K_X)$, $\omega \neq 0$. Let $$A ...
Libli's user avatar
  • 7,300
2 votes
1 answer
342 views

A question on some lemmas in Orlov's "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models" (Exts vanishing)

I'll write the two lemmas I have questions about, and then ask my questions. For reference, I'm using the following definition of Gorenstein: $\mathbf{Definition\ 1.15}$ A local noetherian ring $A$ ...
Marc Besson's user avatar
31 votes
1 answer
3k views

What was the error in the proof of Roos' theorem?

Background: In 1961, Roos (who, sadly, apparently passed away just last month) purported to prove [1] that in an abelian category with exact countable products (AB4${}^\ast_\omega$), limits of inverse ...
Tim Campion's user avatar
  • 63.9k
13 votes
1 answer
375 views

Formality of some inner RHom

Let $Y$ be a scheme and $X$ be a closed subscheme. Consider $\underline{RHom}_{\mathcal O_Y}(\mathcal O_X,\mathcal O_X)$ (inner RHom). We can view this is as a sheaf of dg-algebras on $X$; when both $...
Alexander Braverman's user avatar
11 votes
2 answers
1k views

Is every "nice" abelian category with enough projectives an additive presheaf category?

A "nice" category $\mathcal{C}$ should be (for the purposes of this question) locally presentable at a minimum, and maybe a bit more. One might require $\mathcal{C}$ to be (in roughly order of ...
Tim Campion's user avatar
  • 63.9k
5 votes
1 answer
510 views

General existence theorem for cup products

I'm curious if it is possible to formulate cup products and prove that they exist in a general way which would subsume a lot of examples: e.g. group cohomology, sheaf cohomology for sheaves on ...
dorebell's user avatar
  • 3,058
4 votes
0 answers
235 views

Serre duality graded singularity category

Let $R$ be a local Gorenstein ring of Krull dimension $d$ with an isolated singularity. Defined $D_{sing}(R)$ as the Verdier quotient $D^b(R)/Perf(R)$). Then, a famous result of Auslander says that ...
Libli's user avatar
  • 7,300
14 votes
3 answers
1k views

Counterexamples to gluing complexes of sheaves

Note: I asked the question below last week on MathSE but received no answer. Background: I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This ...
user142700's user avatar
3 votes
1 answer
243 views

Few commutative algebra questions

A few commutative algebra questions for which I have no reference For $P$ = "catenarian", "coherent", " Jacobson": 1- is an arbitrary product of rings satisfying $P$, a ring satisfying $P$? 2- if a ...
user avatar
4 votes
0 answers
432 views

Reference request: sheaf-theoretic operations in the classical topology?

Like many graduate students before trying to learn something about étale cohomology and Deligne's proof(s) of the Riemann hypothesis part of the Weil conjectures, I am hunting for references detailing ...
Student's user avatar
  • 273
10 votes
2 answers
1k views

periodic cyclic homology and tilting in the sense of Scholze

Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost ...
Dmitry Vaintrob's user avatar
3 votes
0 answers
152 views

Derived base change of (nice) topological spaces

Let $f:X\to Y$ be a morphism of (nice) topological spaces. consider the basechange along $g:Y'\to Y$. We write $F:X'\to Y'$ and $G:X'\to X$. Consider some class of sheaves on those spaces, e.g. ...
Rene Recktenwald's user avatar
4 votes
0 answers
350 views

How should one understand shifted Lie algebras, like $T_X[-1]$?

Let $X$ be an algebraic variety, $at(E) \in Ext^1(E, E \otimes T)$ the Atiyah class of a complex $E \in D(Coh X)$ (see Markaryan, $\S$1.1-1.2). Then $at(\Omega[1])$ gives a "shifted Lie algebra" on $...
evgeny's user avatar
  • 1,980
3 votes
0 answers
144 views

Computation of nearby cycles, monodromy action and action of $sl_{2}$ on $\operatorname{gr}(Ψ)$ for Picard-Lefshetz family

Let $f: \mathbb{A}^{2} \rightarrow \mathbb{A}^{1}$ be a map that sends $(x,y)$ to $xy$. Let $U \hookrightarrow \mathbb{A}^{2}$ be the preimage $f^{-1}(\mathbb{A}^{2} \setminus \{0\})$ and $X:=f^{−1}(0)...
Din's user avatar
  • 103
3 votes
1 answer
354 views

Who are the compact generators in the derived category of $\mathcal{D}_X$-modules?

Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators. Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category ...
Saal Hardali's user avatar
  • 7,789
1 vote
0 answers
113 views

Reference request. The adjunction $\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$

We have the adjunction $$\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$$ where $CDGA$ is the category of commutative diffferential graded algebras and $CA$ is the category of ...
Fallen Apart's user avatar
  • 1,615
3 votes
0 answers
154 views

$\operatorname{Ext}^2(O,\omega)$ as a higher extension on $\mathbb{P}^1 \times \mathbb{P}^1$

Let $X = \mathbb{P}^1 \times \mathbb{P}^1$ over a field $k$ and consider $Ext^2(\mathcal{O}_X,\omega_X)\cong H^2(\omega_X) = H^2(\mathcal{O}_X(-2,-2)) = k$ Let $C = \mathbb{P}^1$. By Kunneth $H^2(\...
solbap's user avatar
  • 3,968
4 votes
0 answers
234 views

Is there an analogue to the koszul complex for constructible sheaves?

Given a variety $X$ and a complete-intersection morphism $$ Y \to X $$ is there an analogue of the Koszul complex for $\mathcal{O}_Y \in \textbf{Coh}(X)$ in the setting of constructible sheaves? ...
54321user's user avatar
  • 1,716
7 votes
2 answers
2k views

Local property of split exact sequence

In the module category of a ring $A$, is a short exact sequence split if and only if the localization of this sequence is split for every prime ideal? Thanks!
Jian's user avatar
  • 496
6 votes
1 answer
929 views

Different definitions of derived functors

In principle one uses the notion of derived category, and the other doesn't. Suppose $F: \mathcal A \to \mathcal B$ is a left exact (additive) functor between abelian categories, and suppose the ...
Hang's user avatar
  • 2,789
3 votes
0 answers
538 views

Could we prove the flat base change theorem for cohomology via injective resolution?

Let $X$ be a quasi-separated scheme over a base ring $A$ and $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $A\to B$ be a flat morphism and $X^{\prime}:=X\times_{\text{spec}(A)}\text{...
Zhaoting Wei's user avatar
  • 9,019
7 votes
0 answers
225 views

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 ...
evgeny's user avatar
  • 1,980
7 votes
1 answer
1k views

Can the homological dimension of a coherent sheaf explode along a formal deformation? (is the resolution property hereditary for formal deformations?)

Let $X_0$ be a locally noetherian scheme and $\mathcal{F}_0$ a coherent $\mathcal{O}_{X_0}$-module. Let $C$ be an artin ring with residue field $k$ and let $X \to Spec C$ be a (flat) deformation of $...
Saal Hardali's user avatar
  • 7,789
7 votes
0 answers
555 views

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 ...
Anton Hilado's user avatar
  • 3,309

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