Questions tagged [localization]

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Ample divisors on $T$-varieties

Question: how does one use a torus action to help decide whether a divisor or line bundle is ample? In more detail: I have a (normal, but usually rather singular) $T$-variety $X$, where $T$ is an ...
Geordie Williamson's user avatar
7 votes
0 answers
370 views

Grothendieck Riemann Roch is abelian localisation on loop spaces

Abelian localisation says approximately that for a proper equivaraint map $f:X\to Y$ between schemes with a $\mathbf{G}_m$ action, the pushforward on cohomology $f_*\omega$ can be computed by the ...
Pulcinella's user avatar
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7 votes
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143 views

When do zigzags of weak equivalences detect isomorphisms in the localization?

The usual way to prove that two model categories are equivalent is to construct a zigzag of Quillen equivalences between them, but is it always possible? We can ask a more general question. ...
Valery Isaev's user avatar
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7 votes
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Topological localization of (infinite) inverse limits

The classical localization of topological spaces at a given set of primes $\mathcal{P}$ is a functor $\mathcal{T}\xrightarrow{(-)_{(\mathcal{P)}}}\mathcal{T}$ from a suitable category of topological ...
Tyrone's user avatar
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6 votes
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Does the localization functor $\mathcal{C}\to S^{-1}\mathcal{C}$ preserve finite colimits when $\mathcal{C}$ is not small? (size issues in proof)

$\def\colim{\operatorname{colim}} \def\hom{\operatorname{Hom}}$Let $\mathcal{C}$ be a category and let $S$ be a class of morphisms of $\mathcal{C}$ that is a left multiplicative system. The ...
Elías Guisado Villalgordo's user avatar
6 votes
0 answers
404 views

This sum over partitions has unexpectedly nice denominators

Fix an integer $n\geq0$, a power series $\gamma \in \mathbb Q[[X]]$ with valuation 1, and a symmetric function $f$ (with coefficients in $\mathbb Q$). Now, consider the series $$ S_n = \sum_{\Lambda\...
Drew's user avatar
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6 votes
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176 views

Abelian localisation for K theory?

Let $X$ be a scheme acted upon by $\mathbf{G}_m$ and $K(X)=K_0(\text{Perf}X)$ the Thomason-Trobaugh K theory. Is there a localisation theorem in this context? By this I mean something like $$\text{id}...
Pulcinella's user avatar
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Verdier Quotient a quotient?

This question seems trivial, so hopefully it will be resolved quickly. As pointed out in this question on quotient categories and localization, the two constructions are sometimes related, but in ...
Spencer Leslie's user avatar
6 votes
0 answers
1k views

Localisation of injectives

When working with injective modules, one bad thing is that they do not necessarily behave well with respect to localisation. Consider a commutative ring $R$ and have a look at the following properties:...
Fred Rohrer's user avatar
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Geometric meaning of localization at $(1+I)$?

Let $I\vartriangleleft A$ be an ideal of a commutative ring. Consider the submonoid $1+I\subset A$. What is the geometric interpretation of localization at this submonoid? How does it relate to the ...
Arrow's user avatar
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A conjecture about sums over partitions arising from Hilbert scheme of points

$\DeclareMathOperator{\leg}{\operatorname{leg}}\DeclareMathOperator{\arm}{\operatorname{arm}}$The following situation arose from the study of some localization computations on Hilbert schemes of ...
Drew's user avatar
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5 votes
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132 views

Analog of cellular approximation theorem for $CW_0$-complexes ($CW_\mathcal P$-complexes)

$CW_0$-complexes are analogs of $CW$-complexes, in which the "building blocks" are the rational disks $D^{n+1}_0$ whose boundaries are given by $\partial D^{n+1}_0= S^n_0$, where $S^n_0$ is a ...
Bashar Saleh's user avatar
5 votes
0 answers
75 views

Embedding the Mészáros subdivision algebra in an Orlik-Terao localization

The following is an open question (Question 4.1) from my paper $t$-Unique Reductions for Mészáros's Subdivision Algebra (published version in SIGMA 2018, and slightly updated preprint version with ...
darij grinberg's user avatar
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0 answers
223 views

Weak equivalences of left Bousfield localizations

Suppose C is a complete and cocomplete category with two model structures (C0,F0,W0) and (C1,F1,W1) such that C0⊃C1, F0⊂F1, W0⊂W1. If necessary, the model structures can be assumed to be simplicial, ...
Dmitri Pavlov's user avatar
5 votes
0 answers
115 views

Ring properties which can be checked on sufficiently many Ore localizations

Let $R$ be a non-commutative Ore domain, and let $R[S_i^{-1}]$ be a finite set of Ore localizations. The absence of a geometric theory means there is no obvious candidate for when this set of ...
Greg Muller's user avatar
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Localizations that are endofunctors

Usually when $F: C \rightarrow D$ is a localization functor, the categories $C$ and $D$ are not equivalent. My question is when is it possible for $C, D$ to be abstractly equivalent but $F$ is not an ...
user1077's user avatar
4 votes
0 answers
79 views

Dimension of a positively graded ring after a suitable localization

Quesion- Let $R=\bigoplus_{i\ge 0} R_i$ be a (non-trivial) positively graded commutative Noetherian ring with $1(\not=0)$ of (Krull) dimension $d\ge 0$. Let $S\subset R_0$ be a multiplicative set such ...
Sourjya Banerjee's user avatar
4 votes
0 answers
156 views

When adic completion preserves projectives?

Lets take a ring $R$ and an ideal $\mathfrak p \subset R$, and call them an L-pair (just for brevity) if $\mathfrak p$-adic completion of any projective module is again projective (as R-module); and L-...
Denis T's user avatar
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A variation of the hammock localization

Let $W \subseteq \mathcal{C}$ be a wide subcategory of a category $\mathcal{C}$. The saturation $\overline{W}$ of $W$ is the class of maps of $\mathcal{C}$ which become isomorphisms in $\mathcal{C}[W^{...
Valery Isaev's user avatar
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4 votes
0 answers
274 views

A question related to bousfield localization and nilpotent completion

I am reading Bousfield's paper entitled "The localization of spectra with respect to homology" (MSN). In that paper, Corollary 6.13 states that, if a ring spectrum $E$ has countable homotopy and ...
Surojit Ghosh's user avatar
4 votes
0 answers
105 views

Could we form the homotopy category of a dg-category by inverting homotopic invertible morphisms?

Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but ...
Zhaoting Wei's user avatar
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4 votes
0 answers
166 views

Do we have criteria of strict localization of a Grothendieck category?

Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under ...
Zhaoting Wei's user avatar
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4 votes
0 answers
174 views

What kind of module is this?

Recall that, if $R$ is a commutative ring, then a suitably finite $R$-module $M$ is projective if and only if the localization $M_\mathfrak{m}$ is a direct sum of finitely many copies of $R_\mathfrak{...
Ben Knudsen's user avatar
4 votes
0 answers
208 views

Checking a monad is idempotent

I have a monad $T: \mathcal{C} \to \mathcal{C}$ on a (Grothendieck) abelian category which preserves filtered colimits and direct sums (but is not exact). There is a finite collection $G$ of compact, ...
Dylan Wilson's user avatar
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4 votes
0 answers
418 views

Localization in equivariant cohomology theory for groups other than ($p$-)tori

Recall the following localization theorem, as stated in Hsiang's Cohomology Theory of Compact Transformation Groups: Theorem. Let $G=(S^1)^k$ be a torus, $X$ a paracompact $G$-space with finite ...
user3158840's user avatar
4 votes
0 answers
321 views

Localization of power series and module structure

Let $R=\mathbb{Q}[X,Y]$ be the polynomial ring of two commuting variable. Let $S$ be the multiplicative subset of $R$ generated by homogeneous linear polynomials. Let also $\widehat{R}$ be the ring of ...
e2718's user avatar
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3 votes
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88 views

Cohn's localization for rings with enough idempotents

I am in the following situation: I have a non-unitary (associative) ring $R$ with enough idempotents or, if you prefer, a small pre-additive category. Actually, I even know that $R$ is right coherent (...
Simone Virili's user avatar
3 votes
0 answers
100 views

Localizations of spaces with respect to homology and right properness

Let $E$ be a spectrum (with corresponding homology theory denoted $E_\ast$). In "Localization of spaces with respect to homology", Bousfield constructed a model category structure on the ...
Niall Taggart's user avatar
3 votes
0 answers
264 views

Reduced Noetherian ring is the intersection of its localizations at primes associated to a nonzero-divisor

I shall quote proposition 11.3 of Eisenbud: Commutative algebra If $R$ is a reduced noetherian ring then an element $x\in K(R)$ belongs to $R$ if and only if the image of $x$ in $K(R)_P$ belongs to ...
user301120's user avatar
3 votes
0 answers
124 views

Open Gromov-Witten invariants via lozalization with $\mathbb{C}^{*}$ (not $S^1$) action

Amplitudes of open A-model on a Calabi-Yau 3-fold $X$ with branes are given by the open Gromov-Witten invariants of $X$. It is known how to compute them if there is a toric action on a manifold, which ...
Andrey Feldman's user avatar
3 votes
0 answers
318 views

Localization of the pushforward in equivariant cohomology

I am reading Nekrasov's paper and in page 2 he considers the $G \times T^2$ equivariant cohomology of the (compactified) moduli space $\tilde{M_k}$ of $U(N)$ instantons on $\mathbb{C}^2$. Here $G$ ...
Marion's user avatar
  • 577
3 votes
0 answers
245 views

The multiplicative system in a symmetric monoidal category

Let $\mathcal{C}$ be a symmetric monoidal category. In the 1973 paper "Note on monoidal localisation" by Brian Day, the multiplicative system of morphism in $\mathcal{C}$ has been discussed. See also ...
Zhaoting Wei's user avatar
  • 8,657
3 votes
0 answers
66 views

Computing morphisms in localizations of $K(B)$

Let $B$ be an additive category (a small one; one can assume that it is a $\mathbb{Q}$-category, yet not much else is known about it). Given a set of objects $S$ of $K^b(B)$ (or $K(B)$), I consider ...
Mikhail Bondarko's user avatar
2 votes
0 answers
113 views

About Atiyah-Segal Localization Theorem

In $K$-Theory, actually also in equivariant cohomology theory, there exists a useful theorem as known Borel-Hsiang-Atiyah-Segal Localization theorem. For $K$-Theory Theorem: Let $G$ be a compact Lie ...
Mehmet Onat's user avatar
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2 votes
0 answers
129 views

Some relative GW calculations

I have a question about the $\psi$ class in the following paper of Graber and Vakil: https://arxiv.org/abs/math/0309227 For $k,d\geq 2$, and a partition $d=d_1+\cdots+d_k$ of $d$ into positive ...
Mohammad Farajzadeh-Tehrani's user avatar
2 votes
0 answers
104 views

Localization of the injective hull of a commutative non-Noetherian ring

Let $R$ be a commutative non-Noetherian ring and $m$ a maximal ideal. My question is whether the localization $E(R)_m$ of the injective hull $E(R)$ of $R$ is an injective $R_m$-module. This is true in ...
Michal's user avatar
  • 21
2 votes
0 answers
77 views

Equivalence between integrals over a reduced space

Context: I have been trying to understand this paper from Y. Cho and K. Kim. More precisely, a specific argument in Lemma 2.2 where they say the ABBV localization formula on an integral over a ...
Aaron Maroja's user avatar
2 votes
0 answers
494 views

Irreducibility over the field of fractions of a quotient of a polynomial ring

Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell - 1$ and $z_0, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, ...
Nils Amend's user avatar
2 votes
0 answers
233 views

Flatness of module

$A\rightarrow B$ a ring homomorphism, $N$ a $B$-module which is flat over $A$. $\mathfrak{q}\subset B$ a prime ideal, $\mathfrak{p}\subset A$ its contraction in $A$. Then is it true that $N_{\mathfrak{...
ashpool's user avatar
  • 2,837
2 votes
0 answers
325 views

Localization of module

M an A-module, $S\subset A$ a multiplicative subset. Is it possible for $S^{-1}M$ to have an $S^{-1}A$-module structure satisfying $\frac{a}{1}\cdot\frac{m}{1}=\frac{am}{1}$ other than the "usuall" ...
ashpool's user avatar
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1 vote
0 answers
96 views

Recovering a ring from its localization and completion with respect to a fixed element

Suppose I have a commutative ring $k$ and an element $x \in k$. Then I can form the localization $k[x^{-1}]$ of $k$ at the multiplicative subset $\{ 1, x, x^2, ... \}$, and I can form the completion $\...
Noah Wisdom's user avatar
1 vote
0 answers
41 views

Do local and global symplectic resolutions have same monodromy?

Yoshinori Namikawa associates a Weyl group $ W $ to any symplectic affine complex variety $ X $ with good $ \mathbb{C}^* $-action. He provides a semi-explicit description of $ W $, which requires ...
user341068576's user avatar
1 vote
0 answers
27 views

Construction of a regulariser for the boundary integral operator $\lambda\mathrm{Id} - K'$

$\newcommand\Id{\mathrm{Id}}$Assumptions and Notations : $\Omega$ is a bounded Lipschitz domain in $\mathbb R^2$, $\Gamma$ denotes its boundary and $n$ is the normal vector to the boundary $\Gamma$, ...
SAKLY's user avatar
  • 63
1 vote
0 answers
77 views

How to compute the quotient and localization of the monoid algebra $kG$ for a field $k$

I am given that $k$ is a field and $G$ is the monoid consisting of all monomials $X^iY^j$, where $j$ is between $0$ and $3i$. I am trying to compute the quotient of the monoid algebra $kG$ by the ...
Boris's user avatar
  • 501
1 vote
0 answers
80 views

The localization map for the Mordell-Weil group of elliptic curves over finite Galois extensions

Let $L/K$ be a finite Galois extensions of number fields and $E/K$ be an elliptic curve. Denote by $\mathcal{F}$ the localization map \begin{equation} \mathcal{F}: H^1(G,E(L)) \rightarrow \bigoplus_{v ...
A. Maarefparvar's user avatar
1 vote
0 answers
111 views

Different ways to construct the isogeny category of abelian varieties

Let $k$ be a field and let $\mathbf{AV}_{/k}$ be the category of abelian varieties over $k$. I'm interested in different definitions of the isogeny category of $\mathbf{AV}_{/k}$. Of course, the ...
Lukas Heger's user avatar
1 vote
0 answers
58 views

Intuition behind bound of second moment of Greens function by fractional moment

Consider the Hilbert space $ \mathcal{H} = l^2(\mathbb{Z}^d)$ for some dimension $d$ with basis given by the basisvectors $\{ \vert {x} \rangle \}_{x \in \mathbb{Z}^d} $. Let $A$ be an either self-...
Frederik Ravn Klausen's user avatar
1 vote
0 answers
123 views

Existence of a finite resolution

I have tried to formulate a question in which I was very curious, any hints suggestions are also welcomed. Thanks in advance. Let $M$ be an $R$ module ($R$ commutative ring with unity). It is given ...
user443060's user avatar
1 vote
0 answers
91 views

Does Anderson localisation occur if the potential are equal in pairs?

Consider the Anderson model given by the Hamiltonian $H \in B(l^2( \mathbb{Z}^d)) $ defined by $H = - \Delta + V$ where the potential $V$ acts on a unit vector $ \vert x \rangle  \in l^2( \mathbb{Z}^d)...
Frederik Ravn Klausen's user avatar
1 vote
0 answers
643 views

Completion of localization of completion

Let $(A,m)$ be a noetherian local ring, and let $p \subseteq A$ be a prime ideal. From this data, we can construct two rings: 1. We may localize $A$ at $p$, and then complete, obtaining the $pA_p$-...
Localization's user avatar