Questions tagged [localization]

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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 ...
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Theorem 7.13 (localization theorem) in the book: Heat kernels and Dirac operator

I'm trying to understand theorem 7.13 in the book Heat kernels and Dirac operators which says the following: Theorem 7.13: Let $G$ be a compact Lie group acting on a compact manifold $M$. Let $\alpha$...
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Is hammock localization a localization in the sense of Lurie?

In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant ...
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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 $ \mid {x} \rangle  \in l^2( \mathbb{Z}^...
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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 ...
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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 ...
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This sum over partitions has unexpectedly nice denominators

Fix an integer $n >= 0$, 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_{\...
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1answer
293 views

Can the category of S-local objects be reflective but not a localization by S?

This is cross-posted from MSE (and substantially re-written) after receiving no answers. Suppose $\mathcal C$ is a category and $S \subseteq \operatorname{Mor}(\mathcal C)$ is some collection of ...
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107 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-...
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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 ...
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1answer
155 views

Slices for certain $C_p$-spectrum

By the work of Hill-Yarnall, for the group $G=C_p,$ all the slices for any spectrum, in particular, for $S^V \wedge H\underline{\mathbb{Z}}$, are classified. Here $V$ is a representation of $C_p.$ ...
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What is the extended centroid of a free algebra?

For a prime ring $R$, you can define its "Martindale ring of quotients" $Q(R)$. See for example: Martindale, Wallace S. III, Prime rings satisfying a generalized polynomial identity, J. ...
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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}...
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1answer
207 views

Interesting "epimorphisms" of $E_\infty$-ring spectra

$\newcommand{\Mod}{\mathbf{Mod}} \newcommand{\map}{\mathrm{map}_{E_\infty-A}}$ Suppose $i:A\to B$ is a map of $E_\infty$-ring spectra. It induces a functor of $\infty$-categories $\Mod_B\to\Mod_A$ by ...
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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 ...
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1answer
236 views

Cohn localization examples

I'm working on my master's thesis, part of which involves an exposition on Cohn localization. (nlab discussion) In Free ideal rings and localization in general rings, Sec 7.4, Cohn gives a ...
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1answer
120 views

Geometric meaning of colocalization of modules?

Let $A$ be a commutative ring and $S\subset A$ a subset. A localization of $A$ at $S$ is defined as a ring morphsim $A\to A[S^{-1}]$ which is initial with respect to inverting $S$. Similarly, a ...
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254 views

Commutative ring $R$ with no nontrivial idempotents, with a localization $R_r$ with infinitely many idempotents

I am looking for a commutative ring $R$ with $1$ such that $R$ has no idempotents and there exists $r\in R$ such that the localization ring $R_r$ has infinitely many idempotents.
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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 ...
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Morphisms of hammocks in the simplicial localization

Let $\mathcal{C}$ be a category together with a wide subcategory $\mathcal{W} \subset \mathcal{C}$. In Calculating Simplicial Localizations by Dwyer and Kan, a morphism of hammocks is defined ...
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267 views

When is an $\infty$-categorical localization left exact?

Let $L: \mathcal C^\to_\leftarrow L\mathcal C : i$ be an adjunction with $i$ fully faithful. In ordinary category theory, $L$ is left exact iff the class of $L$-local morphisms is stable under base ...
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What would cohomological localization be good for?

An open problem in algebraic topology is whether arbitrary cohomological localizations of simplicial sets (or, equivalently, topological spaces) can be proven to exist in ZFC. It's provable in ZFC ...
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1answer
148 views

Localization on varieties with toric singularities

Is there a written version of Atiyah-Bott localization formula for varieties/manifolds with toric singularities? More precisely, suppose $F$ is a fixed locus of a torus action $\mathbb{T}={\mathbb{C}^*...
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1answer
107 views

Elementary example of right localization of functor

I am learning about a general framework for derived functors from Hotta et al., D-modules, Perverse Sheaves, and Representation Theory, Appendix B. $\newcommand{\CC}{\mathcal C} \newcommand{\DD}{\...
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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. ...
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212 views

Localization and containment in commutative ring

Let $R$ be a commutative ring with identity and $x, y $ be fixed elements of $R$ such that for each maximal ideal $m$ of $R$ we have $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...
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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^{...
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1answer
157 views

Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)?

Note: Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a (very ...
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1answer
498 views

When does rationalization commute with homotopy fixed points?

Let $X$ be a $G$-space. There are a number of places in the literature where one can find the claim that under certain conditions rationalization and taking homotopy fixed points with respect to a ...
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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 ...
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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 ...
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1answer
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What is classified by generalised Eilenberg MacLane spaces?

Given an abelian group $A$, the Eilenberg MacLane spaces $K(A,n)$ represent the the nth cohomology group in $A$. In a similar vein, given an arbitrary group $G$ and a space $X$, maps to the ...
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1answer
129 views

Special submodules over almost Dedekind domains

An integral domain $R$ is an almost Dedekind domain if for each maximal ideal $m$ of $R$, the ring $R_m$ is a Dedekind domain, where $R_m$ is the localization of $R$ at $m$. Question: Let $M$ ...
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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 ...
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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 ...
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346 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 $pA_p$-...
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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 ...
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Localizing prime ideals over Noetherian rings

Let $R$ be a prime Noetherian ring which is not necessarily commutative. Consider the two natural ways to "extend" $R$: $R[x]$ and $M_n(R)$, polynomials over $R$ and $n$ by $n$ matrices over $R$ ...
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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 ...
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1answer
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Ore localization and model structures

The question is this: Suppose C is a category, with a given multiplicatively closed set of morphisms S ⊆ C. The role of the denominator conditions on S is rather similar to the role of a Quillen ...
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1answer
177 views

modules whose every submodule is a homomorphic image

Let $R$ be a commutative ring with unity. Let us say that an $R$-module $M$ satisfies property $\mathcal P$ if every submodule of $M$ is a homomorphic image of $M$. Can we characterize all ...
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When is this localization map injective, if at all?

Let $K$ be a number field and $E$ be an elliptic curve defined over $\mathbb{Q}$. Consider the localization map $$ E(K)\otimes \mathbb{Q}_p/ \mathbb{Z}_p \rightarrow \bigoplus_{v|p} E(K_v)\otimes \...
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What is the correct definition of localisation of a category?

Disclaimer: I wasn't sure if this was an appropriate question for MathOverflow, and so I've also asked this on StackExchange. There appears to be a discrepancy in the literature regarding the ...
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When a localization of a category is (non-)reflective?

Let $S$ be a class of maps of a category $\mathcal{C}$. The localization of $\mathcal{C}$ at $S$ is reflective when the localization functor $\mathcal{C} \to \mathcal{C}[S^{-1}]$ has a fully faithful ...
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Localization of $\infty$-categories

In ordinary category theory, the localization $C[S^{-1}]$ at a class of morphisms $S$ (with possibly some assumptions on $S$) is a category $C[S^{-1}]$ together with a map $L:C \to C[S^{-1}]$ such ...
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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 ...
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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{...
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1answer
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Factorization of Gabriel-Zisman localization construction?

My question concerns whether the Gabriel-Zisman localization construction $S^{-1}$ for categories admits a known factorization into a pair of commuting constructions $S^l$ and $S^r$. The localization ...
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Acyclic aspherical spaces with acyclic fundamental groups

A space $X$ (by which I mean a CW complex) is acyclic if its reduced singular homology $\tilde H_\ast(X;\Bbb Z)$ is trivial in all degrees. A discrete group $\pi$ is said to be acyclic if its ...
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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 $G$ ...