Questions tagged [localization]

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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
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
4 votes
0 answers
100 views

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
7 votes
1 answer
413 views

Model categories as a tool to resolve size issues for localizing categories

I have a rather basic question about one motivation for introducing model categories in David White's notes, as a possible way to overcome troubles with size issues appearing when localizing a ...
user267839's user avatar
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How to prove a 1-localization of a 1-category is already an $(\infty,1)$-localization?

I don't even know this fits in here or in Mathematics Stack Exchange, but let me ask. I'm new to simplicial stuff, so a good reference would be quite helpful. Let's say $C$ is a certain category, and ...
gksato's user avatar
  • 317
3 votes
1 answer
141 views

Kernels and cokernels in a quotient of an abelian category

I am trying to understand the construction of the quotient of an abelian category called the Serre quotient or Gabriel quotient. From the description here: https://en.wikipedia.org/wiki/...
Ji Woong Park's user avatar
1 vote
0 answers
42 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
5 votes
1 answer
215 views

Hammock localization and free adjoints

The Hammock localization $L^H \mathcal{C}$ of a relative category $(\mathcal{C},\mathcal {W})$ is a simplicial category defined by Dwyer and Kan as a way to compute the $\infty$-categorical ...
Simon Henry's user avatar
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1 vote
1 answer
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On "minimal presentation" of local rings essentially of finite type over a field

Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (https://stacks.math.columbia.edu/tag/07DR). Then, $R$ is the homomorphic image of ...
strat's user avatar
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6 votes
2 answers
364 views

Size issues in localization $\mathcal{C}[\mathcal{W}^{-1}]$ category

When one starts with a locally small category $\mathcal{C}$ and wants to localize it at an appropriate choosen collection of morphisms $\mathcal{W}$, then in general one faces some size issues in the ...
user267839's user avatar
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4 votes
0 answers
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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
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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3 votes
0 answers
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
6 votes
2 answers
551 views

Overloading of the word "local" in category theory

The word "local" in category theory does not seem to have a precise definition in itself but it often appears as part of other terminology. To my understanding, it is then used in the ...
anuyts's user avatar
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1 answer
408 views

When is it possible to localize a scheme along a closed subscheme?

If we have $Z\subset X$ a closed irreducible subscheme of an integral scheme $X$ (which you can take to have various further niceness properties if you want), one can take its generic point $\eta_Z$ ...
xir'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
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
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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
2 votes
1 answer
179 views

How to compute the $G$-theory groups of $k[x,y]/(xy)$ for any field $k$

I am trying to compute the $G$-theory groups of the ring $k[x,y]/(xy)$ for any field $k$. What I have tried so far are two approaches. Approach 1: Use the $G$-theory localization sequence for $k[x,y]/(...
Boris's user avatar
  • 501
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
2 votes
1 answer
168 views

Are hammock localizations locally truncated?

Let us take a relative category $(\mathcal{C},\mathcal{W})$, and consider its hammock localization $L_H \mathcal{C}$. It seems to me that for every two objects $X,Y \in \mathcal{C}$ the mapping ...
Giulio Lo Monaco'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
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
7 votes
1 answer
291 views

Localisation of categories, but instead of "isomorphisms" I want "morphisms with right inverse". Construction via calculus of fractions possible?

Let $\mathcal{C}$ be a category and $S$ a collection of morphisms in $\mathcal{C}$. The localisation $\mathcal{C}[S^{-1}]$ is defined via a functor $F: \mathcal{C} \longrightarrow \mathcal{C}[S^{-1}]$ ...
kevkev1695's user avatar
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5 votes
0 answers
348 views

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
  • 10.3k
7 votes
1 answer
732 views

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 ...
Andrea Marino'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
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
7 votes
0 answers
238 views

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
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
  • 1,469
12 votes
1 answer
377 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 ...
sarahzrf's user avatar
  • 295
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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5 votes
0 answers
151 views

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
  • 1,469
5 votes
1 answer
170 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 $C_p.$ ...
Surojit Ghosh's user avatar
4 votes
1 answer
260 views

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. ...
Nick's user avatar
  • 213
6 votes
0 answers
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
  • 5,506
7 votes
1 answer
275 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 ...
Maxime Ramzi's user avatar
  • 13.3k
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
6 votes
1 answer
335 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 ...
Tyler's user avatar
  • 161
1 vote
1 answer
142 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 ...
Arrow's user avatar
  • 10.3k
8 votes
1 answer
338 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.
Anahita's user avatar
  • 101
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
  • 5,506
8 votes
1 answer
154 views

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 to ...
F.Abellan's user avatar
  • 447
8 votes
1 answer
374 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 ...
Tim Campion's user avatar
  • 60.6k
15 votes
1 answer
536 views

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 ...
Mike Shulman's user avatar
3 votes
1 answer
213 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}^*...
Mohammad Farajzadeh-Tehrani's user avatar
1 vote
1 answer
137 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}{\...
Joshua Mundinger's user avatar
7 votes
0 answers
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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