Questions tagged [localization]

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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
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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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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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1answer
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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
119 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
88 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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111 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. ...
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163 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
137 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 ...
21
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1answer
426 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
530 views

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
127 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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179 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
134 views

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
140 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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1answer
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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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2answers
960 views

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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2answers
220 views

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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4answers
650 views

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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Do twisted group rings of free abelian groups admit universal fields of fractions?

Let $R$ be an associative ring with unit. Recall that an epic $R$-field is a ring epimorphism $\alpha\colon R\to D$ to a skew field/division ring $D$. An epic $R$-field $\alpha$ is called a field of ...
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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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1answer
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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 $G$ ...
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When is genus the same as stable equivalence?

Suppose $M, N$ are two $R$-modules (I had in mind the group ring $R=\mathbb{Z}[G]$ for a finite group $G$). By localizing at a prime $p$ I mean $M_{(p)}\cong M\otimes_R R_{(p)}$. If $M$ and $N$ are ...
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1answer
117 views

On maximal ideals of $k [X_i : i \in I ] $ where $k$ is a field , $I$ is an infinite set with $|k| > |I|$

Let $k$ be a field and $I$ be an infinite set such that $|k| > |I|$ . Let $R := k [X_i : i \in I ] $ and $m$ be a maximal ideal of $R$ ; then is it true that $m \cap k[X_i] \ne 0 , \forall i \in I$ ...
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1answer
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Basic elements and localizations

Let $(R, \mathfrak{m})$ be a local domain and $x$ is a basic element of $\mathfrak{m}$, that is $x \in \mathfrak{m} \setminus \mathfrak{m}^2$. Let $P$ be a prime ideal containing $x$. Is it true that $...
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1answer
83 views

Functions on rings and polynomials with coefficients in a certain kind of localisation

Let $R$ be a commutative ring with unity and let $S$ be a multiplicatively closed subset of $R$ such that $S$ contains no zero divisor . So the canonical map $f : R \to S^{-1}R$ is invective , hence w....
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1answer
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If $\{f\in R[x]\:|\:f\text{ monic}\}$ is a right denominator set, is $\{f^i\:|\:i\geq 0\}$ a right denominator set also?

Let $R$ be a right (and left) Noetherian ring and $T=R[x]$ its polynomial ring. It was shown by Stafford that the set $S=\{f\in T\:|\:f\text{ monic}\}$ is a right denominator set. So my question is, ...
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3answers
510 views

Are localization functors always essentially surjective?

Let $\mathcal{C}$ be a category and $\mathcal{W} \subseteq \text{Arr}(\mathcal{C})$ a set (or class) of arrows. There are (at least) two notions of localization of $\mathcal{C}$ with respect to $\...
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1answer
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Analytic spread of localization of an ideal

Let $J$ be an ideal in a Noetherian local ring $(R,m)$. It is well known that for any prime ideal $p\in Spec(R)$, $l(J_p)\leq l(J)$, where $l(J)$ is the analytic spread of $J$. Q) Are there ...
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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, ...
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1answer
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How to identify localization of categories?

Let $C, D$ be categories with finite limits, $F:C\to D$ be a essentially surjective functor that commutes with finite limits, and let $S$ be the set of morphisms of $C$ that become isomorphisms in $D$ ...
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1answer
143 views

$R$ is $\mathbb{Z}$ graded ring and $0\neq f \in R_1,$ show that $R_f \cong S[X,X^{-1}]$ [closed]

Suppose $R$ is $\mathbb{Z}$ graded ring and $0\neq f \in R_1.$ Then I want to show that $R_f \cong S[X,X^{-1}],$ where $S=(R_f)_0$ and $X$ transcendental over $S.$ I wanted to use the isomorphism $...
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1answer
106 views

Right localization of $R[x,x^{-1}]$ at monic $f\in R[x]$

Let $R$ be a right Noetherian ring and $S=\{f\in R[x]\;|\;f\text{ monic}\}$. It is a result of Stafford that $S$ is a right denominator set in $R[x]$, so in particular we can localize $R[x]$ at any $f\...
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1answer
86 views

Localising a right Noetherian ring at a set of regular elements

Let $R$ be a right Noetherian ring, and $S$ a multiplicative set consisting of regular elements where $1\in S$ and $0\not\in S$. Does the right ring of fractions $RS^{-1}$ exist? This is what I know ...
3
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1answer
164 views

Localization of the pullback diagram

In the paper, Topologically Defined Classes of Commutative Rings, localization of the pullback diagram (with $v,$ surjective) $$ \begin{array} DD & \stackrel{v\ '}{\longrightarrow} & A \\ \...
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295 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, ...