Questions tagged [localization]
The localization tag has no usage guidance.
172 questions
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When localization commutes with arbitrary intersection of ideals
For a commutative ring with identity we know that in general localization does not commute with arbitrary intersection of ideals. I am looking for a paper that considers equivalent condition(s) for ...
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Categories in which isomorphism of stalks does not imply isomorphism of sheaves
Let $\mathcal{A}$ be a locally small category with colimits of small filtered diagrams.
For the purposes of this question, an $\mathcal{A}$-presheaf on a topological space $X$ is a functor $\Omega (X)^...
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Commutativity of pairs of reflective localizations
Suppose there are two classes of morphisms $w_1, w_2$ in $C$ and two
two reflective localizations $L_1: C \overset{\rightarrow}{\hookleftarrow} C^\text{$w_1$-local}: i_1$ and $L_2: C \overset{\...
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Presentable categories as colimits of finitely presentable categories
I am trying to understand the relationship betweeen compactly generated presentable categories, also called finitely presentable categories, and general presentable categories (which I have less ...
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Pullback of localizations
In https://kerodon.net/tag/02LZ, Lurie observes that the pullback of a localization $F$ is not necessarily a localization. If all pullbacks are localizations, then he calls $F$ universally localizing ...
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Equivariant cohomology of fixed points using the localisation theorem
I am trying to understand the Smith-Thom inequality for spaces equipped with an action by a cyclic group and also the case, when it's an equality:
In the following, let $G=\mathbb{Z}/p$, $\mathbb{F}$ ...
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Localization of quasi-excellent rings are quasi-excellent?
If $R$ is a quasi-excellent ring, then is $R_{\mathfrak p}$ also quasi-excellent for every prime ideal $\mathfrak p$ of $R$ ?
I think Matsumura's commutative ring theory book says that localization of ...
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Localization of totally acyclic complex or projective modules remain totally acyclic?
Let $R$ be a commutative Noetherian ring. An acyclic complex $P$ of projective $R$-modules is called totally acyclic if for every projective $R$-module $Q$, the complex Hom$_R(P, Q)$ is also acyclic.
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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/...
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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 ...
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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 $\Gamma$,
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 (...
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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 ...
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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 $\eta_Z$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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]/(...
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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-...
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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 ...
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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 ...
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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 ...
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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 $F: \mathcal{C} \longrightarrow \mathcal{C}[S^{-1}]$ ...
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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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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 $ \vert x \rangle \in l^2( \mathbb{Z}^d)...
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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 $n\geq0$, 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
$$
S_n = \sum_{\Lambda\...
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Can the category of S-local objects be reflective but not a localization by S?
This is cross-posted from MSE (and substantially re-written) after receiving no answers.
Suppose $\mathcal C$ is a category and $S \subseteq \operatorname{Mor}(\mathcal C)$ is some collection of ...
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When adic completion preserves projectives?
Lets take a ring $R$ and an ideal $\mathfrak p \subset R$, and call them an L-pair (just for brevity) if $\mathfrak p$-adic completion of any projective module is again projective (as R-module); and L-...
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A conjecture about sums over partitions arising from Hilbert scheme of points
$\DeclareMathOperator{\leg}{\operatorname{leg}}\DeclareMathOperator{\arm}{\operatorname{arm}}$The following situation arose from the study of some localization computations on Hilbert schemes of ...
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Slices for certain $C_p$-spectrum
By the work of Hill-Yarnall, for the group $G=C_p,$ all the slices for any spectrum, in particular, for $S^V \wedge H\underline{\mathbb{Z}}$, are classified. Here $V$ is a representation of $C_p.$
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What is the extended centroid of a free algebra?
For a prime ring $R$, you can define its "Martindale ring of quotients" $Q(R)$. See for example:
Martindale, Wallace S. III, Prime rings satisfying a generalized polynomial identity, J. ...
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Abelian localisation for K theory?
Let $X$ be a scheme acted upon by $\mathbf{G}_m$ and $K(X)=K_0(\text{Perf}X)$ the Thomason-Trobaugh K theory. Is there a localisation theorem in this context? By this I mean something like
$$\text{id}...
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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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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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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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