# Questions tagged [localization]

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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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1 vote
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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 ...
1 vote
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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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### 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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### 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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### 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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### 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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### 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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### 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 $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...