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Questions tagged [localization]

For questions regarding the process, consequences, and stability of localizing algebraic structures such as rings, categories, and modules. Not for use with local topological spaces.

For questions regarding the process, consequences, and stability of [localizing](For questions regarding the process, consequences, and stability of localizing algebraic structures such as rings, categories, and modules. Not for use with local topological spaces.) algebraic structures such as rings, categories, and modules. Not for use with local topological spaces.

Also for relationships to Spec and quasi-coherent sheaves.

1089 questions
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A question regarding the definition of localization of ring

Let $S$ be a multiplicative system. Then the localization of $R$ is defined by an equivalence relation on $R \times S$. The relation is $(a,b) \sim (c,d)$ if there is an $s \in S$ such $s(ad-bc)=0$. Regarding this, I can't show that transitivity…
Keith
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When is the functor induced by localization an equivalence of categories

I'm reading Soergel's "Derivierte Kategorien und Funktoren" [1]. In 1.2.6, Soergel defines a localization functor as follows. Let $F: \mathcal{C} \to \mathcal{D}$ be a functor and denote by $S$ the set (ignoring set-theoretical issues that may…
fklein
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Limit of localization of integral domain

In the book "Algebraic Geometry I" (Gortz Wedhorn) Example 2.37 they calcul the sections of an arbitrary subset $U\subset Spec(A)=X$, ie $\mathcal{O}_X(U)$, where $A$ is an integral domain ($K=\operatorname{Frac}(A)$) : …
Kuzcohomology
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Why is it the case that $S^{-1}(\mathbb{Z}/(p_i^{e_i})) \cong \mathbb{Z}/(p_i^{e_i})$ for multiplicative set $S \subset \mathbb{Z}$ prime to $p_i$.

Let $S \subset \mathbb{Z}$ denote a multiplicative set, i.e. $1 \in S$ and if $a,b \in S$ then $ab \in S$. Let $p_i \in \mathbb{Z}$ be a prime, and let $e_i \in \mathbb{Z}^+$. Furthermore, let $S$ be such that $\forall s \in S: \text{gcd}(s,p_i) =…
Ben123
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Finding $M[U^{-1}]$

I've been studying localizations, and was given the next excercise: Let $k$ be a field with characteristic 0. Let $R=k[x]$ consider the module $M=R/(x^4-x^2)$ and $U=\{1,x,x^2,\dots\}$. Compute $M[U^{-1}]$. I'm quite lost. I know the definition for…
juaaan
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Ring of fractions

Let $R$ be a commutative ring and let $S \subseteq R$ be a multiplicative closed subset containing 1. Prove that the kernel of the natural homomorphism $\lambda: R\rightarrow R_S$, $r\rightarrow r/1$ is $\ker\lambda=\{x\in R:xs=0$ for some $s$ $\in…
C.Garbrandt
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