All Questions
Tagged with localization ac.commutative-algebra
44 questions
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33
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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 ...
- 11
2
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1
answer
206
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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 ...
- 480
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0
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80
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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.
...
- 480
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0
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125
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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 $\...
- 61
1
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1
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96
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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 ...
- 361
4
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84
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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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118
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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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164
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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 ...
- 167
6
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0
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369
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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 ...
- 10.5k
1
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1
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161
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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 ...
- 10.5k
8
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1
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372
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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.
- 101
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1
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314
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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 $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...
- 11
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215
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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 ...
2
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1
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154
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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 $M$ ...
- 51
3
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313
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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 ...
- 131
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79
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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 ...
- 33.8k
1
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0
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741
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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 $pA_p$-...
- 11
1
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1
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255
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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 ...
- 1,209
4
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0
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177
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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{...
- 985
2
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1
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138
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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 $m \cap k[X_i] \ne 0 , \forall i \in I$ ...
user111492
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120
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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 $...
- 2,773
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1
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102
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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....
user111492
0
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1
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260
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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 ...
- 1,713
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1
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155
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$R$ 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 $...
- 111
3
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248
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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 \\
\...
- 1,355
2
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0
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525
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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, ...
- 61
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1
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695
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What kinds of limits does localization of commutative rings reflect?
Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of ...
- 10.5k
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172
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Local cohomology commuting with fiber
Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$).
Let $M$ be an $A[x]$-module, which is finitely generated as an $A$-...
- 5,562
3
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3
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935
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Dimension of a ring after localization
Let $R$ be a Noetherian domain of dimension $\ge 1$. Let $\mathfrak{p}_i$, $i = 1, 2, ...$ be prime ideals of height one. Let $T = R[[X]]$ with $X$ is a indeterminate. For each $i \ge 1$ we set $\...
- 2,773
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1
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709
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Localisation of $\mathbb{Z}_p[[X]]$ at ideal $(p)$
Let $R=\mathbb{Z}_p[[X]]$ where $\mathbb{Z}_p$ denotes the $p$-adic integers and $p$ is a prime. Then what is $R_{(p)}$ $(R$ localised at the ideal $pR)$ $?$
- 193
3
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1
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587
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is every finitely n-presented (S^{-1})R-module a localization of a finitely n-presented R-module?
Let S be a multiplicative set in a ring R. We can see that every finitely generated $(S^{-1})R$-module is a localization of a finitely generated R-module.
Then, more generally, is every finitely n-...
- 31
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0
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1k
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Localisation of injectives
When working with injective modules, one bad thing is that they do not necessarily behave well with respect to localisation. Consider a commutative ring $R$ and have a look at the following properties:...
- 6,700
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1
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810
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Intersection of localization with finitely generated subalgebra of fraction field
Let $R$ be a (commutative) noetherian integral domain. Let $I$ be a prime ideal of $R$. Let $S$ be a finitely generated $R$-subalgebra of $\mathrm{Frac}(R)$.
Is $S \cap R_I$ necessarily finitely ...
- 101
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2
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8k
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Is primary decomposition still important?
On p.50 of Atiyah and Macdonald's Introduction to Commutative Algebra, in the introduction to the chapter on primary decomposition, it says
In the modern treatment, with its
emphasis on ...
- 15.4k
19
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3
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3k
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Total ring of fractions vs. Localization
Let $R$ be a commutative ring and denote by $K(R)$ its total ring of fractions, the localization of $R$ with respect to $R_{\mathrm{reg}}$. For every multiplicative subset $U \subseteq R$ there is a ...
- 63.1k
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2
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Elements in a localization - category theoretic approach
This question is about the elements in a localization $S^{-1} A$ of a commutative ring $A$. Is it possible to derive $\frac{a}{1} = 0 \in S^{-1} A \Rightarrow \exists s \in S : sa = 0$ only using the ...
- 63.1k
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2
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2k
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Projective modules over semi-local rings
Let $R$ be a semi-local ring, and $M$ a finite projective $R$-module. If the localizations $M_m$ have the same rank for all maximal ideals $m$ of $R$ then $M$ is free.
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1
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364
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Rig of fractions, including zero denominators
For some integral domain $R$, one forms the field of fractions $R^*$ by considering (equivalence classes of) formal pairs {$r/s : r \in R, s\in R\backslash 0$} and defining $+$ and $*$ as you'd expect ...
- 1,872
6
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1
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376
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Checking locally whether a homomorphism is a localization
All rings below are commutative with $1$.
Suppose $A\subset B$ is a subring and that $A\rightarrow A'$ is a faithfully flat ring homomorphism. [You may assume the rings are actually ${\mathbb C}$-...
- 2,083
2
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0
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234
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Flatness of module
$A\rightarrow B$ a ring homomorphism, $N$ a $B$-module which is flat over $A$. $\mathfrak{q}\subset B$ a prime ideal, $\mathfrak{p}\subset A$ its contraction in $A$. Then is it true that $N_{\mathfrak{...
- 2,857
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0
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333
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Localization of module
M an A-module, $S\subset A$ a multiplicative subset. Is it possible for $S^{-1}M$ to have an $S^{-1}A$-module structure satisfying $\frac{a}{1}\cdot\frac{m}{1}=\frac{am}{1}$ other than the "usuall" ...
- 2,857
11
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2
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470
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Localizing at the primitive polynomials?
For any UFD $R$, the concept of a primitive polynomial (gcd of the coefficients is 1) makes sense in $R[x]$. The product of two primitive polynomials is primitive (Gauss's Lemma), and certainly 1 is a ...
- 6,792
0
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3
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2k
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Equality of elements in localization via universal property
I've been studying universal objects of universal algebra in a quite general setting and try to exhibit the structure of their elements just using the universal property. A very nice example for this ...
- 63.1k
0
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2
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1k
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What is the localization of Q[x]/(x) at 0
Q is a rational field. Q[x] is polynomial ring over Q 。(x) is maximal ideal of Q[x].
Take Q[x]/(x) as a module over Q[x]. Then what is Q[x]-module Q[x]/(x) localize at 0??
I think the result is
Q[x]/...
- 25