Questions about commutative rings, their ideals, and their modules.

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22 views

Properties of integral closure and reduction ideals. [on hold]

Definition (Reduction Ideal). Let $ I $ and $ J $ be ideals of $ R $. Then $ J $ is called a reduction of $ I $ iff $ J \subseteq I $ and there exists an $ n \in \mathbb{N} $ such that $ I^{n} = J ...
4
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0answers
48 views

What about $\mathrm{Spec}(\mathbf{Q})$?

I've heard a lot about $\mathrm{Spec}(\mathbf{Q})$ (see for example Minhyong Kim's answer here), but $\mathbf{Q}$ is a field. So isn't $\mathrm{Spec}(\mathbf{Q})$ trivial? What's the point of studying ...
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0answers
31 views

Inclusion of fractional ideals implies equality

Let $R$ be a integral domain and let $\mathfrak U\subseteq\mathfrak B$ two ideals of $R$ such that $\mathfrak UR_\mathfrak p=\mathfrak BR_\mathfrak p$ for all maximal ideals. Then $\mathfrak ...
1
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1answer
47 views

Vanishing set of $\text{Ann} (M)$, where $M$ is a finitely generated $A$ module

Let $M$ be a finitely generated $A$ module, generated by say $x_1, ..., x_n$. Let $V(S)$ denote the set of primes of $A$ containing $S$. I am guessing that $$ V(\text{Ann}(M)) = \cup_{1 \leq i \leq ...
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2answers
25 views

Isomorphism of the completition of polynomial ring modulo second degree polynomial

Let $k$ be a field of characteristic different from $2$, and $A=k[x,y]/(y^2-x^2(x+1))$. Let $\hat A$ be the $(x,y)A$-adic completion. How can I show that $\hat A\simeq k[[u,v]]/uv$? Qing Liu: ...
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0answers
26 views

Invertible element of a $p$-adic integers [on hold]

Let $A=\mathbb Z_p$, $I$ an ideal of $A$ such that $A$ is a complete ring for $I$-adic topology. What would be an example of $n\geq 2$ such that $n$ is invertible in $A$?
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1answer
29 views

Example of a complete ring [on hold]

What would be an example of a commutative ring $A$ with unit and its ideal $I$ such that $A$ is a complete ring with $I$-adic topology?
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27 views

Is submodule of Hilbert module a Hilbert module?

Let $R$ be a commutative ring with identity and $M$ be an $R$-module. $M$ is a Hilbert module if every prime submodule $P$ of $M$ equals the intersection of all maximal submodules of $M$ that contain ...
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1answer
37 views

Going Up Theorem and Affine Sets.

So for an affine scheme, we know that this is true: Suppose that $k$ was algebraically closed. Let $X$ and $Y$ be affine schemes and $\phi: X \rightarrow Y$ be a polynomial map with the corresponding ...
3
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2answers
78 views

Computing the Grothendieck group of affine space.

For a Noetherian scheme $X$ the Grothendieck group $K(X)$ is defined as the free abelian group on coherent sheaves quotiented by the equivalence relation $\mathscr{F}=\mathscr{F}'+\mathscr{F}''$ for ...
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0answers
37 views

Homological criterion for $A(B \cap C) = AB \cap AC$?

Is there a homological criterion for the condition $A(B \cap C) = AB \cap AC$ for ideals in a ring $R$? By "homological" I mean a statement such as "the given equation holds if and only if (some Tor, ...
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1answer
35 views

Ring contained in a R-module finitely generated

Let $R$ be a Noetherian domain with quotient field $K$ and let $b_1,\ldots,b_n\in K$. Suppose that $R'$ is a integral domain, $R\subseteq R'$ and $$R'\subseteq \sum_j Rb_j.$$ Remark: It is ...
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1answer
33 views

Extension of an ideal to a subring of the ring of fractions

Let $A$ be a domain, and $B$ an $A$-algebra inside $\text{Frac}(A)$. Let $x/y\in B$. Then $(yA:_Ax)B\neq B$ if and only if there is a prime ideal $\mathfrak{p}\in \text{Spec}(A)$ such that ...
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1answer
36 views

Integral closure and field of fractions

I have a ring $R = \mathbb{Q}[t^2,t^5] \cong \frac{\mathbb{Q}[x,y]}{\langle x^5 - y^2 \rangle}$ (where the denominator is the ideal generated by $x^5 - y^2$). Now i have to compute the closure of $R$ ...
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2answers
80 views

Relation between intersection and product of ideals

Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...
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0answers
25 views

Embedding of torsion free module into free module [on hold]

Assume $R$ is a Noetherian regular local ring. Can a finitely generated torsion free $R$-module $M$ be embedded in a finitely generated free $R$-module $F$?
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1answer
20 views

Reference for Hilbert function and multiplicities

What is the good book/notes for Hilbert function and multiplicities( except BRUNS and MATSUMURA )? I need basic as well as advance level
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0answers
54 views

Classical algebraic geometry in infinite dimensions?

I ran into this paper of Serge Lang, if I understood him correctly (of which I am doubtful), then since $\mathbb{C}$ has uncountable transcendence degree over its base field $\mathbb{Q}$, Hilbert's ...
2
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1answer
30 views

Finding the kernel of a multiplication map

Consider the ideal $I=(x,y) \subset R=\mathbb{C}[x,y]$ and $\mathbb{C}$ as the $R$-module $R/I$. I am asked to find the kernel of the multiplication map $I \otimes_R I \rightarrow I$ as a submodule of ...
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1answer
56 views

Overring of an integrally closed domain that is not integrally closed

Assume that $A$ is an integrally closed integral domain, and $K$ is its fraction field. Well...this may be a stupid question, but is every overring of $A$ between $A$ and $K$ also integrally closed ? ...
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0answers
39 views

Commutative ring where $r$, $s$ are associates but $r \neq us$ for any $u$ unit. [duplicate]

First of all I think it's important to note that the definition of associates that $r$ divides $s$ and $s$ divides $r$. Secondly, I know that my ring $R$ has to have zero divisors since if $R$ is an ...
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1answer
34 views

Localization of Tor

I have few questions about the second part. (1) I'm not sure why $\operatorname{Tor}_1^A(M,\bar{A}) \otimes_B B_P=\operatorname{Tor}_1^A(M_P,\bar{A})$. (2) I think $\bar{A}$ has a free ...
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1answer
45 views

Theorem 12.3 from Matsumura

Theorem 12.3 (p. 87), Commutative Ring Theory by Matsumura. Let $A$ be a Krull ring, $K$ its field of fractions, and $\mathfrak{p}$ a height $1$ prime ideal of $A$; then if $\mathcal{F} = ...
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1answer
30 views

The functor Tor for $r_R$

Suppose $R$ is commutative ring and $r \in R$. Show that if $r$ is a zero divisor, then $$\text{Tor}^R_n(R/(r),M) \cong \text{Tor}^R_{n-2}(r_R,M)$$ for $n\geq 3$, where $r_R =\{s \in R \ |\ rs =0 \}$. ...
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0answers
27 views

If P(X) is reducible in K[X], show it is reducible in A[X], A integrally closed domain

Let $A$ be an integrally closed ring, $K$ its field of fractions, and $P(X) \in A[X]$ a monic polynomial. If $P(X)$ is reducible in $K[X]$, show that it is reducible in $A[X]$. The hint given is to ...
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81 views

Theorem 16.3, Matsumura. If $a_1, a_2,\dots, a_ n$ is M-quasi-regular then it is an M-sequence [closed]

Theorem: Let $A$ be a noetherian ring. Set $I=(a_1,\dots, a_n)$ and $M$ an $A$-module and $M$ is $I$-adically separated. If $a_1, a_2,\dots, a_ n$ is $M$-quasi regular then it is an $M$-sequence. At ...
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1answer
66 views

Ideal Quotient and Zero Locus.

I stumbled across something I couldn't get while reading. So given two ideals $I$ and $J$ in some ring $R$ where $R = k[\mathbb{A}^n]$. I want to show that $Z(I) - Z(J)$ $\subset$ $Z((I:J))$, where ...
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0answers
44 views

Image of ideal under the isomorphism given by the Chinese Remainder Theorem.

Suppose that $\mathfrak{p}_1,\ldots,\mathfrak{p}_n$ are maximal ideals of a ring $R$. Then $\mathfrak{p}_i+\mathfrak{p}_j=R$ with $i\neq j$ and $\mathfrak{p}_i^a+\mathfrak{p}_j^b=R$ with $a,b$ ...
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0answers
26 views

Is the ring $F(U(R))$ necessarily isomorphic to the ring of all polynomials with coefficients in $R$ and with constant term equal to $0$?

(All my rings are commutative, but not necessarily unital.) I was playing around with the ring freely generated by an Abelian group, and it seems to me that the following holds: letting $U$ denote ...
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1answer
28 views

How to compute singular points on a variety?

Let $H$ be the variety defined by $H = \{(x, y, z, t)\in \mathbb{C}^4: xy = z^2 + t^2\}$. How to compute all singular points on $H$? Thank you very much. My partial solution: it seems that ...
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2answers
75 views

Intersection of ideals in the ring of formal power series

Let $R$ be a commutative ring and $I,J$ ideals in $R$. Denote by $R[[X]]$ the ring of formal power series with coefficients in $R$. If $A\subseteq R$, denote by $A^e$ the ideal in $R[[X]]$ generated ...
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0answers
34 views

Functions on reduced schemes are determined by their values at each point.

This is an exercise in Vakil's Foundations of Algebraic Geometry, namely 5.2.A. Let $X$ be a reduced scheme. If $a\in \mathscr{O}_X(X)$ is such that its image in $\mathscr{O}_{X,p}$ lies in the ...
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0answers
32 views

Example for $f_m(S/I)\ge depth S/I+2$ [migrated]

Let $ R $ be a commutative unital noetherian ring, $ I $ an ideal of $ R $, and $ M $ a finite $R$-module with $\dim M\gt 0$. $f_I(M) = \inf\ \{i : H_I^i(M)$ is not finitely generated$\}$ is defined ...
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1answer
50 views

Rings with same quotient field

Let $R$ be an integral domain and $0 \neq I$ an ideal of $R$. Denote by $\phi: R \rightarrow R/I$ the canonical homomorphism. Let $S$ be a subring of $R/I$ such that $R/I$ is integral over $S$. ...
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1answer
64 views

Matsumura (commutative ring theory) remark, page 177 [closed]

If R is a graded ring and M is a graded R-module (not necessarily finitely generated). The following conditions are equivalent: a) M is R-flat. b) If $\mathcal S$ is an exact sequence of graded ...
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45 views

About integral closure over integral domains.

I need help with this exercise Let $A$ be a integral domain and $A'$ its integral closure. Show that the integral closure of $A[X]$ is $A'[X]$ Any help would be greatly appreciated. Thanks.
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1answer
32 views

Some ideals in $k[[x,y]]$

I have an ideal in $k[[x,y]]$, and I know that it contains $x$ but isn't $\langle x \rangle$ (here $k$ is a field, maybe not alg closed). This means that my ideal must be of the form $\langle x ,y^n ...
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1answer
46 views

Determining whether a ring is a principal ideal ring or not

I have been attempting to attack the following problem off and on for a few weeks now, without much success: Is the ring $R=\mathbb{Z}_{4}[x]$ of polynomials with coefficients in $\mathbb{Z}_{4}$ ...
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1answer
32 views

Showing $S^{-1}(M \otimes_{A} N) \cong S^{1}M \otimes_{S^{-1}A} S^{-1}N$

One of the propositions in Atiyah-MacDonald's Commutative Algebra states $$S^{-1}(M \otimes_{A} N) \cong S^{-1}M \otimes_{S^{-1}A} S^{-1}N.$$ The proof in the text states that one should use that ...
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1answer
27 views

Some question about ring of integers

Let $\mathbb{F}$ be a field and $\dim_{\mathbb{Q}}\mathbb{F} = d<\infty$. If $e_1,\dots,e_d$ is a $\mathbb Q$-basis in $\mathbb{F}$, there's $n \in \mathbb{Z}$ such as $ne_1,...,ne_d \in ...
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1answer
28 views

Associated points of Spec $\mathbb{C}[x,y]/ I$

Suppose we know that the only associated points of Spec $\mathbb{C}[x,y]/ I$ were $[(y-x^2)]$, $[(x-1,y-1)]$ and $[(x-2,y-2)]$. Is there enough information to deduce if this scheme is reduced or not? ...
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1answer
43 views

A commutative diagram of rings

Let $R$ be an integral domain and $\alpha:R\to R'$ an injective ring homomorphism. Let $K$ and $K'$ be the fields of fractions of $R$ and $R'$ respectively. I know that there is a commutative diagram ...
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2answers
54 views

Isomorphism of two $\operatorname{Hom}$ modules

Let $R$ be a ring (associative, commutative, with unity) and $I\subset R$ is an ideal. Let $M$ be an $R/I$-module and $N$ an $R$-module. Is it true that $$\operatorname{Hom}_R(M,N)\cong ...
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0answers
32 views

If $\{f_1,…,f_n\}$ generate $R$ then does $\{f_1^N,…,f_n^N\}$ [duplicate]

Let $R$ be a commutative ring such that $\{f_1,...,f_n\}\subseteq R$ generates $R$. Does this imply that for all integers $N>0$ that $\{f_1^N,...,f_n^N\}$ generates $R$? I would have guessed not, ...
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1answer
79 views

Exercise 5.5.F. on Ravi Vakil's Notes related to associated points

Let $A$ be a Noetherian ring and $M$ a finitely generated $A$ module. In Ravi Vakil's notes he first states that the associated points of $M$ satisfy the following: (A) The associated points of $M$ ...
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1answer
38 views

Irreducible closed subsets of a scheme corresponds to points

I have posted an answer here for the case of an affine scheme, but I got stuck when I tried to generalize the argument to schemes. My thoughts Consider a point $p$ in the scheme, its closure in the ...
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1answer
40 views

what can be said about $spec(R_m)$, where $R_m$ is localization of $R$ at maximal ideal $m$

I've seen how if $p$ is a prime ideal of $R$ and $R_p$ is the localization of $R$ at $P$, then $P_p$ is the unique maximal ideal of $R_p$, but what if we had a maximal ideal $m$ of $R$, then $R_m$ ...
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4answers
201 views

Trouble understanding Eisenbud Exercise 2.19a

I'm working through the "Commutative algebra with a view toward algebraic geometry" book and stumbled onto an exercise I'm struggling to answer. Let $R$ be a ring and let $M$ be an $R$-module. ...
0
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1answer
41 views

Quotient ring of a local ring is also local [closed]

A commutative ring is called local if it has only one maximal ideal. Let $R$ be a commutative local ring and let $I$ be a proper ideal of $R$. Show that $R/I$ is local.
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2answers
71 views

Calculating Spec of the localization $R_P$

I am studying a first course in commutative algebra and I'm currently working through some exercises on calculating $Spec(R_P)$, where $R_P = R[(R\backslash P)^{-1}]$ is the localization of $R$ at a ...