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

learn more… | top users | synonyms (1)

2
votes
0answers
59 views

indecomposable summand

Let $M$ be an $R$ module. Is this true for start a proof that we say "Let $S$ denote the indecomposable summands of $M$"? In fact, I want to know whether any module over a Dedekind domain (or a ...
2
votes
1answer
58 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 n}...
1
vote
2answers
46 views

Isomorphism of the completion 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: ...
2
votes
1answer
86 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 ...
4
votes
2answers
235 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 ...
1
vote
0answers
55 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, ...
1
vote
1answer
46 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 ...
0
votes
1answer
148 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 $R$ ...
0
votes
2answers
137 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 ...
1
vote
1answer
63 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
1
vote
0answers
75 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
votes
1answer
114 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 ...
4
votes
1answer
97 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 ? (...
1
vote
0answers
50 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 ...
0
votes
1answer
61 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 resolution, ...
1
vote
1answer
113 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} = \{R_{\...
4
votes
1answer
94 views

$\varinjlim\operatorname{Hom}_R(N,M_i) = \operatorname{Hom}_R(N, \varinjlim M_i)$

Show that $\varinjlim \operatorname{Hom}_R(N,M_i) = \operatorname{Hom}_R(N, \varinjlim M_i)$ is true when $N$ is finitely generated and $R$ is noetherian. Do you think the noetherian condition is ...
0
votes
1answer
42 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 $r_R =\{s \in R \ |\ rs =0 \}$. ...
1
vote
1answer
83 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 ...
0
votes
0answers
69 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 $a,b$ ...
1
vote
0answers
31 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 ...
0
votes
1answer
49 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 $(x,y,z,t)=...
4
votes
2answers
124 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 ...
0
votes
0answers
61 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 ...
0
votes
1answer
67 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 $S$. ...
0
votes
1answer
39 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 \...
1
vote
1answer
65 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 $\mathbb{Z}_{4}$ ...
0
votes
1answer
40 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 $S^{-...
0
votes
1answer
29 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 \mathcal{O}...
0
votes
1answer
46 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? ...
0
votes
1answer
69 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 ...
0
votes
2answers
86 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 \...
0
votes
0answers
36 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, ...
1
vote
2answers
247 views

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

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 $M$ ...
2
votes
1answer
132 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 ...
0
votes
1answer
63 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 $R_m$ ...
8
votes
4answers
253 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. ...
1
vote
2answers
162 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 ...
1
vote
1answer
84 views

How do I find the spectrum of a ring?

What is $Spec R$ where $R$ is the integers modulo $6$? More generally, what are the techniques to find the spectrum of any commutative ring? (I would also be interested in the non-commutative case but ...
8
votes
1answer
199 views

Is a product of two Noetherian schemes over Spec $\mathbb Z$ a Noetherian scheme?

In Hartshorne's proof of Proposition 6.6 in Chapter 2, he says that if $X$ being Noetherian implies $X\times\mathbf A^1$ is "clearly" Noetherian. I assume this is because $X$ can be covered by affine ...
5
votes
2answers
143 views

Ring with maximal ideal not containing a specific expression

Main question : May there exist an integral domain $R$, with fraction field $K$, that fulfills the following condition: there exists $x\in K$, $x\not \in R$ and a maximal ideal $\frak m$ of $R[x]$, ...
5
votes
1answer
344 views

Example that inverse limit is not exact

Its known that "inverse limit is not exact". Matsumura in his book Commutative Ring Theory, page 272, gives an example for this. I can not understand how he proves that inverse limit of $Z$ is zero. ...
1
vote
0answers
52 views

Tensor product and inverse limit

Let $(R,m)$ be a Noetherian local ring. Let $M$ be an $R$-module, and let $\{N_i\}$ be an inverse system. I am curious to know if there is a condition whereby the natural map $M \otimes_R \varprojlim{...
0
votes
1answer
150 views

Why is the structure sheaf for the spectrum of a ring defined locally?

I am currently learning for a course in algebraic geometry and wonder about the definition of the structure sheaf of the prime spectrum of a ring. Assume, that we have an open set $U$ in $X := Spec(R)...
4
votes
2answers
385 views

Converse to Chinese Remainder Theorem

So as seen on this question Converse of the Chinese Remainder Theorem, we know that if $(n,m) \neq 1$, then $\mathbb{Z} /mn \mathbb{Z} \ncong \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$, ...
2
votes
1answer
71 views

If $I$ is a homogeneous ideal of $A$ contained in $A_+$, then $\sqrt{I} = \bigcap\limits_{I\subset P\in\text{Proj }A} P$?

EDIT: This is from an exercise of Vakil's Foundations of Algebraic Geometry. 4.5.H: Suppose $I$ is any homogeneous ideal of $S$ contained in $S_+$, and if $f$ is a homogeneous element of ...
2
votes
1answer
169 views

direct limit of finitely generated submodule

if $A$ is a module,then the family fin($A$) of all the finitely generated submodules of $A$ is a directed set and direct limit of$M_i$ is isomorphic to$A$. for prove this needed to define to injection ...
3
votes
0answers
94 views

Complete ring and unique continuous homomorphism

Let $n\geq 2$ be an integer, $D=\mathbb Z[1/n]$, and $A$ a complete commutative ring with unit for the $I$-adic topology, where $I$ is an ideal of $A$. Suppose that $n$ is invertible in $A$. Let $x\in ...
2
votes
1answer
38 views

Direct limit of a subdirected system

Let $(M_{i},\phi_{ij})$ be a direct system over a directed set $I$ and let $M$ be its direct limit. Suppose we remove a finite/infinite number of terms from the direct system $(M_{i},\phi_{ij})$. Does ...
0
votes
1answer
46 views

$n$th root of a polynomial

Let $n\geq 2$ be an integer, and $D=\mathbb Z[1/n]$. Consider the polynomial $S=(1+T)^n-1\in D[T]$. How can I show that $D[[S]]=D[[T]]$ and there exists an $f(S)\in SD[[S]]$ such that $1+S=(1+f(S))^n$?...