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

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Is an extension of a discrete absolute value discrete too?

Suppose $L/K$ is a finite extension of fields, suppose $v$ is a non-archimedean absolute value on $L$ such that the restriction of $v$ on $K$ is non-trivial and discrete. Can we say that $v$ is ...
2
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1answer
52 views

The ideal for image of Segre embedding

How to show the ideal $(X_{ij}X_{kl}-X_{il}X_{kj})_{0\le i,k\le m, 0\le j,l\le n}\subset k[X_{ij}]_{0\le i\le m, 0\le j\le n}$ is radical? I can show the zero locus defined by the ideal is the image ...
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0answers
47 views

Generalization of Bezout Theorem to many-hypersurface case in Hartshorne's setting

I try to follow the ideas in Hartshorne's Chapter 1, Section 7. Suppose we have algebraic sets $Y_1,...,Y_l$, I try to define their intersection number $I(Y_1,...,Y_l)$ to be the leading term of the ...
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1answer
40 views

Localization of modules and minimal generating sets.

Let $A$ be a ring and $M$ a finite $A$-module; for $p \in \text{Spec} \space A$, write $\mathcal{K}(\mathfrak{p})$ for the residue field of $A_\mathfrak{p}$, and let $\mu (\mathfrak{p}, M)$ denote ...
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1answer
66 views

Operations with ideals in a commutative ring

Let $R$ be a commutative ring with identity. Let $A$ and $B$ be ideals in $R$. It is true that $(A\cap B)(A+B)$ equals the product $AB$?
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1answer
72 views

Prove that $S$ is an integral domain and $T$ is not an integral domain.

Let $R = \mathbb{C}[x,y]$ $R^i \subset R$ be the abelian subgroup of $R$ generated by elements of $\mathbb{C}$ times monomials of degree at least $i$ $I = (x^3+x^2-y^2)$ $S = R/I$ $S^i$ be the group ...
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0answers
21 views

General differentials operators (Grothendieck definition) and polynomial rings

Let $A$ be an algebra over some field $\mathbb{k}$. A linear map $f:A\to A$ is said to be a differential operator of an order $\le n$ if for all $a_0,a_1,\ldots a_n\in A$ we have ...
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0answers
59 views

Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
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4answers
121 views

Is product of prime ideals prime?

I'm trying to show that the product of ideals $(x_1, x_3)$ and $(x_2, x_4)$ in $\mathbb C[x_1, x_2, x_3, x_4]$ is a radical ideal, but no other way that I can think of works. So, is the product ...
0
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1answer
37 views

Showing local ring isomorphisms

This is a problem in K. Hulek's Elementary Algebraic Geometry. I figured out that $k[X]$ is the collection of polynomials of the form $f(x) + g(y)$ and also the local ring of an affine line at the ...
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0answers
20 views

GCD-Domain and proprieties

Let $A$ be a commutative GCD-domain (not necessary UFD or Bezout) and $a,b,c$ elements of $A$ such that $\gcd(a,b) = \gcd(b,c) = \gcd(a,c) = 1$. Is it true that $\gcd(ab,c) = 1$ ?
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1answer
50 views

An example of Noether normalization

Let $A=k[x_1,x_2]/(x_2^2-x_1^3+x_1)$. As an example of Noether normalization, determine elements $y_1,\ldots,y_m\in A$, algebraically independent over $k$, such that $A$ is a finite ...
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2answers
78 views

$k[x]/(x^n)$ module with finite free resolution is free

How to show a $k[x]/(x^n)$ module with finite free resolution is free? Suppose we have a exact sequence $k[x]/(x^n)^{\oplus n_1}\to k[x]/(x^n)^{\oplus n_{0}}\to M\to 0$, how do we get ...
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2answers
61 views

Logic problem: Atiyah-Macdonald 1.11

Proposition 1.11 in Atiyah-Macdonald's "Introduction to commutative algebra" states the following: "Given an ideal $I$ in a ring $A$ and $p_1, \dots p_n$ prime ideals, then $I \subset \cup_i p_i$ ...
2
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1answer
71 views

Show structure of a commutative ring in a tensor product [closed]

I need some help with this: Let $R$ be a commutative ring and $S$ and $T$ be commutative $R$-algebras. Show that $$ S \otimes T $$ has the structure of a commutative ring with multiplication: $$ (s ...
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2answers
35 views

Units and nilpotents in quotient ring. [closed]

$A$ is a commutative ring and $N(A)$ is the nilradical of $A$. If $A/N(A)$ is a field, show that every $a \in A$ is invertible or nilpotent.
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1answer
46 views

Krull dimension of a finitely generated integral domain over $k$ is equal to the transcendence degree.

This theorem is from Matsumura (p.34) Let $k$ be a field and $A$ an integral domain which is finitely generated over $k$. Then $\dim A = \operatorname{trdeg}_k A$ (where $\operatorname{trdeg}_k ...
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1answer
24 views

About freeness of modules over the coordinate ring of an affine variety

Let $X$ be an irreducible affine variety, $A$ be its coordinate ring, $M$ be an $A$-module. Suppose that for any maximal ideal $m$ of $A$, the localization $M_m$ is a free module of rank $n$ (finite ...
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0answers
32 views

Ideal in power series ring

Let $J$ be an ideal in $k[[x_1,...,x_n]]$ such that $(x_{1},...,x_{n})^{2}\subseteq J$, $\{x_{1},...,x_{r}\}\nsubseteq J$ and $\{x_{r+1},...,x_{n}\} \subseteq J$, for some $1\leq r \leq n$. I want to ...
1
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1answer
22 views

A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is ...
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1answer
37 views

If $0\to M'\to M\to M''\to 0$ is exact, why does $\operatorname{Ass}(M)\subseteq \operatorname{Ass}(M')\cup \operatorname{Ass}(M'')$.

I'm stuck on a proof I'm reading. Let $0\to M'\stackrel{\mu}\to M\stackrel{\sigma}\to M''\to 0$ be a sequence of $A$-modules. Then $\operatorname{Ass}(M)\subseteq \operatorname{Ass}(M')\cup ...
1
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1answer
28 views

Factorization in Dedekind domains

Let $R$ be a commutative, Dedekind (and therefore Noetherian) ring with $1$. Let $I$ be a non-prime ideal of $R$, and let $a,b$ be elements of $R$ such that $a\not\in I,b\not\in I$ but $ab\in I$. Let ...
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2answers
86 views

Hartshorne II Prop 6.8

My weaknesses with commutative algebra are really slowing down my progress through Hartshorne. I hope someone can help me understand some statements in the proof of the proposition below. Prop ...
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2answers
29 views

If $\mathcal{I}(-)$ is the ideal map on subsets of affine space, why does $A\subseteq\overline{B}\iff\mathcal{I}(B)\subseteq\mathcal{I}(A)$?

I think this is a basic property of $\mathcal{I}(-)$, but I'm having trouble seeing it. I denote by $\mathbb{A}^n$ the affine $n$-space over an algebraically closed field $k$, where if ...
2
votes
2answers
79 views

Normalization of a variety

I'm currently in a number theory course and this question popped up. As I'm not super familiar with algebraic geometry, I was wondering if my reasoning is correct: Show that $\mathbb{C}[X,Y]/(Y^2 ...
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2answers
61 views

Show that the ring of continuous functions $f:\mathbb R\to\mathbb R$ is not Noetherian

Prove that the ring of continuous functions $f:\mathbb R\to\mathbb R$ is not Noetherian. I know that to be Noetherian, every ideal is generated by finitely many elements or equivalently R ...
2
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0answers
50 views

UFD and relatively prime elements

I've found the following statement at page 9 of Griffiths, Harris "Principles of Algebraic Geometry": Proposition. If $R$ is a UFD and $u$, $v \in R[t]$ are relatively prime, then there exist ...
2
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1answer
73 views

Commutative Algebra and Game Theory

Is there any relationship between commutative algebra and game theory? For example, have any tools in commutative algebra been applied to game theory? A text or reference would be ideal, but I'd be ...
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2answers
61 views

Sum and product of comaximal ideals

Let $R$ be a commutative ring with unity. If $R=I_{i}+I_{j}$, for all $i\ne j$, where $I_1,I_2,...,I_n$ are ideals of $R$, I want to show that $$R=I_{n}+I_{1}I_{2}\cdots I_{n-1}.$$ I started off ...
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1answer
95 views

Matsumura Example 2 (16.E)

I am reading example 2 (16.E) of Matsumura's Commutative Algebra where he gives an example of a non-CM ring. Let $A = k[x,y]$ and $B = k[x^2, xy,y^2, x^3, x^2y, xy^2,y^3]$. Then $A,B$ have the same ...
3
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1answer
69 views

Isomorphism of ring localized twice - Atiyah Macdonald Exercise 3.3

I studied AM before studying universal properties. When I solved the following exercise, I had a tedious solution that involved dealing with elements. Let $ A $ be a ring with multiplicatively ...
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2answers
48 views

The krull dimension of $\Bbb{Z}$ and artinian rings

On page thirty of Matsumura, it says that $\Bbb{Z}$ has krull dimension 1 because every prime ideal is maximal. I understand this because for any prime p you have $0 \subset p$. However, for artinian ...
2
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2answers
126 views

Do we have $\dim A = \sup_{\operatorname{ht}( \mathfrak{p})=0} \dim A/\mathfrak{p}$?

Let $A$ be a ring (assume Noetherian if necessary). Then it is clear to me that we have $$ \sup_{\operatorname{ht}(\mathfrak{p}) = 0} \dim A/\mathfrak{p}\leq \dim A.$$ However, I can't seem to prove ...
2
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1answer
31 views

Associated primes of the completion of a ring

I am working through a proof somewhere, and I want to use this: Let $(R,m)$ be a local ring (Noetherian commutative) and let $M$ be an $R$-module. If $p$ is an associated prime of $M$, then there ...
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0answers
66 views

When is $A = k[x_1,\ldots, x_n]/I$ integrally closed?

Suppose that it is not easy to determine that $A$ is a UFD (or that it is a local, noetherian dimension 1 domain with principal maximal ideal). Can someone suggest strategies for showing that a ...
1
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1answer
60 views

Why does Proposition 1.8 in Atiyah-Macdonald imply that the smallest prime $\mathfrak{p}$ containing a primary ideal is equal to its radical?

Proposition 4.1 in Atiyah-Macdonal states that the radical of a primary ideal is the smallest prime ideal containing the primary ideal. They start the proof claiming that showing the radical is a ...
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3answers
118 views

Polynomials over $\mathbb{F}_2$ with certain values in $\mathbb{F}_4$

Let $\mathbb{F}_4=\{0,1,u,u^2\}$ be the field with $4$ elements. Is there a polynomial $p \in \mathbb{F}_2[x,y]$ with the following property? (1) For $r,s \in \mathbb{F}_4$, we have $p(r,s)=u ...
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1answer
35 views

Localizing at maximal ideals and the product

Let $D$ be an integral domain, $M_{i}$, $i = 1,...,r$ be some of its mutually distinct maximal ideals, and $e_{i}$be positive integers for all $i$. Is it true in general that the extension of the ...
1
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1answer
32 views

What is wrong with my proof of a step in Artin's construction of algebraic closure?

I'm working through Atiyah & MacDonald, and there's an exercise basically asking you to fill in a certain step in Artin's construction of an algebraic closure for a given field. The question is ...
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0answers
99 views

Hilbert's Basis Theorem - Clever Proof?

So I am studying commutative algebra at the moment and I have come across the proof of the Hilbert Basis Theorem (the proof I have is the same as the one in Reid's "Undergraduate Commutative ...
2
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1answer
49 views

Dense open subsets of schemes

Let $X$ be a scheme. Let $U$ be an open subset of $X$. It is clear that if $U$ contains all the generic points of $X$ (by which I mean the generic points of irreducible components of $X$) then $U$ is ...
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1answer
28 views

Quotient ring of a graded algebra with respect to a graded ideal

An algebra $A$ over $F$ is said to be a graded algebra if as a vector space over $F$, $A$ can be written in the form $$A=\bigoplus_{i=0}^\infty A_i$$ for subspaces $A_i$ of $A$ along with other ...
2
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2answers
86 views

Why does $M\otimes k(\mathfrak{m})=M_\mathfrak{m}/\mathfrak{m}M_\mathfrak{m}$? (From Matsumura, proof of Theorem 4.8.)

Matsumura's Commutative Ring Theory, proof of Theorem 4.8, page 27, says: Let $A$ be a ring, $M$ a finite $A$-module, and $\mathfrak{m}$ a maximal ideal. If ...
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3answers
99 views

Ideal Generated by Three Elements in Polynomial Ring [duplicate]

How would one prove that the ideal $(xy,xz,yz)$ of $k[x,y,z]$ for some field $k$, cannot be generated by two polynomials. In other words, prove: $$(xy,xz,yz) \neq (f,g)\; \forall f,g \in ...
4
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1answer
97 views

Rank of projective module defined as the smallest $n$ such that $P$ is a direct summand of $R^n$

Over a commutative ring $R$, the rank of a projective module $P$ is defined by looking at the map $\text{rank}(P) : \text{Spec}(R) \rightarrow \mathbb{N}_0$ given by $\mathfrak{p}\mapsto ...
0
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1answer
26 views

Decomposition of a polynomial over generators of an ideal

Let $f$ be a polynomial in six variables, say, over complex numbers, and $l_1$, $l_2$ are some linear forms in the same variables. If I know that polynomial $f$ belong to the ideal generated by $l_1$ ...
0
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0answers
22 views

Primality of homogeneous ideal

Let $R$ be the polynomial ring over the finite field $\mathbb{F}_p$ with $n$ variables. Let $I$ be an ideal of $R$ generated by homogeneous polynomials whose coefficients are 1 or -1. Are there any ...
0
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1answer
58 views

Primary decomposition of $(XY,(X-Y)Z)$ in $k[X,Y,Z]$

How to find the primary decomposition of $I=(XY,(X-Y)Z)$ in $R=k[X,Y,Z]$? It has minimal primes $(x,y),(y,z),(z,x)$. I tried to calculate $J=S^{-1}I\cap R$, where $S=R-(x,y)$, but it seems ...
0
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1answer
49 views

Is a graded module over a graded ring zero when all of it's graded localizations at graded primes not containing the irrelevant ideal are zero?

Let $M$ be a graded module over an $\mathbb{N}$-graded ring $S$ and $S_+$ be the ideal of positive degree elements. Is it true that $M=0$ iff the homogeneous localization $M_{(\mathfrak p)}=0$ for ...
0
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1answer
71 views

Hartshorne Chapter II exercise 5.7 on Invertible sheaves

I'm working on part c) which is to prove that for a Noetherian scheme $X$, a coherent sheaf $\mathscr{F}$ is invertible (locally free of rank 1) iff there exists a coherent sheaf $\mathscr{G}$ such ...