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

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When does a homogeneous morphism have only finite fibers?

Suppose that we have a map ${\bf f}:=(f_1,f_2,\cdots ,f_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ given by $$ \mathbb{C}^n\ni {\bf z}:=(z_1,z_1,\cdots,z_n)\rightarrow \big(f_1({\bf z}),f_2({\bf ...
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20 views

Question on a property of $Ass(M)$ for modules over notherian rings

I got stuck reading a proof of the following lemma: Lemma Suppose that $M$ is a module over a commutative noetherian ring $R$ and let $m\neq 0 \in M$. Let $S$ be a multiplicative set in $R$ such ...
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13 views

$K_1(R)$ and splitting

Let $R$ be a commutative ring with unit. Consider the exact sequence $1\to E(R)\to Gl(R)\to K_1(R)\to 1$. Under what conditions does this exact sequence split?
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1answer
87 views

When is an holomorphy ring a PID?

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
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2answers
50 views

Module structure of base extension via tensor product

Let $A,B$ be commutative rings. Defining a product of $B\otimes_{A}B$ as $(b_1 \otimes b_2)\cdot (b_3 \otimes b_4)=(b_1b_3)\otimes(b_2b_4)$, this becomes a commutative ring. Defining $b\cdot(b_1 ...
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48 views

Intersection of valuation rings

Let $F$ be a function field and $S$ be an arbitrary (and non trivial) subset of the set of places of $F$. Let $H=\bigcap_{P\in S} O_P$ where $O_P$ is the valuation ring associated to the place $P$. ...
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0answers
72 views

Differential operators on the polynomial ring

Let $A$ be a commutative algebra over complex numbers. If $a\in A$ we define $m_a$ to be a linear map which sends each $x$ to $ax$. The zero map $A\to A$ is said to be a differential operator of an ...
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1answer
252 views

How to show that differential operator can be defined in terms of certain commutator operators

Let $U$ be any open subset of $\mathbb{R}^n$ (or, more general, of some smooth manifold). Define $\mathcal{D}_{-1}(U):=\{0\}$. For any two linear operators $A$ and $B$, the commutator operator $[A,B]$ ...
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1answer
55 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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1answer
32 views

Relatively prime ideals in Dedekind Domains

I am currently working through Lang's Algebra and have come across an exercise I can not solve (Chapter II, Exercise $19$). Any help would be greatly appreciated. Let $R$ be a Dedekind domain. ...
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49 views

Localization of a regular local ring is regular

Quoting Hartshorne's Algebraic Geometry Definition. We say a scheme $X$ is regular in codimension one if every local ring $\mathcal{O}_x$ of $X$ of dimension one is regular. The most ...
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2answers
41 views

If the localizations of two submodules with respect to any prime ideal are equal then the submodules are equal [on hold]

I want to prove the following: Let R be a commutative ring with 1 and let N and L be two submodules of an R-module M. If the localizations of N and L with respect to any prime ideal of R are ...
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0answers
27 views

About the ramification locus of a morphism with zero dimensional fibers

This question arises from my somewhat frustrating attempts to understand what etale means (in the world of algebraic varieties for now) and marry the more advanced algebraic geometry references and ...
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0answers
19 views

Module length and connection between $\varphi,\det\varphi$ [on hold]

$e_A(\varphi,M) = l_A(\mathrm{coker}\varphi) - l_A(\ker\varphi)$ Let $A$ be domain. I want to prove that $e_{A}(\varphi,A^n) < \infty \iff e_{A}(\det\varphi,A) <\infty$ using the fact that ...
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1answer
80 views

Definition of multiplicity

Q.1. Bruns_Herzog define multiplicity (in the case of graded rings and modules) as My question is that: why multiplicity for $d=0$ it is defined as $\ell(M)$? Is there a kind of ...
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0answers
44 views

Partial derivatives with respect to algebraically independent polynomials

Suppose that $\{f_1, \ldots, f_n\}, \{g_1, \ldots, g_n\}$ and $\{h_1, \ldots, h_n\}$ are algebraically independent polynomials that generates the same algebra of $\mathbb{R}[x_1, \ldots, x_n]$. Then I ...
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24 views

Talking about varieties

hi I was recently reading ideals varieties and algorithms. I ah having problems showing things are not affine varieties. Previously with problems like. $V= \{ (a,a) | a \in R^* \}$ it was much easier ...
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2answers
743 views

Must $k$-subalgebra of $k[x]$ be finitely generated?

Suppose $k$ is a field, $A$ is a $k$-subalgebra of the polynomial ring $k[x]$. Must $A$ be a finitely generated $k$-algebra? Thanks.
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1answer
63 views

About the $k$-subalgebras of $k[x]$

Still in my "commutative algebra marathon", I came across the following exercise: Any $k$-subalgebra $A$ of $k[x]$ is finitely generated as $k$-algebra; also, if $A\ne k$, then $\dim A=1$. ...
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0answers
41 views

Product of schemes and ideal sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be projective schemes over $\mathbb{C}$. Then, 1) Is the structure sheaf of $X \times_{\mathbb{C}} Y$ isomorphic to $\mathcal{O}_X ...
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1answer
33 views

Associated primes of an $R$-module

An associated prime of an $R$-module $M$ is an ideal of the form $Ann_R(N)$ where $N$ is a prime sub-module of $M$ in the sense that $N$ is nonzero and $Ann_R(N)=Ann_R(N')$ for each nonzero sub-module ...
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1answer
43 views

Length of tensor product of finite length modules is finite

Let $R$ be a commutative ring. If $M$ and $N$ are finite length $R$-modules, then $M\otimes_R N$ has finite length, and $l(M\otimes_R N) \le l(M)l(N)$. I know the question has been posted ...
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1answer
77 views

Functorial construction with two integral domains

Motivated by this question: Let $\mathsf{Int}$ be the category of integral domains with ring homomorphisms (perhaps only injective ring homomorphisms, if you need this). Is there a functor ...
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26 views

Let I be an unmixed radical ideal of R. then (I:x) is unmixed

Let $R$ be commutative ring with $1$. One says that an ideal $I$ is unmixed if $I$ has no embedded prime divisors (in other words,􀀀 if the associated prime ideals of $R/I$ are the minimal prime ...
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1answer
64 views

How can I verify that the ideal $(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$ in $\mathbb Q[x,y,z,w]$

I want to show that the ideal $$(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$$ in the ring $\mathbb{Q}[x,y,z,w]$ is prime, how can I?
30
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1answer
3k views

Classification of prime ideals of $\mathbb{Z}[X]$

Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable. My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? (1) $(0)$ (2) ...
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1answer
52 views

If every maximal ideal is finitely generated is the ring Noetherian? [duplicate]

$R$ is a commutative ring with $1$. Suppose every maximal ideal is finitely generated. Is this ring Noetherian? Equivalently, is every prime ideal finitely generated?
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29 views

Subvarieties and finding ideals

Hi guys I am stuck working on this problem. I have a surface $W= V(xz-y^2)$ and we are trying to find an ideal $J \in K[W]$ so that the $V_w(J)=V(y-x^2,z-x^3)$ I showed that the second thing which is ...
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2answers
38 views

Residue class field of coordinate ring

If $X$ is an irreducible affine curve over an algebraically closed field $k$, then its coordinate ring $O(X)$ is a Dedekind domain. Suppose $\mathfrak{p}$ is a prime (hence maximal) ideal in $O(X)$ ...
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0answers
35 views

Property of free submodules for a module over a PID [duplicate]

This question was asked here and remains without solution. It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is ...
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1answer
34 views

A comparison between heights and between grades

I search for noetherian commutative rings having distinct prime ideals $P⊂Q$ with no primes between them, where $grade(Q)≠grade(P) +1$, or $height(Q)≠height(P)+1$. If $R$ is Cohen-Macaulay, are the ...
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1answer
13 views

Residue class ring of Dedekind domain

Zariski and Samuel Commutative Algebra Ch V para 7 makes the following statement: If $R$ is a Dedekind domain with an ideal $\mathfrak{a}=\prod_i\mathfrak{p}_i^{n(i)}$ factored into prime ideals, ...
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1answer
33 views

Describing ideal that vanishes at the variety

We have the following morphism $$\phi(a_1,..a_m;b_1,...,b_n)= \begin{pmatrix} a_1 b_1 & \ldots & a_1 b_n \\ \vdots & \ddots & \vdots \\ a_mb_1 & \ldots & a_m b_n ...
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1answer
35 views

Question on the existence of a prime ideal contained in the $\ker$ of a homomorphism $\mathbb{C}[x,y]\rightarrow\mathbb{C}[t]$.

I found this exercise in a basic algebraic geometry book: Let $f:\mathbb{C}[x,y]\rightarrow \mathbb{C}[t]$ a non-zero homomorphism such that $\ker f$ strictly contains a prime ideal $P\neq0$. Is it ...
2
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1answer
71 views

Dual of polynomial ring

Consider the free $k$-algebra $k[x_i]_{i \in I}$ indexed by $I$. Then is $Hom_{k-Mod}(k[x_i]_{i \in I},k) \cong k[x_i]_{i \in I}$?
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1answer
111 views

Interpretation of sheaf flat over a base

I am trying to get an interpretation of what means for a sheaf to be flat with respect to a base. The definition is that, given $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ is flat over $Y$ ...
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0answers
23 views

tensor product of formal power series

Let $A[[h]]$ be the formal power series algebra over $\mathbb{C}[[h]]$, here $\mathbb{C}$ is the complex number field. Is the canonical map $A[[h]] \otimes_{\mathbb{C}[[h]]} A[[h]] \to ...
2
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1answer
28 views

Gröbner Basis and Division Algorithm

I recently read a lemma on a course in Commutative Algebra that states, If $G$ is a Gröbner Basis for an Ideal $I$ in $k[x_{1},x_{2}...x_{n}]$, then a polynomial $f$ belongs to $I$ if and only if ...
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1answer
101 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 $\mathbb{Z} /mn \mathbb{Z} \ncong \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$, ...
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2answers
159 views

What's the theoretical basis for integration using partial fractions?

Exercises involving integration using partial fractions depend on expressing a rational function $\frac{P(x)}{Q(x)}$ (where the degree of $P$ is less than the degree of $Q$) as a sum of ...
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1answer
38 views

Element in no prime ideal $\iff$ it is a unit

I was working through Atiyah & MacDonald, chapter 1 section 1 problem 17 part iii) where it says Let $R$ be a ring and $f\in R$. Define $V(f)$ to be all elements of ...
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2answers
84 views

The unit group of $\mathbb{Q}[x, y]/(x^2+y^2+1)$

During some calculations, I encountered with the problem of calculating the unit group of the $\mathbb{Q}$-algebra $\mathbb{Q}[x, y]/(x^2+y^2+1)$. I believe it is the unit group of the field of ...
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2answers
38 views

Prime Ideals and multiplicative sets

I am currently studying a course on commutative algebra and came across this statement: An Ideal $I$ in a ring $R$ is prime if and only if $R\setminus I$ is a multiplicative set. I have proved ...
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2answers
64 views

Reference to complete proof that integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$?

Where can I find a complete proof to the fact that the integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$ (the Gaussian integers are the integral closure of $\mathbb{Z}$ in the ...
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Let $R$ be a PID, let $P$ be a nonzero, proper, prime ideal in $R$. Show that $P$ is generated by an irreducible element. Show that $P$ is maximal [closed]

The question comes from Fulton's book, Algebraic Curves, Problem 1.3. Let $R$ be a PID, let $P$ be a nonzero, proper, prime ideal in $R$. Show that $P$ is generated by an irreducible element. Show ...
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3answers
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Nil-Radical equals Jacobson Radical even though not every prime ideal is maximal?

Let's assume we have a commutative ring with identity. Can the Nil-Radical and the Jacobson Radical be equal in a non-trivial case (i.e. not every nonzero prime ideal in said ring is maximal)? Are ...
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2answers
62 views

Problem related to prime ideals of B and A where B is integral over A

Let $ A $ be an entire ring, integrally closed. Let $ B $ be entire, integral over $A$. Let $ Q_1, Q_2$ be prime ideals of $B$ with $Q_1 \supseteq Q_2$ but $Q_1 \neq Q_2$. Let $P_i=Q_i \bigcap A$. ...
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1answer
63 views

In any commutative ring with unity, every prime ideal is finitely generated implies every ideal is finitely generated; can it be prove without A.C.?

Assuming Zorn's lemma, "In any commutative ring with unity, if every prime ideal is finitely generated, then every ideal is finitely generated". Is the converse true, i.e. if in any commutative ring ...
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0answers
23 views

Let $R=k[[ X_1,X_2,X_3]]$ and $S=R/(X_1X_3,X_2X_3)$. Can one compute $\ell_S(S/(a_1^i,a_2^j))$?

Let $R=k[[ X_1,X_2,X_3]]$ and $S=R/(X_1X_3,X_2X_3)$. Let $x_i$ be the natural image of $X_i$ in $S$. Set $a_1=x_1+x_3$ and $a_2=x_2+x_3$. $a_1,a_2$ is a system of parameters of $S$. So ...
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0answers
52 views

Polynomial division, multivariable, indeterminates

Trying to understand something in the proof of Nullstellensatz, if we have a polynomial $p(x_1,...,x_n,t) \in k[x_1,...,x_n,t]$ with $f(t)$ divides $p(a_1,...,a_n,t)$ for all fixed $(a_1,...,a_n) ...