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

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10
votes
3answers
355 views

Geometrically, why do line bundles have inverses with respect to the tensor product?

Geometrically, why do line bundles have inverses with respect to the tensor product? Here my thoughts on the problem so far, please excuse their scatteredness. I know algebraically, it is just ...
10
votes
1answer
132 views

DVR, power series expansion.

Let $A$ be a discrete valuation ring with quotient field $K$, maximal ideal $\mathfrak{m}$, uniformizing parameter $t$. Let $k = A/\mathfrak{m}$, so $k$ is a field. How do I show that there is a ...
5
votes
1answer
52 views

Only DVR's with quotient field $\mathbb{Q}$?

Let $p \in \mathbb{Z}$ be a prime number. I know how to show that $$\{r \in \mathbb{Q}: r = {a\over{b}},\text{ }a,b \in \mathbb{Z},\text{ }p\text{ doesn't divide }b\}$$ is a DVR with quotient field ...
5
votes
1answer
102 views

Complement of open set is finite in Zariski topology

This problem has two parts: a) Let $M$ be a finitely generated module over a Noetherian ring $A$. Prove that $S=\{ P \in\operatorname{Spec}(A) : M_P \mbox{ is a free }A_P\mbox{-module} \}$ is an ...
10
votes
1answer
44 views

Local ring coincides with DVR.

Assume $A$ is a discrete valuation ring with quotient field $K$ and maximal ideal $\mathfrak{m}$. If $S$ is a local ring containing $A$ and contained in $K$ with maximal ideal containing ...
2
votes
1answer
134 views

Regular subrings of a polynomial ring

Let $R=\mathbb{C}[x,y]$. I have the following situation: $\mathbb{C} \subseteq D \subseteq R$ is affine (= finitely generated as a $\mathbb{C}$-algebra), noetherian, has field of fractions ...
10
votes
1answer
128 views

Example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\nsubseteq N$

For an ideal $I\lhd R$ in a commutative ring $R$, let $ann(I)$ denote the annihilator of $\{x\in R\mid xI=\{0\}\}$. A commutative ring $R$ is said to be a dual ring if for every ideal $I$ of $R$, ...
4
votes
1answer
86 views

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$ This is not homework, it is part of an answer of Show that $\mathbb{A}_\mathbb{C}^2 \ncong ...
1
vote
1answer
59 views

Regularity of a quotient ring of the polynomial ring in three indeterminates

Let $I=(f)$ be a prime ideal in $R=\mathbb{C}[x,y,z]$, so $f$ is an irreducible polynomial, and further assume that $f$ is of the following form: $f=z^n+c_{n-1}z^{n-1}+\ldots+c_1z+c_0$, where ...
0
votes
1answer
80 views

A proof for Atiyah-Macdonald Exercise I.21.iii

The following is exercise I.21.(iii) of Atiyah-Macdonald: Let $\phi \colon A \to B$ be a ring homomorphisms. Let $X = \operatorname{Spec} A$ and $Y = \operatorname{Spec} B$ [and let $\phi^\ast ...
3
votes
2answers
78 views

Direct sum of non-zero ideals over an integral domain

Let $R$ be an integral domain. Let $I$ and $J$ be non-zero ideals of $R$. Is this statement always true: $$R\oplus(I\cap J)\cong I\oplus J\ ?$$ I regarded the short exact sequence $0\to I\cap ...
3
votes
1answer
62 views

Showing that $\mathcal{O}(X_f)\cong\mathcal{O}(X)_f$ without schemes language

I have seen this question here in the language of schemes, but I never studied this, so I hope someone can help me to solve this problem without schemes (I'm a beginner in this). The problem is to ...
2
votes
1answer
72 views

Number of generators of prime ideals in $K[x_1,x_2,…,x_n]$

Is there any bound for the number of generators of prime ideals in $K[x_1,x_2,...,x_n]$? (For example in $K[x,y]$.) We know that maximal ideals of $K[x_1,x_2,...,x_n]$ have $n$ generators.
1
vote
1answer
66 views

$\operatorname{Proj}k[x,y,z]/(xz,yz,z^2)$ isomorphic to $\mathbb{P}^{1}_{k}$

While dealing with the Proj construction, I encountered with this seemingly-simple question, but somehow I can't get the point at this moment. Is the scheme ...
3
votes
1answer
67 views

Nilpotents after tensoring with a field

Let $A \to B$ be a homomorphism of commutative rings with unit. Let $A_{\text{red}}=A/ \sqrt{(0)}$ and $B_{\text{red}}=B/ \sqrt{(0)}$ be the corresponding reduced rings. Now let $A_{\text{red}} \to K$ ...
2
votes
2answers
47 views

Conductor of a ring

An easy (possibly trivial) question from Neukirch's Algebraic Number Theory, p.47. Let $A$ be a Dedekind domain, $K$ its fraction field, $L$ a finite separable extension of $K$ and $B$ the integral ...
1
vote
1answer
59 views

Help with computation and Gröbner basis

Hi guys I am learning a new software and a new topic (Gröbner basis) I have this problem $$ \begin{cases} 6-21(x_1x_2+x_1x_3+x_1x_4)=0 \\ 10-21(x_2x_1+x_2x_3+x_2x_4)=0 \\ ...
0
votes
0answers
28 views

Classification of local and semi-local rings in function fields

Let $C$ be a non-singular algebraic curve over an algebraically closed field $k$, and $F$ a function field of this curve. It is well-known that non-trivial discrete valuation rings of $F$ correspond ...
1
vote
1answer
44 views

Existence of induced map on Divisor Class Group?

Let $f: X \rightarrow Y$ be a morphism of noetherian, integral schemes, regular in codimension 1 (so we can talk about Weil divisors). I am wondering whether there is an induced map on divisor class ...
0
votes
0answers
133 views

Singularities in the weighted projective space

Is there an explicit criterion for checking that a hypersurface $f=0$ of degree $d$ and in $\mathbb{P}(a_0,\ldots,a_n)$ is smooth ? I could not convince myself that the criterion $\nabla f\neq 0$ ...
1
vote
1answer
38 views

Characterisation of “projective $k$-algebras”

For my thesis, I'm defining affine $k$-algebras to be reduced, finitely generated $k$-algebras--each of which turns out to be isomorphic to the quotient of a polynomial ring by a radical ideal. I'm ...
1
vote
1answer
44 views

Localization of an integral A-algebra is not always integral. [duplicate]

Let $A$, $B$ rings with a morphism $f : A \to B$ and suppose that $B$ is integral over $A$. Let $\mathfrak{n} \subseteq B$ a maximal ideal, and $\mathfrak{m}$ its preimage under $f$ (so $\mathfrak{m}$ ...
1
vote
1answer
73 views

Can we prove, without axiom of choice, that the set of all zero divisors (including $0$) of a commutative ring with unity contains a prime ideal?

Let $R$ be a commutative ring with unity , I know that assuming axiom of choice , if $A$ is the set of all zero divisors (including $0$ ) then it is a union of prime ideals so it contains a prime ...
4
votes
0answers
68 views

If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?

Let $A$ be a local (not necessarily noetherian) ring with maximal ideal $\mathfrak{m}$ and residue field $k$. Let $M$ be a finitely generated $A$-module such that $\mathfrak{m}\otimes_A M\rightarrow ...
4
votes
2answers
108 views

What are the closed points of $\mathbb{A}_{\mathbb{R}}^2 = \operatorname{Spec}(\mathbb{R}[x,y])$?

I am trying to find all the closed points of $\mathbb{A}_{\mathbb{R}}^2$. After a quick google research, I found that $\mathbb{A}_{\mathbb{R}}^2 = \operatorname{Spec}(\mathbb{R}[x,y])$ and then all ...
1
vote
2answers
45 views

bijection between prime ideals of $R_p$ and prime ideals of $R$ contained in $P$

Given a ring $R$, I want to show that the localization of $R$ at the prime ideal $P$ of $R$(denoted as $R_P$) is isomorphic to the set of prime ideals of $R$ contained in $P$. That is: ...
1
vote
1answer
55 views

Finding primary decompositions of ideals

I have been given this example of the decomposition of an ideal into primary ideals $$ I =⟨x^2,xy,x^2z^2,yz^2⟩$$ Then the primary decomposition of this ideal is: $$⟨x^2,y⟩∩⟨x,z^2⟩⊆K[x,y,z]$$ This ...
0
votes
1answer
19 views

Irreducible decomposition of varieties vs primary decomposition of ideals

I'm new to working with varieties, and the statement mentioned below is left as an exercise, but I'm having some difficulty trying to prove it. Let $R=K[x_1,...,x_n]$. If $X=X_1\cup ... \cup X_n$, ...
0
votes
0answers
37 views

Regularity of simple ring extensions, subrings and quotients

Let $R$ be a regular UFD of zero characteristic, $I=(p)$ a prime ideal of $R$ and $Q(R)$ the field of fractions of $R$. Assume $R[a]$ is integral and flat over $R$, for some $a \notin Q(R)$. Is it ...
2
votes
0answers
47 views

Semiring of formal power series with non-negative coefficients

Has the semiring $\mathbb{Q}_{\geq 0}[[X]]$ of formal power series with non-negative rational coefficients been studied somewhere? For example, I would like to be confirmed that the group of units is ...
4
votes
1answer
54 views

Unique factorization in fields

Suppose $A$ is a commutative $R$-algebra and that is also a field. Define: For $x,y \in A$, say that $x$ divides $y$ iff $xr = y$ for some $r \in R$. Call $x,y \in A$ associates iff each divides the ...
9
votes
1answer
138 views

$p \in C - D$, inflection point for $C$ iff inflection point for $C \cup D$.

Show that if $C$ and $D$ are projective curves in $\mathbb{P}_2$ and $p \in C - D$ then $p$ is a point of inflection for the curve $C$ if and only if $p$ is a point of inflection for the curve $C \cup ...
1
vote
1answer
43 views

polynomial grade

Hamilton-Marley in the paper "Non-Noetherian Cohen–Macaulay rings" have I can't understand highlighted part. my attempt is: $$\text{p-grade} ((x')R',R')=\text{p-grade} ...
2
votes
1answer
54 views

I suppose this is a familiar number-theoretic operation, but what is it?

Define a function $/\!/ : \mathbb{Z}_{\geq 1} \times \mathbb{Z}_{\geq 1} \rightarrow \mathbb{Z}_{\geq 1}$ as follows: given integers $j,k \geq 1$, we have: $$k/\!/j = \min\{n\in\mathbb{Z}_{\geq 1} : ...
3
votes
1answer
31 views

The Dimension Sequence of a Ring

Let $R$ be a commutative ring of finite Krull dimension $n_0$. Let $\dim(R[X_1,\dots,X_m])=n_m$. The sequence $\{n_i\}_{i=0}^\infty$ is called the dimension sequence of $R$. Let $d_i=n_i-n_{i-1}$. The ...
1
vote
2answers
96 views

Extending an automorphism to the integral closure

I need some help to solve the second part of this problem. Also I will appreciate corrections about my solution to the first part. The problem is the following. Let $\sigma$ be an automorphism of ...
1
vote
1answer
27 views

Proving that the three statements on units are equivalent

Let $R$ be a ring and $x\in R$ i) $x$ is a unit ii) $\bar{x}$ is a unit in $R/P$, where $P$ is a prime ideal iii) $\frac{x}{1}$ is a unit in $R_P$, where $P$ is a prime ideal i)$\implies$ ii) if $x$ ...
1
vote
1answer
42 views

how to tell if a ring is noetherian

In general how would i tell if the rings $\mathbb{Z}[\sqrt d]$ and $\mathbb{Z}[\frac{x}{y}]$ are noetherian? I know that the ring $\mathbb{Z}$ is noetherian as all ideals are contained in a finite ...
3
votes
0answers
48 views

Degree of the minimal polynomial of the sum of two integral elements over a UFD

Let $D$ be an integral domain ($D$ is a noetherian UFD, if necessary) and let $a,b$ integral over $D$. Let $f$ be the minimal polynomial of $a$ over $D$ and assume it is of degree $n>1$, and let ...
0
votes
1answer
73 views

Primary and irreducible ideals

I have to verify that the ideal $I = \langle x^3,x^2y,xy^3,y^5\rangle \subset R=\mathbb{C}[x,y,z]$ is primary. I then have to go on to show that it is not irreducible by writing it as an ...
0
votes
2answers
35 views

$\Gamma_I(E)$ is an injective $R$-module? $H^i_I(E)=0;\forall i\gt 0$

1.Let $R$ be a commutative ring, $M$ an $R$-module, $I$ an ideal in $R$, and $E$ an injective $R$-module. Can one claim that $H^i_I(E)=0;\forall i\gt 0$? 2.In the case of noetherian rings we know ...
0
votes
2answers
48 views

definitions of $I$-torsion functor $\Gamma_I$

Let $R$ be a commutative ring, $M$ an $R$-module and $I$ be an ideal in $R$. Bruns-Herzog, Brodmann-Sharp and many other authors define $I$-torsion functor $\Gamma_I$ as: $$\Gamma_I(M)=\bigcup_{n\in ...
0
votes
0answers
55 views

Galois extension over power series fields

Let $K$ be a field, and $L$ be an algebraic extension of $K$. I think it is known that if $T$ is a finite extension of $K((X))$, then $T$ is complete with respect to the $X$-adic valuation, hence if ...
1
vote
2answers
45 views

Ideals in $\mathbb{C}[x,y]/I$ where $V(I)$ is finite set.

$R = \mathbb{C}[x,y]$, $I \subset R$ ideal and $V(I)$ - is finite set. I want to prove that all prime ideals in $R/I$ are maximal. I know that it's true for $R/radI$. Let $\pi$ be natural map $R/I ...
1
vote
0answers
49 views

Showing an Ideal is Irreducible (alternative proof)

Recently, this question was posted regarding the following: Question: Show $$(x^3,y^5,z^2)\subset\mathbb{C}[x,y,z]$$ is an irreducible ideal. I was wondering if the following could be reviewed ...
1
vote
1answer
50 views

Showing an ideal is irreducible

I am currently trying to show that the ideal $\langle x^3, y^5, z^2 \rangle \subset \mathbb{C}[x,y,z]$ is irreducible (i.e.: it cannot be written as the intersection of two larger ideals $J$ and ...
5
votes
3answers
185 views

Hilbert function of a monomial ideal generated by degree two square free monomials

Let $R=K[x_1,...,x_n]$ be a polynomial ring over a field $K$ (one can assume $K$ is the field of complex numbers). Let $I=\langle m_1,...,m_l\rangle= \oplus I_j$ (where $I_j$ is $j^{th}$ graded piece ...
2
votes
1answer
50 views

Is every codimension one subvariety of a projective variety a set-theoretic complete intersection?

Let $X$ be a projective variety over $\mathbb C$ and $D\subseteq X$ some subvariety which is pure of codimension one. In fact, in my case $D$ is the complement of an open affine subvariety $U\subseteq ...
1
vote
1answer
22 views

Quotient of ideals of the ring of rational numbers with denominator prime to p.

Let $R_p=${ $\frac{m}{n} \in \mathbb{Q} $ | gcd(n,p)=1 } and consider the ideals of $R_p$ : $p^{\nu}R_p$ and $p^{3\nu}R_p$. Then $\frac{p^{\nu}R_p}{p^{3\nu}R_p}$ is a cyclic group of order ...
0
votes
2answers
55 views

A short question about the direct limits and direct sum of commutative rings.

By viewing a ring as a $\mathbb{Z}$-module, it is possible to define the direct limit of rings following the same procedure for modules. Let's suppose to work with commutative rings with unity. It ...