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

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4
votes
1answer
58 views

Proof of the Artin-Rees lemma

I am struggling to understand a key step in a proof of the Artin-Rees lemma, which I have put in a red box below. I don't really see how we can pass from a finite direct sum to an infinite one. I've ...
0
votes
0answers
19 views

associated prime of a module

Let $f: A\rightarrow B$ be a homomorphism of Noetherian rings, and $M$ a $B$-module. Question: Is $^af(Ass_B(M))=Ass_A(M)$? If $q$ is an associated prime of the $B$-module $M$, $p=^af(q)$, then from ...
2
votes
1answer
36 views

Zeroes of prime polynomials in the algebraic torus (A Hilbert's Nullstellensatz for Laurent polynomials?)

Let $Q\in\mathbb C[z_1,\dots,z_D]$ be a prime polynomial and let $Z(Q)$ be the algebraic hypersurface of its zeroes. Assume that $P\in\mathbb C[z_1,\dots,z_D]$ is a polynomial which has zeroes at ...
3
votes
1answer
39 views

Algebraic closure for rings

Is there any notion of algebraic closure for commutative rings? I am specifically interested in such a concept for $\mathbb Z_n$, with $n$ not a prime (possibly square-free). Such a concept would be ...
1
vote
1answer
25 views

$F(t)$ as an $F[t]$-algebra and the Weak Nullstellensatz

Sorry if this question has already been answered somewhere, but it's quite hard to find if so, because of the use of the word 'algebra' in the question... In the lead up to a proof of the Weak ...
5
votes
3answers
126 views

Basic application of the Nullstellensatz

Background: I have just started learning basic algebraic geometry. My solution to a simple problem involves an application of the Nullstellensatz and I want to know whether this is overkill (or ...
3
votes
0answers
46 views

Is every reduced $k$-algebra all of whose residue fields are $k$ finitely generated?

Let $k$ be a field (of characteristic zero if you want). Let $A$ be a reduced $k$-algebra with the property that for every prime ideal $\mathfrak{p}$ of $A$ the natural homomorphism $k \to A/ ...
1
vote
0answers
22 views

morphisms of curves and discrete valuation rings

Given a dominant morphism $\varphi\colon C\to C'$ of curves, a nonsingular point $Q\in C'$, such that $\varphi^{-1}(Q) = \{P_1,\ldots, P_m\}$ consists of nonsingular points only. Then it is clear to ...
-1
votes
0answers
52 views

Is this sheaf simple?

Let $S$ be a surface and $C$ be an effective divisor in $S$. That is $C$ is a curve in $S$ and $i:C\longrightarrow S$ is the inclusion morphism. Let $E$ be line bundle over $C$, so $i_*E$ is a ...
8
votes
3answers
247 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
110 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
46 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 ...
0
votes
1answer
35 views

Flat Module finitely generated when over the residue field finite dimensional? [on hold]

Let $(A, \mathfrak{m})$ be a local ring with residue field $\kappa=A/ \mathfrak{m}$. Let $M$ be a flat $A$-module. Assume that $M \otimes_A \kappa$ is a finite dimensional $\kappa$-vector space. Is it ...
5
votes
1answer
88 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 ...
4
votes
1answer
26 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 ...
0
votes
0answers
38 views

Dimension of irreducible components of variety [on hold]

Consider the affine variety $X=\{ (a_1,a_2,a_3,b_1,b_2,b_3) \in \mathbb{C}^6 \mbox{ : }a_1b_2=a_2b_1, a_1b_3=a_3b_1 \}$. Prove that $X$ has two irreducible components, and that both of them are of ...
2
votes
0answers
81 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 ...
4
votes
0answers
41 views
+100

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

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 $R$, ...
3
votes
1answer
49 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
32 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
58 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 ...
0
votes
0answers
19 views

Minimal graded free resolution of $R/I\oplus R/J$ in terms of minimal graded free resolution of $R/I$ and $R/J$.

Let $R=k[x_1,...,x_n]$ be a graded ring over a field. Let $I,J$ be homogeneous ideals. Questions: What is the minimal graded free resolution of $R/I\oplus R/J$ (in terms of minimal graded free ...
3
votes
2answers
51 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 ...
2
votes
1answer
55 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 ...
1
vote
1answer
44 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
54 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
61 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 $A_{\text{red}} \to K$ ...
-1
votes
0answers
76 views

Closedness and going up property: an explicit proof.

Suppose that a ring homomorphism $f: A \to B$ satisfies the going up property (i.e. for any $\mathfrak{q} \in\operatorname{Spec}(B)$ and any $\mathfrak{p}'\in \operatorname{Spec}(A)$ containing ...
1
vote
1answer
25 views

Help with computation and Groebner basis

Hi guys I am learning a new software and a new topic (groebner basis) I have this problem $6-21(x_1x_2+x_1x_3+x_1x_4)=0$ $10-21(x_2x_1+x_2x_3+x_2x_4)=0$ $12-21(x_3x_1+x_3x_2+x_3x_4)=0$ ...
0
votes
0answers
24 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
30 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
49 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
31 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
33 views

Localization of an integral A-algebra is not always integral.

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 $\mathfrak{m}$ ...
0
votes
1answer
66 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
54 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
95 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
36 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
40 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
17 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
27 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
43 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
46 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 ...
7
votes
0answers
83 views
+50

$p \in C - D$, 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
34 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
45 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
22 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 ...
2
votes
2answers
70 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
26 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
32 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 ...