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

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6
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1answer
152 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 ...
6
votes
1answer
237 views

A commutative ring in which every prime ideal is 2-generated

Suppose $R$ is a commutative ring with 1. There are some statements that tells us if prime ideals behave in certain way, then all the ideals will behave in that way. For example, If every prime ...
6
votes
1answer
226 views

Modules $M$ such that the automorphism of $M \otimes M \otimes M$ induced by the permutation $(123)$ is the identity

I've been struggling with the following problem for a couple of days and I don't seem to get any further: Let $R$ be a commutative ring. I would like to get (something like) a classification of all ...
6
votes
2answers
1k views

Rank of free submodules of a free module over a commutative ring. [duplicate]

Free modules over a commutative ring $R$ with $1$ have well-defined rank. I have been wondering if there is a ring $R$ such that there are free modules $M'\subset M$ with $\operatorname{rank}(M')&...
6
votes
1answer
334 views

Are minimal prime ideals in a graded ring graded?

Let $A=\oplus A_i$ be a graded ring. Let $\mathfrak p$ be a minimal prime in $A$. Is $\mathfrak p$ a graded ideal? Intuitively, this means the irreducible components of a projective variety are ...
6
votes
2answers
234 views

homomorphisms of $C^{\infty}(\mathbb R^{n})$

Let $F: \mathbb R^{n} \to \mathbb R^{m} $ be a smooth map, then we have homomorphism of algebras $F^{*}: C^{\infty}(\mathbb R^{m}) \to C^{\infty}(\mathbb R^{n})$. Is it true that any homomorphism of ...
6
votes
2answers
201 views

Is an ideal which is maximal with respect to the property that it consists of zero divisors necessarily prime?

This is in follow-up to this question. Let $R$ be a commutative ring with identity and consider the set $Z \subset R$ of zero divisors. If the ideal $I\subset Z$ is maximal with respect to the ...
6
votes
1answer
133 views

Embedding of free $R$-algebras

Let $R$ be any nontrivial commutative unital ring and $I$ and $J$ any sets with $|I|>|J|$. Does there exist an embedding of $R$-algebras $R[x_i; i\in I]\longrightarrow R[y_j;j\in J]$? When $R$ ...
5
votes
1answer
130 views

Projective resolution of $k$ over $R=k[x,y]/(xy)$

I want to prove that $\operatorname{Tor}_{n}^{R}(k,k)=k\oplus k,\,\,\forall n\ge 1$. I found the projective resolution $$ R^4\stackrel{d_2} \longrightarrow R^3\stackrel{d_1} \longrightarrow R^2\...
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
91 views

Finite Projective Dimension implies non vanishing Ext

Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$? Can't we write the free module as a direct ...
5
votes
2answers
223 views

a flatness criterion

I'm having trouble with part (b) of Exercise 10.5.25 from Dummit & Foote (the goal of the problem is to prove that $A$ is a flat $R$-module iff $A\otimes_R I\to A\otimes_R R$ is one-to-one for all ...
5
votes
1answer
1k views

Finite morphisms of schemes are closed

I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely: Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of ...
5
votes
0answers
610 views

Integral homomorphism induces a closed map on spectra

I'm trying to prove the following: Let $f:A\rightarrow B$ be an integral homomorphism (e.g. $B/f(A)$ is a integral extension). Consider $f^{*}: \operatorname{Spec}B \rightarrow \operatorname{Spec}...
5
votes
2answers
273 views

Localization Notation in Hartshorne

This is a question about notation in Hartshorne's Algebraic Geometry. According to my understanding $k[x_0,\cdots,x_n]_{(x_i)}$, (see e.g. page 18), consists of the elements of degree zero in the ...
5
votes
3answers
926 views

Generators for the radical of an ideal

I am interested in finding a generating set of the radical of an ideal given a set of generators for the ideal itself, but after a lot of thought I cannot figure out a good way to do it. Specifically: ...
5
votes
1answer
388 views

Hilbert's Nullstellensatz without Axiom of Choice

Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951. Can we prove the following theorem without Axiom of Choice? Theorem Let $A$ be a commutative ...
5
votes
1answer
183 views

Dimension inequality for homomorphisms between noetherian local rings

$A$ and $B$ are commutative noetherian local rings with maximal ideals $\mathfrak{m}$ and $\mathfrak{n}$ respectively. If $f\colon A \to B$ is a local ring homomorphism, how do I prove the ...
5
votes
1answer
172 views

The local cohomology modules are Artinian

Let $(R,m,k)$ be Noetherian local ring and $M$ a finitely generated $R$-module. Lemma 3.5.4 of Bruns-Herzog states that the local cohomology modules $H^i_m(M)$ are Artinian and that this ...
5
votes
2answers
388 views

Computing the radical of an ideal

What is the best way to compute $\sqrt{(X^2-YZ,X(1-Z))}$ ? This is after using Nullstellensatz by the way as I thought it would be easier to compute a radical than finding the vanishing ideal.
5
votes
1answer
373 views

Is the integral closure of a Henselian DVR $A$ in a finite extension of its field of fractions finite over $A$?

This question is related to the one here: A question related to krull akizuki In the answers to that question, some examples are given of a discrete valuation ring $A$ and a finite (necessarily ...
5
votes
1answer
128 views

Stable epimorphisms of commutative rings

Recall that an epimorphism $f : A \to B$ in a category with fiber products is called stable (or universal) if for every morphism $C \to B$ the base change $A \times_B C \to C$ is an epimorphism. ...
5
votes
1answer
737 views

Relation between primary ideal and prime ideal

We know that every prime ideal is primary ideal. But can we say, every primary ideal is a power of prime ideal? if it is not correct a counterexample. Thanks.
5
votes
2answers
134 views

Divisors of differentials.

Let $\mathbb{C}$ be the base field. Suppose $C \subset \mathbb{P}_2$ is a nonsingular projective curve of degree $d > 3$. Must it be the case that $D \in \text{Div}(C)$ is the divisor of a ...
5
votes
1answer
461 views

The ring of germs of functions $C^\infty (M)$

Define $C^\infty (M)_x := \{ (U,f) | x \in U $ open $ , f \in C^\infty (U) \} / \sim $ where $M$ is a manifold and $(U,f) \sim (V,g)$ if $\exists W$ open, $x \in W$ such that $W \subset V \cap U$ ...
5
votes
1answer
278 views

Generalizing Artin's theorem on independence of characters

Artin's theorem says that for any field $K$ and any (semi) group $G$, the set of homomorphisms from $G$ into the multiplicative group $K^*$ is linearly independent over $K$. Can this theorem be ...
4
votes
1answer
296 views

Example of non-noetherian algebras which are tensor products of noetherian algebras

We suppose all rings are commutative with unity. I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian $A$-...
4
votes
1answer
95 views

minimal primes of a homogeneous ideal are homogeneous

I am trying to study the proof of this result. It appears as part 3 of the proposition on page 2 of the following document http://math.mit.edu/classes/18.721/projgeom6.pdf I understand everything but ...
4
votes
0answers
144 views

Transversal and complete intersection of hypersurfaces in $\mathbb{P}^{n}$

(a) Let $k<n$ and $F_{1},\dots,F_{k}$ be general homogeneous polynomials of degrees $d_{1},\dots,d_{k}$ in $n+1$ variables. Prove that the corresponding hypersurfaces in $\mathbb{P}^{n}$ ...
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160 views

Why is the completion of the ring of germs of smooth functions $\cong \mathbb{R}[|T|]$?

Let $C^{\infty}$ be the canonical commutative ring on the set $\{f: \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ smooth}\}$. Let $\mathfrak{m}= \{ f \mid f(0)=0 \}$ a maximal ideal. Consider the ...
4
votes
2answers
562 views

Homogeneous ideal and degree of generators

Let $I$ be a homogeneous ideal in a graded local commutative ring $R$, $S$ be its minimal homogeneous system of generators. So, we know that the cardinality of $S$ is unique as the dimension of the ...
4
votes
2answers
470 views

Are projective modules over an artinian ring free?

Quoting a comment to this question: By a theorem of Serre, if $R$ is a commutative artinian ring, every projective module [over $R$] is free. (The theorem states that for any commutative ...
4
votes
1answer
62 views

Does $S = R \cap K$ of a field extension $K \subseteq L = Q(R)$ satisfy $Q(S) = K$?

If $K$ is finite field, then one can easily show that there is no proper subring $R$ with $Q(R) = K$, where $Q(R)$ is the field of fractions of $R$. As a consequence, algebraic extensions $K$ of ...
4
votes
1answer
36 views

Image of element is square of an element, precisely two maximal ideals satisfying condition.

Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us look at the ring $\mathbb{F}_q[x, \sqrt{f}]$....
4
votes
2answers
133 views

$Z(I:J)$ is the Zariski closure of $Z(I)-Z(J)$

Let $(I:J)$ denote the colon ideal (or ideal quotient). It is pretty clear that the Zariski closure of $Z(I)-Z(J)$ is contained in $Z(I:J)$. How can we prove that the the Zariski closure of $Z(I)-Z(J)$...
4
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0answers
63 views

A UFD for which the related formal power series ring is not a UFD

I know that the proposition $$ A \text{ is a UFD } \Rightarrow A[[X]] \text{ is a UFD }$$ is false. Wikipedia states that if $B=K[x,y,z]/(x^2+y^3+z^7)$ then $A=B_{(x,y,z)}$ is a counterexample, but I ...
4
votes
3answers
95 views

ACC on principal ideals implies factorization into irreducibles. Does $R$ have to be a domain?

I am following an argument in chapter zero of Eisenbud's Commutative Algebra book. It is not clear whether or not he is assuming that $R$ is a domain. If I start the proof assuming $R$ is not ...
4
votes
2answers
186 views

Can $\operatorname{Spec}(R[X])$ ever be finite?

I have a quick question. Suppose $R$ is a nonzero commutative ring. Is it possible that $|\operatorname{Spec}(R[X])|<\infty$?
4
votes
0answers
123 views

When will $A[x_1, \ldots, x_n]$ satisfy the dimension formula?

What property should $A$ satisfy so that $A[x_1, \ldots, x_n]$ satisfies the dimension formula, $$\mathrm{dim}(A[x_1, \ldots, x_n]) = \mathrm{dim}(A[x_1, \ldots, x_n]/\mathfrak{p}) + \mathrm{ht}(\...
4
votes
1answer
302 views

In a Noetherian ring every non-zero prime ideal is invertible implies every non-zero proper ideal is invertible.

Suppose that $R$ is a Noetherian integral domain. How to show that if every non-zero prime ideal of $R$ is invertible, then every non-zero ideal of $R$ is invertible? Actually, I am trying to prove ...
4
votes
1answer
148 views

What is the injective hull of a polynomial ring?

The injective hull of a polynomial ring in one variable $K[X]$ (where $K$ is a field) is $K(X)$ since $K(X)$ is a divisible hence injective $K[X]$-module (since $K[X]$ is a PID) and $K(X)$ is an ...
4
votes
2answers
105 views

Power set representation of a boolean ring/algebra

Let $R$ be a finite boolean ring. It's known that there's a boolean algebra/ring isomorphism $R\cong \mathcal P(\mathsf{Bool}(R,\mathbb Z_2))$. I'm trying to get a feel for this. The subsets of $\...
4
votes
2answers
209 views

Is every local ring a valuation ring?

Is every local ring a valuation ring? I know the answer is no and the first example comes to my mind was as following (I started with smallest fields, as $\mathbb{Z}_2$ and $\mathbb{Z}_3$ are not ...
4
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0answers
135 views

Tensor product, Artin-Rees lemma and Krull intersection theorem

I asked another question about tensor product, but can't conclude from the answer, so here is another more concrete question. Let $(A,m)$ be a local ring then by Artin-Rees Lemma $m^k \bigcap I \...
4
votes
3answers
2k views

irreducibility of a polynomial in several variables over ANY field

The irreeducibility of a polynomial $f\!\in\!K[x_1,\ldots,x_n]$ in general depends on what the field $K$ is (for example, if $K=\mathbb{R}$, then $f=x_1^2+1$ is irreducible, but if $K=\mathbb{C}$, it ...
4
votes
2answers
180 views

An overring of a polynomial ring, noetherian or not?

Let $S$ be a commutative domain and let $k$ be a subfield of $S$. Let $R:=k[x,y] \subseteq S$ be the polynomial ring in two variables $x,y$ and suppose that for every $s \in S$ there exists some $0 \...
4
votes
1answer
91 views

Finite intersection of DVRs

Let $K$ be a field and $R_1,\dots,R_n$ DVRs of $K$ with $m_i$ the maximal ideal of $R_i$ and $R_i \not\subseteq R_j$ for $j\neq i$ . Define $A=\bigcap_{i=1}^n R_i$. Then $A$ is semilocal with maximal ...
4
votes
1answer
313 views

The completion of a noetherian local ring is a complete local ring

We have defined the completion of a noetherian local ring $A$ to be $$\hat{A}=\left\{(a_1,a_2,\ldots)\in\prod_{i=1}^\infty A/\mathfrak{m}^i:a_j\equiv a_i\bmod{\mathfrak{m}^i} \,\,\forall j>i\right\}...
4
votes
1answer
388 views

Computing the local ring of an affine variety

Let $W=V(y^{2}-x^{3}) \subseteq \mathbb{A}^{2}$ and $k$ algebraically closed. Clearly the dimension of the tangent space at the origin is $2$. I want to compute this using the definition the fact that ...
4
votes
1answer
215 views

Rees algebra of a monomial ideal

User fbakhshi deleted the following question: Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and $I=(f_1,\ldots,f_q)$ a monomial ideal of $R$. If $f_i$ is homogeneous of degree $d\...