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

learn more… | top users | synonyms (1)

0
votes
0answers
96 views

Injective homomorphism of finite free modules and Nakayama's lemma [duplicate]

Let $f:E→F$ be a homomorphism of modules, finite over a local ring $A$. Assume that $E, F$ are free. Let $\mathfrak m$ be the maximal ideal, $f_{\mathfrak m}: E/\mathfrak mE→F/\mathfrak mF$ be the ...
0
votes
0answers
25 views

Monomial order with all weights equal?

Consider a set of $n$ polynomials $P_i$ with $i=1,2,...,n$ in variables $z_1,z_2,...,z_n\in\mathbb{C[\textbf{z}]}$. Furthermore, let all polynomials $P_i$ be completely homogeneous in all variables ...
2
votes
1answer
158 views

Is every ideal of $R$ a sum of a nilpotent ideal and an idempotent ideal?

I have a question concerning the following local ring: $$R=K[X_1,...X_n,...]/(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...).$$ Is every ideal of $R$ a sum of a nilpotent ideal and an idempotent ideal? ...
0
votes
0answers
20 views

Polynomial rings: if $A \otimes_B A$ is free over $A$, is it a complete intersection?

Let $A = k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic zero and $\{y_j\}_{1 \leq j \leq \ell}$ a family of homogeneous polynomials. Write $B$ for the subring ...
0
votes
0answers
35 views

$\mathcal{T}^i(X/Y,\mathcal{F})$ forms a sheaf

In Hartshorne's Deformation Theory, given an $A$-algebra $B$ and a $B$-module $M$, he defines these functors $T^i$ for $i=0,1,2$ that outputs $B$-modules $T^i(B/A,M)$. In Exercise 3.5, he asks the ...
0
votes
2answers
69 views

Ideal of $(u^3,u^2v,uv^2,v^3)$

Let $k$ be a algebraically closed field, consider $f: \mathbb A_k^2\rightarrow \mathbb A_k^4$ given by $f(u,v)=(u^3,u^2v,uv^2,v^3)$. Let $X=f(\mathbb A_k^2)$. Then how to determine $I(X)$? If I let ...
3
votes
1answer
90 views

An example from Lang's Algebra about primary ideal

On page 421 in Lang's Algebra, the author writes Let $R$ be a factorial ring with a prime element $t$. Let $A$ be the subring of polynomials $f(X)∈R[X]$ such that $$f(X)=a_0 + a_1X + \dotsb $$ ...
0
votes
1answer
38 views

Prime ideal in $R[x]$ lying above $P$ [duplicate]

I am trying to understand this proof, I think "$Q$ is lying above $P$" means $Q \cap P = R$, but I don't know why we can assume "$P=0$" or "$R$ is a field", can someone explain it to me?
1
vote
1answer
28 views

Valuation ring and integral closure

Let $A$ be a one-dimension local noetherian domain and suppose that we know that $K=\text{Frac}(A)$ is a complete discrete valuation field (valuations for me are surjective). Let's denote with ...
2
votes
1answer
21 views

Local subring of a DVR and finite residue field extension

Let $\mathcal O$ be a complete DVR with fraction field $K$, maximal ideal $\mathfrak p$ and residue field $\widetilde K=\mathcal O/\mathfrak p$. Now consider a subring $A\subset \mathcal O$ with the ...
1
vote
1answer
38 views

Geometric generic fibre

I have a question concerning the following exercise in Hartshorne: The inclusion of $k[s]$ into $k[s,t]/(s-t^2)$ induces a morphism of the corresponding affine schemes $X\to Y$. The exercise itself ...
-1
votes
2answers
68 views

Are polynomial rings finitely generated modules over the base ring?

Let $R$ be a commutative ring. Consider $R[X_1,...,X_n]$. Clearly it is a natural $R$-module. Is it true that $R[X_1,...,X_n]$ is a finitely generated $R$-module? If it is, then every its ideal is ...
3
votes
1answer
47 views

Nilpotent or just nil idempotent ideal?

Let $R$ be a ring with zero Krull dimension and $I$ be an idempotent ideal contained in the Jacobson radical $J(R)$ of $R$. Could one infer just with these hypotheses that $I$ is a nilpotent ideal? I ...
10
votes
1answer
109 views

Equality involving Hasse zeta function of commutative ring finitely generated over $\mathbb{Z}$

Let $\mathbb{F}_q$ be a finite field consisting of $q$ elements. Imitating Riemann's zeta function$$\zeta(s) = \sum_{n = 1}^\infty {1\over{n^s}},$$define$$\zeta_{\mathbb{F}_q[t]}(s) = \sum_f ...
1
vote
0answers
23 views

Injective $A$-homomorphism is also surjective? [duplicate]

Let $A$ be a ring and let $M$ be an Artinian module over $A$. Let $f: M \to M$ be an $A$-homomorphism. Assume $f$ is injective. Does it follow that $f$ is surjective? I'm inclined to think yes, and ...
3
votes
1answer
32 views

Is there a name for those elements $x$ of a commutative ring $R$ such that $Rx$ is maximal among all proper ideals?

Ever since learning basic ring theory, I've always felt kind of confused about the fact that: maximal ideals are prime (because every field is an integral domain), but irreducible elements needn't ...
2
votes
0answers
67 views

Visualize the affine to projective map $\mathbb{A}^{n+1}_k-\{O\}$ to $\mathbb{P}^n_k$

Classically, the natural map $\mathbb{A}_\mathbb{C}^{n+1}-\{O\} \rightarrow \mathbb{P}^n_\mathbb{C}$ maps every point in $\mathbb{A}_\mathbb{C}^{n+1}-\{O\}$ to the affine line that contains it. For ...
2
votes
0answers
46 views

Support of an effective Cartier vs Weil divisor

Let $X$ be a Noetherian locally factorial integral scheme so that $\operatorname{Ca}(X)\cong Z^1(X)$. An effective divisor could be thought of as a global section $D\in ...
3
votes
1answer
104 views

Intersection between two integral closures equals an algebraically closed field

Consider an algebraically closed field $k$, a finite field extension $K$ of $k(T)$, the integral closure $A$ of $k[T]$ in $K$, and the integral closure $A'$ of $k[T^{-1}]$ in $K$. Prove that $A ...
1
vote
1answer
52 views

Localization of a valuation ring at a prime is abstractly isomorphic to the original ring

Let $A$ be an integral domain with quotient field $K$. We say that $A$ is a valuation ring if for any $0 \neq x \in K$, $x$ or $\frac{1}{x}$ lies in $A$. Then $A$ is necessarily a local ring. If ...
3
votes
0answers
87 views

Geometric interpretation of algebraic property

Let $X \subset \mathbb{C}^n$ be an irreducible affine algebraic curve with coordinate ring $$\mathbb{C}[X] = \mathbb{C}[z_1, \ldots, z_n] / (f_1, \ldots, f_m ) $$ with each $f_i \in \mathbb{Z}[z_1, ...
0
votes
1answer
34 views

Can a valuation ring properly contains another valuation ring with the same field of fractions?

Definition of valuation ring: Let $R$ be an integral domain with $frac(R)=K$. Then $R$ is said to be a valuation ring if (1) $R \neq K$ (2) $\forall x \in K, x \in R$ or $x^{-1} \in R$. Now my ...
1
vote
2answers
45 views

Sum of Ideals of the Same Type

I have two questions: 1) Is a finite sum of idempotent ideals of a ring $R$ idempotent? 2) Is any sum of nil ideals of a ring $R$ nil? As far as I know, a finite sum of nil ideals of a commutative ...
2
votes
1answer
36 views

Is the tangent bundle of a smooth $\Bbbk$-scheme an infinitesimal extension of it?

An infinitesimal extension of an affine scheme $\operatorname{Spec}R$ is a surjection $\hat R\twoheadrightarrow R$ with nilpotent ideal. The scheme case is defined by globalizing. I read somewhere on ...
2
votes
1answer
41 views

When is the prime spectrum of a ring the finite complement topology on a set?

I have been studying the prime spectrum of different rings recently, and I have noticed that for many "nice" infinite rings, the prime spectrum is precisely the finite complement topology on some set ...
0
votes
2answers
42 views

If $I,J$ are ideals in a polynomial ring over a field, how do I prove that $I = J$ if $\operatorname{in}_<(I)=\operatorname{in}_<(J)$?

If $I\subseteq J$ are ideals in a polynomial ring of $n$ variables, how do I prove that $I = J$ if $\operatorname{in}_{\lt}(I)=\operatorname{in}_{\lt}(J)$, where $\lt$ is any monomial ordering? ...
3
votes
1answer
35 views

completion and heights of prime ideals

Let $A$ be a noetherian, regular local domain of dimension $2$ (for instance the local ring at a smooth point of a surface) and consider its completion $\hat A$ at its maximal ideal. Now let's look at ...
0
votes
0answers
63 views

When is a map $ f : A^{n+1} \to A^n $ injective? [duplicate]

Is there an example of a commutative ring $ A $ with unit $ 1_A $ and an $A$-module map : $ f : A^{n+1} \to A^n $ such that $f$ is injective ? If the answer is yes, when is this enunciation false ? ...
0
votes
1answer
62 views

Commuting square valuation ring — morphism of schemes

Let $f\colon X\to Y$ be a morphism of schemes. For a fixed $x\in X$, let $K=k(x)$. If $y'$ is a specialization of $y=f(x)$, we may choose a valuation ring $R$ of $K$ dominating ...
1
vote
1answer
34 views

How can I show that the Zariski topology is the discrete topology for any finite field?

In other words, that every set is clopen. I know that the closed sets in the Zariski toplogy are the zero sets of sets of polynomials, so they could be points, curves (polynomials, hyperbolas, ...
4
votes
1answer
65 views

A proposed criterion for finding when an homogenous ideal is radical

Let $X$ be a projective variety over an algebraically closed field, and $I$ be the homogenous ideal of $X$ and $J$ be an ideal with the same zero set. Suppose that I know $I=\langle f_1,...f_n ...
1
vote
0answers
36 views

showing that the union and intersection of two affine algebraic sets is still an affine algebraic set

In particular, if R and S are sets of polynomials over a field, then the sets of points where polynomials of R and S are simultaneously zero are Z(R) and Z(S), respectively. Then: Z(R) $\cap$ ...
2
votes
2answers
95 views

When a graded ring is Cohen-Macaulay?

I am trying to solve exercise 19.10 from Eisenbud's Commutative Algebra. I want to show that if $R=k[x_0,...,x_n]/I$ is a graded ring, then $R$ is Cohen-Macaulay iff $R_{\mathfrak p}$ is ...
0
votes
1answer
188 views

radical membership and ideal membership [closed]

Consider the ideal $I=(x^3y-x^2y^2,x^3z+z^2yx,x^2-xz)\subset \Bbb Q[x,y,z].$ Is $x\in I?$ Is $x\in \sqrt I?$ I'm assuming a question like this is quite simple and that there is just a method, if ...
2
votes
1answer
68 views

Structure sheaf of affine variety consists of noetherian rings

Let $X\subseteq \mathbb{A}^n$ be an affine variety. The local ring of $X$ at $p\in X$, given by $\mathcal{O}_{X,p}=\{f\in k(X):f \text{ regular at } p\}$ is noetherian because it is a localization of ...
1
vote
2answers
42 views

Support of a tail of a graded module.

Suppose that $R$ is a non-negatively, graded commutative ring. I have been trying to decide if the following is true for a graded $R$-module $M$ (not necessarily finite over $R$): $$\text{Supp}_R ...
1
vote
1answer
29 views

Does a free $R$-module of countable rank ever arise as a (double) dual of some $R$-module?

The usual counterexample to all vector spaces being canonically isomorphic to their double duals goes something like this: if $F$ is a field, take $F^{\omega}$. Since contravariant hom takes colimits ...
3
votes
0answers
25 views

generating radical of an ideal with small degree polynomials

Say $f_1,f_2,\ldots,f_n$ are a finite set of polynomials in $M$ variables $x_1,x_2,\ldots,x_M$ over a field $k$. Say each $f_i$ has total degree at most $D$. If $I$ the ideal generated by the ...
3
votes
0answers
37 views

When is a polynomial contained in the ideal generated by its partial derivatives?

Let $R = k[x_1,\dots,x_n]$ be a multivariate polynomial ring over a field $k$ of characteristic zero, and let $f\in R$. Is there an easy-to-test necessary and sufficient condition on $f$ such that ...
-3
votes
2answers
42 views

The equivalent conditions for commutative rings with a unique prime ideal [closed]

In a commutative ring $R$ the followig conditions are equivalent. (1) $R$ has a unique prime ideal (2) every nonunit is nilpotent (3) $R$ has a minimal prime ideal which contains all zero divisors, ...
0
votes
0answers
41 views

Localization of $\Bbb k[x]$

Let $R= \Bbb K [x]$ and $S=\{x^n: n \in \Bbb Z, n \geq 0 \} $. Let $D$ be the localization of $R$ in $S$, that is $D = S^{-1}R = \{ \frac{r}{s}: r \in R, s\in S \}$. By using the Universal Property ...
0
votes
0answers
47 views

If $A$ is an integrally closed domain, then is $GrSym A$ (the graded symmetric algebra on $A$) also integrally closed?

Question: If $A$ is an integrally closed domain, which is f.g. k algebra over an algebraically closed field, then is $A'$ (defined in the edit below) also integrally closed? (I am thinking about this ...
1
vote
1answer
30 views

Dimension of an affine scheme, 24.5.7 of Vakil's book

In the remark 24.5.7 of Vakil's book, it claims that the dimension of the scheme $\operatorname{Spec}k(x) \otimes k(y)$ is a $k(x)$-scheme with dimension one, where $k$ is a field and $x,y$ are two ...
4
votes
0answers
65 views

Why are Unique Factorization Domains (UFD's) geometrically significant?

We know that for $A$ a UFD, it's class group is trivial. More generally, for a factorial (stalks are UFD's) scheme $X$ (that is also noetherian and normal), we have an isomorphism between it's Picard ...
3
votes
0answers
51 views

$T^i$ functors in Hartshorne's Deformation Theory

In chapter 3 of Hartshorne's Deformation Theory, he defines functors $T^i$ for $i=0,1,2$ that take as input a ring homomorphism $A\rightarrow B$ and a $B$-module $M$ and outputs $T^i(B/A,M)$, a ...
0
votes
1answer
50 views

Flat schemes over artinian local ring with isomorphic special fibers

I'm sure this is standard, but I don't know where to find it. If $A$ is a local artinian $k$-algebra, $X_1,X_2$ are finite type schemes flat over $A$, and $f:X_1\rightarrow X_2$ is a morphism over ...
1
vote
1answer
27 views

Zero divisors in a finite dimensional Poincaré duality algebra

Let $A= \bigoplus_{i=0}^{n}A_i$ be a finite dimensional algebra over a field $\mathbb{k}$ such that $A_0 \cong \mathbb{k} \cong A_n$. Consider the bilinear form $$\varphi: A_i \times A_{n-i} \to ...
1
vote
0answers
35 views

Questions about flat limits and associate points, Vakil's section 24.4.12

Suppose $(A,m)$ is a discrete valuation ring and $[m]$ is the closed point of $Spec\,A$ and $\eta$ is its generic point, which is also the only nontrivial open set. If we have a morphism $\pi:X ...
0
votes
0answers
20 views

Finitely graded ring and zero divisors

Let $R= \bigoplus_{i=0}^{d} R_i$ be a finitely graded ring such that $R_0$ is a field and $R_d \cong R_0$. I'm trying to understand how zero divisors work in such a ring; when is it true that for $r ...
0
votes
0answers
35 views

Intersection multiplicity inequality problem

I have a question about an inequality that was stated in my class but never proven. We stated p is simple $I_p(F \cap G+H) \ge \min ( I_p (F \cap G) , I_p(f \cap H))$. Where we defined $I_p(F \cap ...