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

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3
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0answers
17 views

$m_p=\{f\in \mathcal{O}_{V,p}| f(p)=0\}$, ideal of $p$ in the local ring. What is $m_p/m_p^2$?

In Section 6.8 of Undergraduate Algebraic Geometry by Reid, the author proved the following Theorem: There is a natural isomorphism of vector spaces $(T_pV)^*\cong m_p/m_p^2$ where $^*$ denotes ...
0
votes
2answers
18 views

properties of homogeneous localization

Let $k$ be a field, $S = k[x_0,\dots,x_r]$, $I$ a homogeneous ideal of $S$ and $R=S/I$. Let $P$ be a homogeneous prime ideal of $R$ and let $R_{(P)}$ be the homogeneous localization of $R$ at $P$. I ...
1
vote
0answers
26 views

Matrices representing a map between free modules of infinite rank and Fitting's Lemma (Eisenbud)

p.497 of Commutative Algebra with a View Toward Algebraic Geometry, Eisenbud: If $\phi: F \rightarrow G$ is a map of free modules, then $I_j\phi$ is the image of the map $$\Lambda^j F ...
2
votes
1answer
46 views

Is this ring extension flat?

Let $k$ be a field of characteristic zero and let $A$ be a finitely generated $k$-algebra. Let $B=A[x_1,\ldots,x_n]$ be the polynomial ring over $A$ and let $I \subseteq B$ be an ideal such that $B/I$ ...
-1
votes
0answers
36 views

Monomial Algebras problem : lcm [on hold]

I am trying to prove the following: Monomial Algebras (Second Edition) Rafael H. Villarreal - Exercise 6.1.22 Let I and J be two ideals generated by finite sets of monomials F and G, respectively, ...
1
vote
0answers
9 views

Algorithm for computing the inverse limit of a finite inverse system

Let $k$ be a field (finite if you'd like), let $(I,\le)$ be a finite directed poset with $|I|=n$, and let $(A_i,f_{ij})_{i\le j\in I}$ be an inverse system of finitely generated, graded, commutative ...
1
vote
0answers
14 views

Algorithm for computing an inverse image

Let $k$ be a field (finite if you'd like), and let $f:A\to B$ be a map of graded, commutative $k$-algebras. Suppose further that $A$ is finitely generated and choose a presentation ...
0
votes
0answers
34 views

Monomial Algebras problem: associated prime ideals

I am trying to prove the following: Monomial Algebras (First Edition) Rafael H. Villarreal - Exercise 1.1.45 or Monomial Algebras (Second Edition) Rafael H. Villarreal - Exercise 6.1.26 Let ...
-1
votes
1answer
43 views

How to use the Universal Coefficient Theorem to determine $H^i(M; \mathbb{Z}_p)$ from $H^i(M; \mathbb{Z})$? [on hold]

Let $M$ be a path-connected finite $CW$-complex. Suppose $$ H^2(M;\mathbb{Z})=\mathbb{Z}_{2k}, \text{ } k\geq 3; $$ $$ H^3(M;\mathbb{Z})=\mathbb{Z}\times\mathbb{Z}_{2}; $$ $$ ...
0
votes
0answers
18 views

Intersection of affine subvarieties [on hold]

If the ideals $I_i$ define irreducible subvarieties of an affine space, can the scheme defined by the ideal generated by finitely many of the $I_i$ contain a embedded component?
0
votes
0answers
32 views

Every non-Noetherian module has a submodule maximal with respect to being not finitely generated. [duplicate]

Let $M$ be a module. Show that if $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated whenever $N<A\leq M$. The question is related to If $M$ isn't ...
0
votes
0answers
27 views

Completion of a polynomial ring [on hold]

Let $R$ be a commutative ring with ideal $I$. Let $J$ be the ideal of $R[x]$ generated by $I$ and $x$. What is the $J$-adic completion of $R[x]$? Is it $S[[x]]$, where $S$ is the $I$-adic ...
4
votes
1answer
46 views

Closed points in projective space correspond to which homogenous prime ideals in $k[x_0,…,x_n]$

I'm trying to think about exercise 4.5.O in Vakil's notes on Algebraic Geometry. Before we defined the scheme $\mathbb{P}^n_k := \operatorname{Proj}(k[x_0,...,x_n])$ and showed that that for $k$ ...
4
votes
2answers
94 views

Does $\operatorname{Spec}$ preserve pushouts?

The spectrum-functor $$ \operatorname{Spec}: \mathbf{cRng}^{op}\to \mathbf{Set} $$ sends a (commutative unital) ring $R$ to the set $\operatorname{Spec}(R)=\{\mathfrak{p}\mid \mathfrak{p} \mbox{ is a ...
1
vote
1answer
70 views

When is an ideal also a ring, and could then be anything said about its relation to the original ring

If $R$ is a ring with unity $1$, then $S \subseteq R$ is called a subring if it is itself a ring with $1 \in S$. A subset $I \subseteq R$ is called an ideal if it is a group with respect to addition ...
-1
votes
0answers
43 views

Field of fractions of $\mathbb{C}[X,Y,Z]/\langle XY-Z^2\rangle$ [on hold]

Let $R=\mathbb{C}[X,Y,Z]/\langle XY-Z^2\rangle$ and let $\mathbb{C}(X,Y)$ be the field of fractions of $\mathbb{C}[X,Y]$. Show that the field of fractions of $R$ can be expressed as ...
0
votes
1answer
44 views

If ideal can be generated by zero divisors, then is the depth of the ideal 0?

Let $R$ be a Noetherian ring and $I$ an $R$-ideal. The number $\operatorname{depth}_I R$ is the length of maximal $R$-regular sequence in $I$. It is well-known that If $\operatorname{depth}_I R = 0$, ...
0
votes
0answers
17 views

Does Magma let you specify primary invariants?

I am cross-posting this question from scicomp.SE. The computer algebra system Magma can calculate primary invariants (i.e. a homogeneous system of parameters) in an invariant ring of a finite group ...
1
vote
1answer
35 views

Example of an integral domain with a non-principal prime ideal of height one [on hold]

Is there an integral domain $R$ with a prime ideal $\mathfrak{p}$ of height $1$ which is not a principal ideal?
1
vote
0answers
24 views

Does the order of a finite group divide the product of degrees of a system of parameters of the invariant algebra?

Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb{F}$, and let $G$ be a subgroup of the finite group $\operatorname{GL}(V)$. Then $G$ acts on the graded algebra $\mathbb{F}(V)$ ...
7
votes
1answer
68 views

Are weakly étale ring homomorphisms of finite presentation étale?

Following [Stacks, 092A], say a ring homomorphism $A \to B$ is weakly étale if both $A \to B$ and $B \otimes_A B \to B$ are flat. Question. Are weakly étale ring ...
0
votes
0answers
70 views

Why is the map from $A^n$ to $M$ a surjective homomorphism?

How can one do the problem 1.3.11 b in Algebraic Geometry and Arithmetic Curves? I have read basics of commutative algebra but this one seems to be too difficult. Let $A$ be a commutative ring with ...
8
votes
1answer
58 views

Is I-adic completion a ring epimorphism?

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For any $n \ge 1$, the ring homomorphism $R \rightarrow R/I^n$ is surjective, hence an epimorphism in the category of rings. What about ...
0
votes
2answers
63 views

Affine varieties and their ideals

I was reading on Wikipedia about quotient ideals. It mentions that if $W$ and $V$ are affine varieties (assume $V$ is) and $I(V)$ and $I(W)$ are the ideals for $V$ and $W$, then $$I(V):I(W) = ...
1
vote
1answer
39 views

$>$ is an elimination ordering for $x_1,\dots,x_t \iff x_i >x_j^m$

Let $R = k[x_1,\dots,x_t,x_{t+1},\dots,x_n]$ and $>$ a monomial ordering on $R$. Then $>$ is an elimination ordering for $x_1,\dots,x_t \iff x_i >x_j^m$ for all $1\leq i \leq t, t+1 \leq j ...
0
votes
1answer
58 views

Finite number of maximal ideals of bounded norm

Suppose that we have an integral extension of rings $R\subseteq S$ and $S$ is finitely generated as $R$-module or as $R$-algebra, and $R/\mathfrak m$ is finite for all maximal ideals and $S/\mathfrak ...
1
vote
1answer
50 views

There exists a zero dimensional ideal I such that $\dim (R/I) - |V(I)| \geq \alpha > 0$

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). I know that ...
0
votes
1answer
21 views

Poincare series and the Hilbert polynomial of $A = A_0[X_1,\dots , X_s]$

Let $A = A_0[X_1, \dots , X_s]$ be a polynomial ring in $s$ variables over an Artin ring $A_0$. This is a graded ring, and can be regarded as a graded module over itself. 1. What are the ...
2
votes
1answer
28 views

Why is $|V(I)| \leq d_1\cdots d_n$?

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). Then if $G$ is a ...
1
vote
1answer
19 views

Poincare series and Hilbert polynomial of some graded modules

Let $k$ be a field, and let $k[X, Y ]$ be the polynomial ring in two variables equipped with the usual grading such that $\deg(X) = \deg(Y ) = 1$. Consider the ideals $I = (X, Y^2)$ and $J = (X^2, ...
1
vote
0answers
25 views

Minimal primary decomposition of the ideal $I = (XY, Y Z, XZ) ⊆ \mathbb C[X, Y, Z]$ [duplicate]

Write out a minimal primary decomposition of the ideal $I = (XY, Y Z, XZ) ⊆ \mathbb C[X, Y, Z]$, and determine the primes belonging to $I$. Determine the dimension of the ring $\mathbb C[X, Y, ...
-1
votes
1answer
53 views

Elimination Ordering for the ring $k[x,y]$

How to show that the only elimination ordering on the ring $k[x,y]$ is the lexicographic ordering? (Ene and Herzog, Gröbner Bases in Commutative Algebra, Problem 3.1.) Definition (Elimination ...
2
votes
1answer
52 views

Is the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ birational?

This is Exercise 5.3 (a) in Undergraduate Algebraic Geometry by Reid. Does the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ define a rational map? Determine ...
2
votes
0answers
15 views

A sufficient condition for factorization in a complete local ring

I think something like the following statement is true, but I don't recall a reference. Suppose $f(x,y)\in k[[x,y]]$ is power series with no constant or linear terms. Then, if the quadratic terms ...
1
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0answers
30 views

Nullstellensatz to prove Noether Normalization

In many commutative algebra texts, Noether Normalization Lemma is proved and then Hilbert's Nullstellensatz is obtained as a corollary. Nullstellensatz and Normalization Lemma seem to be non-trivial ...
-1
votes
0answers
16 views

Proof that maximal ideals in $\mathcal{P}[x_0]$ intersected with $\mathcal{P}$ is a maximal ideal in $\mathcal{P}$ [on hold]

I am trying to show that maximal ideals in $\mathcal{P}[x_0]$ intersected with $\mathcal{P}$ is a maximal ideal in $\mathcal{P}$, where $\mathcal{P}$ is the polynomial ring $K[x_1, \dots, x_n]$ or ...
1
vote
1answer
30 views

$I(X_1 \cap X_2)=\sqrt{I(X_1)+I(X_2)}$

How to prove $I(X_1 \cap X_2)=\sqrt{I(X_1)+I(X_2)}$? Clearly $\sqrt{I(X_1)+I(X_2)} \subseteq I(X_1 \cap X_2)$ But for $f \in I(X_1 \cap X_2)$ $f(x)=0 \forall x\in X_1 \cap X_2$. how to show $f \in ...
0
votes
0answers
47 views

Spectrum and maximal spectrum of a ring

How do the $\mathrm{Spec}(\mathbb{C}\left [ X \right ])$ and $\text{m-Spec}(\mathbb{C}\left [ X \right ])$ look like? I understand the definitions of $\mathrm{Spec}(R)$ and $\text{m-Spec}(R)$ for a ...
-1
votes
0answers
32 views

Buchberger's Algorithm Example

I've been reading Ideals, Varieties and Algorithms and came across an example of Buchberger's algorithm being computed and I am not able to understand how they came to have the final result. The ...
3
votes
1answer
51 views

Suppose A is a principal ideal domain with every ideal of finite index. Must A be a Euclidean domain?

Suppose $A$ is a principal ideal domain with every ideal of finite index (except the zero ideal). Must $A$ be a Euclidean domain? If it's not known, are there any relevant partial results?
1
vote
1answer
49 views

Tensor product of Hom-module and another ring

Let $A$ be a local noetherian ring, $B$ and $C$ are finitely generated $A$-algebras and $M$ is a finitely generated $B$-module. Is the natural morphism $\mathrm{Hom}_B(M,B) \otimes_A C \to ...
6
votes
1answer
65 views

Does the determinant give you the index over $\mathcal{O}_k$ as well as over $\mathbb{Z}$?

It is a standard fact that if $M$ is a nonsingular $n\times n$ integer matrix, the index of the $\mathbb{Z}$-span of its columns as an abelian group in $\mathbb{Z}^n$ is $|\det M|$. What happens if we ...
0
votes
1answer
24 views

where do elements go under multiplication in a graded module?

Assume that $M = \bigoplus_{n = 0}^\infty M_n$ is a graded $A$-module, where $A = \bigoplus_{n = 0}^\infty A_n$ is a graded ring. We have by definition $A_m M_n \subset M_{m + n}$. Does this mean that ...
0
votes
0answers
18 views

Relation between minimal primes of a Noetherian graded ring and its subring

Let $A=⊕A_i$ be a Noetherian graded ring. Is there any relation between minimal primes of $A$ and minimal primes of $A_0$ (its $0$-th component)? In fact, my motivation is tight closure theory. I ...
0
votes
0answers
21 views

Reference request for numerical invariants of modules which are not finitely generated

Suppose that $R$ is an integral domain with subring $S$ and that both rings are finitely generated $k$-algebras ($k$ an algebraically closed field). $R$ is integral over $S$ if and only if $R$ is ...
1
vote
1answer
62 views

Explanation of a proof about graded module structure

Let $\Bbb F$ be a field and $M$ a finitely generated $\Bbb F[x]$-module. The structure theorem for modules over a PID says that $$ M\cong \Bbb F[x]^r\oplus\biggl(\bigoplus_{j=0}^s\Bbb ...
0
votes
1answer
27 views

how to prove an element is non-zero in a tensor-product

I was studying the following example from Atiyah & MacDonald's Introduction to Commutative Algebra: let $x$ be the non-zero element in $N := \mathbf{Z}/ 2\mathbf{Z}$, $M := \mathbf{Z}$, and $M' := ...
0
votes
1answer
28 views

Poincaré series pole at $1$

Let $A$ be a graded ring and $M$ a graded $A$-module. By $P(M,t)$ we denote the Poincaré series for $M$. In Atiyah and Macdonald, theorem 11.1 claims $P(M,t)=\dfrac{f(t)}{\prod _{i=1}^n ...
2
votes
0answers
32 views

The natural numbers form a distributive lattice under gcd and lcm. In arbitrary gcd domains, does gcd distribute over lcm?

Basically what it says in the title. If $A$ is a $\operatorname{gcd}$ domain, for any $x, y, z \in A$, does this identity hold? $$\operatorname{gcd}(x, \operatorname{lcm}(y,z)) = ...
0
votes
1answer
29 views

Does every non-archimedean absolute value on field take value in $\mathbb{Q}$

Let $K$ be a field, a non-archimedean absolute value is defined to be a map $K\to \mathbb{R}$ satisfying $|x|=0\Rightarrow x=0$, $|x|\cdot|y|=|xy|$ and $|x+y|\leq\max(|x|,|y|)$. Is there an example ...