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

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Prove $I(\{(x,y,z) \in \mathbb{C}^3: x^2+y^2+z^2-1=0\}) = (x^2+y^2+z^2-1)\mathbb{C}[x,y,z] $

I got stuck in proving that $$I(\{(x,y,z) \in \mathbb{C}^3: x^2+y^2+z^2-1=0\}) = (x^2+y^2+z^2-1)\mathbb{C}[x,y,z]. $$ Let $X= \{(x,y,z) \in \mathbb{C}^3: x^2+y^2+z^2-1=0\}$, and $I(X) = \{ f \in ...
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1answer
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$φ^∗: \operatorname{Spec}S^{−1}_fA → \operatorname{Spec}A$ is injective and that its image is (Spec A) \ V ((f)) [closed]

Let A be a ring and let Spec A be the set of prime ideals of A. For any ideal I ⊂ A write V (I) for the set of prime ideals p ∈ Spec A containing I. If φ : A → B is a homomorphism of rings, ...
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1answer
37 views

Ideal $(Y^2,X-YZ)$ is $(X,Y)$-primary

Show that the ideal $(Y^2,X-YZ)$ is $(X,Y)$-primary in $K[X,Y,Z]$, where $K$ is a field. I got a hint that I need to use this property: Let $f:A\to B$ be a ring homomorphism. If $q$ is ...
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1answer
65 views

Is the Kernel locally free?

Suppose, on a smooth projective complex variety $X$ that we are given an effective divisor $D$ and $A\in\mathrm{Pic}(D)$ a globally generated line bundle on $D$ with $r$ independent sections. Then we ...
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1answer
35 views

Artinian rings, minimal ideals, and the Hom functor

I was reading the Proof of Theorem 18.1 in Matsumura and there were two places where I was stuck. Theorem 8.1 Let $(A, m, k)$ be an $n$-dimensional Noetherian local ring. Then the following are ...
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1answer
58 views

Blowing up an affine scheme at a regular point

I am reading Liu's Algebraic Geometry and Arithmetic Curves and get stuck at Lemma 8.1.2: Let $A$ be a Noetherian ring an define for an ideal $I \subset A$ the $A$-algebra ...
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1answer
34 views

Canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism

From this MSE question I understand the canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism for $R$ a commutative ring and $I,J$ ideals. I tried proving this directly ...
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29 views

Is there such a concept as a “freest local ring” generated by another?

Suppose I want to make the following argument: Let $(A,m)$ be a local commutative ring. Then $(A,m)$ is a quotient of a local ring $(B,n)$ which is a domain, since there is a "freest local ring ...
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1answer
55 views

Checking quasicoherence on a qcqs scheme

Let $(X,\mathscr{O}_X)$ be a scheme and $\mathscr{F}$ be an $\mathscr{O}_X$-module. It can be shown that $\mathscr{F}$ is quasicoherent iff for every affine open $U = \operatorname{Spec} A$ and $s\in ...
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48 views

Geometric intuition for $R[a^{-1}]$?

The ideal of polynomials vanishing over a point in an affine algebraic variety is maximal, and I think I understand the geometric intuition behind localizing it(s complement). But what about ...
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42 views

Induced morphism on rings of regular functions injective iff original morphism has dense image. [duplicate]

Let $X \to Y$ be a morphism of affine varieties over a field $k$. How do I see that the induced morphism $k[Y] \to k[X]$ on rings of regular functions is injective if and only if the original morphism ...
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32 views

Completions and Localizations

I was wondering if the following is true: Let $(R,m,k)$ be a Noetherian local ring. If $R$ is a complete local ring (with respect to the $m$-adic topology), then $R_p$ is a complete local ring. ...
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1answer
17 views

Radicals: $\sqrt{\sum_k I^{n_k}_k}\supset \sum_k I_k$

Is it true that given ideals $I_1,\dots ,I_n$ of a commutative ring we have $\sqrt{\sum_k I^{n_k}_k}\supset \sum_k I_k$? How can I prove this? I think I can manage if the identity $\sqrt{I^m}=\sqrt ...
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2answers
145 views

When does tensor product have a (exact) left adjoint?

Let $A$ be a commutative Noetherian ring, and let $F$ be a flat $A$-module. We can assume $A$ is local, so $F$ is projective. Question 1. When does $F\otimes_A-$ preserve injective objects? ...
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1answer
76 views

$k[x]\otimes k[x]$ as a right $k[x]$-module

Let $k$ be a commutative ring. Consider the ring map $\varphi:k[x]\to k[x]\otimes_k k[x]$ given by $\varphi(x)=x\otimes 1-1\otimes x$. Now consider $k[x]\otimes_k k[x]$ as a right module over itself. ...
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34 views

Eisenbud exercise 21.9 and the uniqueness of Frobenius structure on Gorenstein rings

I am learning some basic things about Gorenstein rings. Here is exercise 21.9 in Eisenbud's book Commutative Algebra With A View Towards Algebraic Geometry attached: Let $A,P$ be a ...
3
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1answer
49 views

Hilbert series characterization of regular sequences

Let $k$ be a field and $S=k[x_1,\dots,x_r]$ the polynomial ring in $r$ indeterminates. Let $f_1,\dots,f_n$ be a sequence of $n\le r$ forms of degrees $d_1,\dots,d_n$. If $f_1,\dots,f_n$ is a regular ...
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36 views

Polynomial time algorithm for variety of an ideal

Is there a polynomial time algorithm to determine the variety of a zero dimensional ideal in $k[x_1,\ldots,x_n]$? Or is it a NP hard problem?
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63 views

projective dimension over local ring

My question is as below. Let $(R,m)$ be a local ring such that $m$ consists of zero divisors. Prove that if $A$ is a finitely generated R-module, then $pd_R(A)$, the projective dimension of $A$ over ...
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0answers
44 views

Etale morphism from $\mathbb {C}^n$ to itself [closed]

Consider a morphism $f\colon \mathbb{C}^n\rightarrow \mathbb{C}^n$ of the form $(x_1,...,x_n) \mapsto (x_1,...,x_{n-1}, g (x_1,...,x_n))$, for some polynomial $g$. For which polynomial $g$ is $f$ ...
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2answers
50 views

If $R \subseteq k[X]$, then $R$ is a finitely generated $k$-algebra. [duplicate]

Let $k$ be a field, and let $R$ be a subring of $k[X]$ which contains $k$. Does $R$ have to be a finitely generated $k$-algebra? I tried to prove this, but I didn't get anywhere. Would someone ...
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56 views

Generalization of Exercise II.5.9 of Hartshorne. [duplicate]

Here is Exercise II.5.9 of Hartshorne. Let $A$ be a Noetherian ring, let $S$ be a graded ring $S_1$ over $S_0$ over $S_0$ and assume that $S_0 = A$. Let $M$ be a finitely-generated graded ...
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16 views

Is the following a valid argument? Finding the Groebner basis of an elimination ideal

I am asked to find the basis for $I \cap k[x]$ and $I \cap k[y]$ where $I=<x^2+2y^2-3,x^2+xy+y^2-3>$. I will omit the calculation here since it is very long and tedious, but I found one Groebner ...
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23 views

Can I use the elimination theorem in the following case? Elimination Ideals and Groebner basis

I understand the statement of the elimination theorem tells me that, $I \subset k[x_1,...,x_n]$ be an ideal and $G$ a Groebner basis of $I$ with respect to lex ordering. Then, $G_l=G \cap ...
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1answer
41 views

Noetherian local domain $A$ with a prime $P$ so that $\operatorname{ht}P+\dim A/P<\dim A$

Is there a noetherian local domain $A$ with a prime $P$ so that $\operatorname{ht}P+\dim A/P<\dim A$? This is a follow up question to: Does codimension behave weirdly even in local rings?
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Free commutative ring functor

The free commutative ring on a set $X$ is the polynomial ring with variables the elements of $X$. This polynomial ring is the free (additive) abelian group on the free (multiplicative) abelian monoid ...
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1answer
43 views

Does codimension behave weirdly even in local rings?

Is there an example of a finite dimensional local ring $(A,m)$ (maybe Noetherian, preferably not too far away from a ring that would arise when studying algebraic varieties) with a prime ideal $P ...
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0answers
35 views

When does $v \wedge w = w \wedge v$ imply $v \wedge w=0$?

Let $A$ be a noetherian ring such that $2$ is not a zero divisor of $A$. Let $M$ be a submodule of the free module $F=A^r$. Clearly $2$ is not a zero divisor of $M$, i.e. $2x=0$ implies $x=0$ for all ...
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Distinguished open subset of affine variety is an affine variety

I am reading some of Vakil's notes (http://math.stanford.edu/~vakil/725/class6.pdf) and I have got a few questions on the statement and the proof of the theorem (on page 4): $\textbf{Question 1}$: ...
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1answer
92 views

Equivalent conditions for integral element

Let $A\subset B$ be a ring extension and $x\in B$. Then if $A$ is noetherian the following two conditions are equivalent: $i)$ $x$ is integral over $A$. $ii)$ There exists a finitely generated ...
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Can MAGMA determine if a ring is free as a module over a subring?

Suppose I have constructed a pair of rings $R\subset S$ in Magma (the computer algebra system) and I know that $S$ is finitely generated as an $R$-module. Does Magma have a routine to determine if $S$ ...
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1answer
43 views

Is $A/\varphi^{-1}(\mathfrak{m})\subseteq B/\mathfrak{m}$ an integral extension? [duplicate]

Let $\varphi:A\rightarrow B$ be a homomorphism of finitely generated $k$-algebras, and let $\mathfrak{m}$ be a maximal ideal of $B$. We have the injective homomorphism $$ ...
8
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1answer
132 views

$A$-module is free if and only if equation involving Hilbert-Poincaré series holds.

Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector spaces $A_i$ and $M_j$ are ...
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1answer
36 views

Connection of associated primes to the ground ring

Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Then the minimal primes of $M$, i.e. the minimal associated primes of $M$, coincide with the minimal primes of the ring ...
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64 views

(quasi)coherent rings for which $\dim R[T]\neq \dim R+1$

What are some some examples of (quasi)coherent rings for which $\dim R[T]\neq \dim R+1$? Why (hopefully geometrically) should we not always have equality? Notation. Let $I,J$ be two ideals of ...
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1answer
41 views

Is every ring/module the filtered colimit of its finitely presented/coherent/quasicoherent subrings/submodules?

Is every ring/module the filtered colimit of its coherent/quasicoherent subrings/submodules? What about finitely presented subobjects? What's the intuition behind each case? Notation. Let $I,J$ be ...
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1answer
54 views

Non-Noetherian rings satisfying $\dim R[T]=\dim R+1$

What are some examples of non-noetherian rings $R$ for which $\dim R[T]=\dim R+1$ holds?
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1answer
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Let M be an Artinian module over a commutative Artinian unital ring R. Is M necessarily finitely generated?

I was trying to prove this statement as true. Actually I found an identical question here: "Is every Artinian module over an Artinian ring finitely generated?" However, in the proof of the link ...
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1answer
39 views

'Finitely presented' implies 'always finite presented' for algebraic theories

In this MO question it is proven the answer is yes for modules. The proof given relies on the snake lemma, which does not generally make sense in the category of rings, groups, monoids, etc. It seems ...
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1answer
58 views

For any affine open subset $U$ of an irreducible affine variety $X$, why $K(U)\cong K(X)$?

Here is a problem I met while reading Linear Algebraic Groups written by T. A. Springer. The following paragraph is on Page 16 of this book. Let $X$ be an irreducible variety. First assume that $X$ ...
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2answers
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Ring homomorphisms $\mathbb Z[x]\rightarrow \mathbb Q$

I'm supposed to find all the ring homomorphisms $\mathbb Z[x]\rightarrow \mathbb Q$. Here's my attmept: $\mathbb Z[x]$ is generated by $\left\{ x^n \right\} _{n\geq 0}$, so the images of the ...
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1answer
54 views

Equivalent condition for being a regular prime ideal

$\newcommand{\p}{\mathfrak{p}}$ $\newcommand{\tp}{\tilde{\mathfrak{p}}}$ $\newcommand{\tA}{\tilde{A}}$ I have a question about Neukirch, Algebraic Number Theory, page 92. The problem is to show the ...
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2answers
148 views

All prime ideals are invertible $\Rightarrow$ Dedekind domain

$\newcommand{\p}{\mathfrak{p}}$ $\newcommand{\a}{\mathfrak{a}}$ Let $A$ be a one-dimensional Noetherian domain. I am thinking about this claim: If all prime ideals of $A$ are invertible, then $A$ ...
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Computing $\text{Tor}$ for modules over a PID

This is essentially an exercise in Sze-Tsen Hu's "Introduction to Homological Algebra", page 143. Let $R$ be a PID and consider two $R$-modules $X$ and $Y$. Let $S$ denote the subset of the Cartesian ...
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37 views

Are $K[X,Y,Z]$, $K[X,Y,Z]/(Z^2)$ and $K[X,Y,Z]/(XZ,YZ,Z^2)$ flat $K[X,Y]$-modules?

Let $K$ be a field, $A = K[X,Y]$ be polynomial ring in two variables. Are $K[X,Y,Z]$, $K[X,Y,Z]/(Z^2)$ and $K[X,Y,Z]/(XZ,YZ,Z^2)$ flat $A$-modules? I have no idea to check which module is flat ...
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1answer
39 views

Is an ideal which is cyclic as a subgroup always principal?

For $\mathbb Z$, all ideas are principal because they're cyclic as subgroups, hence each $x\in I= \left\langle n\right\rangle$ can be writte $x=mn$ for $m\in \mathbb N$. Luckily, $\mathbb N\subset ...
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1answer
58 views

In stacks project: Polynomial ring over UFD is UFD

Lemma 10.119.8 in this page on Stacks project states that a polynomial ring over a UFD is a UFD. It uses Nagata's criterion for factoriality (10.119.7): If $A$ is a domain and $S\subset A$ a ...
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2answers
58 views

Question concerning commutative algebra

Let $R$ be a commutative ring. Let $P\subset R$ be a minimal prime ideal. Let $S=R-P$. Let $x\in P$ and $s \in S$. Is there any property saying that if $sx=0$, then $x=0$? Should anyone helps me, ...
2
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1answer
46 views

$X$ compact Hausdorff implies $x\mapsto \mathfrak{m}_x$ is a homeomorphism

Let $X$ be a compact Hausdorff space. Denote by $\mathfrak m_x$ the prime ideal of $C(X)$ comprised of functions vanishing at $x$. Topologize $\operatorname{MaxSpec}C(X)$ with the initial topology ...
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2answers
44 views

Divisors and prime elements in rings

Suppose $p$ is a prime element in a commutative ring. Does this imply its only divisors are $1,p$? By definition $a\mid p\implies \exists b:ab=p\implies p\mid a\text{ or }p\mid b$. If $p\mid a$ then ...