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

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2
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82 views

Constructible set in Gieseker's 'Lectures on Moduli of Curves'

I'm reading Gieseker's Lectures on Moduli of Curves, and I have a question concerning a statement made on page 69. We are given a family $p : Z \to U$ of connected curves of genus $g\geq 2$ and ...
3
votes
3answers
236 views

Commutative integral domain does not finitely generate its field of fractions

I want to prove that if we have a commutative integral domain $D$ with field of fractions $F\neq D$ then $F$ is not finitely generated as a $D$-module. (In this question it may be the case that ...
2
votes
0answers
57 views

The multiplicity of $X$ at $x$ does not change when $X$ is cut by a generic hypersurface: what are those generic conditions?

Given an algebraic variety $X$ with a point $x \in X$, the multiplicity of $X$ at $x$ is defined as the multiplicity of the maximal ideal of $x$ in the local ring $\mathcal{O}_{X,x}$. In ...
4
votes
2answers
580 views

In a Dedekind domain every ideal is either principal or generated by two elements.

Prove that in a Dedekind domain every ideal is either principal or generated by two elements. Help me some hints. Thanks a lot.
7
votes
1answer
141 views

Proving normality of affine schemes

One of the exercises in Ravi Vakil's algebraic geometry notes, Ex. $5.4.$I(b), is to show that $$ \operatorname{Spec}\left(k[x_1, \ldots, x_n]/(x_1^2 + \cdots + x_m^2)\right) $$ is normal, ...
5
votes
1answer
84 views

So Gröbner bases lead to nothing new for the Beal polynomial $x^a + y^b - z^c$?

I'm learning about Gröbner bases. And $f(x,y,z) = x^a + y^b - z^c$, is a single monic polynomial in any monomial ordering, $I = (f)$ has Gröbner basis $\{f\}$. So there's nothing interesting to look ...
3
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1answer
87 views

Principal ideal domain is universally catenary …

... actually, even more general statement is true: Theorem. Every regular ring is universally catenary. (see for example Algebraic Geometry by Qing Liu, Corollary 2.16, Chapter 8) Though, the ...
4
votes
2answers
113 views

Regularity of a basis over a local ring

The following Theorem (Thm. 19.9 in Matsumura, CRT) will be necessary for the understanding of my question: Theorem: Let $A$ be a Noetherian local ring and $I$ a proper ideal of $A$ with ...
2
votes
2answers
164 views

What is the ideal of leading terms?

Fix a monomial ordering on the polynomial ring $\Bbb{k}[x_1, \dots, x_n] = R$ over a field. What exactly is $LT(I)$ for an ideal $I$ of $R$? How is it defined and does it form an ideal?
3
votes
1answer
81 views

Krull dimension and graded prime ideals

How can we show that $\dim R/p=0\Leftrightarrow p=(x_{1},\ldots,x_{n})\Leftrightarrow R/p\simeq\mathbb{K}$, where $R=\mathbb{K}[x_{1},\ldots,x_{n}]$ is considered graded with standard grading (i.e. ...
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2answers
56 views

minimal injective resolution and finiteness

Let $M$ be a finite module over a Noetherian ring $A$. Let $0 \rightarrow M \rightarrow I^1 \rightarrow I^2 \rightarrow \cdots $ be a minimal injective resolution of $M$. Question: Is it true that ...
2
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0answers
43 views

On local rings $(R,m)$ having a metrizable $m$-adic topology

If $R$ is a local ring with maximal ideal $m$ and the intersection of powers of $m$ is $0$, then the $m$-adic topology is metrizable. Is there a condition on $R$ assuring that the metric space so ...
1
vote
1answer
172 views

Help with these isomorphisms

Let $X$ be an affine algebraic set and $f\in K[X]$ where $K[X]$ is the coordinate ring of $X$. Suppose $I(X)=\langle G_1,\ldots,G_r\rangle$ and $W=Z(G_1,\ldots,G_r,FT_{n+1}-1)$, where ...
3
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0answers
97 views

Question about complete DVR.

Is there a simple proof that you know to the following statement: If the residue field $k$ of a complete DVR $R$ has the same characteristic as $R$, then $R$ contains a subfield isomorphic to $k$. ...
0
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1answer
44 views

Necessity of an hypothesis in this exercise about integral dependence of $k[x][\frac{1}{f}]$ over $k[x]$

let $k[x]$ be the ring of polynomial over a field $k$. Let $f \in k[x]$ an irreducible polynomial. Then $k[x][\frac{1}{f}]$ is not integral over $ k[x]$. I solved this ex. In this way By absurd, ...
0
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1answer
77 views

Characteristic Polynomial Computation

Let $A_{0}=k[x,y]$, $\mathfrak{m}=(x,y)$. Let $A=(A_{0})_{\mathfrak{m}}$. We wish to compute the characteristic polynomial, $\chi_{\mathfrak{q}}$, of the $\mathfrak{m}$-primary ideals (i) $(x,y)$, ...
3
votes
1answer
52 views

Domain where for each ideal $I$ and each $a \in I$, $I = \langle a,b \rangle$ is Dedekind?

Say $R$ is a domain such that for any nonzero ideal $I$ of $R$ and any $a \in I$, we have $I = \langle a,b\rangle$ for some $b \in I$. Is $R$ a Dedekind domain? If not, what additional assumptions do ...
1
vote
0answers
41 views

Comparing definitions of smooth and unramified

Let $k$ be a base ring, and let $\mathsf{Aff}_k$ be the category of affine schemes over $k$. If $X$ is any presheaf on $\mathsf{Aff}_k$, one defines the associated "de Rham space" to be the presheaf ...
2
votes
1answer
90 views

Computing Hilbert polynomial

We have the following condition: for each $i=2,...,m$ multiplication by $f_{i}$ is injective on $S/(f_{1},...,f_{i-1})$, where $S=k[T_{0},...,T_{n}]$, $m \leq n$, and the $f_{i} \in S_{d_{i}}$ are ...
2
votes
1answer
48 views

Conditions leading to a conclusion.

Let $R$ be a commutative noetherian ring with unity, $M$ a finitely generated $R$-module, $I$ an ideal of $R$ such that $\bigcap_{t\ge 1} I^tM=0$ and $M\cong\underset{t}{\varprojlim}M/I^tM$. Now, ...
2
votes
1answer
71 views

Flat modules on Stacks Project.

I have been reading through a bit of the material from the Stacks project, and there is a statement that I cannot make sense of. Lemma 10.36.19(7) states: Let $R$ be a ring. (7) Suppose ...
0
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1answer
136 views

R ring is noetherian, commutative, unitary and integral domain, is R a field?

This is the question: "let R be a commutative unitary ring that is also integral domain and noetherian, prove that R is a field" I'm having some trouble proving this. For R to be noehterian means ...
0
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3answers
116 views

Local ring with nontrivial prime ideal

I've just learnt the idea of local ring, the only concrete examples in the book are $\{p/q \in \mathbb{Q}: q \mbox{ odd}\}$ and the power series ring $k[[x]]$, but the only prime ideals they have are ...
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vote
1answer
47 views

What is precise definition of $\Gamma_{\mathfrak{a}}(M)$?

For a module $M$ over a ring, $\mathfrak{a} \subseteq A$ any ideal we define $\Gamma_{\mathfrak{a}}(M) = \{m \in M: \mathfrak{a}^nm = 0$ for some $n > 0\}$. What exactly is meant by $\mathfrak{a}^n ...
2
votes
1answer
51 views

Translate a sentence (regarding rings and maximal ideals) from french

Probably this is not a suitable question for this forum but I am stuck reading a paper in French and I cannot understand how "relever" is used in the following part of a theorem: We have that ...
4
votes
1answer
112 views

Associated elements in a ring

Please help me to find elements $a,b$ in a ring $R$ such that $a\mid b$ and $b\mid a$, but there does not exist any unit $u$ in $R$ such that $a=ub$.
2
votes
1answer
100 views

Poincare Series and Hilbert Polynomial of graded $S$-modules

I am trying to find the Poincare series and Hilbert polynomial for graded $S$-modules $I=S \cdot T^m$ and $M=S/I$ where $S=k[T]$ is the graded polynomial algebra and $m \geq 1$. I am not ...
2
votes
1answer
144 views

Prove that if $P$ and $Q$ are projective and finitely generated $R$-modules then $\operatorname{Hom}_{R}(P,Q)$ is projective and finitely generated.

Suppose $R$ is a commutative ring and $P$ and $Q$ are projective and finitly generated $R$-modules. Prove that $\operatorname{Hom}_{R}(P,Q)$ is projective and finitely generated. suppose I proved ...
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0answers
49 views

Vanishing criterion of pure wedges

Let $R$ be a commutative ring, $M$ some $R$-module, and $m,n \in M$. Is there some criterion when $m \wedge n = 0$ in $\Lambda^2(M)$? There are some sufficient criterions, for example that $m \in ...
4
votes
2answers
125 views

Showing that a certain map is not flat by explicit counterexample

I wish to show that the injection $k[y^2, y^3] \rightarrow k[y]$ is not flat. I know of geometric ways to see this, but I wish to see explicitly $k[y^2, y^3]$-modules (or localizations thereof) ...
0
votes
1answer
98 views

Isomorphism between $k$-algebra and its localization at a nonzerodivisor

$k$ is a field. Let $A$ be a $k$-algebra, $f$ a nonzerodivisor in $A$, and the localization $A_f\cong k[x,y]$ as $k$-algebras. Prove that $A\cong k[x,y]$. This is my work: Let $\phi$ be the ...
4
votes
1answer
240 views

Hints about Exercise 4.2 in Miles Reid, Undergraduate Commutative Algebra

$ A \subset B $ is a ring extension. Let $ y, z \in B $ elements which satisfy quadratic integral dependance $ y^2+ay+b=0 $ and $ z^2+cz+d=0 $ over $ A $. Find explicit integral dependance ...
6
votes
2answers
181 views

Zorn's Lemma and Injective Modules

In my study of injective modules over commutative rings, i noticed that Zorn's Lemma is often employed in the proofs. Here are three examples: 1) Baer's Criterion 2) the characterization of injective ...
4
votes
1answer
101 views

Random algebraic numbers are linearly disjoint almost surely?

It is well-known that if one considers a “random” monic polynomial of fixed degree, say $X^n+\sum_{k=0}^{n-1}a_kX^k$ where $(a_0,a_1,\ldots, a_n)$ is drawn from the discrete uniform distribution on ...
1
vote
0answers
22 views

General position for one-parameter family of algebraic numbers

Let $P(x,y)$ be an irreducible twovariate polynomial with rational coefficients such that $P(n,.)$ has degree $>1$ for any $n\in{\mathbb N}$. For any $n\in{\mathbb N}$, one may choose a root ...
4
votes
0answers
100 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 ...
0
votes
1answer
68 views

Question about tensor product of modules and ideals

Trying to prove some properties of tensor product with a given module, I came up with questions some of them I can't prove. Maybe it is also because Im not very used to work with tensor products and I ...
0
votes
1answer
68 views

Why is any power of $(X_1, X_2,…,X_i)$ primary in $k[X_1, X_2,…,X_n]$?

How to prove that all the powers of the ideals $(X_1, X_2,...,X_i)$ are primary in $ k[X_1, X_2,...,X_n]$?
4
votes
2answers
179 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 $|\operatorname{Spec}(R[X])|<\infty$?
5
votes
1answer
320 views

Primary decomposition of the ideal $(XY, X-YZ)$ in $k[X,Y,Z]$

What is a general way to find primary decomposition of an ideal in polynomial ring? Specifically I am trying to see what is the primary decomposition for the ideal $(XY, X-YZ)$ in $k[X,Y,Z]$? The ...
2
votes
1answer
67 views

Determine the kernel of a surjective morphism of differential modules

I am trying to find out how to complete the proof of the following statement. Let $R, S, R', S'$ be commutative rings and $(\psi, \varphi) : (R, S) \to (R', S')$ be a morphism from the algebra ...
6
votes
1answer
133 views

Does $I^{-1}$ invertible imply $I$ invertible?

Now that I have your attention, here are the pertinent definitions: Let $R$ be an integral domain with field of fractions $K$. A $R$-fractional ideal of $K$ is a $R$-submodule $I$ of $K$ such that ...
4
votes
1answer
102 views

Is it true that in a Noetherian ring every descending chain of prime ideals stabilizes?

Is it true that in a Noetherian ring every descending chain of prime ideals stabilizes? It would be good if I had this result. As it would finish off my proof that the minimal primes of an ideal ...
0
votes
1answer
66 views

Krull dimension bound of a Fitting ideal

Given a finitely presented $R$-module $M$ over a ring $R$ one can define, for every integer $k\geq 0$ the $k$-th Fitting ideal of $M$, for instance in this way, using exterior algebra. ...
2
votes
0answers
94 views

Unique factorization in 3-sphere coordinate ring

For $n\geqslant 1$ define $$A_n=\mathbb{C}[X_0,X_1,\dots,X_n]\Bigg/\left(\sum_{i=0}^{n}X^2_i-1\right).$$ I would like to prove that $A_3$ is a unique factorization domain. For $A_2$ it is not true ...
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2answers
73 views

For fractional ideal, why $AB=R$ implies $B=A^{-1}$?

Let $A,B$ be two fractional ideals of $R$ (an integral domain). Could anyone tell me why $AB=R$ implies $B=A^{-1}$?
3
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1answer
142 views

Some questions about Fitting ideals

Let $R$ be a ring and $M$ a finitely presented $R$-module. Given a free presentation $$ R^{\oplus m} \to R^{\oplus n} \to M \to 0 $$ we define $Fitt_k(M)$, the $k$-th Fitting ideal of $M$, to be the ...
5
votes
2answers
177 views

Why is $ \hbox{Ext}_R^* (M,M) = H^*(\hbox{Hom}_R^*(P^*,P^*))$?

Let me first fix some notation and conventions. Let $ R$ be a ring and $ M$ a left $R$-module. Given chain complexes $P^*$ and $Q^*$ in $R$-mod, define $ \hbox{Hom}^*_R(P^*,Q^*)$ to be the graded ...
2
votes
1answer
102 views

How can we form $I/(I + \mathfrak{m}^2)$?

I am trying to do an exercise in FOAG 2013. "Suppose $(A, \mathfrak{m}, k)$ is a regular local ring of dimension $n$, and $I ⊂ A$ is an ideal of $A$ cutting out a regular local ring of dimension ...
2
votes
1answer
93 views

Let $A \subset B$ be faithfully flat and $B$ Noetherian then $A$ is also Noetherian

Let $A \subset B$ be faithfully flat and $B$ Noetherian. Prove that $A$ is also Noetherian. My idea was to construct an exact sequence: $0 \rightarrow A \rightarrow A\otimes B \rightarrow ? ...