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

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5
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1answer
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Flatness under reduction

Suppose that $f : X \to Y$ is a flat morphism of schemes. Is $f_\text{red} : X_\text{red} \to Y_\text{red}$ necessarily flat? Are there any hypotheses that would guarantee this?
2
votes
1answer
33 views

three cubic homogeneous polynomials satisfy a cubic polynomial

Question: How can we show algebraically that three cubic homogeneous polynomials in two variables satisfy a cubic polynomial of three variables? More specifically, let ...
3
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0answers
69 views

Commutative algebra: Integral extension [closed]

Let $R ⊆ S$ be an extension of rings, and let $u$ be a unit in $S$. $(i)$ Prove for $α ∈ R[u] \cap R[u^{−1}]$, there is $n > 0$ such that $αA ⊆ A$, where $$A =\langle 1, u, u^2, . . . , u^n ...
2
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0answers
72 views

Algebra: Integral extension [closed]

Let $F$ be a field, and let $F[[X]]$ be the ring of formal power series over $F$. Show that $R = \{a + X^2f (X) : a ∈ F, f (X) ∈ F[[X]]\}$ is a proper subring of $F[[X]]$ and $F[[X]]$ is integral over ...
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0answers
20 views

Normalization of curve

How do to normalize the curve $ax^2+y^2=1+bx^2y^2$ (hard exercize)? I tried the substitution $t=xy$, $u=y$. But I get $at^2/u^2+u^2=1+bt^2$ can i multiply it on $u^2$.
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1answer
56 views

How to show that an object is a discrete valuation ring? (Fulton, Exercise 2.14)

I need some help to solve the following problem that appears on page 31 of the book of William Fulton entitled Algebraic Curves. Exercise : Let $ V = \mathbb{A}^1 $, $ \Gamma (V) = k[X] $, $ K = ...
0
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1answer
33 views

Associated primes of an $R$-module

An associated prime of an $R$-module $M$ is an ideal of the form $Ann_R(N)$ where $N$ is a prime sub-module of $M$ in the sense that $N$ is nonzero and $Ann_R(N)=Ann_R(N')$ for each nonzero sub-module ...
1
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1answer
30 views

Maximal ideals of commutative Artinian rings

I would like some help on an exercise I thought I had done correctly at first glance, but obviously have doubts about. The question is; Let $R$ be a commutative Artinian ring. Then R has finitely ...
0
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0answers
30 views

Working with affine Varieties

Hi guys I just wanted to hear some input on this, $A=(x^4+y^4-1)$ and $B=(p^2+w^2-1)$ are affine varieties in $C^2$. We want to show that if we apply the map $f(a,b)=(a^2,b^2)$ then $f(A) \subset ...
1
vote
1answer
63 views

About the $k$-subalgebras of $k[x]$

Still in my "commutative algebra marathon", I came across the following exercise: Any $k$-subalgebra $A$ of $k[x]$ is finitely generated as $k$-algebra; also, if $A\ne k$, then $\dim A=1$. ...
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0answers
35 views

Is smoothness of $X\to Y$ for noetherian $X$ a local property on $X$?

Let $X$ be a noetherian scheme. If $X$ is regular, then the scheme $\operatorname{Spec}(\mathcal{O}_{X,x})$ is regular for all points $x\in X$. I wonder if something analog is true for smoothness of a ...
4
votes
0answers
70 views

Property of free submodules for a module over a PID

It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is not free. We can take $R=M=K[x,y]$ , $L=<x>$ , ...
0
votes
1answer
36 views

Decomposition of a maximal ideal as a union of smaller prime ideals

Let $K$ be a field, $S=K[X,Y]$ the polynomial ring in two variables and consider the ideal $M=\langle X,Y\rangle$ (ideal generated by $X$ and $Y$). Show that $M$ is a union of strictly smaller prime ...
2
votes
3answers
86 views

How to show that the ideal $(X^{3},XY,Y^{n})$ of $K[X,Y]$ is primary?

I'm working on a problem in Sharp's Steps in commutative algebra, to be precise exercise 4.28 which states the following: Let $K$ be a field and $R = K[X,Y]$ be the polynomial ring in the ...
4
votes
2answers
76 views

Annihilator of a maximal ideal in a ring

Let R be a ring and M a maximal ideal in R. Prove or disprove: If M is contained in the set of zero divisors of R, then ann(M) is not 0. It is easy to see that the statement is true when M is ...
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0answers
43 views

Covering of $\mathbb{P}^n$ and the complement of a point

Let $p$ be a closed point in $\mathbb{P}^n$ for some integer $n$ and $\{U_i\}$ be an affine open covering of $\mathbb{P}^n\backslash p$. Does there exists an open set in the covering, say $U_0$ for ...
1
vote
1answer
56 views

Every prime is maximal in a Jacobson ring?

In Attiyah commutative algebra page 71, it is given some equivalent definitions of Jacobson ring. One of the definitions are that every prime ideal which is not maximal is equal to the intersection of ...
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vote
2answers
79 views

There's no surjective ring homomorphism from $\mathbb{Z}[x_1,\dots,x_n]$ onto $\mathbb{Q}$.

I'm trying to prove that, if a field $A$ is also a finitely generated $\mathbb{Z}$-algebra, then $A$ is finite. The proof I found for this depends on the fact that $\mathbb{Q}$ cannot be a finitely ...
0
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1answer
34 views

Finding integral dependence for polynomials [closed]

How can I find a integral dependence for each $f\in K[x]$ over $K[x^2]$? For arbitrary field $K$.
3
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1answer
34 views

A noetherian local ring having a height one principal prime is a domain

$A$ is a commutative ring with with $1$. If $A$ is a Noetherian and local ring and $A$ has a principal prime ideal of height $1$ then show that $A$ is a domain. Can anybody give some hint.I tried ...
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0answers
42 views

Locally free sheaf on Cohen-Macaulay scheme and Serre's criterion

Let $X$ be a projective locally Cohen-Macaulay scheme and $\mathcal{F}$ be a locally free sheaf on $X$. If I understand correctly the definition of Serre's criterion $S_k$, $\mathcal{F}$ satisifies ...
3
votes
1answer
67 views

Is $k[x][[h]]$ finitely generated as $k[[h]]$-algebra?

Is $k[x][[h]]$ finitely generated as an algebra over $k[[h]]$, where $k$ is a field, and $xh=hx$.
1
vote
1answer
51 views

Open embedding and localization.

Let $X, Y$ be algebraic varieties. If we have an open embedding $X \hookrightarrow Y$, then we have a map $\mathbb{C}[Y] \to \mathbb{C}[X]$. Is $\mathbb{C}[X]$ a localization of $\mathbb{C}[Y]$? For ...
3
votes
2answers
37 views

On the existence of finitely generated modules with finite injective dimension

Assume $R$ is a commutative local Noetherian ring. It is known that if there is a finitely generated module with finite injective dimension then $R$ is Cohen-Macaulay. My question is: if $R$ is ...
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0answers
28 views

Generic freeness (a Lemma from Matsumura, CRT)

Let $B$ be a Noetherian ring, and $C$ a $B$-algebra generated over $B$ by a single element $x$; let $E$ be a finite $C$-module, and $F\subset E$ a finite $B$-module and $CF=E$. Then $D=E/F$ has an ...
3
votes
0answers
50 views

Geometric interpretation of Ideals in a Prime ideal

I have been told this has a geometric meaning, $I_j \in F[x_1,...x_n]$ be ideals such that $\cap_1 ^n I_J= P$ for P been a prime ideal, then we know that $P= I_j$ for some j=1,..n My Understanding I ...
2
votes
1answer
71 views

Is a surjective $R$-endomorphism over a finitely generated $R$ algebra always bijective?

Let $R$ be a unital commutative ring and $A$ a finitely generated $R$-algebra. I found out that if $R$ is a field, then any surjective $R$-endomorphism over $A$ must be injective, too. Does that hold ...
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1answer
36 views

relations between a set of polynomials

I have a set of polynomials. Is there a computer algebra program that gives all the algebraic relations between them ? I will prefer singular if it has this component.
3
votes
1answer
107 views

Classification of finitely generated multigraded modules over $K[x_1,\ldots,x_n]$?

Let $K$ be a field and $R=K[x_1,\ldots,x_n]=\bigoplus_{a\in\mathbb{N}^n}Kx^a$ the multigraded polynomial ring. Have finitely-generated multigraded $R$-modules been classified? Are they of the ...
1
vote
1answer
55 views

Krull dimension of generic fiber

Let $p$ be a prime and $A=\mathbb{Z}_p[t_1,\ldots,t_n]/I$ a reduced flat and irreducible $\mathbb{Z}_p$-algebra of finite type and of Krull dimension d. Let $e$ be the Krull dimension of the ...
2
votes
1answer
85 views

Definition of multiplicity

Q.1. Bruns_Herzog define multiplicity (in the case of graded rings and modules) as My question is that: why multiplicity for $d=0$ it is defined as $\ell(M)$? Is there a kind of ...
3
votes
1answer
54 views

Is a prime principal ideal which is not maximal among principal ideals always idempotent?

Let $R$ be a commutative ring with identity, $P$ a prime principal ideal of $R$. Suppose that there exists a proper principal ideal $I$ of $R$ which is strictly larger than $P$ (i.e. $R\supsetneq ...
2
votes
1answer
49 views

Surjective morphism of varieties with finite fibers but not “finite”

Let $X$ and $Y$ be affine varieties, and $f : X \to Y$ a dominant regular map. Following Shafarevich, I will call $f$ finite if the induced map on coordinate rings is integral. One consequence of ...
5
votes
1answer
112 views

Interpretation of sheaf flat over a base

I am trying to get an interpretation of what means for a sheaf to be flat with respect to a base. The definition is that, given $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ is flat over $Y$ ...
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0answers
42 views

Spectral sequences and Ext between extension of modules

Suppose $A$ is a commutative ring, $M_1,M_2,N_1,N_2$ are $A$-modules and we have two exact sequences of $A$-modules $$0\to M_1\to M\to M_2\to 0,$$ $$0\to N_1\to N\to N_2\to 0.$$ I want to write a ...
3
votes
0answers
32 views

Tensor product of flat modules - proof verification

Let $A$ be a commutative ring, and let $B,C$ be commutative $A$-algebras. Let $M$ be a flat $B$-module and $N$ a flat $C$-module. I want to show that $M\otimes_A N$ is a flat $B\otimes_A C$-module. ...
3
votes
2answers
50 views

What other classes of commutative rings can be defined by requiring that $\{0\}$ is the only proper ideal satisfying some condition?

A field is just a commutative ring $R$ such that $\{0_R\}$ is the only proper ideal. Interestingly, there's a similar characterization of integral domains. Given a subset $A$ of $R$, let $A^\perp$ ...
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0answers
49 views

Determine whether a regular surjective map is finite

Consider the regular map between affine closed sets $f \colon \mathbb{A}^1 \rightarrow \mathcal{Z}(y^2-x^3) \subseteq \mathbb{A}^2$ given by $f(t) = (t^2,t^3)$. $f$ is obviously a dominant map. I ...
3
votes
1answer
66 views

Does $(a)=(b)$ imply that $a$ and $b$ are associate in a principal ideal ring?

Let $R$ be a commutative principal ideal ring with identity. Suppose that $a,b\in R$ and $(a)=(b)$. I'd like to know if there always exists $u\in R^\times$ such that $a=bu$. I know several ...
0
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1answer
56 views

Atiyah-MacDonald, 3.18. Why is $B_q$ a local ring of $B_p$?

This question is on the hint that the book gives to finish the exercise. Namely, if $f: A \rightarrow B$ a flat homomorphism of rings, $q$ a prime ideal of $B$ and $p = q^c$, then $B_q$ is a local ...
4
votes
1answer
61 views

Projective modules over Dedekind Domains

Show that if $R$ is a Dedekind domain, then every projective $R$-module (not necessarily finitely generated) is a direct sum of ideals of $R$. I have spent a while on this problem and I wonder if it ...
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1answer
44 views

Properties of module length

Let $e_{A}(\phi, M): = l_A(\mathrm{coker}(\phi) ) - l_A(\ker(\phi))$. In my book it is stated that if $IM = 0 \implies e_{A}(\phi, M) = e_{A/I}(\phi, M)$ and this seems to be obvious for the author. ...
3
votes
1answer
78 views

What is a geometric interpretation of regular sequences in various instances?

This question arose from my attempts to understand the inclusion Regular $\subset$ Complete Intersection $\subset$ Gorenstein $\subset$ Cohen Macaulay There are many related questions here and in ...
4
votes
1answer
65 views

Irreducible components of schemes

Consider the scheme $X:=\mathrm{Spec}(k[X,Y]/(X^2,XY))$. According to Qing Liu's "Algebraic geometry and arithmetic curves", the irreducible components are in $1-1$ correspondence with subschemes of ...
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0answers
34 views

Associated graded ring of a quotient

Given a ring A and an ideal $I \subseteq A$ we can form its associated graded ring with respect to $I$ $$ Gr_I(A)= A/I \oplus I/I^2 \oplus I^2/I^3 \oplus \ldots $$ I wondered if there is a way to ...
3
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1answer
74 views

Must a $R$-automorphism on $R[X]$ be of the form $X\mapsto aX+b,\ a\in R^*,b\in R$?

Let $R$ be a commutative ring. I wonder if every $R$-automorphism (that is, a ring automorphism that fix $R$) $\varphi$ of $R[X]$ satisfies $\varphi(X)=aX+b$, where $a$ is an unit in $R$ and $b$ an ...
2
votes
2answers
84 views

The unit group of $\mathbb{Q}[x, y]/(x^2+y^2+1)$

During some calculations, I encountered with the problem of calculating the unit group of the $\mathbb{Q}$-algebra $\mathbb{Q}[x, y]/(x^2+y^2+1)$. I believe it is the unit group of the field of ...
0
votes
1answer
49 views

Primary Ideal and Associated Primes [duplicate]

I'm trying to understand the proof of the following statement: If $R$ is Noetherian, then an ideal $Q$ is $P$-primary for a prime $P$ $\Leftrightarrow$ $Ass(R/Q)=\lbrace P \rbrace$. I can show the ...
3
votes
0answers
36 views

Equivalence of definitions for completion

For the settings on my question, take Atiyah's chapter on completions. Basically we have two definitions of completness (Atiyah's sense, the canonical map $\phi:M\rightarrow \widehat{M}$ is an ...
2
votes
1answer
30 views

An example of a (necessarily non-Noetherian) ring $R$ such that $\dim R[T]>\dim R+1$

What is an example of a non-Noetherian ring $R$ such that the Krull dimension of $R[T]$ is greater than dim$R+1$?