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

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Readings for Noether

I'm studying the theory of Noether but I have only 4 pages of lecture notes with no details or examples. Are there any good lecture notes or chapters you know about? In my lectures the basics of ...
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1answer
40 views

Points lying over a closed point in a separable extension of the base field are rationnal

At the end of the proof of Proposition 4.3.30 In Liu's book we have the following situation: $X$ is an algebraic variety over a field $k$, $x\in X$ is a regular closed point of $X$ with $k'=k(x)$ is a ...
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When does the tensor product consist of elementary tensors only?

The question is: Assume that $R$ is a (commutative) ring. Under what conditions on $R$-modules $M,N$ does the tensor product $M\otimes_RN$ consist of elementary tensors only? That is, every ...
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1answer
115 views

How to calculate $\operatorname{Spec} \mathbb{C}[x,y]/(y^2-x^3)$

Is there a general method for calculating things like $\operatorname{Spec} \mathbb{C}[x,y]/I$ ? Maximal ideals are $ \{(x-\tilde{a},y-\tilde{b}): b^2-a^3=0\}$ because of ...
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1answer
86 views

Localization of Gorenstein ring

Let $R$ be a Gorenstein local ring and $S=R \setminus Z(R)$. I want to prove $S^{-1}R =⊕_{ht\ p=0} R_p$ and $S^{-1}R$ is injective $R$-module. I can see the above $p$'s are minimal, $id_{R_p} R_p=0$ ...
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1answer
61 views

Localization of a ring at one element

Let $A$ be a commutative ring with identity, $f$ an element of $A$, let $g'=g/1$ be the image of the element $g$ of $A$ in $A_f$ under the natural homomorphism $A\rightarrow A_f$. The question is: ...
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1answer
51 views

Criterion for locally free modules of rank $1$

Let $R$ be a commutative ring and let $M$ be a finitely generated $R$-module such that the $R_{\mathfrak{p}}$-module $M_{\mathfrak{p}}$ is free of rank $1$ for every prime ideal $\mathfrak{p}$. Can we ...
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73 views

Graded version of Grothendieck's Non-vanishing Theorem

Is there a graded version of the Grothendieck Non-vanishing Theorem? (Theorem 6.1.4 of the book "Local Cohomology - An Algebraic Introduction with Geometric Applications" written by M. P. Brodmann ...
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1answer
54 views

A statement equivalent to flatness

If $R$ is a ring with identity and $P$ is a flat right $R$-module, it is a fact that any $R$-homomorphism $f$ from a finitely presented right $R$-module $M$ to $P$ factors through a finitely generated ...
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1answer
108 views

How to construct a nonzero homomorphism from a module to a proper submodule?

Let $M$ be a finitely generated module over a commutative ring and $N$ be a non zero proper submodule of $M$. Then is it always possible to have a non zero homomorphism $f$ from $M$ to $N$?
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singular locus and jacobian matrix

Let $R=k[x_1, \cdots ,x_r] / I$ be an affine ring over a perfect field $k$ and suppose that $I$ has pure codimension $c$. Suppose that $I= (f_1, \cdots , f_s)$. If $J$ is the ideal of $R$ generated by ...
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Equations in the semiring of f.g. modules

Let $R$ be a commutative ring. Then we may consider the semiring $G(R)$ of isomorphism classes of finitely generated $R$-modules with $+ = $ direct sum, $* = $ tensor product, $0 = $ zero module, $1 = ...
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62 views

Krull dimension of quotient by principal ideal

Let $R$ be a commutative unitary ring of finite Krull dimension and let $x \in R$. Is it true that $\dim R/(x) \ge \dim R -1$ ?
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Commutative algebra, Krull dimension. Eisenbud, Exercise 10.6

I need help to solve this exercise. If anyone can help, thanks in advance! Let $R=k[x,y,s,t]/(xs-yt)$ and $S=R/(x,y)=k[s,t]$. Let $P=(s,t)\subset R$. Show that $\operatorname{codim} P=1$ but ...
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1answer
36 views

Krull dimension, commutative algebra. Eisenbud, Exercise 10.3

This is the exercise. Let $k$ be a field. Prove that $k[x]\times k[x]$ contains a principal ideal of codimension $1$, although it's not a domain. Now, I have to find a principal ideal prime, such ...
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53 views

Gorenstein, complete intersection

Let $S = k[X_1,...,X_n]$ Example 3.2.11(b) of Bruns-Herzog's book "Cohen-Macaulay Rings", gives a Gorenstein ring that is a complete intersection iff $n \leq 2$. They have proved it as a special case ...
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42 views

Birational equivalence with surfaces

Let $X$ be an algebraic variety of dimension $n$. I'd like to prove that $X$ is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}$. I've already seen algebraic proofs using some ...
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50 views

When the ring of regular functions is a UFD?

Let $X$ be an irreducible affine variety over $\mathbb{k}$. There is the following theorem in algebraic geometry: the algebra $\mathbb{k}[X]$ of regular functions is a UFD if and only if each ...
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3answers
97 views

If $a,b$ is an $R$-sequence, then $ax-b$ is prime (Eisenbud, Exercise 10.4)

This is the exercise mentioned above: Let $a,b$ be regular sequence over a domain $R$. Prove that $ax-b$ is a prime of $R[x]$. Thank you for your answer!
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Noetherian ring with finitely many height $n$ primes

If $R$ is a Noetherian commutative ring with unity having finitely many height one prime ideals, one could derive from the "Principal Ideal Theorem", due to Krull, that $R$ has finitely many prime ...
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57 views

Exercise about Krull Dimension/Eisenbud, Exercise 10.1

I can't solve this exercise. If someone can help me, thanks a lot. Let $R$ be a Noetherian ring, and $x$ an indeterminate. Prove that $\dim R[x,x^{-1}]=\dim R+1$. Thank for your answers!
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23 views

Finding some homogeneous generators of an ideal.

Suppose that $\mathfrak a$ is an homogeneous ideal of $K[T_1,\ldots, T_n]$ where $K$ is a field of characteristic $0$ and $T_1,\ldots,T_n$ are indeterminates. Moreover suppose that $\mathfrak a$ has a ...
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Parametric ideal decomposition

Let $x = \{x_{1},\dots, x_{n}\}$ be a set of variables and let $a = \{ a_{1}, \dots, a_{m}\}$ be a set of parameters. Let $\{f_{1}(a,x), \dots, f_{s}(a,x)\} \subset \mathbb{C}[a,x]$ be a set of ...
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61 views

Grade of a graded ideal, Bruns-Herzog, Exercise 1.5.21

The first part is easy: $\operatorname{grade} I = \dim S - 1 =3$. But I can't prove the second. Can you help please?
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1answer
42 views

Equivalent condition for Jacobson radical

Matsumura, Commutative Ring Theory, page 3, asks this: If $x \in A$ has the property that $1 + Ax$ consists entirely of units, then $x \in \operatorname{rad}(A)$. Prove this. How to show this?
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1answer
43 views

Factorization of an ideal in a number field

The notes I read gives following technique to factor an ideal in a number field without explanation. Can anyone explain how this technique works? To factor the ideal $(2)$ in $\mathbb{Z}[\sqrt{-5}]$, ...
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1answer
109 views

A Question from Algebraic Geometry

For any two disjoint closed subsets $Y_1$ and $Y_2$ of $ \mathbb A ^n$ show that there exists $g \in\mathbb C [x_1, x_2, ..., x_n]$ such that $g(Y_1)=0$ and $g(Y_2)=1$.
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41 views

Number of generators in a tensor product

'Commutative Algebra' by Atiyah and Mcdonald, mentions if $ \lbrace x_{i} \rbrace_{i ∈ I}$ and $\lbrace y_{j} \rbrace_{j ∈ J}$ generate $M$ and $N$ as $A-$modules, respectively, then $x_{i} \otimes ...
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1answer
41 views

Uniqueness of tensor product

The uniqueness property of tensor product $M ⊗ N$ of two $A-$modules $M$ and $N$ specifies the following: for sake of simplicity we will write $M⊗N$ as $T$. A tensor product of $M$ and $N$ is pair ...
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When is $N\otimes_A B \to N$ an isomorphism?

Let $A, B$ be commutative (unital) rings and $f\colon A \to B$ an $A$-algebra. There then exists a canonical functor $f_*\colon \mathbf{Mod}_B \to \mathbf{Mod}_A$ such that, for every morphism of ...
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406 views

Is Frobenius the only magical automorphism?

The Frobenius automorphism is special because the $p$-power map makes sense in any characteristic $p$ ring, which allows us to canonically extend the Galois-theoretic Frobenius to any such ring. I ...
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90 views

Krull dimension of a quotient ring

The question that I am about to ask can be trivial, even if so, I would be grateful for any useful answer. Why $\mathbb C[x,y]/\langle\pi^k\rangle$ is finite dimensional, where $\pi$ is some ...
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2answers
53 views

The Relationship Between Cohomological Dimension and Support

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. The cohomological dimension of $ M $ with respect to $ I $ is defined as $$ \operatorname{cd}(I,M) ...
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26 views

Ring of regular functions on an open set of spectrum of $R$ is a subring of the field of fractions of $R$.

Let $R$ be an integral domain, and let $X=\operatorname{Spec}(R)$. Show that all local rings $\mathcal{O}_X(U)$ - for nonempty open subsets $U\subseteq X$ - are subrings of the field of fractions ...
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22 views

Is every minimal set of generators for a homogeneous ideal composed by homogeneous elements?

An ideal $\mathfrak a$ of a graded ring $A$ is said to be homogeneous if I can find a set of homogeneous generators for $\mathfrak a$. Is it true that every minimal set of generators for a ...
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40 views

Dimension of Kahler differentials of Laurent series over perfect field.

Let $k$ be perfect field, char $k=p>0$. How it can be shown that $\dim_{k((t))}\Omega^1(k((t)))=1$?
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Localization of a polynomial ring at a maximal ideal

Let $R$ be a regular local Noetherian ring, with maximal ideal $M$. Show that $N=R[x]M+(x)$ is a maximal ideal in the polynomial ring $R[x]$, and that the localization $R[x]_N$ is again regular local. ...
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39 views

R is a regular local ring of dimension $d$, and $I$ an ideal. If $R/I$ has depth $d − 1$, then $I$ is principal.

is this true? $R$ is a regular local ring of dimension $d$, and $I$ an ideal. If $R/I$ has depth $d − 1$, then $I$ is principal. if it is true please help me by a hint
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33 views

Isomorphism between two $K$-algebras

Consider a field extension $L\subseteq K$ and suppose that $I=(f_1,\ldots,f_m)$ is an ideal of $L[T_1,\ldots,T_n]$. Denote with $I^e\subseteq K[T_1,\ldots,T_n] $ the extended ideal of $I$ through the ...
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2answers
36 views

intersection multiplicity at non-zero point

Compute the intersection multiplicity of $f=x+y-2$ and $g=x^2+y^2-2$ at $(1,1)$. Is this the same as the intersection multiplicity of $f(x+1)$ and $g(x+1)$ at $(0,0)$ which I have computed to be 2? ...
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72 views

Computing the radical of an ideal

What is the best way to compute $\sqrt{(X^2-YZ,X(1-Z))}$ ? This is after using Nullstellensatz by the way as I thought it would be easier to compute a radical than finding the vanishing ideal.
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$\mathbb{C}[x,y]/(f,g)$ is an artinian ring, if $\gcd(f,g)=1$.

This problem extends the fact that $\mathbb{C}[x,y]/(x^n,y^m)$ is artinian ring. Let $f,g \in \mathbb{C}[x,y]$ such that $\gcd(f,g)=1$. Show that $\mathbb{C}[x,y]/(f,g)$ is an artinian ring.
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Singular points query

I have learnt that a point is singular if all partial derivatives vanish there. However I have recently come across with worked examples where they also require that the function vanishes at the ...
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94 views

Computing irreducible components of algebraic set

Consider the algebraic set $V(X^2-YZ,X-XZ)$. Find the irreducible components of this set and show that $I(V)=(X^2-YZ,X-XZ)$. I reasoned that $X-XZ=0$ iff $X=0$ or $Z=1$. If $X=0$, we get $Y=0$ or ...
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1answer
55 views

Looking for a commutative ring satisfying certain conditions

I'm looking for a commutative ring $R$ (with unit) which is of characteristic 2 and which possesses elements $x$ and $y$ such that the following holds $x^2$ and $y^2$ are inverses of one another but ...
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1answer
34 views

Prove the integral property of $B[x] \cap B[y]$, where $y=x^{-1}$

Let $A$ and $B$ be two commutative rings with a unit element, with $B$ subring of $A$. Suppose $x$ is an invertible element in $A$ such that $xy=1$. Then prove that the intersection of the two rings ...
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A problem involving ideals and prime ideals. [duplicate]

Please help me with a solution to this problem. Let $R$ be a commutative ring. Let $A_1, A_2$ be two ideals of $R$, and $P_1, P_2$ two prime ideals of $R$. Assume that $A_1 \cap A_2 \subseteq P_1 ...
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2answers
53 views

How to show that $(Y- X^2, Z - X^3) \subseteq k[X,Y,Z]$ is a prime ideal?

I suppose that $k$ is an algebraically closed field (actually, my goal is to show $\mathcal{I}(\mathcal{V}(Y- X^2, Z - X^3)) = (Y- X^2, Z - X^3)$). (But I think algebraically closed is not necessary ...
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Continuity of induced map between spectra under constructible topology (Atiyah-Macdonald, Chapter 3, Exercise 29)

Exercise 3.29 in Atiyah-Macdonald asks to show that the map $f^*:\text{Spec}(B)\to\text{Spec}(A)$ (induced by a ring homomorphism $f:A\to B$), is a continuous closed mapping for the constructible ...
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32 views

If $\alpha$ is $R$-linear epimorphism, $\alpha$ is isomorphism.

I am having a difficult time understanding the proof of a corollary to Cayley-Hamilton theorem in Eisenbud's Commutative Algebra. The statement is: Let $R$ be a ring, and let $M$ be a finitely ...