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

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Two definitions for non-singular in codimension 1

I am trying to understand how the following definitions are the same. $\textbf{Shafarevich definition}$ (pg 128) - A variety is $\textit{non-singular in codimension one}$ if the singular locus has ...
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1answer
16 views

Show that $V(\bigcup_{i \in I} E_{i})=\bigcap_{i \in I} V(E_{i})$

This is a part of a problem in Atiyah's Introduction to Commutative Algebra introducing the Zariski Topology. Here we are given that $(E_{i})_{i \in I}$ is a family of subsets of a unital commutative ...
2
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0answers
24 views

Tensoring two short exact sequences

Let $R$ be a commutative ring with $1$ and consider the following short exact sequences of $R$-modules \begin{align} &0 \to M' \to M \stackrel{f}{\to} M'' {\to} 0 \qquad \text{and } \\ &0 \to ...
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1answer
40 views

Maximal ideal in local ring

The maximal ideal in $\mathbb{Z}_{(2)}$ should be $(2)$, but I don't understand this well. Suppose I take $\frac35\in \mathbb{Z}_{(2)}$. It is not in $(2)$ but in $(3).$ But what is the ideal between ...
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3answers
47 views

$\mathbb{Z}_{(2)}$ has one maximal ideal

My lecture notes state that the set $\mathbb{Z}_{(2)}$, defined as $$\mathbb{Z}_{(2)}:=\left\{\frac{a}{b}\in\mathbb{Q}\mathrel{}\middle|\mathrel{}\gcd(a,b)=1\text{ and } 2\nmid b\right\}$$ has a ...
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0answers
20 views

The prime ideals of the ring $K[x]$

I was wondering what the prime ideals of the ring $K[x]$ are, where $K$ is a ring. My guess is that it's any ideal generated by a set of irreducible polynomials over the ring $K$. Have I covered all ...
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0answers
20 views

On finite generation of Certain $Ext$'s

All rings below are commutative. I have the following situation: $A$ is a commutative ring, $B=A/I$, and I know that $B$ is noetherian. I have a $B$-module $M$ which is finitely generated as a ...
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1answer
30 views

Simplify $(y-x^2)\cap(y^2+2y+x^2)$

In the book "Commutative Algebra with a View Toward Algebraic Geometry (Eisenbud, 1995), exercise 1.10 one has to find the ring associated to the union of the circle $C:(y+1)^2+x^2=1$ and the parabola ...
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1answer
16 views

Let $K$ be a field, $A \subset K$, and $p \subset A$. Then $\exists$ a valuation ring $R$ satistfying…

I was stuck when reading a proof of the following theorem (Matsumura p. 72-3, Theorem 10.2), Let $K$ be a field, $A \subset K$ a subring, and $p$ a prime ideal of $A$. Then there exists a ...
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2answers
79 views

What is the Krull dimension of $\mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4]$?

I am studying commutative algebra and saw the following question in one of the tests: What is the Krull dimension of $R=\mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4]?$ I know ...
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0answers
45 views

What does a complex of modules mean?

I try to understand from Qing Liu's book Algebraic Geometry and Arithmetic Curves the problem 1.2.16. It goes as follows: Let $(A,\mathfrak m)$ be a Noetherian local ring, and $$C^\bullet:0\to M'\to ...
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1answer
30 views

What is the injective hull of a polynomial ring?

The injective hull of a polynomial ring in one variable $K[X]$ (where $K$ is a field) is $K(X)$ since $K(X)$ is a divisible hence injective $K[X]$-module (since $K[X]$ is a PID) and $K(X)$ is an ...
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1answer
32 views

Quotient of ring is flat gives an identity of ideals

I have problem to understand and solve the exercise 1.2.14 on Qing Liu's book "Algebraic Geometry and Arithmetic Curves". It goes as follows: Let $A\to B$ be a ring homomorphism, and let $J$ be an ...
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0answers
41 views

Why prime avoidance lemma allows only at most 2 non-prime ideals?

Why prime avoidance lemma allows only at most 2 non-prime ideals? The following is the last part of the proof taken from wikipedia: For the case $n > 2$, choose $z_i \in E \cap (I_i - \cup_{j ...
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0answers
37 views

Integrally closed domain.

Suppose $A$ is a unique factorization domain, $a$ is an element of $A$. Is the ring $A[x,a/x]$ always integrally closed? ($x$ is a variable over $A$) Thanks!
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1answer
49 views

Regularity of $k[X,Y,Z]/(Z^2 - f(X)g(Y))$

Let $R = k[X,Y,Z]/(Z^2 - f(X)g(Y))$, for an algebraically closed field $k$ with char $\not=2$ and $f(X)$ and $g(Y)$ have only simple roots in $k$. Determine the maximal ideals such that the ...
2
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1answer
60 views

Atiyah and Macdonald, exercise 11.7

I am trying to solve the exercise in Atiyah, that $\dim(A[X]) = \dim (A) + 1$ for $A$ noetherian. The very beginning poses a problem, he states in the hint that: for a prime of height $m$ we can ...
2
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0answers
29 views

Extremal Betti numbers of graded ideal with Cohen-Macaulay quotient; Herzog and Hibi, exercise 4.6 [on hold]

Let $I$ be a graded ideal of $S=K[x_{1},...,x_{n}]$ such that $S/I$ is Cohen-Macaulay. Then show that $I$ has only one extremal Betti number. Here, a Betti number $\beta_{i;i+j}\neq 0$ is called ...
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0answers
37 views

Flatness of integral closure over an integral domain

The problem 1.2.10 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves" is the following: Let $A$ be an integral domain, and $B$ its integral closure in the field of fractions ...
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1answer
27 views

how does Macaulay2 computes analytic spread for non-local rings?

Macaulay2 computes analytic spread for R=QQ[a,b,c,d,e,f] which is not a local ring. In the books like ...
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3answers
44 views

Surjectivity of the induced map of affine algebraic sets

For a morphism $f: X\rightarrow Y$ of affine algebraic sets, I want to show that if the induced map $f^*:k[Y]\rightarrow k[X]$ is surjective then $f(X)$ is closed. I am trying to prove that ...
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0answers
30 views

Is my observation correct regarding restriction of scalars?

Let $\alpha: \Lambda\to \Gamma$ be a ring homomorphism, then $ _\Lambda\Gamma_\Gamma$ is a bimodule. We have the following pairs of adjoint functors $$ \mathbf{Mod_\Lambda} \xrightarrow{\cdot\; ...
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1answer
24 views

Primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$

I am looking for a minimal primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$. I realize that this is a similar question to some of the previous ones, but the ring is different than in ...
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1answer
37 views

Tensoring the exact sequence by a faithfully flat module

I have problems to do the exercise 1.2.13 from Liu's "Algebraic Geometry and Arithmetic curves". First, Liu defines that if $M$ is a flat module over a ring $A$, then $M$ is faithfully flat over $A$ ...
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1answer
21 views

Is the (Krull) dimension of a semi-local Jacobson ring equal to zero? [duplicate]

Let $R$ be a commutative ring with identity element. If $R$ is semi-local (number of maximal ideals of $R$ is finite) and a Jacobson ring (this means that every prime ideal of $R$ is equal to the ...
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1answer
64 views

Best schools for commutative algebra [on hold]

I will be applying to graduate programs this fall and I was curious which schools have the best commutative algebra groups. I know berkeley and michigan are up there, but what are others?
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2answers
39 views

Flat algebra over a Dedekind domain

Let $B$ be a flat algebra over a Dedekind domain $A$. Let $f\in B$ be such that for every maximal ideal $\mathfrak m$ of $A$, the image of $f$ in $B/\mathfrak mB$ is not a zero divisor. How can I show ...
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42 views

Simplicial homology [on hold]

Let $\Delta$ be the simplicial complex on vertex set [5] whose Stanley-Reisner ideal is $I_{\Delta}=(x_{1}x_{4},x_{1}x_{5},x_{2}x_{5},x_{1}x_{2}x_{3},x_{3}x_{4}x_{5})$. Write the augmented oriented ...
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0answers
34 views

The annihilator numbers of $S/I$ [on hold]

Let $S=K[x_{1},x_{2},...,x_{n}]$ and $I$ be a strongly stable ideal of $S$. Compute the annihilator numbers of $S/I$ with respect to the almost regular sequence $x_{n},x_{n-1},...,x_{1}$. ...
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1answer
59 views

Finding conditions for $\mathbb Z[i][X,Y]/(Y^2 - aX)$, $a \in \mathbb Z[i]$ to be regular

I am trying to find the dimension and the necessary and sufficient conditions under which $A[X,Y]/(Y^2 - aX)$ is regular, that is, the localizations of $A[X,Y]/(Y^2 - aX)$ at all maximal ideals are ...
3
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2answers
67 views

Finding an ideal such that quotient is Cohen-Macaulay

Let $R$ be a commutative local Noetherian ring which is not a domain and not Cohen-Macaulay. Can we find an ideal $I$ in $R$ such that $R/I$ is Cohen-Macaulay, and $\dim R/I=\dim R$?
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1answer
60 views

Is $\{3,5,6\}$ a $2\mathbb{Z}$-sequence?

I have just started reading about the concept of $M$-regular sequences on my own and to understand the definition I asked myself the following question: Is $\{3,5,6\}$ a $2\mathbb{Z}$-sequence? ...
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1answer
54 views

Example of $I$-adic topology of submodule not matching subspace topology?

I'm reading about the $I$-adic topology on $M$ for $R$ a commutative ring, $I$ an ideal of $R$ and $M$ an $R$-module. The references I'm reading don't provide examples, but they say that if $N$ is a ...
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1answer
28 views

Flatness of module over field of fractions

This is from Liu 1.2.9. Let $A$ be an integral domain, and $K$ its field of fractions. Let $M$ be a finitely generated sub-$A$-module of $K$. Why do $M$ is flat if and only if $M_{\mathfrak p}$ is ...
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0answers
16 views

Commutative version of hyper operators.

As I understand it, addition and multiplication are defined on the reals as having identity elements 0 and 1 and being commutative and associative. Multiplication is also distributive over addition. ...
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1answer
54 views

Let $f: U \rightarrow W$ be a morphism of affine algebraic sets and $f': k[W] \rightarrow k[U]$ be the k-algebra morphism of coordinate rings.

Prove if $f'$ is surjective then $f$ is a homeomorphism of $U$ onto the closed subset $W$. Well, it's the first time I've seen this word "homeomorphism" but I read online that a map is a ...
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0answers
88 views

useful exact sequences [closed]

There are some exact-sequences or long-exact-sequences that are great help in proving to prove some surprising theorem, or have some interesting applications. What's your favorite exact ...
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0answers
17 views

Direct sum of ideals over Dedekind domain [duplicate]

I'm trying to show that Let $\frak{a},\frak{b}$ be two ideals of a Dedekind domain $\cal{O}$. Show that there is an isomorphism \begin{equation*} ...
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1answer
46 views

Canonical homomorphism and free module, Liu 1.2.8 c

How can I do the problem 1.2.8 c in "Algebraic Geometry and Aritmetic Curves". Namely, let $A$ be a Noetherian ring, $M$ a finitely generated $A$-module, and $N$ an $A$-module. Let $B$ be a flat ...
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2answers
59 views

Total ring of fractions of a Noetherian reduced ring is artinian

I'm doing the preparation to an exam, and I'm stuck in the following: If $R$ is a Noetherian ring with zero nilradical ($N(R) = 0$), and $S$ is the set of regular elements of $R$ ($r \in S$ if $rs ...
2
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1answer
56 views

How to prove that a ring is not flat over $k[t,s]$? [duplicate]

Let $k$ be a field, $A=k[t,s]$, and $C=A[z]/(tz-s)$. How can I prove, using the ideals $tA$ and $sA$, that $C$ is not flat over $A$? I know that if $A$ is a Dedekind domain then $A$-module is ...
2
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1answer
39 views

Flatness and intersection of ideals

This is Liu 1.2.6 a Let $B$ be a flat $A$-algebra. Show that for any finite family $\{I_\lambda\}_{\lambda\in \Lambda}$ of ideals of $A$, we have $\cap_{\lambda\in\Lambda}(I_\lambda ...
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1answer
42 views

$H^i_I(M)$ is finitely generated iff the support of $Ext^{d-i}_S(M, S)$ has dimension zero

$(R,m)$ is a local Noetherian ring. $M$ is a finite $R$-module. Here, using dualizing complex, Karl Schwede says that if $R=S/I$ where $S$ is regular of dimension $d$, then we have: "$H^i_m(M)$ is ...
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0answers
58 views

How cannot localization of any integral domain respect to maximal ideal not be integrally closed?

Suppose that there is integral domain $I$. Now we take localization $I_m$ of $I$ respect to its maximal ideal $m$. $I_m$'s elements will consist of $a/b$ where $a \in I$ and $b \in m$. But integral ...
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0answers
52 views

How to check if a polynomial is inside an ideal using a Groebner basis

I'm given that an ideal $I=\langle F_1, F_2, F_3, F_4, F_5, F_6, F_7\rangle$ $F_1=a+b+c-d-e-f$ $F_2=a+b+c-g-h-i$ $F_3=a+b+c-g-e-c$ $F_4=a+b+c-a-e-i$ $F_5=a+d+g-a-e-i$ $F_6=a+d+g-c-f-i$ ...
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1answer
35 views

Noetherian ring with infinite Krull dimension.

I just started to read about the Krull dimension (definition and basic theory), at first when I thought about the Krull dimension of a noetherian ring my idea was that it must be finite, however this ...
2
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1answer
47 views

Showing that for every monomial $x^u\in\operatorname{in}_{<}(I)$, there exists $f\in I$ s.t. $\operatorname{in}_<(f)=x^u$

Given an ideal $I\subset R=K[x_1, ...,x_n]$ and let $<$ be a term order on the ring $R$. I must show that $\forall x^u\in\operatorname{in}_<(I)$, $\exists f\in I$ s.t. ...
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1answer
25 views

Tensor product and localization

This is from Liu, problem 1.2.2. Let $\rho:A\to B$ be a ring homomorphism, $S$ a multiplicative subset of $A$, and $T=\rho (S)$. Show that $T^{-1}B\simeq B\otimes_AS^{-1}A$ as $A$-algebras. I ...
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1answer
42 views

Question on Algebraic Hartogs Lemma for locally Noetherian normal schemes

I am reading the proof by Götz-Wedhorn Algebraic Geometry I Theorem 6.45, and also Liu, Theorem 1.14. One thing that I do not understand is this: For easier cases, we assume $X=\text{Spec A}$ and let ...
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1answer
48 views

Question about split monomorphisms of free modules over local rings

In May's notes on Cohen-Macaulay and Regular Local Rings, during the proof of Serre's theorem on page 9, he claims that if $R$ is a local ring and $\phi\colon F\to F'$ is a map of finitely ...