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

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86 views

Question on rank function.

In a previous question I asked about the fiber $M(P)=M_P / PM_P$ where $M$ is an $A$-module and $P$ a prime ideal of $A$. Later I introduced the rank function $$rk_M : \text{Spec} A \to \mathbb{N} ...
10
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2answers
396 views

Is every prime element of a commutative ring “veryprime”?

Let $R$ denote a commutative ring. Define a function $$\| : R \times R \rightarrow \mathbb{N} \cup \{\infty\}$$ such that $a \| b$ is the number of times $a$ divides $b$ (and include $0$ in ...
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1answer
204 views

What is the algebraic tangent cone really?

Let $A$ be a (commutative unital) ring, let $\mathfrak{a} \subseteq A$ be an ideal, and let $B = A / \mathfrak{a}$. Then we have a descending filtration $$\cdots \subseteq \mathfrak{a}^3 \subseteq ...
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1answer
80 views

$IJ$ is the set of nilpotent elements

Let $R$ be a commutative ring with identity which is Noetherian. Let $V(A)$ denote the set of all prime ideals of $R$ containing the ideal $A$. Suppose that $V(0) = V(I) \cup V(J)$ and $V(I) \cap V(J) ...
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1answer
60 views

what inequalities can one have between $depth\ R$ and $depth\ M$?

Let $(R,m)$ be a commutative Noetherian local ring which is not CM. Let $M$ be a finite $R$-module. what inequalities can one have between $depth\ R$ and $depth\ M$? Obviously there are ...
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2answers
85 views

David Eisenbud, Hilbert theorem

I just started reading D. Eisenbud Commutative algebra with a view towards algebraic geometry and I wonder about a theorem on page 42: If $M$ is a finitely generated graded module over ...
0
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1answer
111 views

Fiber as vector space over residue field.

Let $A$ be a commutative ring with identity and let $M$ be an $A$-module. The fiber of $M$ at $P \in \text{Spec}A$ is the module $M(P):=M_P / PM_P$, which is a vector space over the residue field ...
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1answer
85 views

Integral closure of a DVR in finite extension of fraction field

Let $(K,|\cdot|)$ be a complete valued field and let $L$ be a field extension with $[L:K]<\infty$. Let $\mathcal{O}_K$ be the valuation ring in $K$ and let $\mathcal{O}_L$ be the integral closure ...
2
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1answer
60 views

Notation for the number of times one element divides another.

Let $R$ denote a commutative ring with unity. Consider elements $a,b \in R$. Is there an accepted notation (like $a \| b$ or some such) for the number of times that $a$ divides $b$? Explicitly, we can ...
4
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2answers
166 views

A question on Mumford's drawing of $\text{Spec}\,\mathbb{Z}[x]$

This might seem like a really silly question, but what are those weird curves connecting $(x^2 + 1)$ and $(5, x+2)$ in Mumford's picture of $\text{Spec}\,\mathbb{Z}[x]$?
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1answer
101 views

Factorization of Artinian ring by its nilradical

Theorem. Let $R$ be a commutative Artinian ring with 1 over a field $k$ and $\mathfrak n$ be its nil radical. If char $k \ne 2$ then $R/\mathfrak n$ is isomorphic to a direct sum of fields. My proof. ...
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1answer
134 views

Direct-Sum Decomposition of an Artinian module

Let $R$ be a commutative Noetherian ring. Suppose $M$ is a finitely-generated non-zero Artinian $R$-module. Question: How can we prove that there are maximal ideals $m_1 , m_2 , \ldots , m_n$ such ...
2
votes
2answers
62 views

Units in a ring of fractions

Let $R$ be a UFD and $D \subseteq R$ multiplicative set. What are the units in $D^{-1}R$? I assume the answer should be $D^{-1}R^{\times}$, but I get stuck: If $a/b$ is a unit, then there ...
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1answer
45 views

example of an ideal $I$ in an integral domain $A$ for which there is a prime in $\text{Ass}(A/I)$ that is not in $\text{Ass}(A)$

What is an example of an ideal $I$ in an integral domain $A$ for which there is a prime in $\text{Ass}(A/I)$ that is not in $\text{Ass}(A)$? I've tried constructing one, but all my attempts have ...
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2answers
52 views

Annihilator of extension of scalars vs. the extension the annihilatar

Let $A,B$ be commutative rings with 1, $f:A\to B$ a morphism of rings, $M$ an $A$-module, and $M_B=B\otimes_AM$ the extension of scalars. Then is it the case that $\text{Ann}(M)^e=\text{Ann}(M_B)$? ...
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1answer
61 views

Calculate the support of module

Let $A=k[x,y]$ where $k$ is an algebraically closed field and let $M=A/(xy)$ be an $A$-module. I am supposed to calculate $\text{Supp}(M)= \{ P \in \text{Spec}(A) : M_p \not= 0 \}$ where $M_p = ...
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3answers
222 views

Find the field of fractions and the integral closure of a subring of $\mathbb Z[x]$.

Let $R$ be a subring of $\mathbb{Z}[x]$ consisting of polynomials such that the coefficients of $x$ and $x^2$ are zero. Find the field of fractions of $R$. Find the integral closure of $R$ in it's ...
6
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1answer
189 views

Splitting of an exact sequence

Let $(R,\mathfrak m)$ be a Noetherian local ring. Suppose that $x \in \mathfrak m \setminus \mathfrak m^2$. Is it true that $$ \frac{\mathfrak m}{x\mathfrak m} \cong \frac{\mathfrak m}{(x)} ...
3
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1answer
88 views

Proving an Equivalent Definition of Shellability

Bruns & Herzog (Cohen-Macaulay Rings) give the following definition of a pure shellable simplicial complex: I am stuck in their proof that condition $(b) \implies (c)$: In the argument ...
3
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1answer
96 views

$R$ noetherian, $I$ injective $R$-module $\Rightarrow$ $S^{-1}I$ is injective over $S^{-1}R$

I am trying to prove that if $R$ is a noetherian ring, $S$ a multiplicative part and $I$ an injective $R$-module, then $S^{-1}I$ is an injective $S^{-1}R$-module. So far I thought: I reduce to check ...
6
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1answer
74 views

For which countable successor ordinals $\alpha$ is the reverse order isomorphic to the ideals of a PID ordered by inclusion?

Let $\alpha$ be a countable successor ordinal and $\alpha^{\mathrm{op}}$ the reverse order. For which $\alpha$ is there a commutative principal ideal ring $R$ such that the ideals of $R$ form a chain ...
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2answers
107 views

Why would a field *not* be considered a discrete valuation ring?

There are two theorems in Matsumura (p. 78-9) Theorem 11.1 Let $R$ be a valuation ring. Then the following conditions are equivalent: (1) $R$ is a DVR (2) $R$ is a PID (3) $R$ is ...
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1answer
163 views

Prove that some canonical homomorphism is injective.

Let $A \not= \{0 \}$ be a Noetherian commutative ring and let $M$ be an $A$-module. Prove that the canonical homomorphism $$M \to \bigoplus_{P \in \text{Ass}(M)} M_p$$ is injective. My question is, ...
3
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0answers
85 views

Can $\operatorname{Spec}(A)$ be expressed as an inverse limit?

We know that given a ring $A$ such that $A/\mathfrak{R}$ is absolutely flat, then $\operatorname{Spec}(A)$ is Hausdorff (it's an equivalence). So $Spec(A)$ becomes a quasi-compact, Hausdorff and ...
2
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0answers
53 views

Constant projective dimension of $R/I^i$ for all $i$.

Let $R$ be a local Noetherian ring and $I$ an $R$-ideal. What can we say about the ideal $I$ if the projective dimension of $R/I^i$ for $i \ge 1$ is a finite number which is independent of $i$, i.e., ...
6
votes
1answer
206 views

Most general version of Hensel's Lemma

Roughly speaking, Hensel's Lemma states that a polynomial $f \in O[X]$ over a certain local ring $(O,\mathfrak{m})$ which factors over the residue field $O/\mathfrak{m}$ into coprime polynomials also ...
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0answers
55 views

Spectrum of $\mathbb R[X,Y]$ [duplicate]

Let $A=\mathbb R[X,Y]$. Is it easy to classify the $\operatorname{Spec}A$? I guess it contains at least $(0)$ and $(p)$ for primes $p\in A$ but maybe some else sets. Is it easy to classify those? ...
0
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1answer
144 views

Zero dimensional Gorenstein ring

Let $(R,\mathfrak m)$ be a zero dimensional Gorenstein ring and $\mathfrak q$ be an $\mathfrak m$-primary ideal of $R$. Then TFAE: 1) $\mathfrak q$ is irreducible, 2) $(0:\mathfrak q)$ is principal, ...
3
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1answer
105 views

uniqueness of morphism $Spec(K) \to X$ of schemes

let $K$ be a field and $X$ a scheme. I'd like to understand the bijection $Hom_{Sch}(Spec(K), X) \cong \{x \in X | \exists \kappa(x) \to K \}$ That map is given by sending a morphism $f: Spec(K) \to ...
0
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3answers
134 views

Localisation commutes with taking quotients.

If $A$ is a ring, $S$ a multiplicative set and $I$ an ideal, write $T$ for the image of $S$ in $A / I$. Then $T^{-1}(A/I) \cong S^{-1}A/S^{-1}I$ and in particular, for a prime ideal $P$ we have that ...
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2answers
72 views

Liu, exercise 2.1.4: Minimal prime ideals and nilpotents

In the book "Algebraic Geometry and Arithmetic Curves" Liu wrote in errata that there is a mistake in this problem: Let $A$ be a commutative ring with unit. (a) Let $\mathfrak p$ be a minimal prime ...
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1answer
52 views

A construction in the Proof of Theorem 4.4.9 in Bruns&Herzog

Consider the following theorem and the part of its proof shown: So let $R$ be a homogeneous Cohen-Macaulay $k$-algebra with canonical module $\omega_R$. Let $b$ be the smallest degree for which ...
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0answers
103 views

Computing Hodge numbers of a complete intersection

The situation is this: I have a 5-dimensional irreducible projective variety $Y$ embedded in $\mathbb P^{13}$. This variety is singular, the singularities being a disjoint union of two curves. I have ...
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1answer
53 views

Tensor product of quotient and kernel

In my problem I have a PID $R$, elements $0\neq a,b\in R$ and a map $\phi_a:R\rightarrow R$ where $r\mapsto ar$. Assuming I have done all my previous calculations right I need to prove that ...
0
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1answer
77 views

Question about the support of a module $M$

Let $A \not= \{0 \}$ be a commutative ring and let $M$ be an $A$-module. Define $$\text{Supp} (M) = \{ P \in \operatorname{Spec} A : M_P \not= 0 \}$$ My first question is if $0$ is the element $0/1$? ...
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1answer
74 views

$k$-point after base change

If $X$ is a variety over $k$, is it true that there exists a finite separable extension $k'$ of $k$ such that $X$ has a $k'$-point? What if we can assume $X$ is a smooth projective curve? This seems ...
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1answer
109 views

Prove, that if the commutative ring has no zero divisors, then it is a field [duplicate]

Let $R$ be a commutative finite ring in which $ab = 0$ implies either $a = 0$ or $b = 0$ for any $a,b \in R$. Then, $R$ is a field. I do not understand how I should act. I tried different ways, but ...
2
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1answer
58 views

Existence of Hilbert's polynomial

I heard that Hilbert's syzygy theorem can be used to show the existence of Hilbert polynomials. How does the construction works? Namely, why do every coherent $O$-module $\mathscr F$ the ...
4
votes
1answer
83 views

Open Set of Non-zero Divisors of a Module

Let $R=k[x_1,\dots,x_r]$ be the polynomial ring over the field $k$. Denote by $R_1$ the vector space of linear forms, i.e. all the degree-$1$ elements of $R$. Let $M \neq 0$ be a finitely generated ...
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1answer
104 views

Hilbert Polynomial vs Hilbert Quasi-Polynomial

Let $R$ be an $\mathbb{N}$-graded ring with $R_0$ Artinian and $R = R_0[x_1,\dots,x_r]$, where the degree of $x_i$ is $d_i > 0$. Let $M$ a finitely generated $\mathbb{N}$-graded $R$-module with ...
2
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1answer
98 views

A non flat $R$-module $M$ with $\operatorname{Tor}_{n}^R(k,M)=0$ for all $n\ge 1$

I want to find a non-flat $R$-module $M$ with $\operatorname{Tor}_{n}^R(k,M)=0 \,\, \forall n\ge 1$, where $R=k[x,y]/(xy)$ and $k$ is field.
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0answers
75 views

Derivation (Matsumura: Commutative algebra)

I am reading Masumura, Commutative algebra, Chapter 10: Derivation. The following is in pages 177, 178. Two extensions $(C, \varepsilon, i)$ and $(C_1, \varepsilon_1, i_1)$ are said to be isomorphic ...
5
votes
1answer
130 views

Projective resolution of $k$ over $R=k[x,y]/(xy)$

I want to prove that $\operatorname{Tor}_{n}^{R}(k,k)=k\oplus k,\,\,\forall n\ge 1$. I found the projective resolution $$ R^4\stackrel{d_2} \longrightarrow R^3\stackrel{d_1} \longrightarrow ...
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3answers
142 views

Question about localization

If $A \not= \{0 \}$ is a commutative ring and $P \subset Q$ are prime ideals of $A$ then of course $P \cap (A \setminus Q) = \varnothing$ so that $PA_Q = S^{-1}P$ is a prime ideal of $A_Q$ where $S=A ...
0
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1answer
47 views

An ordered group $G$ is Archimedean if and only if the following holds…

Let $G$ be an ordered group; then $G$ is Archimedean if and only if the following condition holds: $$\text{if} \space a, b \in G \space \text{with} \space a>0, \space \text{ there exists a ...
1
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1answer
76 views

Integral closure of a PID is torsion free

Can anyone explain me why the integral closure of a PID $A$ in a separable finite extension of its fraction field is a torsion free $A$-module? I know that it is a finitely generated A-module ...
2
votes
1answer
107 views

Equivalent conditions for an ideal to be prime

Let $R$ be a commutative ring. An ideal $I$ is called prime if whenever $ab\in I$ then $a\in I$ or $b\in I$. I want to show that $I$ is prime if whenever $JK\subseteq I$, then $J\subseteq I$ or ...
2
votes
1answer
108 views

Why are minimal irreducible closed sets in $A^n$ single points?

In Hartshorne's Algebraic Geometry example 1.4.4, he says A maximal ideal $m$ of $A = k[x_1,\cdots,x_n]$ corresponds to a minimal irreducible closed subset of $A^n$, which must be a point ... I ...
3
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1answer
138 views

Canonical Module and Socle of an Artinian $k$-Algebra

Let $R$ be an Artinian $k$-algebra generated by elements of degree $1$. Denote the canonical module of $R$ by $\omega_R$. By Theorem 3.6.19 in Bruns and Herzog (CMR), we have that $\omega_R = ...
1
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1answer
81 views

Symmetric algebra

If $V$ is a vector space over the field $K$ with basis ${v_1, v_2,…,v_n}$, then the symmetric algebra $S(V)= K[v_1,v_2,..,v_n]$. The question is: If $K$ is a commutative ring, then this equality is ...