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

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5
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4answers
131 views

Show that $(A / \mathfrak{a}) \otimes_A M \cong M / \mathfrak{a} M$ for a ring $A$, ideal $\mathfrak{a}$, $A$-module $M$.

This is Atiyah-Macdonald Exercise 2.2 Exercise: Let $A$ be a ring, $\mathfrak a$ an ideal, $M$ an $A$-module. Show that $(A/\mathfrak a) \otimes_A M$ is isomorphic to $M/\mathfrak aM$. [Tensor the ...
2
votes
1answer
65 views

Nontrivial example of an artin algebra R such that R is pure-injective as an R-module

Give a nontrivial example of an artin algebra $R$ such that $R$ is pure-injective as an $R$-module. Clearly $0$-Gorenstein (self-injective) artin algebra has this property. Can anyone give me ...
0
votes
2answers
62 views

When a prime ideal is restricted to a basic open subset of projective space, is it still prime?

Suppose $I\subset k[x_0,\ldots,x_n]$ is a prime ideal. Now restricted on the basic open subset $\mathbb{P}^n_{x_i}$ of $\mathbb{P}^n$, is $I$ still prime? Note: 1. Here $\mathbb{P}^n_{x_i}$ is ...
6
votes
0answers
51 views

$E \to S$ surjective in degrees $\geq 1$ implies $\widetilde{E} \to \widetilde{S}$ surjective

In the proof of Theorem II.8.13 in Hartshorne (which is the Euler sequence), the author writes: Let $S = A[x_0, \ldots, x_n]$. [...] The exact sequence $$0 \to M \to E \to S$$ of graded ...
2
votes
1answer
46 views

Restriction and extension of scalars between flat algebras and their completion over a DVR and ideals.

So, in a proof I am currently reading I have stumbled upon the following. Let $R$ be a discrete valuation ring, $\hat{R}$ its completion and $t$ a uniformizing parameter for $R.$ Let $A$ be a flat ...
1
vote
2answers
49 views

pictorial illustration of simplicial complexes

Consider the following two complexes (Bruns&Herzog p.215): By just looking at the complex on the left, i am not sure how to read its faces. Surely its vertices are $v_1,v_2,v_3,v_4,v_5$. The ...
3
votes
2answers
127 views

Invertible matrices in commutative rings

Let $A$ be a square matrix over a commutative ring $R$. Then $A$ has a left inverse iff it is invertible. Does there exist a elementary proof of this fact? (i.e. without using the determinant!)
1
vote
1answer
32 views

local PID that is not a field is a DVR

I would be very happy if someone would check my proof of the fact that a local PID that is not a field is a DVR: Let $A$ be a local PID that is not a field. Since irreducibles generate maximal ideals ...
1
vote
2answers
75 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} ...
0
votes
0answers
54 views

Picard group of affine scheme of a UFD

In which book/notes can I find proofs of the following facts? 1) Pic(Spec$A)$ is $0$ where $A$ is a UFD. 'Pic' is the Picard group. 2) The invertible sheaves on projective space P$^n(k)$ for $k$ a ...
9
votes
2answers
341 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 ...
4
votes
1answer
70 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 ...
1
vote
1answer
34 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) ...
1
vote
1answer
32 views

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

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 ...
0
votes
2answers
73 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
votes
1answer
82 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 ...
1
vote
1answer
21 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
votes
1answer
50 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
votes
2answers
88 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 $\text{Spec}\,\mathbb{Z}[x]$?
3
votes
1answer
33 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. ...
1
vote
1answer
84 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
44 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 ...
0
votes
1answer
42 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 ...
0
votes
2answers
30 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 $\text{Ann}(M)^e=\text{Ann}(M_B)$? ...
0
votes
1answer
51 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 = ...
0
votes
3answers
86 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 ...
3
votes
0answers
154 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)} ...
2
votes
0answers
63 views

If every maximal ideal of $R$ is principal, is every ideal of $R$ principal? [duplicate]

Let $R$ be a commutative ring and every ideal maximal of $R$ is principal (generated by only one element). Is every ideal of $R$ principal? Please help me my friends. It's necessary for me.
1
vote
1answer
68 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 condition (c): In the argument ...
3
votes
1answer
76 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 ...
3
votes
0answers
27 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 ...
1
vote
2answers
58 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 ...
0
votes
1answer
117 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
votes
0answers
67 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
votes
0answers
30 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., ...
5
votes
1answer
108 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 ...
1
vote
0answers
41 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
votes
1answer
79 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, ...
2
votes
1answer
55 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 ...
1
vote
3answers
71 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 ...
1
vote
2answers
58 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 ...
0
votes
1answer
41 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 ...
6
votes
0answers
66 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 ...
0
votes
1answer
40 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
votes
1answer
64 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 $0/1$? ...
1
vote
1answer
63 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 ...
1
vote
1answer
79 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 ...
1
vote
0answers
53 views

A minimal prime ideal consists of zerodivisors [duplicate]

Let $A$ be a unital commutative ring (I do not assume $A$ to be Noetherian). Let $\mathfrak{p} \subset A$ minimal prime ideal. Question: Are all elements of $\mathfrak{p}$ zero divisors? Comment: I ...
2
votes
1answer
41 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
80 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 ...