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

learn more… | top users | synonyms (1)

2
votes
0answers
105 views

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 ...
0
votes
1answer
43 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 ...
1
vote
1answer
65 views

Choice of ideal so that localisation of a module must be free?

If $R$ is a Noetherian (commutative) ring and $M$ is a finitely generated $R$-module, then what choice of nonzero ideal $I$ of $R$ is such that $M_P$ is a free $R_P$-module for any prime $P$ in $R$ ...
4
votes
2answers
174 views

Applications of powerful theorems in Bruns -Herzog's book “Cohen-Macaulay Rings”

It seems that theorem 1.4.13 and it's corollary of Bruns and Herzog's book Cohen-Macaulay Rings, are powerful tools but I don't see any example that shows the power of it. My original question was an ...
3
votes
1answer
64 views

Inequality amongst projective dimensions!?

Assume $φ : R\to S$ is a ring homomorphism between commutative rings sending unity to unity. Taking any $S$-module $M$ as an $R$-module, is it true that always $$\operatorname{pd}_R (M)\le ...
0
votes
2answers
65 views

Zero direct summand

I want a suggestion for the following: Let $M$ be a finitely generated $R$-module, where $R$ is a commutative Noetherian local ring with $1_R$, and $N$ a direct summand of $M$ such that $N⊆mM$, ...
4
votes
0answers
101 views

Application of Zariski's Main Theorem

Suppose $f: A \to B$ is a local homomorphism, $B$ is isomorphic to a localization of an $A$-algebra of finite type. Let $L$ be the field of fractions of $B$, and suppose that $B$ contains the normal ...
0
votes
0answers
80 views

Alternative proof for localization isomorphism

Let $f$ be an $A$-module morphism and $\operatorname{res}_{A_m}^A$ be the restriction of scalars functor from $A_m$-mod to $A$-mod. I'm curious if you have proven that for every maximal ideal ...
1
vote
1answer
99 views

Projective dimension zero or infinity

Let $R$ be a commutative Noetherian local ring (having unity) with the maximal ideal $m$, which consists only of zero-divisors. Then for any finitely generated $R$-module $M$, the projective ...
0
votes
0answers
50 views

Discrete Valuation for Local Rings

Let t be the uniformizing parameter of the local ring $O_{C,P}$, where $C \subseteq A^2_k$ is an irreducible variety, k a field, and $P \in C$. Let $\alpha \in O_{C,P}$. Prove that there exist ...
2
votes
1answer
76 views

Prove that $m^2$ is primary

Let $m$ be a maximal ideal. I'm having a hard time proving that $m^2$ is primary. Let ${xy\in m^2}$ so $xy=t_{1}s_{1}+...+t_{n}s_{n}$ where the $t_{i},s_{i}$ are in $m$.
1
vote
1answer
79 views

Proving A Discrete Valuation Ring

I am given the definition that $R$, an integral domain, not a field is a DVR if either: R is a local Noetherian ring such that its maximal ideal is principal. There exists an irreducible element $t ...
2
votes
1answer
90 views

Product of ideals generated by linear forms that has zero dimension

Let $k$ be a field, $R=k[x_1,\dots,x_r]$ the polynomial ring in $r$ indeterminates and $I_1,I_2\dots,I_d$ homogeneous ideals of $R$ generated by linear forms. Define $J = I_1 I_2 \cdots I_d$ and ...
1
vote
2answers
267 views

$Spec(R)$ Hausdorff and totally disconnected

I've found the following claim. If $R$ is a commutative ring, and every prime ideal is maximal, then $Spec(R)$ with the Zariski topology is Hausdorff and totally disconnected. Is it true ? Why ?
0
votes
2answers
677 views

When is nilradical not a prime ideal

Atiyah gives this criterion for nilradical to be a prime ideal.Nilradical is the intersection of prime ideals.Is nilradical prime iff there is only one prime ideal? ie Intersection of distinct prime ...
6
votes
2answers
467 views

Exercise 2.27 Atiyah-Macdonald, absolute flatness

A commutative ring $R$ is absolutely flat if every $R$-module is flat. Prove that the following are equivalent: 1) $R$ is absolutely flat 2) Every principal ideal of $R$ is idempotent 3) Every ...
2
votes
0answers
53 views

Krull dimension of a direct limit of modules

Suppose that $\left\{M_{\lambda}\right\}$ is a directed system of $R$-modules, all of them with finite Krull dimension (specifically, all of them of dimension $n$). Is it true that $\dim\varinjlim ...
1
vote
0answers
107 views

Resolution of module over polynomial ring

The problem is: Let $F$ be a field, and let $R = F[x_1, \ldots, x_r]$, the polynomial ring over $F$. Consider the $R$-module $M = R/(x_1, \ldots, x_r) \cong F$. Find a resolution of $M$ by free ...
7
votes
1answer
200 views

The analytic spread of an ideal

Let $(R,\mathbb{m})$ be a Noetherian local ring and $I$ an ideal of $R$. Let $t$ be an indeterminate over $R$. The analytic spread $l(I)$ of $I$ is defined to be the Krull dimension of the ring ...
0
votes
2answers
102 views

Make ring in natural way

Let $S$ be a subset in a commutative ring $R$, such that: $1 \in S$ $\forall x,y \in S \qquad xy\in S$ Define a relation $\sim$ on the Cartesian product $R\times S$ through ...
1
vote
1answer
202 views

Singular points and tangent lines

Find the singular points and the corresponding tangent lines of $X^4+Y^4-X^2Y^2$. I get that the singular points are those of the form $(a,a)$ and $(a,-a)$. The corresponding tangent lines are ...
2
votes
1answer
172 views

Tor for graded modules over a graded ring

I am confused about how this Tor is defined. Suppose $R$ is a graded ring, $M,N$ graded modules over $R$. What is $\operatorname{Tor}_{st}^R(M,N)$? I am confused about the subscripts. I realize ...
4
votes
1answer
124 views

Intersection multiplicity

Let $f=y^2-x^3$ and $g=y^3-x^7$. Calculate the intersection multiplicity of $f$ and $g$ at $(0,0)$. I know the general technique for this (passing to the local ring) but I having difficulty with ...
3
votes
0answers
80 views

Modules with finite injective dimension have $\omega_R$-resolutions

Let $(R,m,k)$ be a local Noetherian ring, $M$ a finitely generated $R$-module. How can I prove that $M$ has finite injective dimension if and only if it has a $\omega_R$-resolution? ($\omega_R$ is the ...
1
vote
1answer
135 views

Tor functor for the quotient of a Gorenstein local ring

Let $(R,m)$ be a Gorenstein local ring, $I\subset R$ a perfect ideal of grade $g$ and $S = R/I$. Prove that $S$ is Gorenstein iff $\operatorname{Tor}_g^R(S,S)=S$. This question is Exercise 3.3.25(c) ...
5
votes
1answer
177 views

Rational singularities for normal varieties

On page 17 of this paper there is the following claim. For $f: Y \rightarrow X$ a proper birational map with $Y$ smooth (i.e. a desingularization of $X$) and $X$ is a normal variety, $R^i f_* ...
3
votes
2answers
69 views

Question about the set of all $\mathfrak p\in\operatorname{Spec}R$ such that $M_{\mathfrak p}=0$.

Is the following subset of $\operatorname{Spec}(R)$ always open? $R$ is a commutative, unitary ring and $M$ is an $R$-module.$$\{\mathfrak p \in\operatorname{Spec}(R) \colon M_{\mathfrak p} = 0\}$$
3
votes
1answer
104 views

Castelnuovo-Mumford regularity of a product

Let $k$ be a field, $R = k[x_1,\dots,x_n]$ the polynomial ring, $\mathfrak m = (x_1,\dots,x_n)$ and $M$ a finitely generated graded $R$-module. How can we see that $\operatorname{reg}(\mathfrak mM) ...
3
votes
1answer
149 views

Minimal primary decomposition

Let $m$ be an integer ${\geq}3$ and $f(x,y,z)=y^m(x+y^3)-z^3$ in $k[x,y,z]$. Find the singular points of $f$ and find a minimal primary decomposition of the jacobian of $f$. I find the set of ...
2
votes
1answer
91 views

a proposition on formal smoothness

Proposition: Let $A\to B$ be a local homomorphism of noetherian complete local rings. Assume $A$ is regular of dimension $d$, with residue field $k$. Assume $\dim B=d+r$ and $B\otimes_A k$ is formally ...
2
votes
0answers
101 views

Proof of a corollary of the Noether normalisation lemma

I can't understand the proof of a corollary of the Noether normalisation lemma in Undergraduate algebraic geometry by Miles Reid. Noether normalisation lemma Let $k$ be an infinite field, and ...
3
votes
0answers
108 views

When does the inverse limit preserve the localisation?

Question When is the following true? $$\varprojlim(S_\alpha^{-1}A_\alpha)\cong(\varprojlim S_\alpha)^{-1}(\varprojlim A_\alpha)$$ (For details, consult the next part.) Notations One can ...
1
vote
1answer
94 views

Find intersection multiplicities

Let curves $A$ and $B$ be defined by $x^2-3x+y^2=0$ and $x^2-6x+10y^2=0$. Find the intersection multiplicities of all points of intersection of $A$ and $B$. If we let $f=x^2-3x+y^2$ and ...
0
votes
0answers
50 views

Some basic definitions

Let $k$ be a field and $f,g$ be algebraic affine curves in $A_{k}^2$. 1) What does $f,g$ algebraic mean? I know that a set is algebraic if it is equal to $V(I)$ for some ideal $I$ of the ...
5
votes
1answer
232 views

Integral dependence and (faithfully) flat ring extensions

Let $R\to S$ be a flat ring extension. By theorem 9.5 of the book Commutative Ring Theory written by Matsumura the going-down theorem holds between $R$ and $S$. Is it true (or not) about these ...
4
votes
1answer
148 views

In a faithfully flat ring extension, is $\operatorname{ht}I=\operatorname{ht}IS$ right?

For Noetherian rings $R$ and $S$, let $R\rightarrow S$ be a faithfully flat ring extension and $I$ an ideal of $R$. Does $\operatorname{ht}I=\operatorname{ht}IS$ hold? Is it a conclusion in some ...
9
votes
1answer
126 views

Converse of fundamental theorem of finitely-generated modules

Let $R$ be a ring. If every finitely-generated $R$-module $M$ is isomorphic to a finite direct product of quotients of $R$ by ideals then call $R$ a wheel ring. For a domain $R$ we have the ...
4
votes
1answer
164 views

Localization in formal power series

I saw in a textbook the following assertion: Let $R$ be a commutative ring with unity, and $R[[X]]$ be the ring of power series in one indeterminate $X$. If the homomorphism $\phi∶ R[[X]] \to R$ ...
2
votes
1answer
128 views

Exercise 2.26 Atiyah-Macdonald, flatness

I'm stuck on this exercise. $A$ is a commutative ring with unit. $N$ is an $A$-module. Then $N$ is flat $\Longleftrightarrow $ $\text{Tor}_{1}(A/a, N ) = 0 $ for every finitely generated ideal $a$ of ...
1
vote
1answer
41 views

$\operatorname{Hom}_\Lambda (B, \operatorname{Hom}_{\Bbb Z}(\Lambda, X)) = \operatorname{Hom}_\Bbb Z (B, X)$

Let $G$ be a group, $B$ a $G$-module and $X$ an abelian group. Let $\Lambda:=\Bbb Z[G]$. Serre states in his book local fields that we have the equality: $$\operatorname{Hom}_\Lambda (B, ...
4
votes
1answer
271 views

Exercise 2.11 Atiyah-Macdonald [duplicate]

This is part of the exercise, I'm stuck with it. $A$ is a commutative ring with unit. 1) Suppose we have an homomorphism $\phi : A^{m} \to A^{n}$ surjective. Is true that $m \geq n $ ? 2) Suppose ...
0
votes
1answer
76 views

Equality with powers of an ideal

Let $A$ be an arbitrary (commutative with an identity) ring. Suppose $\alpha$ is an ideal. Is it true that $$\alpha(\alpha\cap\alpha^2\cap\alpha^3\cap…)=\alpha\cap\alpha^2\cap\alpha^3\cap…?$$ ...
3
votes
0answers
79 views

Approximating modules over complete local rings

Let $A$ be a complete local noetherian ring with maximal ideal $\mathfrak{m}$. Is the canonical functor $$\mathsf{Mod}(A) \to \varprojlim_n ~ \mathsf{Mod}(A/\mathfrak{m}^n),~ M \mapsto ...
1
vote
1answer
70 views

Exercise from Atiyah about flatness

This is an exercise from Atiyah. Let $N$ be a flat $B$-module, and $B$ a flat $A$-algebra where $A$ is a commutative ring with unit. Then $N$ is flat as $A$-module Any hint ?
0
votes
1answer
55 views

kernel of k-homomorphism

Let's $f_1,\ldots,f_m \in k[x_1,\dots,x_n]$ and the $k$-algebra homomorphism: $$g:k[x_1,\ldots,x_n,y_1,\ldots,y_m] \rightarrow k[x_1,\ldots,x_n]$$ that sends $y_i \mapsto f_i$ and $x_j \mapsto x_j$ ...
0
votes
1answer
92 views

Is the homomorphic image of a G-domain is G-domain?

I have no idea how to prove this if it is true or to give a counter example if it is not true. Is the homomorphic image of a G-domain is a G-domain? A G-domain is an integral domain $R$ with ...
1
vote
2answers
92 views

Example of Localization and Prime Ideals

For each $n\in \mathbb Z^+$, give an example of a localization of $\mathbb Z$ with exactly $n$ prime ideals. Justify your answer. Could an example have something to do with a UFD or Noetherian ...
0
votes
2answers
386 views

Spectrum of a product of rings isomorphic to the product of the spectra

I've found in an exercise this statement: If $A$ is a commutative ring with unit and $A = A_{1} \times \dots \times A_{n}$ then $$\def\Spec{\operatorname{Spec}} \Spec(A) \cong \Spec(A_{1})\times ...
12
votes
1answer
265 views

What is the intuition behind the name “Flat modules”?

I am studying Atiyah and MacDonald's book "Introduction to Commutative Algebra" and I have just read the definition of a flat module. It seems to me that if they have called that kind of modules ...
1
vote
1answer
101 views

Fraction field of $R/P$

It may be a simple question seeming too easy, but I seek a help: If $P$ is a prime ideal of a commutative ring $R$, could one say that $R_P/PR_P$ is the field of fractions of $R/P$? Thanks a ...