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

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3
votes
1answer
47 views

Maximal linearly independent sets in a f.g. module

Suppose $M$ is a finitely generated module over a commutative unital ring $R$. Is it true that every maximal linearly independent set in $M$ has the same size? What is the most general condition ...
0
votes
1answer
51 views

Characterization of the kernel and cokernel of the natural homomorphism between a module and its double dual. [closed]

Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Suppose $$ G \overset{\varphi}{\rightarrow} F \to M \to 0$$ is exact where $F,G$ are finite free modules. Suppose ...
0
votes
1answer
55 views

A finite module over a Noetherian ring is torsionless if and only if it is a submodule of a finite free module

Let $R$ be a Noetherian ring, and $M$ a finite $R$-module. Then $M$ is torsionless if and only if it is a submodule of a finite free module, where torsionless is defined here. (Bruns and Herzog, ...
3
votes
2answers
166 views

What to study from Eisenbud's Commutative Algebra to prepare for Hartshorne's Algebraic Geometry?

I surveyed commutative algebra texts and found Eisenbud's "Commutative Algebra: With a View Toward Algebraic Geometry" to be the most accessible for me. The book outlines a first course in commutative ...
6
votes
2answers
76 views

If $\{M_i\}_{i \in I}$ is a family of $R$-modules free, then the product $\prod_{i \in I}M_i$ is free?

If $\{M_i\}_{i \in I}$ is a family of free $R$-modules, then $\bigoplus_{i \in I}M_i$ is free. Is this true for the product $\prod_{i \in I}M_i$ too?
1
vote
1answer
56 views

Conditions for a quotient module to be Noetherian

I'm solving this problem from "Introduction to Commutative Algebra" of Atiyah and Macdonald. Here is the problem: Let $M$ be an $A$-module and let $N_1, N_2$ be submodules of $M$. If $M/N_1, ...
0
votes
1answer
34 views

Depth zero module and $R$-regular element

Let $(R,m)$ be a commutative Noetherian local ring with $\operatorname{depth}(R)>0$ and $M$ be a finitely generated $R$-module with $\operatorname{depth}(M)=0$. Then can we take an $R$-regular ...
0
votes
1answer
40 views

Another non-regular C-M ring

Among other examples of Cohen-macaulay rings which are not regular I am run into $R=F[x]/(x^2)$ with $F$ a field. It is clear that it is not regular, since every regular local ring must be a domain ...
2
votes
0answers
111 views

Reduced Gröbner basis and extension of scalars

Consider a field extension $L\subseteq K$, and let $\mathfrak a\neq 0$ be an ideal of the polynomial ring $L[T_1,\ldots,T_n]$. Suppose that a monomial order is fixed, so there exists a unique reduced ...
1
vote
0answers
67 views

Cohen structure theorem [closed]

Let $R$ be a one-dimensional noetherian complete local ring and $I$ an ideal of it. Are there any conditions that satisfy $R\cong S/J$, where $S$ is a regular local ring with ideal $J$, such that $J$ ...
7
votes
1answer
97 views

An example of an $m$-primary ideal in noetherian local domain

Is there any example of a $m$-primary ideal $I$ in a noetherian local domain $(R, m)$ such that $I^2=mI\not=m^2 $?
0
votes
1answer
33 views

Both $R$ and $R/I$ are regular local rings

Let $R$ be a Noetherian local ring and $I$ is an ideal of $R$ such that both $R$ and $R/I$ are regular local rings. Could we deduce that $I$ is generated by an $R$-sequence? I know that a ...
1
vote
1answer
32 views

Issue in the first French edition of Serre's local fields

I've been reading Serre's Corps Locaux, and I believe my copy is a first edition, as there's only one copyright date listed, 1968. I believe I found an issue on page 57, which (if you're looking at ...
0
votes
1answer
60 views

Resolution three noncollinear points [closed]

Let $p_1$, $p_2$ and $p_3$ three noncolinear points, and let $R$ be the homogeneous coordinate ring. Show that $R$ have a resolution $$0 \to S^{ \oplus 2}( - 3) \to S^{ \oplus 3}( - 2) \to S \to R ...
0
votes
1answer
72 views

Generalisation of a result on Kahler differentials

Let $B$ be a local ring which contains a field $k$ of characteristic zero, isomorphic to its residue field $B/\mathfrak{m}$. We know that the map $\delta:\mathfrak{m}/\mathfrak{m}^2 \to \Omega^1_{B/k} ...
5
votes
1answer
72 views

Modules over local artinian rings

What is known about the structure of finitely generated modules over local artinian commutative rings $R$? Any information is appreciated. Let us denote by $\mathfrak{m}$ the maximal ideal and by $k$ ...
1
vote
1answer
43 views

Can zero divisors be in the denominator when we localize rings?

Can we localize rings with zero divisors? Can those zero divisors be in the denominator? I thought defining $$\frac{a}{b}=\frac{c}{d} \text{iff }t(ad-bc)=0 \text{ where $b,d,t$ belong to the same ...
1
vote
0answers
61 views

Local complete intersection scheme, conormal sheaves and differentials

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $Z \subset X$ be a local complete intersection subscheme in $X$. Denote by $I_Z$ the ideal sheaf of $Z$ in $X$ and $\Omega^1_X$ the sheaf ...
3
votes
0answers
57 views

Isomorphism between Ext groups in Huybrechts and Lehn's book Geometry of Moduli Spaces of Sheaves

On p.46 (or p. 43 in the 1st edition) of Huybrechts and Lehn book Geometry of Moduli Spaces of Sheaves, 2nd ed., they write: Since $K$ is $A$-flat and $I \otimes_k F_0$ is annilated by $m_A$, ...
6
votes
1answer
58 views

Hilbert Nullstellensatz and ring of continuous functions

Is there any relation between Hilbert's Nullstellensatz and the fact that the maximal ideals in $\mathcal C([0,1])$ correspond to a point in $[0,1]$ (which can be generalized to compact hausdorff ...
1
vote
1answer
44 views

What is $\overline{Y}$ in $\text{Spec}A$?

Consider a subset $Y$ of $\text{Spec}(A)$. (Here $A$ is a commutative ring.) What is the closure of $Y$ (or $\overline{Y}$)? I have been under the impression that $\overline{Y}$ is the set of ...
-1
votes
1answer
35 views

Relation between $\dim R_P/QR_P$ and $\operatorname{height}(P/Q)$ [closed]

If $P⊇Q$ are two prime ideals of a commutative ring $R$ is it true that $\dim R_P/QR_P$ equals $\operatorname{height}(P/Q)$? I appreciate any cooperation.
3
votes
1answer
82 views

If $\mathbb{Z}$ satisfies an identity $\eta$, then every **commutative** ring satisfies $\eta$? And related questions.

Assume all rings have unity and that ring homomorphisms preserve unity. Now by general principles, if every free object in the category of rings satisfies an identity $\eta$, then every object in the ...
1
vote
1answer
37 views

If a proper ideal contains some power of a maximal ideal then the maximal ideal is the only prime ideal that contains the ideal.

Let $R$ be a commutative ring with $1$ and $\mathfrak{m}\subset R$ be a maximal ideal. Show that if $I\subset R$ is a proper ideal containing $\mathfrak{m}^n$ for some $n\geq 1$, then $\mathfrak{m}$ ...
3
votes
1answer
50 views

Use of Zorn's Lemma in showing nilradical equals intersection of primes

I'm very confused as to the use of Zorn's lemma in showing that the nilradical of a ring is the intersection of all the prime ideals. Namely, we let $a \notin N$, where $N$ is the nilradical. Then we ...
1
vote
0answers
64 views

Reid, Undergraduate Commutative Algebra, Exercise 0.23

Let f $\in$ A; if f is reducible then the principal ideal (f) is contained in a bigger principal ideal (f$_1$). Consider the following conditions on a ring A. (a) A is a UFD; (b) every incresing ...
0
votes
0answers
49 views

Proof that $\mathbb{Z}[\sqrt{-5}]$ is integrally closed

There are demonstrations on the Internet saying that the polynomial $$\left(x-\frac{a}{c}-\frac{b}{d}\sqrt{-5}\right)\left(x-\frac{a}{c}+\frac{b}{d}\sqrt{-5}\right)$$ is monic if and only if ...
0
votes
1answer
33 views

an apparent contradiction regarding the local ring at a point

I have encountered an apparent contradiction: Let $Y$ be an affine variety of $\mathbb{A}^n$ and $P$ a point of $Y$. Then i have proved that $\mathcal{O}_P$ is an integral domain and it is also not an ...
0
votes
1answer
43 views

Localization of a direct product

Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.
0
votes
1answer
40 views

Grade of maximal ideals in polynomial rings over Artinian local rings

If $R$ is a commutative Artinian ring it is well-known that $R$ is Cohen-Macaulay. Also, if $S$ is a Cohen-Macaulay ring, then any polynomial ring $S[X_1,\dots,X_n]$ is so. Now if $R$ is a commutative ...
1
vote
2answers
66 views

Can $R[[x]]$ contain constants?

Consider the ring $R[[x]]$ of formal power series $\sum_{n=0}^\infty a_nx^n$ with coefficients in $R$. I was wondering whether $R[[x]]$ contains elements of $R$ (polynomials of degree $0$). I'm ...
-2
votes
1answer
58 views

Is $R[X]/(f)$ Cohen-Macaulay if $R$ is so?

Let $R$ be a commutative (Noetherian) Cohen-Macaulay ring, and $f \in R[X]$ be monic. I guess that $R[X]/(f)$ is also Cohen-Macaulay. Is my hunch valid? Thanks for any help.
1
vote
0answers
98 views

Flat base change preserves the property of being non-degenerate

We say a homomorphism $f:A\rightarrow B$ of noetherian rings is non-degenerate if the induced map $f^*:{\rm Spec}(B) \rightarrow {\rm Spec}(A)$ maps ${\rm Ass}(B)$ into ${\rm Ass}(A)$. Let $f:A ...
0
votes
1answer
52 views

Let $A= \mathbb{Z}[x]$ and $ m=(2,x)$. Find the Krull dimension of $A_m$.

Let $A=\mathbb{Z}[x]$ and $m=(2,x)$. $1$. Then what is the Krull dimension of $A_m$? $2$. If $B=A_m/(x^2-125)$ what is the Krull dimension of $ B $? Any suggestions?
0
votes
1answer
38 views

About weakly associated primes

Let $A$ be a commutative ring, and $M$ an $A$-module. A prime ideal $\mathfrak{p}\subset A$ is said to be weakly associated to $M$ if it is minimal over some $\operatorname{ann}m$, where $m\in M$. I ...
2
votes
1answer
88 views

The family of schemes $\operatorname{Spec} A[x]/(x^n)$

Consider the family $S_n:=\operatorname{Spec} A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for ...
1
vote
0answers
24 views

Why is $A/I^n A \cong \hat{A}/I^n \hat{A}$?

Let $A$ be a commutative ring, $I \subset A$ a finitely generated ideal. Define $\hat{A} := \varprojlim A/I^n$. What is the best way to proof that $A/I^nA \cong \hat{A}/I^n \hat{A}$ for all $n$?
0
votes
2answers
30 views

Question related to integrality of field of fractions

This is actually not a problem, but it's a statement which is taken for granted and I don't know how to prove it. Hope some one can help me. I really appreciate: Suppose $A$ is subring of ...
0
votes
2answers
71 views

A quotient of a Cohen-Macaulay ring

Let $S$ be a Cohen-Macaulay local ring, $I$ an ideal of $S$ and $R=S/I$. If we know that $I$ is generated by $\dim S-\dim R$ elements could we infer that $R$ is Cohen-Macaulay? Thanks in ...
0
votes
0answers
120 views

Algebraic proof of Ehrhart's theorem

Let $P \subset \mathbb{R}^d$ be a $d$-dimensional polytope, where all vertices lie on integral coordinates, and let $L(P,n)$ denote the number of integral lattice points contained in the scaled ...
3
votes
2answers
45 views

ACC on principal ideals implies factorization into irreducibles. Does $R$ have to be a domain?

I am following an argument in chapter zero of Eisenbud's Commutative Algebra book. It is not clear whether or not he is assuming that $R$ is a domain. If I start the proof assuming $R$ is not ...
0
votes
0answers
37 views

Grade of an ideal greater than the projective dimension of quotient of another one

We know that the grade of an ideal $I$ in a Noetherian ring $R$ is the infimum of the set of all $i$ with $Ext^i(R/I,R)$ nonzero. Also, the projective dimension of an $R$-module $M$ is at most $s$ if ...
5
votes
1answer
60 views

From a vector bundle to a Koszul complex

Let $k = \mathbb C$. Given a commutative $k$-algebra $A$, an $A$-module $M$ and a homomorphism of $A$-modules $s:M \to A$, we can construct the Koszul dg algebra. $$K(A,M,s) = \wedge^{-\!*}_A(M)$$ ...
2
votes
3answers
115 views

$S^{-1}A \cong A[x]/(1-ax)$ [duplicate]

If $A$ is a commutative ring with unit, $a \in A $ and $S = \lbrace a^n \mid n \geq 0 \rbrace $ then there is an isomorphism $$S^{-1}A \cong A[x]/(1-ax).$$ In fact we can consider the ...
0
votes
1answer
58 views

If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ satisfies $S_1$

Let $I$ be an ideal of polynomial ring $R=K[x_1,\ldots,x_n]$ and $x$ be a non-zero divisor of $R/I$. Is the following statement true? If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ ...
2
votes
1answer
58 views

What is $\overline{\{ x \}}$ in Atiyah-Macdonald?

On pg. 13 of Atiyah-Macdonald's "Introduction to Commutative Algebra": 18.ii) Prove that $\overline{\{ x \}}=V(p_x)$ What is $\overline{\{ x \}}$? Is it the closure of prime ideal $x$? I assumed ...
1
vote
2answers
69 views

Noetherian Jacobson rings

One of the equivalent forms in definition of a Noetherian Jacobson ring $R$ is that $R$ has no prime ideals $P$ such that $R/P$ is a 1-dimensional semi-local ring. When $R/P$ has dimension 1, it ...
3
votes
1answer
40 views

Rings of algebraic integers

A basic question on algebraic numbers. If $L/K$ is a finite extension of number fields with respective rings of integers $\mathcal O_L$ and $\mathcal O_K$ then is it true that $\mathcal O_L$ is ...
1
vote
1answer
91 views

Morphism of rings and localization

Let $ \varphi : A \to B $ be a morphism of rings. Why are the two following assertions equivalent: $ 1) $ There exists a multiplicative subset $ S $ of the ring $ A $, and an ideal $ I $ of $ A $, ...
2
votes
1answer
73 views

what are the “points” of the scheme $\mathbb{Z}_8[x] /(x^2 + 7)$

I noticed modulo 8 the quadratic $x^2 + 7$ is zero for four separate values $x = 1,3,5,7 \in \mathbb{Z}_8$. The number of zeros exceeds the degree. I would like to define the "variety" ...