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

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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 $k$ ...
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1answer
44 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 ...
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0answers
67 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 ...
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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 $m_A$, ...
5
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1answer
59 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 ...
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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 ...
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1answer
38 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.
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1answer
83 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 ...
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1answer
40 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
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1answer
52 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 ...
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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 ...
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0answers
53 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 ...
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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 ...
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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.
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1answer
43 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 ...
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2answers
68 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 ...
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1answer
64 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.
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0answers
102 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
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1answer
53 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?
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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
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1answer
93 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 ...
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0answers
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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 $n$?
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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 ...
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2answers
83 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 ...
3
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1answer
185 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
46 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 ...
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0answers
40 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 ...
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1answer
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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. $$K(A,M,s) = \wedge^{-\!*}_A(M)$$ ...
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votes
3answers
125 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
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1answer
63 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 $R/(I,x)$ ...
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1answer
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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 ...
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2answers
79 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 ...
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1answer
41 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 ...
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1answer
93 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 $, ...
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1answer
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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" ...
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1answer
49 views

A local PID is a Euclidean domain

Studying commutative algebra I've encountered this statement: A PID which is also a local ring is a Euclidean domain. Is it true ? Why ?
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1answer
42 views

A condition for a homogeneous ideal to be prime

The following is the problem 11 of Chaper 8 Section 4 of Ideals, Varieties, and Algorithms by Cox, Little and O'Shea. A homogeneous ideal is said to be prime if it is prime as an ideal in ...
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1answer
48 views

Commutative ring can be homomorphically mapped onto field

During my algebra lecture, my lecturer used the fact that any commutative ring can be homomorphically mapped onto a field. Is the statement true? How to show that? Thanks
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2answers
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Why is $ \mathrm{Frac} ( A / \mathfrak{p} ) = A_{\mathfrak{p}} / \mathfrak{p} A_{\mathfrak{p}} $? [duplicate]

$ A $ is a commutative ring, $ \mathfrak{p} \in \mathrm{Spec} A $, $ A_{\mathfrak{p}} = ( A \backslash \mathfrak{p} )^{-1} A $, $ \mathrm{Frac} ( A / \mathfrak{p} )$ is the field of fractions of $ A / ...
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0answers
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$M_{\mathfrak{p}} \otimes_{R_{\mathfrak{p}}} N_{\mathfrak{p}} = 0$ implies $M_{\mathfrak{p}} = 0$ or $N_{\mathfrak{p}} = 0$ [duplicate]

Studying commutative algebra I've found this statement: If $M$ and $N$ are finitely generated $R$-modules, with $R$ a commutative ring, and $\mathfrak{p} \subset R $ is a prime ideal, then ...
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1answer
53 views

Computing a regular sequence of generators for an ideal

Let $R = \mathbb{C}[x_1,\ldots,x_n]$. Let $I$ be an ideal, and suppose we know a finite list of generators for $I$, say $I = \langle f_1,\ldots,f_k\rangle$. Is this information enough to compute a ...
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0answers
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Finitely generated idempotent ideal must be generated by an idempotent [duplicate]

Let $A$ be a commutative but not necessarily unital ring. How can we show that a finitely-generated ideal $I$ of a ring $A$ satisfying $I=I^2$ is generated by an idempotent element?
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1answer
71 views

Let I, J ideals. Are they equal?

Let $$I= \langle 11x^5y+7xy^6+9,8xy^4+6xy+9 \rangle$$ $$J= \langle 7x^5y^2+17x^2y^5+29,13xy^4+62xy^3+19 \rangle$$ ideals. Examine whether those two ideals are equal. By seeing their 3D plots I ...
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1answer
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Noether normalization and surjectivity (revisited)

Let $Y$ be an affine variety of dimension $d$ inside the affine space $\mathbb{A}^n$. Then $A(Y) = k[x_1,\dots,x_n]/I_Y=:k[\bar{x}_1,\dots,\bar{x}_n]$. By the Noether normalization theorem, there ...
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1answer
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(Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
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4answers
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Maximal ideal in the ring of polynomials over $\mathbb Z$

Let $\mathbb Z[x]$ the ring of polynomials with integers coefficients in one variable and $I =\langle 5,x^2 + 2\rangle$, how can I prove that $I$ is maximal ideal. I tried first see that $5$ and ...
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1answer
41 views

Relation between faithfully flatness and map of $Spec$

I'm stuck on this exercise ( from Bosch ) : Let $\phi :R \to R' $ a flat ring morphism. Show that $\phi$ is faithfully flat if and only if the associated map $Spec(R') \to Spec(R)$ , ...
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1answer
28 views

Canonical homomorphisms $R_{\mathfrak{p}_i} \to R/\mathfrak{p}_i^n$ are isomorphisms when $R$ is artinian

I'm doing this exercise (from the book of Bosch): Let $R$ be an Artinian ring and let $\mathfrak{p}_1, \ldots \mathfrak{p}_n $ be its (pairwise different) prime ideals. Show that: a) The ...
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1answer
42 views

Common equivalent conditions for Cohen-Macaulayness of a ring

I know the fact that a local ring $(R,m)$ with $\dim(R)=d$ is Cohen-Macaulay (C-M) if and only if any one of the following holds: 1) $\operatorname{grade}(m)=\operatorname{height}(m)$ 2) ...
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1answer
51 views

Rings which are finitely generated and free over Cohen-Macaulay rings are also Cohen-Macaulay

Let $S$ be a Cohen-Macaulay (C-M) ring, and $R$ a ring containing $S$ such that as an $S$-module is finitely generated free. Could we deduce that $R$ is also C-M? I guess probably we could use ...