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

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47 views

Why is this intersection supported on the closed point?

Let $R$ be a (commutative unitary) local ring. Let $M$ and $N$ be finitely generated $R$-modules such that $\mathrm{length}(M\otimes_R N)$ is finite. Let $x$ be the closed point of $X=\operatorname{...
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1answer
35 views

A good reference for irreducible and noetherian spaces

I am looking for a comperhensive reference for irreducible and noetherian topological spaces. Also, a reference for prime spectrum of a commutative ring.
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1answer
24 views

Essential Prime Ideal

I search for an example of a commutative ring $R$ with unity having a prime ideal $P$ and some element $r\in R$ such that the annihilator of $r$ is both contained in $P$ and essential in $R$. By ...
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2answers
77 views

Quotient of a polynomial ring localized

Question: Prove that $\mathbb{R}[x,y]/(xy)$ localised at $(x-a)$ is isomorphic to the ring $\mathbb{R}[x]$ localised at $(x-a)$. Related question: What is the local ring at the point $(0,0)...
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1answer
40 views

Are formal power series rings over Dedekind domains formally smooth?

Let $A$ be a Dedekind domain. Consider the ring of formal power series $A[[t]]$ over $A$. Now let $B$ be any $A$-algebra, and let $N\subset B$ be a nilpotent ideal. Then, can any homomorphism $$A[[t]...
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1answer
49 views

Prove that every maximal ideal of a commutative ring $R$ with $R^2=R$ is prime

Prove that every maximal ideal of a commutative ring $R$ (not assumed to have $1$) with $R^2=R$ is prime. If $M$ is a maximal ideal of $R$, I am trying to prove that for all $a,b,ab \in M$ implies $...
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1answer
44 views

Intersection of flat submodule with direct summand

Let $R$ be a (commutative) domain, $M$ a flat $R$-module which decomposes as $M=A\oplus B$ and $N$ a (not necessarily pure) flat submodule of $M$. Is it the case that $N \cap A$ is always a pure ...
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2answers
35 views

Top exterior product of exact sequence

Let $M,N,P$ be free $R$-modules of rank $a,a+b,b$ respectively, and that they fit into an exact sequence $0\to M\to N\to P \to 0$. Is it true that $\Lambda^{a+b}N=\Lambda^aM \otimes \Lambda^bP$? (...
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1answer
32 views

$\frac{\mathbb{Z}} {p^k\mathbb{Z}}$ as $\mathbb{Z}$-module localized at $(p)$

Consider $\frac{\mathbb{Z}} {p^k\mathbb{Z}}$ as $\mathbb{Z}$-module, where $p\in \mathbb{Z}$ is a prime. What is, up to isomorphism, the localized $(\frac{\mathbb{Z}} {p^k\mathbb{Z}})_{(p)}$ as a $\...
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1answer
78 views

Lower bound on dimension of fibres of a dominant mophism of irreducible affine varieties

Whilst doing exercise $11.4.B$ of Ravi Vakil's "Foundations of Algebraic Geometry", I got stuck with the following problem (although I think that many of the hypotheses are unnecessary and a more ...
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1answer
25 views

Why the ideal norm is multiplicative

Let $I\subseteq B$ be an ideal, we define the ideal norm of $I$ as the ideal in $A$ generated by the elements $N_{E/K}(\alpha)$ where $\alpha \in I.$ We denote it by $N_{E/K}(I).$ If $\mathfrak{p}$ ...
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3answers
80 views

What does Hom(M,N) mean? Atiyah Macdonald proposition 2.9

In Atiyah Macdonald, "Introduction to commutative Algebra" it says: Proposition 2.9:i) Let $M' \xrightarrow[]{u}M \xrightarrow[]{v} M'' \rightarrow 0$ be a sequence of A-modules and homomorphisms. ...
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1answer
59 views

Do two rational parametric curves intersect only finitely many times?

Suppose there are two rational parametric curves $f = (f_1, \ldots, f_n)$ and $g = (g_1, \ldots, g_n)$ in $\mathbb{R}^n$. I read somewhere that such a parametric expression can always be transformed ...
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0answers
29 views

Finite module over Noetherian ring faithfully flat?

If I have Noetherian rings $B=A^G\subset A$ (for some action of finite group $G$, maybe not relevant) and $A$ is finite as $B$-module. Is it always true that $A$ is faithfully flat over $B$? EDIT: ...
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0answers
32 views

Krull's intersection theorem and closedness of submodules

In the book "Formal and rigid geometry" by Bosch, page 70, there is a claim that if $\varphi :\operatorname{Sp}A\to \operatorname{Sp}A^{\prime}$ is a map of affinoid spaces, $x\in \operatorname{Sp}A^{\...
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2answers
45 views

Auslander-Buchsbaum formula without minimal/finite resolutions

Does anybody know a proof of Auslander-Buchsbaum's formula that uses only projective/injective/flat resolutions and homological functors Ext and Hom without using minimal/finite resolutions?
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43 views

Certain map of modules is iso [Mumford Abelian Varieties]

I have trouble showing the following in the proof of Prop. 2 in Abelian Varieties (pg.70 my edition, Chapter about quotients by finite groups): Suppose you have a Noetherian ring $B=A^G$ as ...
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1answer
35 views

There always exists a finite, increasing chain of R-submodules of M isomorphic to R/P. Can we describe P?

So I've been studying some commutative algebra and I came across the following theorem Theorem : Let R be a Noetherian ring. Let $M$ be a non trivial $R$-module, finite over $R$. There exists a ...
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0answers
80 views

How do ring theorists think about square roots?

Let $R$ denote a commutative ring. Then it seems to me that we can adjoin to $R$ a square-root of $4$ as follows: $$R[\sqrt{4}] = R[x]/(x^2-4)$$ This defines a functor $\mathbf{CRing} \rightarrow \...
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2answers
84 views

Give an example of a commutative von Neumann regular ring which is not a product of fields

One knows that every commutative von Neumann regular ring with a finite Boolean algebra of idempotents is a product of fields. Give an example of a commutative von Neumann regular ring which is ...
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1answer
39 views

Well-definedness of homogeneous coordinate ring of projective scheme [closed]

Let $I,J \subset S=k[x_0,...,x_n]$ be homogeneous ideals. How can I show that $\operatorname{Proj}S/\bar{I}=\operatorname{Proj}S/\bar{J}$ iff $\bar{I}=\bar{J}$? (Here $\operatorname{Proj}R$ is the ...
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1answer
145 views

Algebraically Closed Quotient Fields

It is well-known that if the quotient field of a commutative noetherian integrally closed domain $R$ is algebraically closed, then $R$ is a field. The proof is easy: let $r_0 \in R$ and choose $r_i ...
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1answer
42 views

Saturation of homogeneous ideal

Let $I \subset S=k[x_0,...,x_n]$ be a homogenous ideal. The saturation of $I$, $\bar{I}$ is defined to be $\{s \in S: \exists m \; s.t. \; \forall i \; x_i^m s \in I\}$ Is it true that $\bar{I}=(s \...
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48 views

What is extension of scalars used for in algebraic geometry?

Given a ring homomorphism $f:A \rightarrow B$ and an $A$-module $M$, one can construct and $A$-module with the tensor product: $M_B=B \otimes_A M$ which has a $B$-module structure. This is said to be ...
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1answer
25 views

Commutative ring which is essential extension of each of its non-zero ideals

Let $R$ be a commutative ring with unit. Assume $R$ is an essential extension of each of its non-zero ideals. I feel that there should be something in the literature about this, but I could not find ...
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2answers
63 views

Showing the ideal $\left \langle yz,xz,yx+ay,x^2+ax \right \rangle$ is radical for all $a\neq 0$

Let $I_a = \left \langle yz,xz,yx+ay,x^2+ax \right \rangle$ be an ideal of $k[x,y,z]$, where $a \neq 0$. Show that $I_a$ is radical. What is the geometric meaning of the elements in $\sqrt{I_0}\...
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1answer
43 views

Zeros of specialization of a family of polynomials [closed]

Let $k$ be an algebraically closed field, and $K\supset k$ be an algebraically closed extension. Let $a\in K^n$ be a tuple, we call $a^\prime\in k^n$ a specialization of $a$ if for any $f(X)\in k[X]$ ...
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1answer
36 views

$B/I$ and $B/J$ flat $A$-algebras; does $I=J$ hold?

Let $A\to B$ be a ring homomorphism. Consider $I$ and $J$ ideals of $B$ such that $B/I$ and $B/J$ are flat $A$-algebras. We know furthermore that there exists a non zero-divisor $t\in A$ such that $(B/...
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2answers
43 views

Flat module and finite intersection of submodules

Let $R$ be an integral domain, $F$ be a flat $R$-module, and $A$ and $B$ are two $R$-submodules of $Q$, where $Q$ is the quotient field of $R$. How can we show that $F\otimes (A \cap B) = (F\otimes A) ...
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0answers
19 views

Integral basis of an extension of complete fields

Let $\mathcal{O}_K$ be a complete discrete valuation ring with quotient field $K = \text{Quot}(A)$. Let $L | K$ be an arbitrary finite field extension. Because $K$ is henselian, the integral closure $\...
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1answer
27 views

Singular ideals and rings

In Lam's book, Corollary (7.4)(2) says that for a nonzero ring $R$ we have $Z(R_R)≠ R$, where $Z(R_R) $ stands for the singular ideal of $R$.. But, some nonzero commutative rings are "singular" in the ...
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3answers
89 views

Recommended books on commutative algebra stressing links with algebraic geometry

Can someone recommend some books on commutative algebra stressing links with algebraic geometry? My concern is this. It seems to me that most of commutative algebra was formulated at least initially ...
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2answers
60 views

Help in showing that the cusp $(y^2-x^3)\subset \mathbb{C}^2$ is not isomorphic to $\mathbb{C}$

Let $X:=(y^2-x^3)\subset \mathbb{C}^2$ be the vanishing of the polynomial $f(x,y)=y^2-x^3.$ I have proved an exercise in Hartshorne: If $\varphi:\mathbb{C} \to X, \ t \mapsto (t^2,t^3)$ is the ...
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1answer
157 views

normalization of a curve, simplest example

I am learning about normalization of nodal curves and I am trying to understand the simplest example: $xy=0$ As far as I understand its coordinate ring is $k[x]\oplus k[y]$ (let $k$ be an ...
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0answers
55 views

Non-zero ideal in algebraic integers generated by two elements

I've been doing past questions for my exams next week and would like to check an answer: Let $I$ be a non-zero ideal of the algebraic integers and let $0\neq a \in I$. Show that $\exists b \in I$ ...
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1answer
55 views

What does $(0:x)$ mean?

The following excerpt is from Eisenbud's "Commutative Algebra with a view toward Algebraic Geometry" on pg. 424 We can decide whether an element $x\in R$ is a nonzerodivisor from the homology of ...
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93 views

Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.

I would like to have a result similar to "Traces of all positive powers of a matrix are zero implies it is nilpotent". Namely: Let $R$ be a commutative $\mathbb{C}$-algebra, $A \in \mathcal{M}_n(...
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2answers
74 views

Integrally Closed domain and Principal Ideal

Let $R$ be an integrally closed local domain. Suppose there is a $y\in I^n$ such that $yI^n=I^{2n}$ for some $n$. I would like to prove that $I^n=(y)$. Source: The above question comes from the ...
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0answers
50 views

Conjugation in algebraic number theory

Let $K$ be an algebraic number field of deg $n$ over $\mathbb Q$, then given $\alpha \in$ $O_k$ its ring of integers, we can choose a $\mathbb Q$-basis $\omega_1, \omega_2, ...,\omega_n$ of $K$ s.t. $\...
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1answer
102 views

Is every “prefield” a field?

Definition 0. Call a poset $P$ well-ranked iff it is well-founded, and for all $x \in P$, we have that any two maximal subchains in the lowerset generated by $x$ have the same length. Definition ...
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35 views

Analytical isomorphism implies same multiplicities [duplicate]

I want to prove the following problem in Robin Hartshorne's Algebraic Geometry Chapter 1 exercise 5.14 If $P\in Y$ and $Q\in Z$ are analytically isomorphic plane curve singularities, show that the ...
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1answer
26 views

Regular element of a Noetherian ring [duplicate]

Let $R$ be a Noetherian ring and $x\in R$ an $R-\mathrm{regular}$ element. Show that $\mathrm{Ass}_R(R/(x^n))=\mathrm{Ass}_R(R/(x))$ for every $n\geqslant 1$. Let $M$ be an $R-\mathrm{module}$. An ...
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0answers
31 views

Characterization of prime homogeneous ideals

Let $R$ be a graded ring and $I$ ideal in $R$ and homogeneous. $I$ is prime if and only if for all $a, b\in R$ homogeneous such that $ab\in I$ then $a\in I$ or $b\in I$. Let $ab\in I$ and $a = ...
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1answer
46 views

Existence of homogeneous non-unit non-zero divisor in a particular graded ring.

Let $R$ be a finitely generated $k$-algebra of dimension greater than $1$, let $Q$ be any maximal ideal of $R$. It is claimed by my lecturer that one can find a homogeneous, non-unit, non-zero divisor ...
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1answer
28 views

Exact sequence of graded modules and localization

I know that a sequence of modules is exact iff the localization at each prime ideal is exact What happens in the case we are working with graded modules? Can we say that a sequence is exact iff the ...
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33 views

Local ring of an affine curve $K$ at a point $p\in K$

I'm reading A Royal Road to Algebraic Geometry by Holme. The book defines the local ring as follows: The local ring of $K$ at $P=(a,b)$ is the ring $$\mathcal{O}_{K,P}=\Gamma(K)_{\mathfrak{m}(a,...
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1answer
41 views

Flatness of quotient rings

The following is Exercise 2.4, in Chapter 1 of Liu, Algebraic Geometry and Arithmetic Curves: Let $I$ be a finitely generated ideal of $A$: $A/I$ is flat. $I^2 = I$. $I = (e)$ where $e^2=e$. I ...
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1answer
38 views

Hilbert function and Hilbert polynomial

I have largely studied Hilbert function and Hilbert polynomial for polynomial rings over fields of characteristic zero. Is it possible to extend the theory also for polynomial rings over fields of ...
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1answer
82 views

In $A$-Mod, $M\oplus A\cong A\oplus A$ implies $M\cong A$

(Exercise from an introductory course in homological algebra) Whenever $A$ is a commutative ring with unit and $M$ an $A$-module, the following holds: $$M\oplus A\cong A\oplus A \Rightarrow M\...