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

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The ideal of the intersection of two algebraic subsets

If $V, W$ are algebraic subsets of $\mathbb A^n(k)$. Show that $I(V∩W)=\sqrt{I(V)+I(W)}$
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164 views

One to one correspondence of ideals in $R$ and $S^{-1}R$?

I proved the following statement, but I am very unsure that it is correct, since this proposition is not stated in my books for general ideals but only for prime ideals. Please point out where the ...
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109 views

Prime ideals in the same connected component of Spec

Let $R$ be a commutative Noetherian ring with unit. Is it true that two prime ideals $p$ and $q$ are in the same connected component of $\text{Spec} R$ iff there exists a series of minimal primes ...
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218 views

Injective Cogenerators in the Category of Modules over a Noetherian Ring

Let $R$ be a Noetherian ring and let $\mathcal{A}$ be an injective $R$-module. The injectivity of $\mathcal{A}$ is equivalent to the exactness of the functor $Hom_R(-,\mathcal{A})$, i.e. whenever we ...
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1answer
84 views

Maps of maximal ideals

Prove that $\mu:k^n\rightarrow \text{maximal ideal}\in k[x_1,\ldots,x_n]$ by $$(a_1,\ldots,a_n)\rightarrow (x_1-a_1,\ldots,x_n-a_n)$$ is an injection, and given an example of a field $k$ for which ...
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2answers
210 views

Atiyah's book Introduction to Commutative Algebra

I would like to ask a question on localisation. It says $A-\mathfrak{p}$ is a multiplicative closed subset if and only if $\mathfrak{p}$ is a prime ideal. But how do I show that $\mathfrak{p}$ is a ...
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242 views

Why is zero this map between Exts?

Let $(R,\mathfrak m,k)$ be a local ring of depth $d$ and $u:F_1\rightarrow F_0$ a homomorphism of finite free modules such that $\operatorname{Im}u\subset \mathfrak mF_0$. Then this map induces the ...
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49 views

One construction about sheafs

Let $(X,O_X)$ be a ringed space, $E$ - finite locally free $O_X$-module. Let $E^*=Hom_{O_X}(E, O_X)$. How to show, that $E^{**} = E$? It's clear, that locally $E|_U = O_X^n|_U$, and then $E^*|_U = ...
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44 views

Classification of commutative Frobenius algebras

I would like to know if there is a pedagogical reference that explains the classification of all commutative Frobenius algebras.
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145 views

Proj construction and fibered products

How to show, that $Proj \, A[x_0,...,x_n] = Proj \, \mathbb{Z}[x_0,...,x_n] \times_\mathbb{Z} Spec \, A$? It is used in Hartshorne, Algebraic geometry, section 2.7.
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Extension of some properties of $\mathbb{R}$ to other fields and subrings.

We know that the only non-zero ring homomorphism from $\mathbb{R}$ to $\mathbb{R}$ is identity. From this some questions came in to my mind as follow: Question $1$: Can we characterize all fields ...
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120 views

The Gorenstein dimension of a ring

I'm studying on these notes. I have a question about page 64, the remark. A local ring is Gorenstein if and only if the Gorenstein dimension of the residue field is finite. Of course if the ...
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164 views

Topologically nilpotent elements of a linearly topologized ring

In what follows all rings are commutative topological rings. An element $x$ of a ring is called topologically nilpotent if $lim_{n\rightarrow \infty} x^n = 0$. If a ring $A$ has a fundamental system ...
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Why is $\langle x^2, xy,y^3\rangle$ primary in $k[x,y,z]$?

Can someone tell me a quick reason as to why $\langle x^2, xy,y^3\rangle$ is primary in $k[x,y,z]$? I'm trying to read a solution and I don't get this.
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138 views

What is a generic element in a ring

I am now reading a commutative algebra paper, in which the name "generic element" of a commutative ring appears, however, I can not find the definition in that paper, and also my commutative algebra ...
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308 views

$A[x,y] \not \simeq A[x^2, xy, y^2]$

Is it true that for any (commutative, unital) ring $A$ that $A[s,t], A[x^2, xy, y^2]$ cannot be isomorphic as rings? This is mentioned in passing in Eisenbud-Harris, Geometry of Schemes Exercise ...
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119 views

Linear compact module

Can you tell me why: Finite module on a complete local ring is linear compact module? I am sturdy about linear compact module, can you tell me some concerned paper. Thanks you very much!
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Ex. 2.6 from Eisenbud: localization and infinite intersections

Can someone please explain to me why, given an infinite field $k$, we have that $$\left(\bigcap_{a \in k}{(x-a)}\right) [U^{-1} ] \neq \bigcap_{a \in k}{\left((x-a)[U^{-1}] \right)},$$ where $U$ is ...
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2answers
126 views

Finitely generated quotients in Noetherian local rings

If $A$ is a Noetherian local ring with maximal ideal $\mathfrak m$, how do you show that $\mathfrak m^{i}/\mathfrak m^{i+1}$ is a finitely-generated $A/\mathfrak m$-module/vector space? I know each ...
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71 views

If $M$ is locally free of finite rank, is the symmetric part of its tensor power?

Let $R$ be a commutative ring with unity, and let $M$ be locally free of rank $n$. (Specifically, suppose there exist $\{r_i: i\in I\}\subset R$ such that $\sum_i r_iR=R$ and $M_{r_i}$ is free of ...
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95 views

Is a module an inverse limit of finitely generated modules?

Every module is the direct limit of finitely generated modules. Is it true that every module is the inverse limit of finitely generated modules?
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A theorem due to Gelfand and Kolmogorov

For any topological space $X$, we can define $C(X)$ to be the commutative ring of continuous functions $f\,:\,X\rightarrow \mathbb{R}$ under pointwise addition and multiplication. Then $C(-)$ becomes ...
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133 views

Direct limits of injective modules

Is it true that the direct limit of injective modules is injective?
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309 views

Direct limits and $\rm Hom$

I read that $\lim\limits_{\longleftarrow}\mathrm{Hom}(N_j,M)\cong\mathrm{Hom}(\lim\limits_{\longrightarrow}N_j,M)$. I was wondering if we can write $\lim\limits_{\longrightarrow}\mathrm{Hom}(N_j,M)$ ...
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300 views

Tensor product of a module and a projective module.

Suppose that i have a commutative ring $R$ and $R$-modules $M$ and $N$ such that $M$ and $M\otimes_R N$ are finitely generated projective modules. Suppose also that $ {\rm rk}_R(M)={\rm ...
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Non-commutative integral extensions?

In Commutative algebra there is a notion of an integral extension: Let $P$ be a subring of $R$. Then $R$ is the integral extension of $P$ if each element of $R$ is a root of a monic polynomial with ...
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67 views

Limits of subrings and surjectivity

Let $A$ be a ring and let $\mathcal{F}$ be the inductive system of subrings of $A$ which are of finite type over $\mathbb{Z}$: $$ \mathcal{F} = \{ \mathbb{Z}[a_1,\dots,a_n] \subseteq A \mid n \geq 0, ...
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136 views

The dual of a finitely generated module over a noetherian integral domain is reflexive.

As posted by navigetor23 in this question the dual of a finitely generated module over a noetherian integral domain is reflexive. Could you tell me how to prove it?
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A question regarding solutions of polynomials in a field

Let $F$ be a field and $\langle a_1,...,a_n \rangle \subset F$. Then given a non-zero polynomial $f \in F[X_1,...,X_n]$ is it true that if $f(a_1,...,a_n)=0$ then $(X_i - a_i)$ divides $f$ for some ...
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123 views

Ring homomorphism and affine scheme

How to describe all ring homomorphisms $f: A \rightarrow B$, such that corresponding affine scheme morphism $f: Spec \, B \rightarrow Spec \, A$ is open immersion?
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145 views

A question regarding tensor product and isogenies of elliptic curves

Let $E_1$ and $E_2$ be elliptic curves and $T_l(E_i)\cong \mathbb{Z}_l \oplus \mathbb{Z}_l$ the $l$-adic Tate module. Given $ \varphi \in Hom(E_1,E_2)$ this induces $\varphi_l \in ...
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111 views

Geometric explanation of primary ideals

I would like to know how to think about primary ideal geometrically. Vaguely speaking, I think it's an irreducible closed subscheme with some "infinitesimal" data - however, I am not sure how to make ...
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286 views

Injectivity of Homomorphism in Localization

Let $\alpha:A\to B$ be a ring homomorphism, $Q\subset B$ a prime ideal, $P=\alpha^{-1}Q\subset A$ a prime ideal. Consider the natural map $\alpha_Q:A_P\to B_Q$ defined by ...
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1answer
84 views

Number of generators of ideals in a PID.

Let us fix a field $K$, and let us consider the principal ideal ring $K[x]$. If I is an ideal, since $K[x]$ is a PID, we can write $I = (p(x))$ for some polynomial $p(x)$. Now, let us say that a set ...
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193 views

isomorphism between quotients of polynomial rings

It is fairly known that $\mathbb C[x,y,z]/(xy+z^n) \cong \mathbb C[x,y,z]/(z^n+x^2+y^2)$. This appears, for example, in the study of singularities of type $A_n$. But, unfortunately, I am not able ...
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108 views

Functional sheaf (Hartshorne, Cartier divisors)

In Hartshorne there is the following description of the sheaf $K$ on the scheme. For each open $U = Spec \, A$ we define $K(U) = S^{-1} A$, where $S$ is the set of non-zero-divisors. Why is it a ...
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90 views

Fix points of group action under base change

Let $A$ be a local noetherian domain. Let $M$ be a torsion free $A$-module equipped with an $A$-linear action of a group $G$. Let $\mathfrak{m}$ be the maximal ideal in $A$. Is the natural map ...
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1answer
89 views

Prime ideals of height less than the dimension

Let $A$ be a noetherian local ring with maximal ideal $\mathfrak{m}$ of height $d$, and suppose $\mathfrak{p}_1, \ldots, \mathfrak{p}_s$ are prime ideals of height $i - 1 < d$. It's quite clear ...
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372 views

Krull dimension and transcendence degree

What is the simplest proof of the fact that an integral algebra $R$ over a field $k$ has the same Krull dimension as transcendence degree $\operatorname{trdeg}_k R$? Is it possible to use only Noether ...
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130 views

An equivalent condition for a ring to be Cohen-Macaulay

I want to prove that a local ring $A$ is Cohen-Macaulay if and only if for every $A$-module $M$ we have $\mathrm{grade}\;M+\mathrm{dim}\;M=\mathrm{dim}\;A$. If that equation holds we just take $M=k$ ...
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81 views

On equivalent definitions of Gorenstein local rings

I'm studying these notes, in particular page 17 theorem 1. I have some problems in the implication iii $\Rightarrow$ ii: Let $A$ be a local ring of dimension $n$ and maximal ideal $m$ ii) $A$ is ...
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214 views

Transcendence degree for a $k$-algebra which is an integral domain

Let $R$ be an integral domain over a field $k$. Is it true, that $\deg.\mathrm{tr}_k \ \mathrm{Frac}(R)$ is the greatest number of elements of $R$ algebraically independent over $k$?
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252 views

Tensor products commute with inductive limit

How to prove, that tensor products commute with direct limits, if the main ring is not the same? For every $i$ we have modules $L_i$ and $M_i$ over a ring $A_i$, and for every $i \geq j$ homomorphisms ...
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157 views

Principal ideals in a commutative ring R

Given $A$ and $B$ principal ideals with the sum $A+B$ also principal. How to show $A\cap B$ is principal? If $A+B$ happens to be the unit ideal then I see that $A\cap B=AB$ which is principal. I tried ...
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1answer
358 views

Finitely generated modules over noetherian rings isomorphic to their double duals

Let $R$ be a noetherian ring and $M$ a finitely generated $R$-module. Suppose that $M$ is isomorphic to the double dual, how can I prove that $M$ is reflexive? (i.e. it is isomorphic to the double ...
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2answers
126 views

Understanding proof of a corollary leading up to Nakayama's Lemma

I would appreciate help on what should be an easy concept in the proof of a corollary leading up to Nakayama's Lemma. This link to mathoverflow.com (in the green highlighted section) gives the ...
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135 views

Does every zero-dimensional commutative ring have a bounded index of nilpotency?

A commutative ring is called zero-dimensional if all its prime ideals are maximal, and a ring is said to have a bounded index of nilpotency if there is a positive integer $n$ such that $x^n = 0$ for ...
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81 views

A commutative ring which has a group acting locally finitely on itself

The following definitions and proposition are motivated by this question. All rings are assumed to be commutative and have identity elements. Definition 1 Let $B$ be a ring. Let $A$ be a subring of ...
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140 views

If the complement of a subring is closed under multiplication, then the subring is integrally closed.

Let $A\subset B$ be rings, and suppose that $B\setminus A$ is closed under multiplication. I am trying to show that $A$ is integrally closed in $B$. I tried localizing at $B\setminus A$, but this did ...
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70 views

Imbedding of a reduced ring into a direct sum - Why reduced?

This pertains to Ex. 1.13 (self-studier) in Reid's "Undergrad. Commutative Algebra": If $A$ is a reduced ring and has finitely many minimal prime ideals $P_i$ then $A\hookrightarrow ...