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Questions tagged [commutative-algebra]

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

Commutative algebra is the area of mathematics that deals with commutative rings and their ideals, as well as modules over commutative rings.

Many results and tools of commutative algebra are cornerstones of algebraic geometry. Important tools of commutative algebra include localization and completion of rings and modules.

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Please just explain how the sequence are exact?

$\text{Ramification Lemma}:$ Let $A$ be a ring, $\alpha$ an ideal of $A$, and $M$ an $A$-module. Then prove $M/ \alpha M \cong (A/\alpha) \otimes_A M $. Proof: We have the exact sequence: $$0 \to \alpha \to A \to A/\alpha \to 0.$$ (Why ?) Tensoring…
MAS
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Relation between M-regular sequence and Ext

Given a notherian ring $A$ and a $A$-module $M$, let $I$ be an ideal of $A$ with $IM\neq M$. Then in Matsumura's, there is a statement that: If there exists an $M$-regular sequence $a_1,a_2,...,a_n$ of length $n$ in $I$, then $\mathrm{Ext}^i_A…
Yuyi Zhang
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Local Gorenstein ring and unique factorization domain

Is every local Gorenstein, integrally closed, integral domain, going to be an unique factorization domain?
Chen
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Hypothesis of the Going-Down Theorem

In Atiyah-Macdonald, the Going-Down Theorem (listed as Theorem 5.16) assumes the integral extension to be a domain, which is due to the usage of a lemma (listed as Proposition 5.15). However, it seems to me nowhere in the proofs of both theorems…
Zhenyi Chen
  • 41
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two generated height one ideals

Suppose $R$ is a regular local ring of dimension 3 and $I=(a,b)$ is of height one. Is it always true that $I_p$ is not principal for all height two primes $p$? Is there an example? I suspect that this may be false in general. But somehow the ring…
dongrugose
  • 189
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The quotient of an ideal generated by idempotents of a prime ideal does not contain any idempotents

Let P be a prime ideal in a ring R (commutative with identity) and I be the ideal generated by the idempotents of R, contained in P. Show that R/I contain no idempotents other than 0 and 1. An approach suggested is to show that for $x$ being one…
Harun rashid
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Hilbert Theorem of zeros

Use the Hilbert Nullstellensatz Theorem to prove the following result: Given $F_1, F_2, F_3 \in \mathbb{C} [X_1,\dots,X_n]$ polynomials checking the following conditions: $F_1$ is irreducible; $F_2$ is not a multiple of $F_1$; For every element $…
rgl4
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Free algebra implies free module

Let $R$ be a commutative unital ring. Let $n\geq 1$ be an integer. Suppose that $R[x_1, \dots, x_n]$ has a $R$-subalgebra $A$ such that $R[x_1, \dots, x_n]$ is a finitely generated $A$-module. Is it true that $R[x_1, \dots, x_n]$ is a free…
user693936
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An uncountable PID with countably infinitely many prime ideals that is not a localization

Does there exist a characteristic $0$ uncountable principal ideal domain $R$ that has countably infinitely many prime ideals, that is not a localization of a PID with uncountably many prime ideals at a multiplicative set? This question without the…
user693936
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Finite generated domains are finite domains

Let $R\subseteq S$ be two integral domains. $S$ is finitely generated over $R$. $Frac(S)$ is finitely generated over $Frac(R)$ (hence, finite by Zariski's lemma). Is it true that $S$ is finite (as module) over some $R[1/f]$ where $0\neq f\in R$? I…
CO2
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what is the basic difference between an algebra over a field and a vector space over a field?

We know $\text{an algebra over a field is a vector space}$. So what is the basic difference between an algebra over a field and a vector space over a field ? what is the difference between an algebra over a field and a group algebra ?
MAS
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Prove that the sum of the length of modules is alternate

Take an exact sequence of $A$-modules $$0 \rightarrow M_r \rightarrow M_{r-1} \rightarrow \dots \rightarrow M_0 \rightarrow 0$$ Prove that $\sum_{0\le i\le r} (-1)^i$ len$_A M_i = 0$. I know how to prove the result if it's a short exact sequence,…
José
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Nilpotent elements.

Let $f=\sum_{n=0}^{\infty}a_nx^n$. If $f$ and $a_0$ is nilpotent how I can prove that $f-a_0$ is nilpotent? Or if $f^n=0$ and $a_0^n=0$ how can I prove that $(f-a_0)^n=0$, where $n\in \mathbb N$.
someone
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Krull-Akizuki Theorem on David Eisenbud's book

On page 267 of Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud is Theorem 11.3 (Krull-Akizuki Theorem). If $R$ is a one-dimensional Noetherian domain with quotient field $K$, and $L$ is a finite extension field of…
user682705
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finitely generated subrings of number fields

How far from integrally closed can a finitely generated subring of a number field be? For example, is it possible to have a finitely generated subring of a number field that has infinite index (as additive group) in its integral closure? And is it…
Peter Kropholler
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