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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.

16857 questions
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Characterization of an invertible module

Let $B$ be a commutative ring. Let $A$ be a subring of $B$. If $M$ and $N$ are $\mathbb{Z}$-submodules of $B$, we denote by $MN$ the submodule of $B$ generated by the subset $\{ab\mid a \in M, b\in N\}$. If $M$ and $N$ are $A$-submodules of $B$,…
Makoto Kato
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Counter Example for Exact sequence

I encountered this question when I am working on Proposition 2.9 from Atiyah-Macdonald. Let $M_1 \xrightarrow{f} M_2 \xrightarrow{g} M_3$ be a sequence of $R$-module. Let $D$ be any $R$-module, and $Hom(M_3,D)\xrightarrow{\tilde{g}}Hom(M_2,D)…
GradStudent001
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Rees Ring, Associated Graded Ring and Dimensions

This question refers to the Rees ring, and can be posed from first principles as follows. Let $A$ be a Noetherian ring, $t$ an indeterminate over $A$ and define $u = t^{-1}$. Let $I$ be a proper ideal of $A$ generated by some $a_1, \dots,a_n \in…
Manos
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A question from Eisenbud, Commutative Algebra

On page 35, the proof of corollary 1.8: If k is an algebraically closed field and A is a k-algebra, then A = A(X) for some algebraic set X iff A is reduced and finitely generated as a k-algebra. In the proof, it says: "... Conversely, if A is a…
user2690457
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On the multiplicity of complete intersections

Suppose $R$ is a complete intersection. How can I prove that $\operatorname{mult}(R)\geq2^{\operatorname{codim}(R)}$, where $\operatorname{mult}(R)$ is the multiplicity and $\operatorname{codim}(R)=\operatorname{edim}(R)-\dim(R)$.
Chris
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Hom between adically complete modules

Suppose $A$ is a commutative noetherian ring, $\mathfrak{a} \subseteq A$ is an ideal, and $M,N$ are two $\mathfrak{a}$-adically complete $A$-modules (complete = complete and separated). Is the $A$-module $\operatorname{Hom}_A(M,N)$ also an…
the L
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proving that $\dim A[X] = \dim A + 1$ (Matsumura)

Let $A$ be a Noetherian ring and $X$ an indeterminate over $A$. I am having trouble understanding Matsumura's proof (Commutative Ring Theory, Theorem 15.4) that $\dim A[X] = \dim A + 1$. Below, i provide all the necessary details. As an auxiliary…
Manos
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was flatness really used in this argument? (Matsumura, Theorem 7.2)

Let $A$ be a ring and $M$ an $A$-module. Then $M$ is faithfully flat over $A$ $\Leftrightarrow$ $M$ is flat over $A$ and $M \otimes N=0 \Rightarrow N=0$. This is part of theorem 7.2, p. 47 in Matsumura's Commutative Ring Theory. Let's consider the…
Manos
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increasing union of finitely generated submodules of M need not be finitely generated

Show by an example that an increasing union of finitely generated submodules of M need not be finitely generated. I was thinking about $R[x_1,x_2,x_3,....]$. Then if we consider the ideal $$ , does it form a…
Germain
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Is every ring with a homomorphism from $R$ an $R$-algebra if and only if $R$ is a solid ring?

Given a commutative ring $R$, is every ring with a homomorphism from $R$ an $R$-algebra if and only if $R$ is a solid ring? A ring $R$ is said to be solid if the unique homomorphism $\mathbb{Z} \to R$ is an epimorphism in the category of rings, or…
Geoffrey Trang
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Elementary method for finding $I(Y)$ for the curve $Y$ defined parametrically by $x=t^{3}$, $y=t^{4}$, $z=t^{5}$

In order to motivate some of the theory we will be learning in a computational commutative algebra course, my professor assigned a number of computational problems that are [seemingly] quite difficult without more advanced methods. The problem I…
Max
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What are the localizations of $\mathbb{Z}/n\mathbb{Z}$ at prime ideals in general?

(This self-answer post was extracted from an answer I posted in this question; see this meta thread.) Is there a general rule for computing the localization of $\mathbb{Z}/n\mathbb{Z}$ at a prime ideal? This computation is useful, for example, in…
Anakhand
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Is $A[x]$ ever a Goldman ring for nontrivial $A$?

There is this notion of a Goldman domain which is a domain $A$ satisfying $Q(A)=A[u^{-1}]$ for some $u\in A$, where $Q(A)$ is the field of fractions of $A$. This is equivalent to saying $Q(A)$ is finitely generated as an $A$-algebra. We can extend…
P-addict
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How to compute $I(Y)$ for the curve $Y$ defined parametrically by $x=t^{3}$, $y=t^{4}$, $z=t^{5}$?

Let $Y\subset\mathbb{A}^{3}$ defined parametrically by $x=t^{3}$, $y=t^{4}$, and $z=t^{5}$. I want to compute $I(Y)$. I think that $$I(Y)=(x^{20}-z^{12},x^{20}-y^{15},y^{15}-z^{12}),$$ but I can only get inclusion $\supset$. Any ideas on how to…
TheNumber23
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Application of Noether's Normalization lemma.

$\newcommand{\Q}{\mathbb{Q}}$ Problem: Let $A = \Q[x,y,w,z]/(xy-wz)$. Show that $\Q[x,y,w] \subseteq A$ is not an integral extension. Exhibit $A$ as an integral extension of $\mathbb Q[a,b,c]$ for some $a,b,c$ algebraically independent…
Irving Rabin
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