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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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Finding the field of fractions of $k[x,y,z]/(xy^2-z^2)$

The exercise is to find the field of fractions of the ring $k[x,y,z]/(xy^2-z^2)$ where $k$ is a field. I'm not exactly sure where to begin, and would appreciate some help/hints.
user221483
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In what generality does the following isomorphism involving tensors and homs hold?

Let $R$ be a CRing and let $M,N$ be $R$-modules. Let $M^*:=Hom_R(M,R)$. I have seen the following isomorphism asserted in the case where $R$ is a field and $M$ and $N$ are f.g. vector spaces: $M^*\otimes_R N\cong Hom_R(M,N)$. I can give a proof…
dtripleez
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Definition of Grothendieck group

I'm reading the Wiki article about the Grothendieck group. What's the reason we define $[A] - [B] + [C] = 0 $ rather than $[A] + [B] - [C] = 0 $ (or something else) for every exact sequence $0 \to A \to B \to C \to 0$? What is the property we…
Rudy the Reindeer
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Is $\mathbb{Z}[\sqrt{2},\sqrt{3}]$ flat over $\mathbb{Z}[\sqrt{2}]$?

Is $\mathbb{Z}[\sqrt{2},\sqrt{3}]$ flat over $\mathbb{Z}[\sqrt{2}]$? The definitions doesn't seem to help. An idea of how to look at such problems would be helpful.
rola
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Generalized Nakayama over a local ring with an almost nilpotent ideal

Let $(A,\mathfrak{m})$ be a local ring. Let us call $\mathfrak{m}$ almost nilpotent if for every sequence $a_1,a_2,\dotsc$ in $\mathfrak{m}$ there is some $n \geq 1$ such that $a_1 \cdot \dotsc \cdot a_n = 0$. Let $M$ be an $A$-module with $M =…
Martin Brandenburg
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Height one prime ideal of arithmetical rank greater than 1

Let $R$ be a Noetherian local domain which is not a UFD and let $P$ be a height one prime ideal of $R.$ Can we find an element $x\in P$ such that $P$ is the only minimal prime ideal containing $x$?
messi
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Localization arguments in Dedekind domains

I am reading Serre's Local Fields, and have questions about the text. Specifically, pages 11 and 12. 1) Consider a Dedekind domain. We want to show that all fractional ideals are invertible. Serre claims that because the image of a fractional ideal…
Potato
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If $S$ consists of units then $S^{-1}R \cong R$

I want to show that if $S$ consists of units then $S^{-1}R \cong R$. Can you tell me if my proof is correct? Since $S$ consists of units, $S$ is zero-divisor free and hence $f: R \to S^{-1}R$, $r \mapsto \frac{r}{1}$ is injective. So we have an…
Rudy the Reindeer
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Hilbert series characterization of regular sequences

Let $k$ be a field and $S=k[x_1,\dots,x_r]$ the polynomial ring in $r$ indeterminates. Let $f_1,\dots,f_n$ be a sequence of $n\le r$ forms of degrees $d_1,\dots,d_n$. If $f_1,\dots,f_n$ is a regular sequence, then it is easy to see that the Hilbert…
Manos
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What's so "shrieky" about this shriek map?

On page 88 of Atiyah-Macdonald's "Introduction to Commutative Algebra" there is an exercise about the Grothendieck group $K(A)$ of a noetherian ring $A$. In this context to every finite ring homomorphism $f: A \rightarrow B$ of noetherian rings…
Nils Matthes
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radical of an ideal

Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. We define $L$-radical of $I$, denoted by $\sqrt[L]{I}$, the intersection of all primary ideals of $R$ containing $I$. It is clear that $\sqrt[L]{I\cap J}\subseteq \sqrt[L]{I}…
m. sam.
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Isomorphisms involving localisation of graded rings

I have been trying to establish an isomorphic concerning graded rings, and there is a last step that I'm confused about. Let $R$ be a $\Bbb{Z}$ - graded ring. Let $f$ be a homogeneous non-nilpotent element of degree $1$ in $R$. Let $R_f$ denote the…
user38268
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What do ideals of a ring say about its inner structure?

Could people with knowledge in Commutative Rings elaborate on this sentence from the Wikipedia article (Ideals and Factor Rings subsection, first sentence): The inner structure of a commutative ring is determined by considering its ideals, i.e.…
anon2328
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Converse of "localization at a prime is local"

Suppose $S^{-1}R$ is the localization of a ring R at a multiplicative subset S, and is local. Must S be the complement of a prime ideal?
LCL
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when is the cokernel of a map of free modules free?

Let $R$ be a commutative ring (noetherian if needed) and $n,m$ be two nonnegative integers. Consider a map $\varphi: R^n\rightarrow R^m$ Is there a characterisation, e.g. in terms of the matrix representation of $\varphi$ of the cokernel of this map…
jorst
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