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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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Is there a mistake in Hartshorne Proposition II.6.6?

This is Proposition II.6.6 in Hartshorne. We assume that $X$ is a Noetherian, integral, and separated scheme which is regular in codimension 1, i.e. every local ring of dimension one is regular. For this question, I am solely interested in the Type…
Born to be proud
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First fundamental Exact sequence

I am reading matsumura, in that if $k \xrightarrow{f} A \xrightarrow{g}B$ are $k$ algebra morphisms. Then $$\Omega_{A/k}\otimes_A B \xrightarrow{\alpha} \Omega_{B/k} \xrightarrow{\beta} \Omega_{B/A} \to 0$$ is an exact sequence of $B$ modules. Where…
Yossarain
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About the definition of the dualizing module for a Noetherian local ring.

The page 329 of the book "Modular Forms and Fermat Last Theorem" says that: I have two questions about this equivalence. In the cited reference, the definition further requests that the set of associated prime idals of $I$ is $\{\mathfrak{M}\}$…
Phanpu
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Is a field a GCD domain?

I know this is probably a dumb question. Since a field is a PID then it should be a GCD domain according to the chain of domains. However, take $\mathbb{Q}$ for an example, I don't think it is a GCD because I can't find the greatest common divisor…
Coco
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why is $K[[x_1^{1/p},...,x_n^{1/p}]]$ flat over $K[[x_1,...,x_n]]$

I am studying Kunz theorem using the notes by Karen smith, in the forward part regularity implies flatness, i am having difficulty verifying that $K[[x_1^{1/p},…,x_n^{1/p}]]$ is free over $K[[x_1,…,x_n]]$. Here $R$ is a complete local ring of…
Yossarain
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Exercise 4.30 of Eisenbud: how to connect a maximal ideal in a localization with the original ring's property of being finitely generated?

I am doing Exercise 4.30 in Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. I will copy the problem statement here: Exercise 4.30: Suppose that $k$ is a Noetherian ring such that $*)$ for every finitely generated $k$-algebra…
my2cents
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Gathmann commutative algebra Exercise 1.13

I'm doing Exercise 1.13 on Gathmann's commutative algebra notes: Show that the equation of ideals $$(x^3-x^2,x^2y-x^2,xy-y,y^2-y)=(x^2,y)\cap (x-1,y-1)$$ holds in the polynomial ring $\mathbb C[x,y]$. Is this a radical ideal? What is its zero locus…
Dasheng Wang
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Simple question for case $k=0$ in Artin Rees Lemma

In AM, Artin-Rees lemma, there is no constraint on integer $k$ but in most sources, Artin-Rees lemma is stated: "there exists an integer $k>0$ s.t. ..." My question is: is $k=0$ not a possibility?
metalder9
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Is the reasoning correct for showing $\frak{R}$ $(A[x])\subset$ $\frak{R}$ $(A)[x]$?

Let $\frak{R}$ $(A)$ denote the nilradical of the commutative ring $A$ i.e. the ideal containing all nilpotent elements of $A$, which is also the intersection of all prime ideals of $A$. For $A$ commutative ring with $1$,…
frelg
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How to show the existence of central element here?

The following question is from my exercise sheet in Commutative Algebra and I am not making any sufficient progress on it. Question: Let $A$ be a ring and let $I$ be a finitely generated two-sided ideal of $A$ such that $I^2 =I$. Show that there…
user775699
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Does the ring extension $\mathbb{Z}\subset\mathbb{Z}[\frac{1}{10}]$ satisfy the going up property?

Does the ring extension $\mathbb{Z}\subset\mathbb{Z}[\frac{1}{10}]$ satisfy the going up property? Going up property: For a ring extension $A\subset B$, if for any prime ideals $p\subset p'$ of $A$ and prime ideal $P$ in $B$ that lies over $p$,…
Anish Ray
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why is $(x,y) \to \phi(x) (y )?$ why not $(x,y) \to \phi(x, y)?$

This theorem proof confuses me a little. $Hom(M \otimes N, P) \cong Hom(M, Hom(N,P))$ In Atiyah book(page no :$28$) it is written that Any $A$- homomorphism $\phi :M \to Hom_A(N,P)$ defines a bilinear map , namely $(x,y) \to \phi(x) (y)$ My…
jasmine
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When is $R\cong \Pi_{Q\in \max(R)}R_Q$?

Problem: Let $A$ be a semilocal Noetherian ring and let $\text{max}(A)$ be the finite set of maximal ideals of $A$. I am trying to prove the following statement (which I hope to be true but am not certain about) $A\cong \Pi_{Q\in…
Countable
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Non-zero fractional ideals are invertible implies DVR

There’s a theorem saying that for local ring $A$, DVR $\Leftrightarrow$ every non-zero fractional ideal of $A$ is invertible. I am now in the $\Leftarrow$ direction. I managed in proving that the unique maximal ideal $m$ is principal and that $A$ is…
julian
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When is the formal power series ring a valuation ring?

If $K$ is a field, the formal power series ring in $1$ variable $K[[X]]$ is a discrete valuation ring. What about the many variable case? Is $K[[X_1, \ldots, X_n]]$ a valuation ring? Instead if we consider a formal power series ring $R[[X_1,…
esk
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