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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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Is this graded ring problem corrected?

let $B=\oplus_{i\geq0}B_i$ be a graded ring with $B_d=B_1^d$ for every $d\geq1$. Suppose $B_1$ is a finitely generated $A$-module for some ring $A$. Then, is $B$ an $A$-algebra in some canonical way? In Qing Liu's Algebraic Geometry and Arithmetic…
M.N.
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Going down theorem with modification.

Going-down Thm: Let $A\subseteq B$ be an integral extension. Assume that $B$ is an integral domain and that $A$ is integrally closed. Then going down holds for the above extension. Question1: Can we remove the hypothesis that $B$ is a domain and…
messi
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Is there such a concept as a "freest local ring" generated by another?

Suppose I want to make the following argument: Let $(A,m)$ be a local commutative ring. Then $(A,m)$ is a quotient of a local ring $(B,n)$ which is a domain, since there is a "freest local ring generated by it," which is a domain. (In my particular…
Elle Najt
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Calculating an example of the tensor product

I would like to show that $\mathbb{Z}/8 \otimes_{\mathbb{Z}} \mathbb{Z}_{\langle 2 \rangle} \cong \mathbb{Z} / 8$. If we let $S = \mathbb{Z} \setminus \langle 2 \rangle$, then $$\mathbb{Z}/8 \otimes_{\mathbb Z} \mathbb{Z}_{\langle 2 \rangle} =…
Paul Slevin
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Finite extension of residue fields of DVR's

Let $R$ be a DVR with $K = Quot(R)$ and residue field $k$. Let $k'/k$ be a finite field extension. I would like to have a reference for the following statement (or to see, that it is not true): There exists a finite field extension $K'/K$ s.t. the…
boxdot
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Explain the explanation of the Rabinowitsch trick

Math Overflow offers this explanation of the Rabinowitsch trick in proving the strong Nullstellensatz from the weak Nullstellensatz. https://mathoverflow.net/questions/90661/the-rabinowitz-trick "Perhaps the "Rabinowitz trick" is more clear if one…
AnatolyVorobey
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A homomorphism between finite free modules over a local ring

The following lemma is stated in a book without a proof. How can this be proved? Lemma Let $A$ be a local ring. Let $k$ be the residue field of $A$. Let $E$ and $F$ be finite free modules over $A$. Let $f:E → F$ be a $A$-homomorphism. Let…
Makoto Kato
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Confusion about local properties + trying to show a finite morphism is quasi-finite

Let $A$ be a commutative ring and let $B$ be a finite $A$-algebra. Let $f:A \to B$ be a ring homomorphism. I want to show that whenever $\mathfrak{p} \subseteq A$ is a prime ideal, then there are finitely many prime ideals $\mathfrak{q} \subseteq B$…
Paul Slevin
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Flatness of closure of subring

Assume we are given Noetherian local rings $(A,\mathfrak{m})$ and $(B,\mathfrak{n})$ such that: $A \subset B$ and $\mathfrak{m} = A \cap \mathfrak{n}$, $B$ is a finite $A$-module. It is known that the $\mathfrak{m}$-adic completion $\hat{A}$ is…
Elvorfirilmathredia
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Infinite many curves passing through finite points?

Let $R$ be a Noetherian domain of dimension two. Let $\mathfrak{m}_1,\mathfrak{m}_2$ be two distinct maximal ideals of height two. Are there always infinitely many prime ideals contained in $\mathfrak{m}_1\cap\mathfrak{m}_2$? I have done the…
wxu
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Proof of the Artin-Rees lemma

I am struggling to understand a key step in a proof of the Artin-Rees lemma, which I have put in a red box below. I don't really see how we can pass from a finite direct sum to an infinite one. I've tried writing out both sides of the equality to…
Tim
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Cancelling summands in a direct sum decomposition

Let $M$ be a Noetherian and Artinian module. Suppose that: $$\bigoplus_{i=1}^{q} A_{i} \oplus \bigoplus_{i=1}^{t} B_{i} \cong \bigoplus_{i=1}^{q} A_{i} \oplus \bigoplus_{i=1}^{r} C_{i}$$ where all $A_{i},B_{i},C_{i}$ are indecomposable submodules…
user10
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Poincaré series and short exact sequences

For an additive function $\lambda$ and an exact sequence of modules $0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0$, we have $\lambda(M_2) = \lambda(M_1) + \lambda(M_3)$ by definition. If the modules are graded and every morphism…
jonny
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Transcendental degree and dimension

I do not fully understand the proof of Lemma 5.6 in the book A Course in Commutative Algebra of Gregor Kemper (you can find it here) The lemma states that : If $A$ is an algebra over a field $k$, $S\subset A$ be a generating set of $A$ as a…
Arsenaler
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Extending and contracting an ideal by a faithfully flat homomorphism

Let $ B $ be a faithfully flat $ A $-algebra. Let $ I \subset A $ an ideal. Shows that $ IB \cap A = I $. This is the second item of Exercise 2.6, Chapter 1, of the Qing Liu's book Algebraic Geometry and Arithmetic Curves. If I suppose that $ I $ is…
Miguel F D
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