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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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Castelnuovo-Mumford regularity of products of ideals

Let $R=k[x_1,\ldots,x_n]$ be a usual graded ring. Let $I$ , $J$ be monomial ideals. Definition: $reg(I)=\max\{j-i| \beta_{i,j}(I) \neq 0 \}$. We know that $reg(I+J) \le reg(I)+reg(J)-1$. Can we write $reg(IJ)$ in terms of $reg(I)$ and $reg(J)$ ?
user177523
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Lying Over for Algebraic Ring Extensions

Let $B$ be a finitely generated algebraic $A$-algebra (but not necessarily integral). Is it true that for any prime in $A$ we can find a prime in $B$ which contracts to $A$? What if we also allow the map $A\to B$ to be inclusion?
Nicolas Bourbaki
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Isomorphic Affine Schemes

If $f:A\to B$ is a homomorphism of rings such that $f':\text{spec} B \to \text{spec} A$ is a homeomorphism does it follows that the spectra are isomorphic as schemes? I was able to reduce this problem to trying to show that given a prime…
Nicolas Bourbaki
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Atiyah-MacDonald: proof of Proposition 7.9, weak Nullstellensatz.

Proposition 7.9 in Atiyah & MacDonald's Introduction to Commutative Algebra states: Let $k$ be a field and $E$ a finitely-generated $k$ algebra. If $E$ is a field, then it is finite algebraic extension of $k$. The proof begins with the line: ''Let…
Steve88
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Theorem 31.7 of Matsumura, Commutative Ring Theory

Theorem: If A is a Noetherian local ring and A[x] catenary, then A is formally catenary. In the proof, it is assumed that A is integral domain and A* (the completion of A) is not equidimensional and then Lemma 3 is used. * (Lemma 3. Let (R,m) be a…
azna
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$R/I$ satisfies $S_2$ condition

Let $R=k[x_1,...,x_{n},y_1,...,y_n]$ be a polynomial ring over a field $k$ and $I=\langle \{x_iy_j|$ for some $i,j \in\{1,...,n\}\}\rangle$ be ideal of $R$ and there are $r,s\in\{1,...,n\}$ such that $x_ry_s\notin I$. Assume $R/I$ is a Buchsbaum…
user177523
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System of parameters for a local ring

I need some help to solve this problem. This is the kind of problem that makes me stuck at the very beginning. Let $K$ be an algebraically closed field, $X = \{(x,y)\in\mathbb{A}^2_K: \ y^2-x^3=0\}$ an affine variety and $\mathfrak{m}$ the maximal…
diff_math
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Flatness and normality

I have just read: Direct proof of non-flatness and wondered what is exactly the claim that Alex Youcis is referring to: "...but are you aware of the fact that flatness preserves normality. In your case $A$ is non-normal and $B$ is normal, so $B/A$…
user237522
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Doubt about completeness

May I refer you to page 3 of: http://www.math.iitb.ac.in/atm/caag1/balwant.pdf Proof that $\hat{M}$ is complete, where it says "We choose $n(m)$ such that $n(m+1) \geq n(m)$ for every $m$". Question: Why is this possible?
user10
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Question on complete intersection ideal.

Let $R$ be a Noetherian commutative ring with unity and let $I$ be an ideal of $R$. Suppose I want to know if $I$ is a complete intersection, I know that $I$ is finitely generated but I am unable to find its generators, how can I determine if $I$ is…
jeff
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radical of an ideal in the polynomial ring $k[x,y]$

How can I compute the radical of an ideal? I suppose that there no exist an algorithm for compute it. But in the case of polynomials rings? there exist an algorithm? I need to compute the radical of the ideal generated by $ (x^2y , xy^3) $ in the…
Pilot
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tensor product of formal power series

Let $A[[h]]$ be the formal power series algebra over $\mathbb{C}[[h]]$, here $\mathbb{C}$ is the complex number field. Is the canonical map $A[[h]] \otimes_{\mathbb{C}[[h]]} A[[h]] \to (A\otimes_\mathbb{C} A)[[h]]$ injective? Thanks!
wsq
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Is $\mathbb{Q}_p$ a $\mathbb{Z}_p$-algebra of finite type?

Let $p$ be a prime. The p-adic numbers $\mathbb{Q}_p$ are an algebra under the $p$-adic integers $\mathbb{Z}_p$ via the localization $\mathbb{Z}_p\to \mathbb{Z}_p[\frac{1}{p}]=\mathbb{Q}_p$. Is $\mathbb{Q}_p$ a $\mathbb{Z}_p$-algebra of finite type?…
user228077
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Maximal $R$-sequences in non-noetherian rings

If $a_1,...,a_n$ is a maximal $R$-sequence in an ideal $I$ of a noetherian commutative ring $R$ then $I⊆∪P$, where the union is taken over all the associated prime ideals of the $n$-generated ideal $(a_1,...,a_n)$. Does this result also hold for…
karparvar
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Every prime is maximal in a Jacobson ring?

In Attiyah commutative algebra page 71, it is given some equivalent definitions of Jacobson ring. One of the definitions are that every prime ideal which is not maximal is equal to the intersection of prime ideals which contains it strictly. How is…
harajm
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