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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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Finite free rings over complete local rings are direct products of local rings

I came across the following statement: Let $R$ be a complete local Noetherian commutative ring. If $A$ is a commutative $R$-algebra that is finitely generated and free as a module over $R$, then $A$ is a semi-local ring that is the direct product of…
only
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Complete DVR with finite residue field is compact?

How do I go about proving this? Do I have to show total boundedness (I don't know how to use the finiteness of the residue field, and this seems like something that it might pertain to).
Ethan Parris
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$A[[T]]$ Noetherian $\Rightarrow$ $A[T]$ Noetherian without Hilbert's Basis Theorem

Let $A$ be a commutative ring (with unit) such that $A[[T]]$ is Noetherian. Can we conclude that $A[T]$ is Noetherian without invoking Hilbert's Basis Theorem (or a similarly heavy argument)? In other words, if we know Noetherianity of the power…
M.G.
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Decomposition of finite algebra over field

Let $A$ be a finite algebra over field i.e. $A=\sum_{i=1}^{n}ke_i$, $\{I_\alpha\}$ set of all maximal ideals of $A$. I know that $\#\{I_\alpha\}<\infty$ and $r(A)$ is nilpotent i.e. $r(A)^n=0$ for some $n\in\mathbb{N}$. Am I right…
Aspirin
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map of a finite extension

The following seems be well-known but I am not able to prove it. Please give me a help. Let $A \hookrightarrow R$ be an extension of Noetherian domains such that $R$ is a finitely generated $A$-module. Then there exists an $A$-linear map $\varphi : …
Pham Hung Quy
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Concrete syzygy generators

Consider $g_1=x^2, g_2=y^2, g_3=xy+yz\in k[x,y,z]$ with a field $k$. We consider the reverse lexicographic order, and put $x>y>z$. I want to find the generators of the syzygies. Eisenbud CA book, p739, exercise 15.27, says that it is…
Tom
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Infinitely generated modules

Can you give me some examples of infinitely generated modules over commutative rings, other than $A[x_1,\ldots,x_n,\ldots]$? Thanks a lot!
Aspirin
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Determining a rule which assigns each ring a unit element

Probably this is easiest, but as I am somehow stuck I would be pleased about some comments. What I give myself is a rule $f$ which does the following: To every commutative ring $A$ with $1$ the rule $f$ assigns a unit $f(A)\in A^*$, and this…
Cyril
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Non trivial representation of zero in a module-finite extension

Suppose, $S$ is a ring, module-finite over a subring $R$. Let $s_1,...,s_n$ be a minimal set of generators of $S$ as an $R$-module. Can we have, $0=t_1s_1+...+t_ns_n$ for some $t_1,...,t_n\in R$, not all zero?
Sharath Sampat
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How to show $X=Spec(A)$ has maximal components $V(p)$?

This is from Atiyah Macdonald, Chapter 1, Execrise 20. If $A$ is a ring and $X=Spec(A)$, then the irreducible components of $X$ are the closed sets $V(p)$, where $p$ is a minimal prime ideal of $A$. It is not difficult to prove the closed sets…
Bombyx mori
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Proposition 7.8 Atiyah Macdonald

Proposition 7.8 Atiyah Macdonald says that: Let $A\subseteq B\subseteq C$ be rings. Suppose that $A$ is Noeterian, that $C$ is finitely generated as an $A$-algebra and that $C$ is either (i) finitely generated as a B-module or (ii) integral over…
Soulostar
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Tensor product of projective modules

(All rings are commutative) Let $A$ be a noetherian ring. Let $B$ be a noetherian $A$-algebra (not nessecerily f.g!) Suppose $M$ and $N$ are finitely generated projective $B$-modules (for my application I can assume that $M=N$ are of rank…
the L
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symmetric algebra

Suppose we have the following situation: $I = \oplus_{i=1}^{p-1}I_1$ is a decomposition of an $R$-ideal in invertible modules of rank $1$, $I_iI_j \subset I_{i+j \pmod{p-1}}$ and $I_1^i = I_{i \pmod{p-1}}$. Then it is clear that $\text{Sym}_RI_1$…
user5262
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understanding $\mathrm{Spec}\mathbf{Z}[\mu_{p-1},\frac{1}{(p-1)p}] \cap \mathbf{Z}_p$

What's $\mathrm{Spec}\mathbf{Z}[\mu_{p-1},\frac{1}{(p-1)p}] \cap \mathbf{Z}_p$, where $\mu_{p-1}$ denotes the $(p-1)$-th roots of unity and $\mathbf{Z}_p$ the $p$-adic integers? It should be $\mathbf{Z}[\mu_{p-1}]$ without (the places above the…
user5262
  • 1,863
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Why isn't $\mathbb{Z}[\sqrt{5}]$ integrally closed?

The book I'm reading casually mention that $\mathbb{Z}[\sqrt{5}]$ isn't integrally closed without explaining why. Could someone show me why this is the case?
Sam Spiro
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