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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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Prime Avoidance Application

I have a question about a step in the proof of following statement: Let $A$ be an integrally closed domain with the field of fractions $K, L$ a finite normal extension of $K, B$ the integral closure of $A$ in $L$. Then the group $G=\operatorname…
user267839
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Hilbert functions of polynomial growth

Let $R=k[x_1, \dots , x_n]$. Let $M$ be a graded $R$-module whose hilbert function $h(i):=\text{dim}_kM_i$ is well-defined and grows like a polynomial. Is $M$ a finitely generated $R$-module?
dOuUuq3podCuoArv
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Example of reduced ring which is not integral domain

So, a reduced ring has no nonzero nilpotent elements, while integral domain has no nonzero zero divisors. Of course, the latter is way stronger then the first condition, implying that every integral domain needs to be reduced ring. But, having…
nikola
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Unramified implies local-etale

I'm trying to understand the proof of the following lemma in Freitag/Kiehl, "Etale Cohomology...": 1.5 Lemma. Let $A \rightarrow B$ be a finitely generated local homomorphism. We assume that it is injective and that $A$ is a normal ring. If $B$ is…
rj7k8
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Koszul Complex of Powers of Elements

For a sequence of elements $\underline{a}= a_{1}, ..., a_{r} \in R$, let $K^{\bullet}(\underline{a};R)$ be the Koszul complex generated by $\underline{a}$. Let $\underline{a}^{v} = a_{1}^{v_{1}}, ..., a_{r}^{v_{r}} \in R$. My question: if…
do_math
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Is there a difference between being Artinian/Noetherian as an $R$-module, versus as an $R/I$-module?

A standard fact in commutative algebra: Let $A$ be a ring in which there exists some finite number of maximal ideals whose product is $0$. Then $A$ is Artinian iff $A$ is Noetherian. The usual proof involves noting that each…
SSF
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about graded free $R$-module

Let $R$ be a graded ring and $M$ a graded$R$-module. $M$ is called a graded free module if $M$ has a Basis consisting of homogeneous elements Now in my lecture note there is the following Theorem without a proof. $M$ is a graded free $R$-module if…
user562724
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Length and dimension of a module

Let $(R,m,k)$ be a local commutative ring and $M$ be a $k$-module. I want to prove that $$\mathrm{length}_R(M)=\dim_k(M)$$ Since it is so obvious, I doubt my own proof. Let $n=\dim_k(M)$ (could be infinity). Then $M\cong k^n$ as $k$-module (vector…
T C
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isomorphism of some rings and intuition

I proved by routine check that: let $R$ be a commutative ring with unit, $f$ be $R$-regular, $m$ be a maximal ideal s.t. $f\in m$. Then $(R/fR)_m\cong (R/fR)_{m/fR}$ as rings with the isomorphism $\overline{r}/{s}\mapsto \overline{r}/\overline{s}…
T C
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$M$ is flat $\Leftrightarrow S^{-1}M$ is a flat $S^{-1}R$-module

Hi I have the following problem: Let $R$ be a ring and $S\subset R$ multiplicatively closed and $M$ be an $R$-module. Show: $M$ is flat $\Leftrightarrow S^{-1}M$ is a flat $S^{-1}R$-module. I think I can show $"\Rightarrow$" but for $"\Leftarrow"$…
Tobi92sr
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Computing ideal from its radical

This question is motivated by the following question: What is a primary decomposition of the ideal $I = \langle xy, x - yz \rangle$? I'm wondering under what circumstances we can compute an ideal from its radical (assuming the ideal is not already…
cofnmarol
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equivalence of two statements

I have the following problem: Let $R$ be a ring and $a\subset R$ an ideal with $a\subset J(R)$, where $$J(R):=\bigcap_{m\,\in\,\operatorname{mSpec} R} m.$$ Let $M$ be an $R$-module and $N$ a finitely generated $R$-module, $f:M\rightarrow N$ an…
Tobi92sr
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The multiplication of rank for finite projective modules

Assuming all rings are commutative, suppose $M$ is finite projective module over $R$ of rank $m$ which is itself an $R'$-algebra that is as a module finite projective of rank $n$. How can we show that $M$ is of rank $mn$ over $R'$? Where the rank…
davik
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A question about localization of integral domains

Let $R$ be an integral domain and $P,Q$ be proper prime ideals of $R$. Let $R_P,R_Q$ be localizations of $R$ at $P,Q$. If $R_P\subseteq R_Q$, is $Q\subseteq P$?
Mary
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If $A^{\wedge}$ is the completion of a local ring, do we always have $\mathfrak{m}^{\wedge}=\mathfrak{m}A^{\wedge}$?

Let $A$ be a local ring with maximal ideal $\mathfrak{m}$, and let $A^{\wedge}$ be the $\mathfrak{m}$-adic completion of $A$. Then $A^{\wedge}$ is a local ring with maximal ideal $\mathfrak{m}^{\wedge}=\ker(A^{\wedge}\to A/\mathfrak{m})$. In the…
user501746
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