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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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Computing the Hilbert-Poincaré series of a quotient

I am preparing for an exam of commutative algebra, and I am at loss about how to compute Hilbert-Poincaré series of rings. In particular, I have some preparation exercises I can't solve. Mainly they involve computing the Poincaré series of quotient…
Daniel Robert-Nicoud
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proving the existence of a regular element in the quotient of a completion

Let $(R,m)$ be a local Cohen-Macaulay ring and $p$ a prime ideal of $R$. Denote by $\hat{R}$ the completion of $R$ with respect to the $m$-adic topology. Take $q \in \operatorname{Ass}(\hat{R}/p \hat{R})$ and suppose that $q \cap R \neq p$.…
Manos
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An example of a $P$-primary ideal $I$ satisfying $I^2 = IP$

Give some examples of a $P$-primary ideal $I \not=P $ in a noetherian domain $R$ such that $I^2=PI $.
lina lie
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graded modules have enough projectives

In the bottom of page 32 in Bruns and Herzog, Cohen-Macaulay Rings, the authors write that if $R$ is a $\mathbb{Z}$-graded ring and $M$ a $\mathbb{Z}$-graded $R$-module, then $M$ is the homomorphic image of an $R$-graded module of the form…
Manos
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relation between projective dimension (pd) of ideal and pd of the quotient

Let $R$ be a Noetherian ring and $I$ a proper ideal of $R$. Suppose that the projective dimension of $I$ is equal to $n$. Let $0 \rightarrow P_n \rightarrow \cdots \rightarrow P_0 \rightarrow I \rightarrow 0$ be a projective resolution of $I$.…
Manos
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Little question about Nakayama's Lemma

Let $M$ and $N$ be finitely generated modules over a local ring $A$ with residue field $k$ and $f:M\rightarrow N$ a A-homomorphism, such that the induced morphism $M\otimes_{A}k \rightarrow N\otimes_{A}k$ is an isomorphism. Can I conclude with…
Descartes
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Nakayama's Lemma

When we prove Nakayama's Lemma, which states that if $M$ is a finitely generated $R$-module, where $R$ is a commutative ring and if $I$ is an ideal of $R$ contained in the Jacobson radical of $R$, if $IM=M$ then $M=0$. We take the contradiction that…
S786
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what classes of modules admit finite free resolutions?

As i understand, finitely generated graded modules over Noetherian graded rings admit a finite free resolution (FFR). What are other classes of modules that admit a FFR? How about finitely generated modules over local Noetherian rings?
Manos
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Regular sequence in local rings

Assume that $(R,m)$ is a local ring and $J\subset I$ are proper ideals of $R$. If $I/J$ is generated by regular sequence in $R/J$, I want to show that $$J\cap I^t=JI^{t-1}\ \forall t\geq1.$$
Sahar
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an equivalence of functors involving tensor and hom

Let $\phi:(R,m,k) \rightarrow (S,n,l)$ be a morphism of local Noetherian rings. Let $M$ be a finite $R$-module and $N$ a finite $S$-module that is flat over $R$. Question: Why is it true that $Hom_R(k,M) \otimes_R N \cong Hom_S(k \otimes_R S, M…
Manos
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Atiyah-Macdonald Ex8.6

Is there anybody can give a proof? I can prove "finite" only, but I cannot prove "bounded". Here is the exercise: Let A be a Noetherian ring and Q a P-primary ideal in A. Consider chains of primary ideals from Q to P. Show that all such chains are…
Chan Yat-Fay
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Conditions leading to a conclusion.

Let $R$ be a commutative noetherian ring with unity, $M$ a finitely generated $R$-module, $I$ an ideal of $R$ such that $\bigcap_{t\ge 1} I^tM=0$ and $M\cong\underset{t}{\varprojlim}M/I^tM$. Now, let $U\subseteq M$ be a nonzero submodule. From the…
Q.TL
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Ideal structure of a ring, flat over a PID

I know that the following question is somewhat vague; I'm trying to understand what flatness "buys". Suppose $A, B$ are rings (commutative with identity) with $A$ a PID. Suppose we are given a flat homomorphism $\rho: A \to B$. What can be inferred…
Tom Bachmann
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height of ideal generated by regular sequence

Let $(A,m)$ be a local Noetherian ring, $M$ a finitely generated non-zero $A$-module and $a_1,\cdots,a_r$ an $M$-sequence. If $M=A$, then by using the Hauptidealsatz we can prove that $\operatorname{ht}(a_1,\cdots,a_r)=r$. For a general $M$, a…
Manos
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surjection and injection of modules over local ring

If I have a finitely generated module $M$ over a local noetherian ring $(A,m)$, where $m$ denotes the maximal ideal, and $\operatorname{supp}{(M)}=m$, then there exists a surjection $$M\rightarrow A/m$$ and an injection $$A/m\rightarrow M$$ How can…
Descartes
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