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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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Isomorphism of tensor product

Let $k$ be a field and $A$ and $B$ be two commutative $k-$algebras. Furthermore, let $I$ be an ideal in $A$ and $N$ be a $A\otimes_kB$-module. Then is it true that $((A/I) \otimes_k B) \otimes_{A\otimes_k B} \ N)$ as a $B-$module is isomorphic to…
Cyril
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Product of ideals generated by linear forms that has zero dimension

Let $k$ be a field, $R=k[x_1,\dots,x_r]$ the polynomial ring in $r$ indeterminates and $I_1,I_2\dots,I_d$ homogeneous ideals of $R$ generated by linear forms. Define $J = I_1 I_2 \cdots I_d$ and suppose that $\dim R/J = 0$. Question: Is it true…
Manos
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showing that a set of linear forms is closed (Bruns and Herzog, Theorem 4.2.12)

Let $k$ be an infinite field and $R$ a homogeneous $k$-algebra, i.e. a $k$-algebra that is generated by linear forms. Let $s = \sup\left\{\dim_k h R_{n-1} : h \in R_1\right\}$, where $R_i$ denotes the homogeneous component of $R$ of degree $i$. Let…
Manos
  • 25,833
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How is an onto map implies $N+mM=M$ in Commutative Algebra?

I am having hard time understanding some details in Proposition 2.8 which is on page 22 of Atiyah and Macdonald's book: Introduction to Commutative Algebra. How the writers are claiming that being the composite…
user54992
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Problem 10.5 in Atiyah's book

Here is the problem: Let $A$ be a Noetherian ring and $a$, $b$ be ideals in $A$. If $M$ is any $A$-module, let $M^a$, $M^b$ denote its $a$-adic and $b$-adic completions respectively. If $M$ is finitely generated, prove that $(M^a)^b\cong…
WWK
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Factoring polynomials in several variables over algebraically closed fields

This is a follow-up to Projective Spectrum of $K[X,Y]$ . I see why the given ideals are prime or even maximal, however, I have yet to prove that they in fact make up the entire spectrum of $K[X,Y]$. Why is it that any polynomial other than the…
Wendelin
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What is the unmixedness theorem?

I want to show that $R$ is Noetherian local ring and $S$ is Gorestein local ring s.t. $R=S/J$ and $f$ is an $R$-regular element, if for every $p\in\operatorname{Ass}_{S}(R),\operatorname{ht}_{S}(p)=\operatorname{ht}_{S}(J)$, then for every…
user141617
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A question about the depth of a ring with respect to some ideal

So here is my question: I want to compute the depth of $k[x,y]$ with respect to the ideal $(x,y^2)$ where $k$ is a field. The depth $t_{(x,y^2)}(k[x,y])$ is defined as follows, $$ t_{(x,y^2)}(k[x,y]):=\sup\{r\in\mathbb N:\exists(x_i)_{i=1}^r…
Thorben
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About injective hull of residue field

Let $(A,\mathfrak{m})$ be a noetherian local ring, and $E(A/\mathfrak{m})$ the injective hull of $A/\mathfrak{m}$. I'm pretty sure that $E(A/\mathfrak{m})$ doesn't automatically extend to an $A/\mathfrak{m}$-module via the projection map…
ashpool
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Why is $f'(x)$ the annihilator of $dx$?

Let $B=A[x]$ be an integral extension of a Dedekind ring $A$ where $x$ has minimal (monic) polynomial $f(x)$. Then the module of Kahler differentials $\Omega_A^1 (B)$ is generated by $dx$. Why is its annihilator $f'(x)$? $\Omega_A^1 (B)=I/I^2$,…
Rodrigo
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Construct a counterexample of a primary ideal which ....

Let $A$ be a Noetherian local ring of dimension $d$, $\mathfrak{m}$ its maximal ideal. Suppose $\mathfrak{q}=(x_1,\ldots,x_d)$ is an $\mathfrak{m}$-primary ideal. Suppose $f(t_1,\ldots,t_d)\in A[t_1,\ldots,t_d]$ is a homogeneous polynomial of…
user119882
  • 1,442
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If $P$ is a prime ideal in a ring, then its only associated prime is the ideal itself?

This seems obvious to me, but I don't see it explicitly mentioned, so I may be wrong here, but since a prime ideal is primary, we have a primary decomposition for a prime ideal consisting of just the prime ideal and it's radical is itself. So this…
Sanjoy Pagg
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When can a ring homomorphism to the integers modulo 2 be lifted to a homomorphism to the integers?

Let $A$ be a commutative ring with unity. Let $f: A \to {\mathbb{Z}}/2$ be a ring homomorphism to the integers modulo 2. When does there exist a lift $g: A \to {\mathbb{Z}}$ to the integers such that $f(x)$ is the residue of $g(x)$ modulo 2 whenever…
user17982
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Question about proof that every f.g. projective module over a local ring is free.

I'm reading the proof here. I'm at the line where they say $$ \psi\pi(f)=\psi(f+FR)=\varphi(f)+PR.$$ Since $\psi\pi$ is surjective, it should follow that $\{\varphi(f)+PR:f\in F\}=P/PR$. I don't understand the notation $\mathrm{im}(\varphi)+PR=P$. I…
cdfd
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a "paradox" regarding regular and complete intersection rings

The following "paradox" arose as i was studying the proof of Theorem 2.3.3 in Bruns and Herzog, CMR. My question is self-contained but i could expand on details upon request. Let $(S,\mathfrak{n})$ be a regular local ring and let $I$ be an ideal of…
Manos
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