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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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Irredundant primary decomposition of a submodule transferred to irredundant primary decomposition of an ideal

Let $N=\bigcap_{i=1}^nN_i$ be an irredundant primary decomposition of a submodule $N$ of the $R$-module $M,$ where $(N_i:M)$ is a $P_i$-primary ideal of $R$. (Irredundant means that all the $P_i$'s are different and $\bigcap_{i\neq j}N_j\nsubseteq…
harajm
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Direct Sum Decomposition of Certain Quotient Algebras

Consider a weighted homogeneous polynomial $f \colon (\mathbb{C}^{n}, \mathbf{0}) \to (\mathbb{C},0)$ with an isolated critical point at the origin and satisfying $\lambda f(z_1, \dots, z_n) = f(\lambda^{\omega_1} z_1, \dots, \lambda^{\omega_n}…
user02138
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Why $\dim \operatorname{supp} M_{m}\ge \dim \operatorname{supp} M_{m'}$?

Jacob Lurie made the following claim during his lecture: If $R\rightarrow R'$ is a morphism that makes $R'$ a finitely generated $R$-module (in particular, integral over $R$). Let $m'\subset R'$ be maximal. Let $m$ be the pull back to $R$, which is…
Bombyx mori
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Preprint by W. Danielewski, "On the cancellation problem and automorphism groups of affine algebraic varieties," Warsaw, 1989?

Where can I get a copy of the 8-page preprint by W. Danielewski, "On the cancellation problem and automorphism groups of affine algebraic varieties," Warsaw, 1989? It appears to have never been published. Everyone cites the preprint.
Tri
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Why we need $R$ to be radical at here?

Jacob Lurie used one example for the theorem of Scholism: Let $R=k[X]$ be the coordinate ring of a variety $X$ in $\mathbb{C}^{n}$. Assume $X$ is reduced. Then $MaxSpecR$ is a union of irreducible components $X_{i}$, which are the closures of the…
Bombyx mori
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If $B$ is integral over $A$ and there is only one $P$ over $\mathfrak p$, then $B_P=B_\mathfrak p$?

I am stuck in the Exercise 5.3 of Matsumura's Commutative Algebra: $\newcommand{\p}{\mathfrak p}$ $\newcommand{\sp}{\operatorname{Spec}}$ Let $B$ be a ring, $A$ be a subring and $\p\in\sp(A)$. Suppose that $B$ is integral over $A$ and that there is…
josephz
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If $g=(f\cdot a+g\cdot b) G-J\cdot g\mod m^{i+1}[x]$ is $g$ a multiple of $G$?

Let $R$ be a local ring with maximal ideal $m$. If we have polynomials $f,g\in m^i[x]$ $a,b, G\in R[x]$ $J\in m[x]$ such that $$g=(f\cdot a+g\cdot b) G-J\cdot g\bmod m^{i+1}[x].$$ Then is it correct that $g$ is a multiple of $G$ in $m^{i+1}[x]$? I…
KJA
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Prove that the following module is injective

Let $R$ be a commutative Noetherian ring with identity. Prove that if $I$ is an ideal of $R$ and $E$ an injective $R$-module, then $\bigcup_{n\geq 1}(0:_{E}I^{n})$ is an injective $R$-module. Please help me.
Aliakbar
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Special Case of Lying Over Theorem

In Pete Clark's commutative algebra lecture notes which can be found here. He proves the following lemma (14.12) Let $R$ be a local ring with maximal ideal $\mathfrak{p}$ and $S/R$ an integral ring extension. Then the pushed forward ideal…
JSchlather
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A step in the proof of Atiyah' s exercise 2.26

Let $N$ be an $A$-module. The exercise want to prove $N $ is flat iff $\operatorname{Tor}_{1}(A/\alpha,N)=0$ for all finitely generated ideals $\alpha$ in $A$ From the hint ，I know $N $ is flat iff $\operatorname{Tor}_{1}(M,N)=0$ for all…
Mike
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Faithful Flat after Base Change

Let $k \to k'$ a field extension / injective morphism and $A$ a $k$-algebra. Obviously $k \to k'$ is faithful flat (so flat and corresponding map $\operatorname{Spec}(k') \to \operatorname{Spec}(k)$ is surjective). Why is $k \otimes_k A \to k'…
user267839
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Question on Integral Closure

I'm trying to prove this fact: given $A$ an integral domain and an element $f\in A$ such that $A/fA$ has no nilpotents, then $A$ is integrally closed if and only if $A_f$ is integrally closed ($A_f=S^{-1}A$ with $S=\{1,f,f^2,...\}$). One implication…
Corra
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Are these two subrings of $\mathbb{C}\lbrack t\rbrack$ isomorphic?

Are the subrings $\mathbb{C}\lbrack t^2,t^3\rbrack$ and $\mathbb{C}\lbrack t^3+3t^2,t^2+2t\rbrack$ of $\mathbb{C}\lbrack t\rbrack$ isomorphic? I do not think that they are, but I was not able to prove it. My attempt was focused on comparing their…
TheoPatr
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Hilbert-Samuel polynomial computation tutorial

I'm trying to understand how to compute the Hilbert-Samuel polynomial of a specific example. Could someone help me with an elaborate computation so that I get it... For example, what is the HS-polynomial of $\mathbb Z[x,y,z]_{(2,x,y-1,z-2)}$ (i.e.…
jiaji
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filtered colimit of $Hom_{A_i}(M_0\otimes_{A_0} A_i, N_0\otimes_{A_0} A_i)$

Let $I$ be a small filtered category. Let $F\colon I \rightarrow \textbf{CRng}$ be a functor, where $\textbf{CRng}$ is the category of commutative rings. We write $A_i = F(i)$ for $i \in I$, $A =$ colim $A_i$. Suppose $I$ has an initial object…
Makoto Kato
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