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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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The square of the maximal ideal in a local ring of dimension $2$ over a field.

I found the following assertion at page 62 in Geometry of schemes by Eisenbud and Harris: Let $R$ be a local $K$-algebra of vector-space dimension $2$, where $K$ is an algebraically closed field, and let $\mathfrak m$ be its maximal ideal. Then…
awllower
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Minimal number of generators of $Hom_R(M,M)$

Let $(R,m,k)$ be a local ring, and $M$ be an $R$-module of finite length. Suppose the minimal number of generators $\mu(M)$ of $M$ is $n$. What is the relation of the minimal number of generators $\mu(Hom_R(M,M))$ of $Hom_R(M,M)$ with $n$? Since…
Flying pencil
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a problem involving a homogeneous ideal and an infinite field (Matsumura, CRT, 13.1)

I am trying to solve the following problem (this is 13.1 from Matsumura's Commutative Ring Theory): Prove the following: (i) Let $R= \bigoplus_{n\ge0}R_n$ be a graded ring. Then for any $u \in R_0^*$ the map $T_u(\sum x_n) = \sum x_n u^n$ is an…
Manos
  • 25,833
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On a certain colon ideal

There's a remark in Infinite Integral Extensions and big Cohen--Macaulay Algebras by Hochster and Huneke which I cannot verify. Let's first define some notation. Let $I$ be an ideal of a ring $R$ of characteristic $p>0$ prime. Let $I^{[p]}$ be the…
Alex Scheffelin
  • 356
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Atiyah-Macdonald Ex7.22: Equivalent condition for $E$ to be open in a Notherian space

Atiyah-Macdonald Ex7.22 Let $X$ be a Notherian topological space and let $E$ be a subset of $X$. Show that $E$ is open in $X$ if and only if, for each irreducible closed subset $X_0$ in $X$, either $E \cap X_0 = \emptyset$ or else $ E \cap X_0 $…
Gobi
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Applying a theorem about $\mathbb{Z}[x_1,\dots,x_n]$ to arbitrary rings

Apparently there is this general proof technique where one takes advantage of the fact that objects are usually specified by a finite amount of data. In commutative algebra one example is proving a result over $\mathbb{Z}[x_1,\dots,x_n]$ then…
P-addict
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Finitely generated ideal

We say that an ideal $\mathfrak a$ of $A$ is finitely generated if $\mathfrak a =(x_1,\cdots,x_n)=\sum_{i=1}^{n} Ax_i$, i.e. finitely generated as an $A$-module. Is there a name for when $\mathfrak a$ is generated by all the finite products of the…
Gobi
  • 7,458
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Depth formula of Stanley-Reisner rings

Let $\Delta$ be a simplicial complex with $r$-skeleton $\Delta_r$ (i.e. the set of all faces $F\in \Delta$ such that $\dim F\leq r$). Show $\operatorname{depth}k[\Delta]=\text{max}\{r:\Delta_r \text { is Cohen-Macaulay over $k$}\}+1$. This is…
Bromelain
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rings in which all ideals are radical

If $R$ is a commutative Hilbert ring then each of its prime ideals is an intersection of maximal ideals. Is there a similar class of commutative rings for which every ideal is an intersection of prime ideals (that is, in which all ideals are…
Chris Leary
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Finite dimensional commutative local algebras -- reference request

What can be said about the structure of a finite dimensional, commutative, associative, unital local algebra over an algebraically closed field of characteristic zero?
Frank
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What do elements in a localized quotient ring look like?

Given a ring $R$ and a prime ideal $P$, then we can define localized ring denoted $R_p$ effectively giving all elements not in the ideal the property of unit. I understand that $R_p/I_p\cong (R/I)_p$ for any ideal $I$ of $R$. But I'm having trouble…
Student Lv 1
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Reducing the dimension of a finitely generated $k$-algebra by $1$

Let $k$ be a field and $Q$ an ideal of $k[x]=k[x_1,\dots,x_s]$ such that $R=k[x]/Q$ has Krull dimension equal to $d>0$. Define $V=x_1 k + \cdots +x_s k$ to be the vector space of linear forms over $k$. Let $P_1,\cdots, P_t$ be the minimal prime…
Manos
  • 25,833
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Units of Witt vectors

Let $R$ be a perfect ring of characteristic $p$, $W(R)$ its Witt vectors. We know that any element $x\in W(R)$ has a unique Teichmuller expansion: $$x=[c_0]+[c_1]p+[c_2]p^2+\cdots$$ here all $c_i$ are in $R$. I want to know whether the following…
Richard
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Theorem 26.10 (the Cartier Equality) in Matsumura's Commutative Ring Theory

I am trying to understand the proof of Theorem 26.10 in Matsumura's Commutative Ring Theory, and I cannot understand the last step. To expand a bit, let $k\to A\to B$ be ring homomorphisms, then we have a natural map $\Omega_{A/k}\otimes_A B\to…
Alex Scheffelin
  • 356
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showing that the power series in two indeterminates over a field has dimension 2

Let $k$ be field and consider the power series $A=k[[x,y]]$. What is the simplest way (in the sense of using the least "heavy" theorems) to show that $\operatorname{dim} A=2$, where $\operatorname{dim} A$ means dimension as a ring (Krull…
Manos
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