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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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Localisation commutes with taking quotients.

If $A$ is a ring, $S$ a multiplicative set and $I$ an ideal, write $T$ for the image of $S$ in $A / I$. Then $T^{-1}(A/I) \cong S^{-1}A/S^{-1}I$ and in particular, for a prime ideal $P$ we have that $A_P/PA_P$ is isomorphic to Frac$(A/P)$. My…
user117449
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Hilbert Polynomial vs Hilbert Quasi-Polynomial

Let $R$ be an $\mathbb{N}$-graded ring with $R_0$ Artinian and $R = R_0[x_1,\dots,x_r]$, where the degree of $x_i$ is $d_i > 0$. Let $M$ a finitely generated $\mathbb{N}$-graded $R$-module with Hilbert function $H(M,n)$. Then it is known that there…
Manos
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on the proof of Theorem 4.3.2 in Bruns & Herzog ``Cohen-Macaulay Rings" (Gotzmann's regularity theorem)

The theorem and the first part of its proof is shown below: In particular, the authors conclude (2 lines below equation (2)) that $(i): P_R(n) = {n + a_1 \choose a_1}+\cdots+ {n+a_r -(r-1) \choose a_r} + c$. Question 1: Is there an immediate way to…
Manos
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Extension of R-linear derivation to localization

Let $R$ be a commutative ring. Given a commutative $R$-algebra $A$, a multiplicative subset $S \subset A$, and a $R$-linear derivation $D: A \rightarrow M$, where $M$ is an $S^{-1}A$-module, $D$ can be uniquely extended to a $R$-linear derivation…
baltazar
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tensor of Hom-sets.

Let $k$ be a field. Let $A$ be a commutative $k$-algebra. Let $M,N$ be $A$-modules, and assume that $M$ is finite over $A$. Is it true that the map $$\mathrm{Hom}_{A}(M,N) \otimes_{k} \mathrm{Hom}_A(M,N) \to \mathrm{Hom}_{A\otimes_k A}(M\otimes_k…
just me
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If $A$ is a semilocal ring and $f:A\rightarrow B$ is a surjective homomorphism, then $f(\operatorname{rad}A)=\operatorname{rad}B$

If $A$ is a semilocal ring and $f:A\rightarrow B$ is a surjective homomorphism, then $f(\operatorname{rad}A)=\operatorname{rad}B$. I know that if $A$ is a semilocal ring and if $I_{1},\dots, I_{n}$ are all of its maximal ideals, then…
user23505
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Why does passing to the reduced ring not change the number of primes ideals?

I'm reading a note of Hochster's, and I don't follow something. He writes as the Corollary on page 9, Let $K\subseteq S$, where $K$ if a field, and $S$ is a finitely-dimensional $K$-vector space of dimension $\leq n$. Passing to $S_\mathrm{red}$…
Brach
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Eisenbud Corollary 6.7

Let $k$ be a field, $R=k[t]$ the polynomial ring in one variable, let $S$ be a Noetherian ring flat over $R$, If the fiber $S/tS$ over $t$ is a domain, and $U$ the set of elements of the form $1-ts$ for $s\in S$, then $S[U^{-1}]$ is a…
user93417
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1 answer

Quotient of a local regular ring

How can I prove this: Let $A$ be a local regular ring with maximal ideal $\mathfrak m$ and $x \in \mathfrak m-\mathfrak m^2$. Then $A/(x)$ is a regular ring. Prove also that if $x\in\mathfrak m^2$, $x\ne 0$, the result does not hold anymore. I…
balestrav
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a basic question about the locus of a polynomial

First a little introduction and notation. I have a question with a words that the book says. Let k be a field. Let F $\in $ $k[x_1,...,x_n]$ We define the locus $V(F)$ of F , by $$ V\left( F \right) = \left\{ {P = \left( {a_1 ,...,a_n }…
August
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Cohen structure theorem for artinian local rings

Let $(R,m)$ be an artinian local ring. Since $m^n=0$ for some $n$, it is clear that $R$ is complete with respect to $m$-adic topology. Now i want to know that how do we state the Cohen structure theorem for $R$?
e-r
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$Hom_{R-Alg}(\prod R, S)$

Given an $R$-algebra $S$, is $Hom_{R-Alg}(\prod_{i=1}^n R, S) = Hom_{S-Alg}(\prod_{i=1}^n S, S)$?
user3267
  • 420
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Krull dimension in finite ring extensions

Let $K$ be a field and $R=K[a_1, \dots, a_n]$ a finite ring extension. Suppose that the degree of transcendence of $R$ over $K$ is $r$. Then the Krull dimension of $R$ is at most $r$. I would like to prove it is exactly $r$. If ${t_1, \dots , t_r}$…
Andrew Chiriac
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question about $Spec(A)$ in Atiyah's book Introduction to Commutative Algebra

Let $A$ be a ring and $X=spec(A)$, the prime spectrum of $A$. Prove that $X$ is quasi-compact. Definition of quasi compact: each open covering of $X$ has a finite subcovering of $X$. It is enough to considering the covering in the basis…
claire
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An ideal that contained in finitely many maximal ideals but all of its elements contained in infinitely many maximal ideals

Is it possible that an ideal I in an integral domain D is contained in only finitely many maximal ideals but each element of I is contained in infinitely many maximal ideals? I am quit sure that it is possible but I need an example. If some one have…
Shafiq Ur Rehman
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