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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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Krull dimension of generic fiber

Let $p$ be a prime and $A=\mathbb{Z}_p[t_1,\ldots,t_n]/I$ a reduced flat and irreducible $\mathbb{Z}_p$-algebra of finite type and of Krull dimension d. Let $e$ be the Krull dimension of the $\mathbb{Q}_p$-algebra $A[\frac{1}{p}]$ whose spectrum is…
user228077
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Relation between the inverse of finitely generated ideals and the inverse of their powers.

Let $D$ be an integral domain, $K$ its field of fractions, and $J_1,...,J_n$ are ideals of $D$ such that $(\sum_{i=1}^{n} J_i)^{-1}=D$. How can we prove that this implies $(\sum_{i=1}^{n} J_i^m)^{-1}=D$ for all $m\ge1$ ? I know that the converse…
Marcos
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Question about the polynomial grade of a finitely generated ideal

Let $R$ be a commutative ring (not necessary Noetherian), $Q$ its total ring of fractions, and $I$ a finitely generated ideal of $R$ such that $\forall$ $a \in I$ we have $(a:_R I) = a$. My question is how to prove that this implies $(R:_Q I) = I$…
Med S.
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$((I \cap J)^{-1})^{-1} = (I^{-1})^{-1} \cap (J^{-1})^{-1}$ for ideals of an integral domain?

Let $A$ be an integral domain and $I$, $J$ be non-zero ideals. Is $((I \cap J)^{-1})^{-1} = (I^{-1})^{-1} \cap (J^{-1})^{-1}$? For an ideal $I$, we define $I^{-1} = \{x \in K \mid xI \subset A\}$, where $K$ is the field of fractions of $A$. The…
Makoto Kato
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Line Bundles on Local Rings

Let $A$ be a local ring and $L$ a module over $A$ which is projective and of rank one. Does it follow that $L$ is isomorphic to $A$?
Nicolas Bourbaki
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"Going between" property

Let $A \subset B$ be an integral ring extension and assume that $A$ is a finitely generated $K$-algebra over some field $K$. Let $P_1\subsetneq P_3$ be prime ideals of $A$ and let $Q_1\subsetneq Q_3$ be prime ideals in $B$ lying over $P_1$ and…
user181158
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Isomorphism of the completion of polynomial ring modulo second degree polynomial

Let $k$ be a field of characteristic different from $2$, and $A=k[x,y]/(y^2-x^2(x+1))$. Let $\hat A$ be the $(x,y)A$-adic completion. How can I show that $\hat A\simeq k[[u,v]]/(uv)$? Qing Liu: Algebraic Geometry and Arithmetic Curves, ex. 1.3.10.
Jaakko Seppälä
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Is the ring $F(U(R))$ necessarily isomorphic to the ring of all polynomials with coefficients in $R$ and with constant term equal to $0$?

(All my rings are commutative, but not necessarily unital.) I was playing around with the ring freely generated by an Abelian group, and it seems to me that the following holds: letting $U$ denote the functor that takes a ring and returns its…
goblin GONE
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Calculating Spec of the localization $R_P$

I am studying a first course in commutative algebra and I'm currently working through some exercises on calculating $Spec(R_P)$, where $R_P = R[(R\backslash P)^{-1}]$ is the localization of $R$ at a prime ideal $P$. Unfortunately, I'm not sure if…
Mystery_Jay
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Tensor product and inverse limit

Let $(R,m)$ be a Noetherian local ring. Let $M$ be an $R$-module, and let $\{N_i\}$ be an inverse system. I am curious to know if there is a condition whereby the natural map $M \otimes_R \varprojlim{N_i} \to \varprojlim{(N_i\otimes_R M)}$ is…
thebowler_300
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Existence of a canonical isomorphism of completions

How one can do the problem 1.3.8 from Qing Liu's Algebraic Geometry and Arithmetical Curves. Namely, Let $A$ be a Noetherian ring, and $I,J$ ideals of $A$. Let $\widehat{A}$ be the $I$-adic completion of $A$ and $(A/J)^\widehat{\hskip1ex}$ the…
Jaakko Seppälä
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Tensorproduct of $R$-module and $R$-algebra

let $M$ be an $R$-module and let $S$ be an $R$-algebra through the ring homomorphism $\phi$. I can make $M\otimes S$ into a $R$-module in several different ways. Either by defining $r. (m\otimes s)=rm\otimes \phi(r)s$ or $r. (m\otimes…
harajm
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a f.g., projective, non free $R-$module

I know that if $R$ is a PID ring, then a projective $R-$module is free. Now, i want an example of a f.g., projective, non free $R-$module where $R$ is a non PID ring.
Aliakbar
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nonNoetherian ring of countable cardinality

I recently asked myself if I could find a nonNoetherian ring (commutative w/ one) of countable cardinality. I could not. My wealth of nonNoetherian rings is small and usually relies on taking $k[x_{1}, ....]$ modulo something, a nonfinite direct…
do_math
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Let $R$-algebra $A$. If $P⊂A$ is a minimal prime ideal then $p=P \cap R$ consists of zerodivisors for $A$?

We have: Let $R$ be a Noetherian commutative ring. Suppose $P⊂R$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors. But how to prove this? The $R$-algebra $A$ is assumed to be finitely generated as an $R$-module. For…
hakuna
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