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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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Verifying universal property of Grothendieck group

I'm trying to verify the universal property of the Grothendieck group. Let $\overline{C}$ be the set of isomorphism classes of finitely generated $R$-modules over say a Noetherian ring $R$ and let $C$ denote the set of all f.g. $R$-modules. Let $A$…
Rudy the Reindeer
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Integral dependence and rings of fractions

I have a question on a Proposition in Atiyah and MacDonald's text. It concerns Proposition 5.12 ($A$ and $B$ are commutative rings with an identity) pictured here: Here's my concern: After multiplying the equation of integral dependence in the ring…
John Myers
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The localization is localization of some affine domain.

Let $A$ be a finitely generated $K$-algebra, and let $\mathfrak p$ be a prime ideal of $A$ such that $A_{\mathfrak p}$ is an integral domain. Then have to show that $A_{\mathfrak p}$ is a localization of a finitely generated $K$-algebra which is a…
user185640
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Need a reference/proof for computing the regularity of ideal of points in $\mathbb P^d$?

In a lecture notes on 'Cohomology modules' i read the following remark: Given a set $X$ of points in $\mathbb P^d$,using the Local Cohomology modules one can easily compute the reg$(S_X)$ where $S_X=\frac {S}{I_X}$,and $I_X$ is the homogeneous…
Arpit Kansal
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Minimal free resolution of ideal generated by three homogeneous polynomials

I am trying to solve the following exercise; Let $R=k[x_0,x_1,x_2]$ and $f_i$ homogeneous polynomials of degree $d_i, 0\leq i \leq 2$. Suppose $f_0,f_1,f_2$ have no common roots in $\mathbb P^2$. Construct a minimal free resolution of…
Arpit Kansal
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Why is $B[x]/M$ algebraic over $B/m$?

Let $B$ be a subring of some field $K$, $x$ some element in $K$, $m$ a maximal ideal in $B$ and $m[x]$ the extension of $m$ in $B[x]$ and $M$ a maximal ideal in $B[x]$ such that $m[x] \subset M $ and $M \cap B = m$. Why is $B[x]/M$ algebraic over…
Rudy the Reindeer
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Example of non-finitely generated $R$-algebra

By definition, an $R$-algebra is a ring homomorphism $f: R \to S$. For example, if $R=\mathbb Z$ and $S= \mathbb Z / n \mathbb Z$ then the projection $k \mapsto k \mod n$ is a ring homomorphism so that $\mathbb Z / n \mathbb Z$ is a $\mathbb…
Rudy the Reindeer
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A question about the proof that for an integral domain $D \subset K$ there is a valuation ring

I have some questions about a proof in Atiyah-Macdonald. It starts on page 65 where they write "...We want to prove that $B$ is a valuation ring of $K$." Let me modify the claim a tiny bit and present it as follows: If $A$ is an integral domain with…
Rudy the Reindeer
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Curve has a point which is either singular or has a tangent line parallel to the y-axis.

Suppose $k$ is an algebraically closed field with characteristic 0. Suppose $f(x,y)\in k[x,y]$ is irreducible and viewing $f(x,y)$ as a polynomial over $k[x]$ which is monic in $y$ and of degree>1 in $y$. We want to prove that the ideal…
wxu
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Relation between the localization $R_f$ and the polynomial ring $R\left[\frac{1}{f}\right]$?

Let $R$ be a commutative ring with $1$. (If required, assume also that $R$ is an integral domain.) Consider the localization $R_f$ at $0\neq f \in R$ where the multiplicative set is $S=\{f^n\}_{n \geq 0}$. Is there any relation between this ring…
user166467
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What is the notation behind the $k[x_1,\ldots,x_n]$?

I don't understand the proof of Noether Normalization Lemma in "Algebraic Geometry and Arithmetic Curves" . Liu considers first the case $k[X_1,X_0]/I$, then $k[X_1,X_1]/I=k[X_1]/I$, then again $k[X_1,X_2]/I$. It feels strange that induction proves…
Jaakko Seppälä
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Localization of regular sequence is still regular sequence

If $x_1,...,x_n$ is an $M$-sequence, then for prime ideal $P$ of $R$, can we localize the $M$-sequence to an $M_P$-sequence? $I_PM_P$ is not $M_P$ by Nakayama's lemma. Then how can I prove after localization, $x_i$ is non-zero divisor in…
user198206
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System of Parameters.

Let $R=k[X,Y,U,V]/(XV-YU)$, where $k$ is field of characteristic $0$. Consider $S=R_m$, where $m$ is the maximal ideal $(X,Y,U,V)/(XV-YU)$. How can we find a system of parameters for $S$ and what are they? We know that there are 3 elements in any…
messi
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A question on modules over Noetherian ring

If $G$ is a module over the non-trivial commutative Noetherian ring $R$ then is it possible that for all maximal ideal $M$ of $R$ we have $MG=G$ ? I guess the answer is no.
pritam
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Existence of associated prime

For a Noetherian ring $R$, it is well known that $R$ has at least one associated prime. In particular, minimal primes of $R$ are associated primes. My questions is Question 1: For a commutative ring $R$ with 1 not necessarily Noetherian, is a…
Youngsu
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