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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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Meaning of the Rank of a Map of Free Modules?

I am reading the section on differentials in Eisenbud's book (Commutative Algebra), and I'm just wondering what he means in sentences like this one: "Suppose that $J:R^t \rightarrow R^r$ is a map of free modules over a ring $R$ whose rank is less…
Cass
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Doubt in Atiyah-Macdonald in understanding Prop 5.6

In Atiyah-Macdonald I came across the following theorem: Let $A \subseteq B$ be rings, $B$ integral over $A$. If $\frak{b}$ is an ideal of $B$ and $\frak{a} = \frak{b}^c= \frak{b} \cap$$A$, then $B / \frak{b}$ is integral over $A/ \frak{a}$. If…
Ajin Shaji Jose
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Finite ring extension and field of fractions

There is a proposition: Let $A$ be a Noetherian integral domain, $B$ be an integral domain and $A \subset B$ a finite extension. Then $A\subset B\cap Q(A)$ is also a finite extension where $Q(A)$ is field of fractions of $A$. The proof is…
George
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Vanishing of higher $\text{Tor}$ in Cohen-Macaulay rings

Let $R$ be a Cohen-Macaulay local ring. Let $M$ be a finitely generated $R$-module with $\text{pd }M<\infty$. Let $N$ be a maximal Cohen-Macaulay $R$-module. Then $\text{Tor}_i^R(M,N)=0$ for all $i>0$. If this statement is true. It suffices to show…
Bromelain
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Reduction to the case $n=1$ in the proof of Proposition 5.23 of Atiyah, Macdonald

I don't understand the first sentence in the proof of Proposition 5.23 of Atiyah, Macdonald, Introduction to Commutative Algebra. Assume we know the result true for all $A$-algebras that can be generated by $n$ elements. Suppose…
Elías Guisado Villalgordo
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Example of a ring $A$ such that $\text{nil}(A)^n\neq 0 \forall n\in \mathbb{N}$

I understand the somewhat default example with $k[X_1…]/(x_1,x_2^2…)$. My question is, would $k[X_1…]/(x_1^2,x_2^2,x_3^2,…)$ work as well? Since for any $n$ I could just take the product of $x_1\cdot…\cdot x_n\neq 0$? Or is my understanding of ideal…
average math enjoyer
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How to prove the number of distinct prime divisors?

Could someone explain me the following Lemma 4.3.1 from the book "Field Arithmetic", 3rd edition by Michael D. Fried and Moshe Jarden: Let $F/K$ be a function field and denote the unique extension of $K$ of degree $r$ by $K_r$. Also, denote the…
Jaakko Seppälä
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For a finite locally free $A\to B$ when does the different equal the Noether different?

(Cross-posted to MO.) All rings commutative with $1$. Let $A\to B$ be an $A$-algebra which is finite projective, meaning $B$ is finitely generated projective as an $A$-module, so there is the trace map $\operatorname{tr}_{B/A}:B\to A$ where…
P-addict
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Is this generalization of Hilbert's basis theorem already known (and is it even true)?

Hilbert's basis theorem states that if $R$ is a commutative noetherian ring (i.e. every ideal $I\subseteq R$ is finitely generated) then $R[x]$ is noetherian as well. I immediately thought that this could be proven using the fact that $R[x]/(x)\cong…
Li__ON
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Explicit Construction for Cohen’s $p$-ring with imperfect residual field

Let $p>0$ be a prime. Recall that a $p$-ring is a complete DVR $A$ with residual filed $k$ of characteristic $p$ such that $p$ is a uniformizer. It is known (e.g. Ch 29. of Matsumura’s Commutative Ring Theory) that for any field $k$ of…
aaa acb
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Deduce Stone's theorem in commutative algebra

I am working on Ex.1.25 of Atiyah's "Commutative Algebra", which asks to deduce Stone's Theorem: every Boolean lattice is isomorphic to the lattice of open-and-closed subsets of some compact Hausdorff topological space. By the previous exercises, I…
Yuheng Shi
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Deducing purity from existence of Cohen Macaulay modules

Let $R$ be a regular local ring, $S$ be a normal extension of $R$ that is finite over $R$ and is unramified at height $1$ primes of $R$. Griffith's article normal extensions of regular local ring (Comment before Theorem 1.6) states that the purity…
fyx1123581347
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The integral closure of a power series ring over a field

Let $k$ be a field of characteristic $p$ and $K$ the field of fractions of the formal power series ring $k[[X_1,\dots,X_n]]$. Let $L$ be a finite purely inseparable field extension of $K$, then there exists an integer $m$ such that $L\subseteq…
nick
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If an ideal contains a "toric" polynomial, does then also the Groebner basis contain such polynomial?

Suppose $k$ is an algebraically closed field, $I \subset k[x_1, \ldots, x_n]$ is an ideal that contains a polynomial of the form $x_1^{m_1} \cdots x_r^{m_r}+cx_{r+1}^{m_{r+1}}\cdots x_n^{m_n}$, where $c \in k$, and $m_i \in \mathbb{N} \cup\{0\}$ for…
K.J.
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Kernel of a matrix with polynomial entries

What is the most direct method to calculate the kernel of a matrix $A\in k [x_1,\dots,x_l]^{n\times m}$, $k$ being a field? What is the most direct method to calculate its Hilbert series? Is one obliged to use Gröbner bases?
HCH
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