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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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Faithful modules in integral dependence

In Atiyah-Macdonalds, Proposition 5.1, for a ring $B$ and a subring $A$, we have There exists a faithful $A[x]$-module $M$ which is finitely generated as an $A$-module $\Rightarrow$ $x \in B$ is integral over $A$. The proof uses the fact that…
nekodesu
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In commutative ring, flat is equivalent to locally free

In wikipedia https://en.m.wikipedia.org/wiki/Flat_module , particularly Case of commutative rings, they say that "In a commutative ring, a finitely generated module is flat if and only if it is locally free, i.e. $M_P$ is free for all prime…
T C
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Integrality of an element of the quotient field of a domain

Let $R$ be a Noetherian domain with quotient field $K$, $I \subseteq R$ a finitely generated ideal, $I \neq (0)$ and $x \in K$ such that $x \cdot I \subseteq I$. I want to show that $x$ is integral over $R$. Supposedly, this is also true when…
Ben
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Ring containing a Dedekind ring

Suppose I have two domains, $A\subset B$, where $A$ is Dedekind and $\operatorname{Frac}(A)=\operatorname{Frac}(B)$. I also know that $B$ is both integrally closed and has height $1$. Is $B$ necessarily Dedekind? If not, I'd love to see a…
Brett Frankel
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Compactness of a complete DVR with finite residue field

Let $(K, |.|)$ be a non-archimedean field, with valuation ring $\mathring{K}$ and residue field $\tilde{K}$. I want to show that $\mathring{K}$ is compact $\Leftrightarrow$ $K$ is complete, $\mathring{K}$ is a DVR and $\tilde{K}$ is finite. I did…
LeLoupSolitaire
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Localization at Maximal Ideals

Suppose $A$ is a commutative ring with $1\neq0$ satisfying the property that $A_\mathbf{m}$ has no nonzero nilpotent elements for any maximal ideal $\mathbf{m}$, where $$A_\mathbf{m}=S^{-1}A\quad \text{ and }\quad S=A-\mathbf{m}.$$ Prove that $A$…
Clayton
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Local $k$-algebra with residue field $k$

Let $k$ be a field, $A$ a local $k$-algebra with maximal ideal $\mathfrak m$. Suppose furthermore that the residue field $A/\mathfrak m$ is isomorphic to $k$ as a ring. Can we then deduce that $$ k\to A \to A/\mathfrak m $$ is an isomorphism, i.e.…
user501746
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Formal implicit function theorem

I have seen a lemma about formal series in the book Complex Projective Varieties by D. Mumford. Let $$f(x)=\sum_{i=1}^n a_iX_i+(\text{higher order terms})\in \mathbb{C}[[X]]$$ and assume $a_1\not=0$. Then $f$ can he factored…
Intoks Liobein
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Localization of finite modules, or: compatibility of ideal norms with localization at a prime number

Let $A$ be an integral domain and $M$ be a finite $A$-module (where "finite" means "finite cardinality"). Let $S\subset A$ be a multiplicative subset. Is the cardinality of $S^{-1}M$ the same as that of $M$? I'm interested in the case where…
user46225
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On the definition of integral closure

I'm self-learning Integral Closure of Ideal via the book of I.Swanson &C.Huneke. In the begining of this book they say that: A next attempt can be an asymptotic version. Let $v_n(r)$ be the least power of $r$ in $I^n$. If the limit of $\frac…
Anh_Rose 1210
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Picard group of $\Bbb R[x_1,\dots,x_n]/(x_1^2+\dots+x_n^2-1)$ and $\Bbb C[x_1,\dots,x_n]/(x_1^2+\dots+x_n^2-1)$

What is the Picard group of $\Bbb R[x_1,\dots,x_n]/(x_1^2+\dots+x_n^2-1)$, i.e. the coordinate ring of real sphere $S^{n-1}$, and $\Bbb C[x_1,\dots,x_n]/(x_1^2+\dots+x_n^2-1)$? As $\Bbb R[x_1,x_2]/(x_1^2+x_2^2-1)$ is not a UFD while $\Bbb…
user395911
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When does a ring have a flat noetherian extension?

The title says it all: Let $A$ be a commutative ring. Are there any interesting known conditions on $A$ (other then being noetherian of course...) to ensure existence of a (non-zero) commutative noetherian ring $B$, and a flat ring map $A\to B$?
the L
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projective dimension

All rings and algebra in this question are commutative and contains unity. Suppose $M$ is an $A$ module and $A$ a $R$ algebra. If $pd_R(M) < \infty$, then will that imply $pd_A(M) < \infty$? In particular if $M = \frac{R[X_1, X_2,…
user7451
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Existence of a simple submodule

Let $(R,\mathfrak{m})$ be a local ring and $N$ be a $R-$module such that every element of $N$ is killed by a power of $\mathfrak{m}$. Show that $N$ has at least a simple submodule, that is $$soc(N)\ne0.$$ This implies a question : When does a…
tlquyen
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Uniqueness of prime factorization of ideals

According to Wikipedia (and some other resources), if every ideal in a ring (or domain) $R$ can be decomposed into a product of prime ideals, then the factorization is unique up to the order of factors. Is there a direct proof of that? (i.e. Do not…
Eclipse Sun
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