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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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Embedding $\mathbb Q$ in $\mathbb Z^n$

I have read that the $\mathbb Z$-module $(\mathbb Q,+)$ is not isomorphic to a submodule of $\mathbb Z^n$, for any $n\in \mathbb Z^+$. I come up with a proof (below) that seems a bit facticious, and likely, there is a cleaner way to do it. What do…
CRinge
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Elements in exterior product vanish.

Let $R$ be a commutative ring with $1\in R$. Let $M$ be an $R$-module. The exterior product of degree 2 is defined as $$M\wedge M:=\frac{M\otimes_RM}{\langle m\otimes m:m\in M\rangle}.$$ Let $x,y\in M$ be two elements. If there exists $m\in\mathbb…
Display Name
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Showing that a module has depth zero in the proof of the Buchsbaum-Eisenbud criterion of exactness of finite free resolutions

Let $R$ be a Noetherian ring with a maximal ideal $P$. Assume $\operatorname{depth} R_p=0$. Let $\varphi : F\to G$ be a homomorphism of free modules of finite ranks. Let $M:=\operatorname{coker}\varphi$. Then how do we show that…
Plantation
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$k$ Field. Prove that $(XW-YZ,Y^2-XZ,Z^2-YW)\in k[X,Y,Z,W]$ is an ideal generated by a regular sequence

$k$ is a field, then $k[X,Y,W,Z]$ is a domain, so $XW-YZ$ is a non-zero divisor. $XW-YZ$ is irreducible, then $(XW-YZ)$ is a prime ideal and $k[X,Y,W,Z]/(XW-YZ)$ is a domain, so $Y^2-XZ$ is a non-zero divisor. That's the easy part. I'm failing to…
user8785084
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ideals that become principal in extension

Let $A,B$ be two Dedekind domains such $A\subset B$. On denote by $K$ and $L$ their quotient field respectively. One assumes que $[L:K]$ is finite. Let $I$ be an ideal of $A$. Suppose that $IB=a B$ with $a\in A$. Can one assert that $I=aA$? Thanks…
joaopa
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Diagonal F-threshold

Let $(R, \mathfrak{m},K)$ be a standard graded $K$-algebra of characteristic prime $p$. We denote the diagonal F-threshold by $c^{ \mathfrak{m}}( \mathfrak{m})$. It is known that $\dim(R) \geq c^{ \mathfrak{m}}( \mathfrak{m})$. If $R$ is also a…
Wágner Badilla
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Is every simple extension of a local domain inside its fractions again local?

Let $A$ be a local domain with fractional field $K$. Let $x\in K\setminus A$. My question is that, is the ring $A[x]$ also local? (It is easy to see that the anwser is yes, if $A$ is further a valuation ring.)
J.Li
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A lower bound of projective dimension of modules

Let $R$ be a Noetherian ring. For any nonzero $R$-module $M$, and any prime $P\in \operatorname{Ass}M$, do we have $\operatorname{pd}M\geq\operatorname{depth}P$? When $M$ is finitely generated, this can be shown by first reducing to the local…
user782932
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The first step in a proof of the general form of the Nullstellensatz

I'm reading an article about the proof of the general form of nullstellensatz : http://www.math.uwaterloo.ca/~jpbell/nullstellensatz.pdf. Recall that a ring $R$ is called Jacobson if the Jacobson radical $J(R/P) =0 $ for every prime ideal $P$ of…
Plantation
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$K[x]/QK[x] [b^{-1}] \cong S[b^{-1}]$ (where $S=R[t]$, $R$ domain) ? (Commutative Algebra)

I have a question. Let $R$ be a jacobson ring, integral domain with quotient field $K$. And, let $S=R[t]$ (i.e., $R$-algebra generated by just one element). So we can write $S \cong R[x]/Q$ for some prime ideal $Q\subseteq R[x]$. And assume furthur…
Plantation
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A detail in proposition 2.4 of Atiyah-MacDonald

I understand everything, but no how the equation $\displaystyle \sum_{j=1}^n(\delta_{ij}\phi-a_{ij})x_j=0$ is obtained. I guess it's just a multiplication by $\delta_{ij}$ but I ever get a different equation. Thanks for any explanation about this
Gyadso
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Why does equality between radicals imply this?

Let $R$ be a commutative ring. Given an ideal $I\subset R$, we define the radical $\mathfrak{r}(I)$ to be the intersection of all prime ideals $J\subsetneq R$ where $I \subseteq J$. Given an ideal $I \subset R$, we define $V(I)$ to be the set of…
Zach Hunter
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Integral dependence of an algebraic element

Let $A$ be a UFD, $K$ its field of fractions, and $L$ an extension of $K$. Then, let $\alpha \in L$ and let $f_\alpha \in K[x]$ be its minimal polynomial over $K$. Is it true that $\alpha$ is integral over $A$ if and only if $f_\alpha$ has…
Francesco Genovese
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A grade $0$ module with finite projective dimension has annihilator $(0)$

Let $M$ be a finitely generated module of finite projective dimension over a noetherian local ring $A$. Then if $M$ is of grade $0$, the annihilator of $M$ is $(0)$. A sketch proof of the above says that one takes a projective resolution of $M$.…
Flying pencil
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Concept of Free module in Polynomial ring

I'm studying Atiyah's commutative algebra. I have a question with free modules and the kind of thing in polynomial ring. I wrote the following so it cannot be true facts. A free $A$-module is $\bigoplus_{i\in I} A$. Then we know that every module…
Gobi
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