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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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Zariski tangent vectors, dual numbers

Let $k$ be a field, $A$ be a Noetherian local $k$-algebra, $m$ its maximal ideal, and an isomorphism $i:A/m \to k$ . Let $v:m/m^2 \to k$ be a $k$-linear map (i.e. a Zariski tangent vector). I believe there exists a $k$-local homomorphism $f:A \to…
usr0192
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Behavior of associated primes under inclusion

Let $A \subset B$ be an inclusion of commutative rings inducing $f: \text{Spec}(B) \rightarrow \text{Spec}(A)$. Must it be the case that $\text{Ass}(A) \subset f(\text{Ass}(B))$? If this isn't true in general, is it true when $A,B$ are Noetherian…
Tony
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Noether Normalization of $\mathbb{C}$-algebra $\mathbb C[x,y,z]/(xy+z^2,x^2y−xy^3+z^4−1)$.

I'm trying to find a Noether normalization of the $\mathbb{C}$-algebra $$\mathbb C[x,y,z]/(xy+z^2,x^2y−xy^3+z^4−1).$$ The proof of Noether Normalization Theorem is not constructive. I choose $x = {x_1} - {y_1},$ $y = {x_1} + {y _1}$ and $z = {z_1}$…
taubetasigma
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Are formal power series rings over Dedekind domains formally smooth?

Let $A$ be a Dedekind domain. Consider the ring of formal power series $A[[t]]$ over $A$. Now let $B$ be any $A$-algebra, and let $N\subset B$ be a nilpotent ideal. Then, can any homomorphism $$A[[t]]\rightarrow B/I$$ (of $A$-algebras) be lifted to…
oxeimon
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What can $\operatorname{Hom}\left(\prod_i M_i, N\right)$ look like?

It's easy to see that $\operatorname{Hom}\left(\bigoplus_i M_i, N\right) = \prod_i \operatorname{Hom}(M_i, N)$. However, there are a couple of ways this can conceivably fail if we replace the coproduct on the left with a product: we could have a…
Daniel McLaury
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Local rings and flatness

Let $A \rightarrow B$ be a flat and local homomorphism of commutative local rings. Let $M,N$ be two $B$-modules which are free of finite rank as $A$-modules. Consider the product $M \otimes_B N$ as an $A$-module. Is this $A$-module flat?
Cyril
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Proposition 11.4 in Atiyah-MacDonald

I'm not seeing a line in the proof of Proposition 11.4 in Atiyah-MacDonald. Here is a link to some notes I found online which contain the proof: http://folk.uio.no/fredrme/Kommalg.pdf It is also 11.4 in these notes. Specifically, I don't understand…
Mike B
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Localization and Noetherian property

From page 101 in Atiyah-MacDonald: "Two of the important properties of localization are that it preserves exactness and the Noetherian property...." I remember proving that it preserves exactness, it's proposition 3.3. on page 39. But what is meant…
Rudy the Reindeer
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Is it true that if $\alpha \in \operatorname{Frac}(A)$ and $s\alpha \in A$, then $\alpha \in S^{-1}A$?

In the proof of Proposition 1.9 in Chapter VII of Algebra by Serge Lang, it seems to me that the following property is used. Let $A$ be a commutative entire ring, $S$ a multiplicative subset of $A$, $0 \not \in S$. Let $\alpha$ be an element of the…
Aki
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If $\phi: R \rightarrow S$ satisfies lying over, then $\textrm{ht } IS \leq \textrm{ht } I$.

Let $\phi: R \rightarrow S$ be a homomorphism of Noetherian rings with prime spectra $X, Y$, and suppose the contraction map $\phi^{\ast}: Y \rightarrow X$ is surjective. I'm trying to show that for any ideal $I$ of $R$, the height of $IS$ (that…
D_S
  • 33,891
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Zero Dimensional Commutative Ring

Let $R$ be a commutative ring with unity. I want a proof of the fact that $R$ is zero-dimensional (in the sense that all prime ideals are maximal) if and only if $R/J(R)$ is von Neumann regular and $J(R)$ is nil, where $J(R)$ is the Jacobson…
karparvar
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If ideal can be generated by zero divisors, then is the depth of the ideal 0?

Let $R$ be a Noetherian ring and $I$ an $R$-ideal. The number $\operatorname{depth}_I R$ is the length of maximal $R$-regular sequence in $I$. It is well-known that If $\operatorname{depth}_I R = 0$, then any $x \in I$ is a zero divisor. Is the…
Youngsu
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Regular sequence in the local ring $k[x, y]_{(x, y)}$

Let $R = k[x, y] $ and $Q = k[x, y]_{(x,y)}$, the localization of $k[x, y]$ at $(0, 0)$. Let $I$ be an ideal of $Q$ generated by a regular sequence of length $2$. Assume additionally $I$ is $(x, y)$-primary, i.e., $(x, y)^n \subset I \subset (x, y)$…
Alex Miller
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Can we use Atiyah-Macdonald Proposition 1.11(ii) to prove the Lagrange interpolation theorem?

Two Theorems: Theorem 1 (Atiyah-Macdonald, Proposition 1.11(ii)): Let $A$ be a commutative ring, and let $\mathfrak p_1,\dots,\mathfrak p_n$ be prime ideals in $A$, and let $\mathfrak a$ be an ideal of $A$ such that $\mathfrak a\subset\mathfrak…
John Gowers
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$M/IM \to N/IN$ an isomorphism $\Longrightarrow M \to N$ an isomorphism?

given a ring $R$, a nilpotent ideal $I$ and a morphism $\phi$ of $R$-modules $M \to N$, such that $M/IM \to N/IN$ is an isomorphism. It is easy to see that this implies $\phi$ surjective, but what about injectivity?
user5262
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