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Questions tagged [nonnoetherian]

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5
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0answers
170 views

Slightly noncommutative Nakayama's lemma?

Nakayama's lemma asserts the following. If $R $ is a commutative ring with an element $s$, and $M$ is a finitely generated module such that $sM = M$, then there exists $r \in R$ such that $rM =0$ and $...
7
votes
1answer
261 views

An infinite dimensional local domain whose chains of primes are finite

Does there exist a local domain of infinite dimension in which every chain of prime ideals is finite? Of course, such a ring must be neither noetherian nor catenary. (This question arose while ...
3
votes
1answer
231 views

Local ring of infinite dimension

Short version: Let $R$ be a commutative ring such that all chains of primes of $R$ with the same extremities have the same finite cardinality. Is $R$ locally finite-dimensional? Longer version: Let $...
3
votes
1answer
212 views

Non-noetherian cohomology and base change

Let $S$ be a connected scheme, let $\pi : \mathbb{P}_{S}^{r} \to S$ be projective $r$-space over $S$, and let $\mathcal{E}$ be a flat and locally finitely presented $\mathcal{O}_{\mathbb{P}_{S}^{r}}$-...
2
votes
1answer
148 views

Heights of contracted ideals

Let $R$ be a non-noetherian domain. $S$ be a multiplicatively closed subset of $R$. Let $S^{-1}R$ be a localisation of $R$, where all element of $S$ is invertible. Suppose we have an ideal $I$ of $R$. ...
3
votes
0answers
130 views

On rings for which given an ideal , over it every minimal prime ideal is finitely generated

Let $R$ be a commutative ring with unity. If for every ideal of $R$, the minimal prime ideals over it are all finitely generated, then there are finitely many minimal prime ideals over every ideal of $...
1
vote
1answer
92 views

Integral domain satisfying a.c.c. on radical ideals and with algebraically closed fraction field

If $R$ is an integral domain satisfying acc on radical ideals (i.e. Noetherian spectrum) and if the fraction field of $R$ is algebraically closed, then is $R$ a field ? If $R$ is normal (integrally ...
4
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0answers
103 views

Do the transfinite powers of an ideal in the radical always reach 0?

This is true in the Noetherian case by Nakayama's lemma. Is it true in general? Less tersely, let $R$ be a commutative ring and $I \subseteq \mathrm{rad}(R)$ be an ideal contained in the Jacobson ...
3
votes
0answers
103 views

Assassins in zero-dimensional local rings

During a study of the behaviour of assassins and torsion functors (cf. this paper), I met the following problem about assassins in $0$-dimensional local rings. Let $R$ be a commutative ring, and let $...
1
vote
0answers
217 views

On the coherence of $K[[X_1,X_2,…]]$

Recall that a commutative ring is coherent if every finitely generated ideal is finitely presented, or equivalently if every submodule of every finitely generated module is finitely presented. Let $A ...
0
votes
1answer
203 views

Proof of the coherence of ${\Bbb F}_q[[X_1,\dots,X_{\infty}]]$

We shall define the infinitely-many-variable formal power series ring $A = {\Bbb F}_q[[X_1,\ldots,X_{\infty}]]$ over a finite field ${\Bbb F}_q$ as the following$\colon$ $A \colon= \underset{n \geq ...
0
votes
1answer
203 views

Relative Bertini Theorem

Let $A \colon= {\Bbb C}[S_1,\ldots,S_n]$ with $1 \leq n < \infty$ $B \colon= A[X_1,\ldots,X_d]$ with $2 \leq d < \infty$. $O \colon= (0,\ldots,0)$ be the origin of ${\mathrm{Spec}}\,B$. ...
3
votes
1answer
134 views

Torsion submodules of non-noetherian modules

Let $R$ be a commutative ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. The $\mathfrak{a}$-torsion submodule of $M$ is defined as $$\Gamma_{\mathfrak{a}}(M)=\{x\in M\...
1
vote
0answers
250 views

An infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$

We shall define a infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$ as follows$\colon$ ${\Bbb F}_p[[X_1,\ldots]]\colon= \underset{n \geq 1}{\varprojlim}\, {\Bbb F}_p[[X_1,\...
1
vote
0answers
89 views

Torsion functors and weak assassins

Let $R$ be a commutative ring, and let $\mathfrak{a}\subseteq R$ be an ideal. For an $R$-module $M$, we set $\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\exists n\in\mathbb{N}:\mathfrak{a}^nx=0\}$, and we ...
4
votes
1answer
127 views

On the relation between two definitions of torsion functors

Let $R$ be a commutative ring, and let $\mathfrak{a}\subseteq R$ be an ideal. For an $R$-module we consider the sub-$R$-modules $$\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\exists n\in\mathbb{N}:\mathfrak{...
4
votes
1answer
184 views

Is $Hom_R(S_X^{-1}R, E)$ the minimal injective cogenerator of $S_X^{-1}R$?

Assume that $R$ is a commutative Noetherian ring with minimal injective cogenerator $E$. For a finite set of maximal ideals $X$ of $R$, define the multiplicative set $$S_X=R-\bigcup_{\mathfrak{m}\in X}...
9
votes
2answers
383 views

Algebras whose subalgebras are finitely generated

Let $k$ be a commutative ring. Is there a name for those commutative $k$-algebras with the property that every subalgebra is finitely generated? (Equivalently, the partial order of subalgebras is ...
0
votes
2answers
646 views

Tensor products of two domains

Let $R$ be an integral noetherian regular local ring. Let $S$ be a noetherian integral domain such that $S/R$ is finite. That is, $R \subset S$ and the surjection $R^{\oplus n} \twoheadrightarrow S$ ...
1
vote
1answer
365 views

Prime Ideal of $A[X_1,…,X_d]$

Let $A$ be a UFD domain, i.e. integral and for any height one prime ${\frak p}$ of $A$, we have ${\frak p} = (u_{\frak p})$ for some $u_{\frak p} \in A$. Once and for all, we fix the algebraic ...
5
votes
0answers
337 views

Why are formal schemes assumed to be (locally) noetherian?

All sources that I know that study formal schemes seem to assume that they are locally noetherian. For instance, in Hartshorne "Algebraic Geometry", the author states: "For technical reasons we will ...
1
vote
3answers
509 views

Krull-dimension of local domain

Let $(R,{\frak m}_R)$ be a local domain (not necessarily Noetherian). That is, $R$ is integral and ${\frak m}_R$ is the unique maximal ideal of $R$. Suppose that ${\frak m}_R$ is finitely generated. ...
4
votes
1answer
326 views

Can K[[T_1,…,T_∞]] be embedded into K[[X,Y]]?

In the MathOverflow question about common false beliefs, the following answer teaches us that there is an embedding $\iota_n \colon K[[T_1,...,T_n]] \hookrightarrow K[[X,Y]]$. Now let us define the ...
1
vote
0answers
139 views

Popescu-Neron Desingularization for K[[T_1,…,T_∞]]

Let $K[[T_1,...,T_n]]$ be a finitely many variables formal power series ring over a field $K$. Dorin Popescu proved that there are smooth algebras $A_{\lambda}$'s which are of finite type over $K$ ...
3
votes
1answer
255 views

Let $f:R\to S$ be a local finite monomorphism .If $M$ is an Artinian $S$-module, is it an Artinian $R$-module?

$(R,m)$ and $(S,n)$ are local rings (commutative Noetherian with 1). Let $f:R\to S$ be a local homomorphism/monomorphism ($f(m)\subseteq n$), such that the natural induced homomorphism $R/m\to S/n$ ...
1
vote
0answers
343 views

On the coherence of formal power series ring

Let $A = {\Bbb F}_p[[X_1,X_2,...]]$ be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$ $A$ consists of such formal sum elements as $\sum c_{e_1,.....
4
votes
1answer
392 views

Structure sheaf of affine variety consists of noetherian rings (again)

Let $X\subseteq \mathbb{A}^n$ be an affine variety. The local ring of $X$ at $p\in X$, given by $\mathcal{O}_{X,p}=\{f\in k(X):f \text{ regular at } p\}$ is noetherian because it is a localization of ...
2
votes
0answers
231 views

Structure theorem for non-Noetherian local rings

Is there a structure theorem (like Cohen 's structure theorem) for non-Noetherian local rings? I am adding what I am looking for as someone asked in the comment. If $R$ is a local domain (not ...
3
votes
0answers
341 views

Are prime ideals of finite height in the powers series ring in infinitely variables finitely generated?

Let $A:= {\mathbb F}_p[[X_1,...,X_∞]]$ be the infinitely many variables formal power series ring over ${\mathbb F}_p$, which is UFD. Consider an arbitrary prime ideal $P$ of $A$ such that the height ...
1
vote
0answers
188 views

Number of minimal primes for UFD

Let $R$ be a UFD which is NOT noetherian. It is well-known that $R$ is a Krull ring. Let $I$ be an ideal of $R$ such that the height of $I$ is $d$ which is finite. Question. Is the number of minimal ...
6
votes
1answer
315 views

non-Noetherian r-Noetherian ring with Noetherian total quotient ring

A commutative ring is said to be r-Noetherian if every regular ideal is finitely generated, where an ideal is said to be regular if it contains a non-zerodivisor. Does there exist a non-Noetherian r-...
62
votes
1answer
3k views

$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$

Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$? This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
4
votes
0answers
127 views

rings with 'flat functions'

Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat elements, i.e. $\mathfrak{m}^\infty\neq\{0\}$. (The prototype is of course $C^\infty(\Bbb{R}^p,0)$, or a quotient of it, ...
9
votes
2answers
488 views

Strategies for proving a category is Noetherian?

Let $C$ be a small linear category over a commutative ring $R$. A representation of $C$ is an $R$-linear functor $C \to \mathrm{Mod}(R)$. For example, for each $c\in C$, there is a representation $[...
0
votes
1answer
146 views

$\inf\{i\in \mathbb N \cup \{0\}\cup\infty\mid Ext^i_R(R/I,R)\neq 0\}=0 ?$

Let $R := k[x_1, \cdots, x_n, \cdots]/(x_1^1, \cdots , x^n_n, \cdots),$ where $k$ is a field. Set $I:=(x_1, \cdots, x_n, \cdots)$. the questions are: Is $\inf\{i\in \mathbb N \cup \{0\}\cup \{\...
3
votes
0answers
272 views

Weak assassins and essential morphisms

Let $R$ be a commutative ring and let $M\rightarrow N$ be an essential morphism of $R$-modules. Then, $M$ and $N$ have the same associated primes. Over non-noetherian rings the notion of associated ...
6
votes
2answers
2k views

Condition for a local ring whose maximal ideal is principal to be Noetherian

The statement "a local ring whose maximal ideal is principal is Noetherian" is (I think) false. The ring of germs about $0$ of $C^\infty$ functions on the real line seems to be a counterexample since ...
2
votes
1answer
369 views

quotients of varieties as non-noetherian schemes?

Let $X$ be a variety (i.e. a reduced scheme of finite type over a field) and let $G$ be an abstract group, finitely generated, acting of $X$ algebraically freely. The example I have in mind is $\...
2
votes
1answer
230 views

Is the pushforward via a proper map of a finite presentation module of finite presentation?

It's true that the pushforward of a coherent sheaf is coherent via a proper morphism: but do proper morphisms preserve a finite presentation? Under some assumptions perhaps? Does it change if we are ...
10
votes
2answers
784 views

Is the support of an Artinian module finite?

Let $R$ be a commutative Noetherian ring, $M$ is an Artinian $R$-module. Is the set $Supp_R(M)$ finite? Thanks.