Questions tagged [nonnoetherian]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4
votes
0answers
113 views

Removing Noetherian condition from cohomology and base change

This question is related to a question I asked a few days ago. Since there seems to be no (at least for me) satisfying reference for cohomology and base change as stated by Vakil in his script in ...
6
votes
1answer
268 views

Cohomology and base change without Noetherian assumption

In the "The Rising Sea" by Vakil one can find the base change theorem for proper morphisms over a locally Noetherian base (28.1.6). He later indicates (28.2.M) how one could exchange the ...
1
vote
1answer
156 views

An example of a special $1$-dimensional non-Noetherian valuation domain

I am looking for a $1$-dimensional non-Noetherian valuation domain $R$ such that there exists a sequence $\{a_i\}_{i=1}^\infty$ of elements of $R$ such that $\langle a_1\rangle \subsetneqq\langle a_2\...
2
votes
0answers
111 views

if $I$ is finitely presented nilpotent and $M/IM$ is finitely presented, then $M$ is finitely presented

Let $R$ be a commutative ring, and let $I \subseteq R$ be a nilpotent ideal. Let moreover $M$ be an $R$-module, and let $IM$ be the submodule generated by the products $xm$ with $x \in I$ and $m \in M$...
0
votes
0answers
144 views

Determinant of a special matrix in characteristic $p$

Let $K$ be a field of characteristic $p > 0$. Choose $p^i$ numbers of elements $c_1,\ldots,c_{p^i} \in K$ and consider the determinant $D$ of the following matrix$\colon$ \begin{pmatrix}\label{...
2
votes
0answers
91 views

On a certain radical of the formal power series ring $K[[X_1,X_2,\ldots,X_{\infty}]]$

Let $K$ be a field of characteristic $p > 2$ and $A_{\infty} \colon= K[[X_1,X_2,\ldots,X_{\infty}]]$ be an infinitely-many-variable formal power series ring over $K$ (the symbol $X_{\infty}$ is to ...
10
votes
2answers
553 views

Krull dimension of a local ring and completion

Let $A$ be a local ring (not noetherian) of finite Krull dimension such that its maximal ideal $\mathfrak{m}$ is of finite type. Let $\hat{A}$ be its $\mathfrak{m}$-adic completion. Do we have that $\...
1
vote
1answer
171 views

Power series ring $\Theta[[X_1,\ldots,X_d]]$ and prime ideals

Let $\Theta$ be a domain. We shall choose $d$ elements $\theta_1,\ldots,\theta_d \in \Theta$ such that any chosen $j$ elements $\theta_{i_1},\ldots,\theta_{i_j}$ form a prime ideal $(\theta_{i_1},\...
4
votes
1answer
304 views

Is a universally closed monomorphism a closed immersion?

The question is essentially in the title: $f\colon X \rightarrow Y$ is a monomorphism of schemes that is universally closed; does this imply that $f$ is a closed immersion? Any such $f$ is quasi-...
0
votes
2answers
200 views

Power series ring and monomials

Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a formal power series ring over a field $K$ of characterisc $p > 0$ in $n$ variables. For a given positive number $\epsilon > 0$ we call a monomial $X_{...
2
votes
0answers
82 views

Reference request : $I$-adic smoothness

The following result has been know for a while now: Let $f:\mathfrak X \rightarrow \mathfrak Y$ be an adic morphism of (locally) Noetherian formal schemes and $K\subset \mathcal O_{\mathfrak Y}$ and ...
6
votes
1answer
407 views

Algebras such that the tensor product with any Noetherian algebra is Noetherian

Let $R$ be a Noetherian commutative unital ring. It is generally speaking not true that the tensor product of two Noetherian $R$-algebras is Noetherian (e.g. take $R$ to be a field, and consider the ...
10
votes
1answer
781 views

“Noetherian” and “finitely generated” for polynomial algebras

Let $k$ be a field. Does there exist a positive integer $n$ such that there is $k$-subalgebra of $k[x_1, \dots, x_n]$ which is Noetherian but not finitely generated?
3
votes
1answer
266 views

Moduli space of almost complex structures as an algebro-geometric object

Let $M$ be a closed real-analytic manifold of dimension $2n$. Is it possible to make sense of the moduli space of real-analytic almost complex structures on $M$ as an algebro-geometric object (...
5
votes
0answers
333 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
318 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
330 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 $R$...
3
votes
1answer
306 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
202 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
256 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
144 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
votes
0answers
108 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
121 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
235 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
229 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
269 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
161 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
364 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
91 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
136 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
248 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
621 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
1k 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
386 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
418 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
626 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
341 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
151 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
265 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
389 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,.....
5
votes
1answer
472 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
1answer
384 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
375 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
195 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
394 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-...
74
votes
1answer
4k 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
145 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
621 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
156 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
307 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 ...