Questions tagged [nonnoetherian]

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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 ...
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6 votes
1 answer
358 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 ...
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1 vote
1 answer
165 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\...
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2 votes
0 answers
118 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 $m \in M$...
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0 votes
0 answers
147 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{...
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2 votes
0 answers
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 ...
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  • 555
10 votes
2 answers
726 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 $\...
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1 vote
1 answer
179 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},\...
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  • 555
4 votes
1 answer
381 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-...
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  • 2,573
0 votes
2 answers
213 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_{...
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2 votes
0 answers
85 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 ...
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  • 121
6 votes
1 answer
458 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 ...
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11 votes
1 answer
983 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?
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3 votes
1 answer
298 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 (...
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5 votes
0 answers
377 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 $...
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7 votes
1 answer
343 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 ...
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3 votes
1 answer
383 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 $R$...
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  • 6,504
3 votes
1 answer
341 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 $\mathcal{O}_{\mathbb{P}_{S}^{r}}$-...
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  • 1,912
2 votes
1 answer
229 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 $R$. ...
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  • 555
3 votes
0 answers
281 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 $...
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1 vote
1 answer
166 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 ...
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4 votes
0 answers
110 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 ...
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3 votes
0 answers
122 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 $...
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  • 6,504
1 vote
0 answers
238 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 ...
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  • 555
0 votes
1 answer
236 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 ...
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  • 555
0 votes
1 answer
295 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 ${\mathrm{Spec}}\,B$. ...
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  • 555
3 votes
1 answer
178 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\...
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  • 6,504
1 vote
0 answers
404 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,\...
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  • 555
1 vote
0 answers
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 ...
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  • 6,504
5 votes
1 answer
145 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{...
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  • 6,504
4 votes
1 answer
259 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}...
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9 votes
2 answers
744 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 ...
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  • 5,294
0 votes
2 answers
2k 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 $R^{\oplus n} \twoheadrightarrow S$ ...
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1 vote
1 answer
394 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 ...
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5 votes
0 answers
458 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 ...
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1 vote
2 answers
681 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. ...
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4 votes
1 answer
346 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 ...
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1 vote
0 answers
152 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 $K$ ...
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3 votes
1 answer
268 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 $R/m\to S/n$ ...
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  • 1,291
1 vote
0 answers
427 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,.....
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5 votes
1 answer
522 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 ...
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  • 4,939
2 votes
1 answer
449 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 ...
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  • 281
3 votes
0 answers
381 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 ...
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1 vote
0 answers
197 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 ...
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6 votes
1 answer
410 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-...
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74 votes
1 answer
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 ...
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4 votes
0 answers
148 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, ...
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9 votes
2 answers
669 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 $[...
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0 votes
1 answer
159 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 \{\...
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  • 1,291
3 votes
0 answers
320 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 ...
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