# Questions tagged [nonnoetherian]

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $\inf\{i\in \mathbb N \cup \{0\}\cup \{\... 0answers 272 views ### Weak assassins and essential morphisms Let$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 ... 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 ... 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$\...
Let $R$ be a commutative Noetherian ring, $M$ is an Artinian $R$-module. Is the set $Supp_R(M)$ finite? Thanks.