rings with ‘flat functions’

Let (R,m) be a local ring over a field. Suppose the ring has flat elements, i.e. m∞≠{0}. (The prototype is of course C∞(Rp,0), or a quotient of it, by some finitely generated ideal.) 1. For which rings and ideals, J⊂R, the following holds. If the completions satisfy ˆJ⊇(ˆm)N then J⊇mN+n, for some finite n. At … Read more

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⊆rad(R) be an ideal contained in the Jacobson radical. Define the transfinite powers Iα for α an ordinal inductively as (I1=I and) Iα+1=I⋅Iα, and Iα=∩β<αIβ at limit ordinals. This is a … Read more

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 exercise 28.2.M (or below), I would like to record a proof in this post. But I am … Read more

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 ιn:K[[T1,…,Tn]]↪K[[X,Y]]. Now let us define the infinitely many variables formal power series as follows: K[[T1,…,T∞]]:=lim←n≥1K[[T1,…,Tn]]. For example, ∑∞i=1Ti=T1+T2+T3+…∈K[[T1,…,T∞]]. Then I would like to ask Q. Can K[[T1,…,T∞]] be embedded into K[[X,Y]]? That is, does the embedding ι∞:K[[T1,…,T∞]]↪K[[X,Y]] … Read more

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 limit our discussion to noetherian schemes”. What are these technical reasons? what basic facts about formal schemes fail if they are not noetherian (or … Read more

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 $r-1 $ belongs to the ideal generated by $s$. Is the following known (or false)? Question: … Read more

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 $k[X]$. If $U\subseteq X$ is open, let $\mathcal{O}_X(U)=\bigcap_{p\in U}\mathcal{O}_{X,p}$. Is this ring noetherian as well? Note This question has already been asked here … Read more

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-Noetherian commutative ring whose total quotient ring is Noetherian? EDIT: A commutative ring is Dedekind if every regular ideal is invertible. (A … Read more

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 tensor square of some crazy field extension). What is true is that a tensor product of a finite type $R$-algebra and … Read more

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 $e^{-1/x^2}\in \left(x^n\right)$ for all $n\geq 1$. If I add to the hypothesis that the ring is a domain, then (I … Read more