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

What kind of module is this?

Recall that, if R is a commutative ring, then a suitably finite R-module M is projective if and only if the localization Mm is a direct sum of finitely many copies of Rm for every maximal ideal m⊆R. Consider the following alternate condition for an R-module M: each Mm is a direct sum of finitely … Read more

Sections of smooth morphisms over henselian rings

Let (A,m) be a henselian local ring. Let R and S be A-algebras of finite type and f:R→S be a smooth morphism. Assume that the induced morphism R/mR→S/mS has a section. Does it imply that f has a section? If A is m-complete, this seems to be true: one can lift f to R/mnR inductively … Read more

Structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ with $A$ a local integral domain

I am trying to see the structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ where $A$ is a local integral domain with small residue field. Let $A$ be a local integral domain with maximal ideal $M$, residue field $k$ and its group of units $A^{\times}$. Now we have that $A$ is an $A^{\times}$-module where the action is multiplication by a … Read more

Reference on the classification of (low rank) Gorenstein rings over $\mathbb{C}$

I am interested in the question of the classification of (low rank) Gorenstein rings over $\mathbb{C}$. The socle of a local algebra is the annihilator of its maximal ideal. A commutative local ring is called Gorenstein iff its socle is 1-dimensional. Here, “rank” means the dimension of the algebra as a vector space. A classification … Read more

Canonical module of a semigroup ring

Let $S$ be a numerical semigroup and $k[S]$ is the associated semigroup ring. I would like to compute canonical module $\omega$ of $k[S].$ I want to show that $\omega=k[t^{-n}:n\in\mathbb Z\setminus S]$. I have shown that $H^1_{m}(k[S])=k[t^{n}:n\in\mathbb Z\setminus S]$ where $m$ is the maximal homogeneous ideal of $k[S].$ Using duality I tried to compute canonical ideal … Read more

injective hulls in mixed characteristic

Let R=lim be a complete local ring, with residue field k=R/\mathfrak m, and let’s assume that R is Noetherian. If R is a k-algebra, then I believe that the following is correct: The injective hull of k can be described as the set of continuous homomorphisms E(k) = \hom_{\mathrm{cts}}(R,k)=\underrightarrow\lim (\hom(R/\mathfrak m^i,k)), where continuity is with … Read more

Given a non-field local domain RR, finding a dominating Valuation ring whose residue field is algebraic/finite extension of the residue field of RR

Let (R,m) be a non-field local domain with fraction field Q(R) . Let kR:=R/m. We know that there is a Valuation ring (V,mV) such that R⊆V⊊ such that \mathfrak m_V \cap R=\mathfrak m (in fact any local ring between R and K maximal w.r.t. “dominance” is a valuation ring). Then k_V:=V/\mathfrak m_V is an extension … Read more

Artin approximation theorems over non-regular rings/non-Noetherian rings

In Artin1968 he considers $\underline{analytic}$ equations, but over the ring $R=k\{x_1,..,x_n\}$. In Artin1969 he works with $R=k\{x_1,..,x_n\}/I$, not necessarily regular, but considers $\underline{polynomial}$ equations. Is there some version like this: “Let $R$ be a local Noetherian Henselian ring(not necessarily regular), over a normed field. Given an arbitrary (possibly countable) system of analytic equations over $R$, … 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