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Let $F$ be an algebraically closed field of characteristic $p$ equipped with a nonarchimedean dense absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete. Let …

Let $F$ be an algebraically closed field of characteristic $p$ equipped with an absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete. Define $|\cdot|_{prod} …

Let $R\subset S$ be commutative rings, $I\trianglelefteq R$ an ideal and $M$ be an $R$-module. Suppose that 1) $R$ is Noetherian and $I$-adically complete. 2) $M$ is a finite $R$-module (hence $M$ i …

While working through a proof of this paper,1 at the middle of page 45, the author's claim of a short exact sequence seems to amount to the following problem: Let $A$ be a commutative ring and let …

Suppose that we have the Laurent series fields $F_1:=\mathbb F_p((X))$ and $F_2:=\mathbb F_p((Y))$. Equip $F_1$ with the $X$-adic multiplicative absolute value $|\cdot|_1$, i.e. define $|X|_1=\dfrac{1 …

While working through a proof of this paper, at the end of page 46, the author seems to claim along the lines that the following is true: Let $A\rightarrow B$ be an etale map of rings. Then the un …

While working through a proof of this paper, at the middle of page 46, the author introduces a dimension notion which seems to claim that the following is true: Let $A\rightarrow B$ be an etale ma …

While working through a proof of this paper, at the middle of page 46, the author seems to claim the following is true: Let $A\rightarrow B$ be an etale map of rings. Suppose that for every prime …

While working through a proof of this paper, at the beginning of page 42, the author seems to claim the following is true: Let $R\subset S$ be rings, where $R$ is a finite type algebra over $\math …