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For any $A$-algebra $B$ ( commutative ring with 1 ), we have the existence of $\Omega_{B/A}$, the module of relative differentials of $B$ over $A$, which can be defined by an universal property. In th …

Let $k$ be a value field (archimedean), for example $k = \mathbb{Q}_p$, the p-adic field. The free Tate algebra is $$ T_n := \left\{ \ \sum a_I X^I, \ a_I \in k, \ a_I \rightarrow 0 \text{ as } |I| \r …

I got an idea. Let $ \overline{k} $ be the algebraic closure of $ k $ and extend everything to be over $ \overline{k} $, i.e extend the $ k $-derivations $ d_2 $ to $ D_2: T_n \otimes_{k}\overline{k} …

Let $k$ be a field and $C/k$ be an affine plane curve over $k$, namely $C = \mathrm{Spec}(A)$ for some $A = k[x,y]/(f(x,y))$, here $f(x,y) \in k[x,y]$ is an irreducible polynomial. Let $B$ be the int …

In someone's note, I have seen such an example, but I can't show that it is not universally open. Here is the example: Let $k$ be a field and $A = k[T]_{(T)}$, the discrete valuation ring obtained fr …

Let $X/k$ and $Y/k$ be two smooth affine varieties over a field $k$ with $\mathrm{char}(k) = 0$ and $\varphi: X \rightarrow Y$ be a morphism. I would like to know under what conditions, the induced ma …

After reading Fulton's book "Algebraic Curves", I know how to do resolution of singular points on curves. Given an affine equation, I can get it's non-singular affine model, i.e the normalization of i …

The Newton Method over $\mathbb{R}$ has the property that the precision is doubled (under some continuous differentiable assumption) in each iteration. For the ring $\mathbb{Z}_p$ of $p$-adic integers …