Let $f:X\rightarrow S= Spec(k[[\pi]])$ a finite type faithfully flat morphism. Let $U\subset X$ be an open subset such that $U$ is smooth and surjective on $S$. We consider the $k$-scheme ...
Let $f: X\rightarrow Y$ a smooth morphism over a field $k$. We assume that $X$ is locally of finite type, does it imply that $Y$ is also locally of finite type?
Based on Possible formal smoothness mistake in EGA, let $X$ and $Y$ $k$-schemes ($k$ a field), let $f:X\rightarrow Y$ a fpqc morphism such that $f$ is formally smooth and $X$ formally smooth, do we ...
Let $f:X\rightarrow Y$ a morphism between schemes and $Y'\rightarrow Y$ a fpqc morphism such that the base change $f'$ of $f$ to $Y'$ is formally smooth, does it imply that $f$ is formally smooth?
Let a cartesian diagram Let $X'\rightarrow X$ be a rational resolution of singularities of $k$-schemes of finite type and $Y$ a closed subscheme. Let $Y'\rightarrow Y$ be the base change to $Y$, we ...
Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
A morphism of schemes is formally smooth and locally of finite presentation iff it is smooth. What happens if we drop the finitely presented hypothesis? Of course, locally of finite presentation is ...
Let f:X -> Y be a morphism of schemes over a field k. Can one check that f is formally smooth using only Artin rings of the form k'[t]/t^n, where k' is also a field? Considering cuspidal curves one ...