# Tag Info

Accepted

### Translation between formal geometry and rigid geometry

No, these are not the same thing. Formal schemes are to rigid-analytic spaces as $\mathbf{Z}_p$-schemes are to $\mathbf{Q}_p$-schemes. The book Lectures in Formal and Rigid Geometry by Bosch is an ...
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### Intuition about $\mathrm{Spec}\mathbb{C}[[t]]$ versus $\mathrm{Spf}\mathbb{C}[[t]]$ versus $\mathrm{Specan}\mathbb{C}[[t]]$ (and similar objects)

In a nutshell: $\mathrm{Spec}~\mathbb{C}[[t]]$ Is a “trait” i.e. the spectrum of a discrete valuation domain, with a generic (open) point and a closed point. It might be imagined as a refinement of ...
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### Vector bundles on complete rings

I will assume that $A$ is Noetherian (if you wish to work in the non-Noetherian setting, I'll see what I can do to modify my answer). Without loss of generality, we may assume that $A$ is in fact $I$-...
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### Smooth loci and formal neighborhoods

No. Take for $R$ a discrete valuation ring with fraction field $K$, and for $f$ any non-smooth morphism of smooth $K$-algebras (viewed as a morphism of $R$-algebras).
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### What is the essential image of $AbVar$ in $p-div$?

In short, I still don't know over a general ring, but for $\mathbb{F}_q$, we have an answer: The idempotent completion of the essential image of $AbVar_{/\mathbb{F}_q}$ in $FG_{/\mathbb{F}_q}$ is ...
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### Formal Schemes Methods: Applications

Instead of trying to answer the question in full, let me give some further appearances of the notion of formal scheme. There are several situations when a consideration of a topology in certain rings ...
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