14 votes
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 ...
  • 2,898
13 votes

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 ...
  • 8,341
10 votes
Accepted

Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence?

You need a characteristic zero assumption, and you need some additional axioms on the homorphism to make it a bijection. The map from derivations $D$ to homomorphisms sends a function $y \in \mathcal ...
  • 122k
10 votes
Accepted

Connectedness, loops and formal moduli problems

The presentation of the formal moduli problems story in Gaitsgory-Rosenblyum A Study in Derived Algebraic Geometry, Vol 2 may be what you are looking for. We review it here (in the case over $\mathrm{...
9 votes
Accepted

Symmetric powers of curves and completion along the diagonal

$\operatorname{Sym}^{d-1}(C) $ is ample, and $\Delta $ is not unless $C\cong \mathbb{P}^1$. To see this, consider the finite map $\pi :C^d\rightarrow \operatorname{Sym}^{d}(C) $, and the projections $\...
  • 34.8k
8 votes
Accepted

A derived category of formal sheaves

For a start, you have an approach to ind-coherent sheaves and its derived category in Duality and Flat Base Change on Formal Schemes Contemp. Math. 244 (1999), pp. 3-90. On line version, with some ...
  • 8,341
7 votes
Accepted

Basic example of a formal affine scheme, functorial point of view

It might be illuminating to first work the example of (ordinary) affine space $\mathbb{A}^1_\mathbb{Z}$ over the integers. As a functor, $\mathbb{A}^1_\mathbb{Z}$ is the forgetful functor $\mathit{...
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6 votes
Accepted

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 $I$-...
6 votes

Intuition about $\mathrm{Spec}\mathbb{C}[[t]]$ versus $\mathrm{Spf}\mathbb{C}[[t]]$ versus $\mathrm{Specan}\mathbb{C}[[t]]$ (and similar objects)

One good way of seeing the difference between $\mathrm{Spec}\,\mathbb{C}[[t]]$ and $\mathrm{Spf}\,\mathbb{C}[[t]]$ is to look at what functors they represent on affine schemes. In fact we have $$\...
  • 15.8k
6 votes
Accepted

about morphisms of affine formal schemes $\mathrm{Spf}(B)\to \mathrm{Spf}(A)$

See EGA I, sec. 10. For the specific question you mention, see paragraph 10.2. There is further issues on the cohomology of formal schemes in EGA III, sect 3.4. Also, the existence theorem is treated ...
  • 8,341
6 votes
Accepted

Basic questions about formal schemes

For 1) The functorial interpretation is developed by Strickland in Formal schemes and formal groups. Homotopy invariant algebraic structures (Baltimore, MD, 1998), 263–352, Contemp. Math., 239, Amer....
  • 8,341
6 votes
Accepted

GAGA for henselian schemes

OK, it turns out that $H^1((P^1_A)^h, O^h)$ is nonzero in general. A counter example can be found in a blog post on the very blog you mention in your post. Here is a link.
  • 76
5 votes

Complete subring of F_p[[X]]

No : just take $A = \mathbb{F}_p$. This is the only counterexample : any complete local sub-algebra $\iota_A \colon A \hookrightarrow {\Bbb F}_p[[X]]$, with $A \neq \mathbb{F}_p$, is noetherian of ...
  • 7,129
5 votes
Accepted

References on topological rings

Here are 3 references that haven't been mentioned yet. I am not sure if the latter two would be of any use to you, but probably are worth a look. Arnautov, Glavatsky, Mikhalev - Introduction to the ...
  • 5,708
5 votes

Intuition about $\mathrm{Spec}\mathbb{C}[[t]]$ versus $\mathrm{Spf}\mathbb{C}[[t]]$ versus $\mathrm{Specan}\mathbb{C}[[t]]$ (and similar objects)

To complement the other answers, I would like to add a word on the analytic spectrum $\mathrm{Specan}(\mathbb{C}[[t]])$. First, let me say that I am not sure what $\mathrm{Specan}$ means and have no ...
5 votes
Accepted

Is the formal completion of an affine group necessarily a formal group?

The universal map is not a map of formal groups without some extra condition. An easy class of counterexamples comes from completions of an affine group along a closed subscheme that does not contain ...
  • 43.4k
5 votes
Accepted

Does $R\hat{f}_*\mathcal{F}=\hat{f}_*\mathcal{F}$ hold for affine adic morphisms?

I never liked the definition of formal completion (as defined in EGA or Hartshorne), so I'll use the following definition from Brian's formal GAGA paper. Definition: Let $X$ be a locally Noetherian ...
4 votes

Motivation for Henselian rings in algebraic geometry

First about the ralation between henselian and formal smoothness property, I think a good idea is to look at what was the first version of Hensel lemma: it says that if $f(\bar{a})=0,f'(\bar{a})\, ...
  • 1,146
4 votes
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Cohomology of Formal Groups

Lazarev is doing calculations with this "cohomology theory" in his paper "Deformations of Formal Groups" Among the thing he shows is that this cohomology is the E2 term of the bar spectral sequence ...
4 votes

Functorial point of view for formal schemes

Although this doesn't cover every example, formal completions $\hat{X}_Y$ of a scheme $X$ along a closed subscheme $Y$ have a particularly nice functorial presentation. We have $$\hat{X}_Y \simeq Y_{...
  • 2,700
3 votes
Accepted

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).
2 votes

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 ...
2 votes

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 ...
  • 8,341
2 votes
Accepted

Formal neighbourhood of a closed subscheme

If you look at a family of elliptic curves like $y^2 = x^3 - x-t^n$ as a surface $X$ mapping to a curve $C$ with parameter $t$, and $Y$ the fiber over $0$, then the neighborhood modulo $\mathcal I_Y^n ...
  • 122k
1 vote
Accepted

When is a formal group smooth?

I think an adaptation of Schlessinger's argument from Functors of Artin rings should work. Let $\Lambda$ be a Noetherian ring, and suppose we have a connected formal group $\mathcal{G}$ formally ...
  • 1,570
1 vote
Accepted

A translation between formal and rigid geometry

Using your notation: it corresponds, like in the "classic" case of polynomials over a field, to the ideal generated by $\zeta_1-x_1,\dots,\zeta_n-x_n$, where $x=(x_1,\dots,x_n)\in \mathbb B^n(K)$. In ...
  • 1,321
1 vote

Coherent modules over complete adic rings: counterexamples

Suggestion: read the paper "On Hausdorff completions of commutative rings in rigid geometry", by Fujiwara, Gabber, and Kato.
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1 vote

generic fibre functor for relative rigid spaces

Yes. You will find what you are looking for in Ahmed Abbes' 'Elements de Geometrie Rigide $I$', where rigid geometry a la Raynaud is developed in a relative setting. It is written in French, but it ...
  • 7,129
1 vote

Formal completion of a complex normal bundle along the zero section

Instead of trying to adapt Hartshorne's definition for formal completion of noetherian schemes to the difficult case of analytic schemes, which are never noetherian, we could instead consider the ...
1 vote
Accepted

Representability of deformation functors via SGA

It turns out that the only thing Böckle is using is the smoothness of $\widehat{\mathrm{PGL}}_d$. I claim that the following more general result is true. Suppose $R\rightrightarrows X$ is an ...

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