# Questions tagged [formal-schemes]

The formal-schemes tag has no usage guidance.

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### Reference request : $I$-adic smoothness

The following result has been know for a while now:
Let $f:\mathfrak X \rightarrow \mathfrak Y$ be an adic morphism of (locally) Noetherian formal schemes and $K\subset \mathcal O_{\mathfrak Y}$ and ...

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### Special fibers and cohomology groups of different Raynaud formal models

Let $K$ be a $p$-adic field with residue field $k=\mathbb F_q$. We know any qcqs (quasi-compact and quasi-separated) rigid space $X$ over $K$ admits a formal model, and different formal models are ...

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### GAGA for henselian schemes

In this paper, F. Kato recollects basic facts on henselian schemes and proves some partial results towards GAGA in the context of henselian schemes.
Let $I$ be a finitely generated ideal in a ...

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### Smooth loci and formal neighborhoods

Let $R$ be a Noetherian local ring with maximal ideal $I$.
Suppose we have a morphism of smooth $R$-algebras $f : A\to B$ such that its reduction modulo $I^n$
$$f_n : A/I^n \to B/I^n$$
is an ...

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### A translation between formal and rigid geometry

The following lemma is in Bosch's book "Lectures on Formal and rigid geometry" p198.
Lemma Let $K$ be a non-archimedean field and $R$ its valuation ring. Let $X= \mathrm{Spf}A$ be an affine ...

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

It is well known that there is a correspondence between homomorphism of rings $A\to B$ and morphism of affine schemes $\mathrm{Spec}(B) \to \mathrm{Spec}(A)$.
Question: (1) In analogy, is there ...

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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)

The first one $\mathrm{Spec}\mathbb{C}[[t]]$ is a scheme, the second one $\mathrm{Spf}\mathbb{C}[[t]]$ is a formal scheme. In my mind they both realize an "infinite order infinitesimal neighbourhood ...

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### Coherent modules over complete adic rings: counterexamples

Let $A$ be a coherent ring, complete with respect to the adic topology generated by a finitely generated ideal $I$.
Define the category $Coh(A,I)$ whose objects are inverse systems $\{M_n\}$ of $A$-...

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### Absolute approximation of formal schemes

Let $\mathfrak{X}_j$ be an inverse system of qcqs $p$-adic formal scheme, flat over $\mathbf{Z}_p$, with affine transition maps, and assume $\mathcal{O}_{\mathfrak{X}_j}$ is a coherent sheaf of ...

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### Basic questions about formal schemes

I have some questions related to formal schemes. Essentially I would like to understand how much formal schemes are different from usual schemes. First of all, let me specify the definition I prefere ...

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### Smoothness in formal GAGA

Suppose $X$ is a noetherian scheme over a complete noetherian local ring $(A,\mathfrak{m})$, and assume $X$ is projective.
If $\widehat{X}$, the $\mathfrak{m}$-adic formal completion, is smooth in ...

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### Algebraization of open formal subschemes

Let $\mathfrak{X}$ be a locally noetherian adic formal scheme over $\text{Spf}(A)$, with $A$ an $I$-adically complete and separated noetherian ring.
Suppose the mod $I$-fiber of $\mathfrak{X}$ is an ...

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### Ring of functions of generic fiber of affine special formal schemes

Fix $R$ a complete DVR. Recall from Berkovich's Vanishing Cycles for Formal Schemes II paper that we have a class of special formal schemes which are not topologically of finite type over $\...

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### What is the essential image of $AbVar$ in $p-div$?

Given an abelian variety $A$ over a base scheme $\text{Spec } \mathcal{O}_{K_p}$, we define the functor $P$ as taking $A \mapsto \text{colim}_n A[p^n]$, its associated $p$-divisible group. What is the ...

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### Algebraizability of formal schemes

Let $R$ be a complete DVR with quotient field $K$ and let $f:\mathfrak{X} \to \mathrm{Spf}(R)$ be a smooth proper formal scheme.
If the (rigid analytic) generic fibre of $f$ is (the analytification ...

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### References on topological rings

What is a good book on topological rings and modules?
I'm interested in topological rings and modules typically endowed with non-linear topologies, e.g.. non-linearly topologized normed rings.
I ...

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### generic fibre functor for relative rigid spaces

The classical theory of formal models of rigid analytic spaces due to
Raynaud introduces the category of admissible R-formal schemes for $R$ a
discretely valued ring, which includes locally ...

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### Formal completion of a complex normal bundle along the zero section

The total space of a complex normal bundle over a submanifold $Y$ in a complex manifold $X$ can be seen as an analytic scheme. If one blindly uses Hartshorne's definition about "formal schemes" from ...

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### A derived category of formal sheaves

Grothendieck (EGA I 0.7 & 1.10) defined a category of (topologically Noetherian) "formal rings" and a corresponding global category of formal schemes. Roughly, a formal ring is a topological ...

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### Compatibility of formal completion and rigid analytic generic fiber

Let $R$ be a complete valuation ring of rank $1$ (e.g., a complete discrete valuation ring) and let $K$ be its field of fractions. Consider a proper $R$-scheme $X$ that is, say, normal (if needed). ...

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### Why are formal schemes assumed to be (locally) noetherian?

All sources that I know that study formal schemes seem to assume that they are locally noetherian. For instance, in Hartshorne "Algebraic Geometry", the author states: "For technical reasons we will ...

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### Approximating formal surfaces by analytic surfaces

Let $C$ be a smooth projective curve contained in a smooth projective $3$-fold (everything defined over $\mathbb C$). Let $\widehat X$ be the formal completion of $X$ along $C$ and suppose there ...

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### Complete subring of F_p[[X]]

Pointed out on famous disbelief, I know now that there is an embedding
$\iota_n \colon {\Bbb F}_p[[T_1,...,T_n]] \hookrightarrow {\Bbb F}_p[[X,Y]]$
for any $n \leq \infty$. Then I would like to ask ...

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

Main Question: What Is the correpondence between flows and vector
fields in algebraic geometry?
Here is a more precise statement could be an answer If it was true (I have no idea it is):
"...

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### Dieudonne modules and Cartier-Dieudonne module of a formal group

As far as I understand, there are Dieudonne modules defined through the homomorphisms to Witt covector scheme and Cartier-Dieudonne modules defined by curves. Am I right that the latter sometimes (for ...

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### Representability of deformation functors via SGA

I'm trying to understand Böckle's proof of Theorem 2.1.1 in his notes on deformation theory.
Let's start with some motivation. Let $\Gamma$ be a profinite group (I'm thinking of an absolute Galois ...

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### Formal group law is a group object in …?

A formal group law over a commutative ring $R$, (by nLab) is a sequence of power srires
$$
f_1,...,f_n\in R[[x_1,...,x_n,y_1,...,y_n]]
$$
such that, using the notation
$$
x=(x_1,...,x_n),y=(y_1,.....

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### Formal vs analytic trivializations of line bundles

Let $X$ be a smooth complex projective variety.
Let $Y$ be a smooth divisor on $X$, and let $\mathfrak X$ be the formal completion of $X$ along $Y$.
Question.
If $\mathcal L$ is a line bundle on $...

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### Possible no standard use of replacement axiom

The idea is to build in ZFC using replacement, a set REPLACEMENT(x ∈ A: TERM(x))
from a set and a term in the same way the set {x ∈ A: FORMULA(x)} is built
using specification from a set and a formula....

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### composition of Puiseux series?

Formal series over a field (or ring) k can be composed in the following sense: given $y \in k[[x]]$, $y$ lying in the maximal ideal, there exists a unique map of topological rings $k[[x]] \to k[[x]]$ ...

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### Does the following first order approximation of the Kapranov-Vasserot infinitesimal loops still do any job?

Let $X$ be a scheme over, say, a field $k$. Let us denote $\mathrm{Spec}(k[\varepsilon])$ by $T$ and its (unique) $k$-point by $0\in T$. Call the first order infinitesimal cone $C_{T,0}(X)$ over $X$ ...

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### Cohomology of Formal Groups

Lubin and Tate, in discussing moduli of 1-dimensional formal groups construct a cohomology theory of formal groups, at least in degrees 0,1 and 2. Does their result about deformations actually follow ...

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### Unfolding subspace algebraic space

Let $ Y \subset X$ be a closed subspace of an algebraic space of finite type over $\mathbb{C}$. Let $p : Y' \rightarrow Y$ be a proper map of algebraic spaces. Artin proved that there exists a ...

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### Are formal completions along a subvariety only dependent on the normal bundle?

In his paper "Mukai flops and derived categories", Namikawa reduces a general Mukai flop of a smooth projective $2n$-dimensional variety $Z$ along a subvariety $W\cong \mathbb P^n$ with $N_{W/Z}\cong \...

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### Rigid analytic spaces vs Berkovich spaces vs Formal schemes

I wonder if someone could explain briefly what is the relation between these 3 formal models, of a Berkovich space, a rigid analytic space and a formal scheme?
I have been working with formal schemes ...

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### Realization of a formal duality isomorphism via integration

This is a more precise version of my previous question. Let $X$ be a smooth variety of dimension $n$ over $\mathbb{C}$ and $Z$ a proper sub-scheme. We denote by $\tilde{X}$ the formal completion of $X$...

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### Flattening techniques of Raynaud and Gruson

Suppose $R$ is an adic valuation ring with a finitely generated ideal of definition. Let $A$ be an $R$-algebra of topologically finite type, i.e. $A$ is isomorphic to $R<\zeta_1,\zeta_2,...,\zeta_n&...

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### Dimension of formal fiber

The question comes from my attempt to understand the following question.
height of contracted prime ideals in power series rings
$\bullet$ My original question: Let $(R,m)$ be a Noetherian local ring ...

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### working with local rings: “abstract” vs “geometric” proofs

Let $R$ be a local ring (commutative, Noetherian, over an algebraically closed field; if needed Henselian). Suppose one wants to prove some statement.
Suppose $R$ happens to be the ring of "functions"...

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### quotients of varieties as non-noetherian schemes?

Let $X$ be a variety (i.e. a reduced scheme of finite type over a field) and let $G$ be an abstract group, finitely generated, acting of $X$ algebraically freely. The example I have in mind is $\...

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### Stein factorization and thm. of formal functions

If $f: X \rightarrow Y$ is a proper morphism of locally noetherian schemes with $f_* \mathcal{O}_X = \mathcal{O}_Y$ then the thm. of formal functions tells us that $f$ has connected fibers, since ...

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### algebraization theorems

One of the fundamental properties that distinguishes schemes among all contravariant functors $\mathrm{Sch}^\circ \rightarrow \mathrm{Sets}$ is algebraization: a functor $F$ satisfies algebraization ...

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### Galois cohomology of generic points of formal completions (of components of a hypercovering of the subvariety): examples or general statements?

Let $Y$ be a closed smooth subvariety in a (smooth) affine variety $X$. What can one say about the etale cohomology of the generic points of the formal completion of $X$ along $Y$ i.e. about the ...

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### Formal Schemes Mittag Leffler

Here is a question that I'm just copying from Math Stack Exchange that I asked awhile ago. It has just been sitting there unanswered, and although I haven't really thought about it since I posted it, ...

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### coherent sheaves on affine formal schemes

Let $\hat{X} = \text{Spf} \hat{A}$ be obtained as the formal completion of an affine scheme $X = \text{Spec} A$ where $A$ is an adic noetherian ring. Given a coherent sheaf $\mathfrak{F}$ on $\hat{X}$,...

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### Formal lifts of schemes in finite characteristic

Let X and Y be smooth varieties over a finite field F. Let R be a complete DVR of unequal characteristic with residue field F. I have the following question:
If f is a morphism from X to Y, is it ...

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### How does Schlessinger's criterion sit with Grothendieck Existence (aka GFGA)?

Grothendieck Existence, which I imagine is the less well known result among the two, states the following:
Let $A$ be a noetherian ring that is complete w.r.t. a proper ideal $I$. Let $V$ be a proper $...

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### Non-algebraizable Formal Scheme?

What is an example of a formal scheme that is not algebraizable?
Recall that, if $X$ is a locally noetherian scheme and $Z$ is a closed subset (of the underlying topological space), then one can ...

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### Flat connections with logarithmic poles and p-adic parallel transport under choice of branch of logarithm

Consider the following simple situation: We work over the ring $R=\mathbb{Z}_p[[t]]$, over which we consider a rank $2$ free module $M$ with basis $(e,f)$. On $M$, we define a flat (topologically ...

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### Definition of formal schemes as formal direct limits

My advisor showed me a definition of formal schemes as follows (acknowledging that these hypotheses may not be minimal): A formal Noetherian scheme is a sequence
$$Y_1 \hookrightarrow Y_2 \...