Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [formal-schemes]

The tag has no usage guidance.

1
vote
0answers
64 views

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 ...
4
votes
0answers
93 views

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 ...
14
votes
1answer
592 views

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 ...
0
votes
1answer
107 views

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 ...
2
votes
1answer
222 views

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 ...
5
votes
1answer
175 views

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 ...
20
votes
3answers
658 views

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 ...
4
votes
1answer
238 views

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 $A$-...
2
votes
0answers
124 views

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 ...
4
votes
1answer
419 views

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 ...
0
votes
0answers
75 views

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 ...
1
vote
0answers
155 views

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 ...
5
votes
0answers
111 views

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 $\...
8
votes
1answer
274 views

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 ...
9
votes
0answers
204 views

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 ...
3
votes
1answer
305 views

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 ...
3
votes
1answer
201 views

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 ...
3
votes
1answer
326 views

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 ...
5
votes
1answer
548 views

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 ...
4
votes
0answers
474 views

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). ...
5
votes
0answers
341 views

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 ...
2
votes
0answers
93 views

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 ...
1
vote
1answer
169 views

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 ...
7
votes
1answer
482 views

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): "...
3
votes
0answers
172 views

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 ...
1
vote
1answer
323 views

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 ...
14
votes
2answers
967 views

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,.....
5
votes
0answers
232 views

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 $...
0
votes
1answer
213 views

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....
1
vote
1answer
141 views

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 $k[[x]] \to k[[x]]$ ...
3
votes
0answers
209 views

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 $X$ ...
12
votes
1answer
581 views

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 ...
1
vote
0answers
56 views

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 ...
4
votes
1answer
487 views

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 \...
21
votes
2answers
3k views

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 ...
1
vote
0answers
123 views

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 $X$...
3
votes
0answers
417 views

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&...
2
votes
1answer
612 views

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 ...
3
votes
1answer
533 views

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"...
2
votes
1answer
372 views

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 $\...
1
vote
0answers
240 views

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 ...
17
votes
3answers
2k views

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 ...
1
vote
0answers
157 views

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 ...
17
votes
2answers
965 views

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, ...
7
votes
2answers
930 views

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 $\hat{X}$,...
2
votes
1answer
196 views

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 ...
5
votes
1answer
573 views

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 $...
15
votes
3answers
2k views

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 ...
6
votes
0answers
557 views

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 ...
4
votes
0answers
730 views

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 \...