Questions tagged [formal-schemes]

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1answer
192 views

Formal neighbourhood of a closed subscheme

Let $X$ be a variety and $Y \subset X$ a closed subvariety. Edit: Assume they are both smooth. Denote $N_{Y / X}$ the normal bundle of $Y$ in $X$. The formal neighbourhood of $Y$ in $X$ is the ...
8
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1answer
388 views

Connectedness, loops and formal moduli problems

Let $k$ be an algebraically closed field of characteristic zero. Formalizing a classical folk concept, Pridham and (in a different way,) Lurie defined a formal moduli problem (over $k$) to be a ...
3
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1answer
311 views

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

Notation and Setting: let $\operatorname{Aff}$ denote the category of affine schemes whose objects are covariant representable functors $\operatorname{X}:\operatorname{Ring}\rightarrow\operatorname{...
4
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1answer
215 views

Formal Schemes Methods: Applications

Possibly this question is bit too broad but up to now I was not able to find a satisfying answer. Let $X$ be a locally Noetherian scheme and $X' \subset X$ be a closed subscheme of $X$ which is ...
7
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1answer
613 views

Translation between formal geometry and rigid geometry

I'm reading a paper that translates between formal geometry and rigid geometry. In particular, this paper begins with two rigid analytic spaces $A$ and $C$ (each coming from a scheme over $\mathbb{Z}...
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0answers
165 views

Theorem on Formal Schemes

I have few questions about the proof of Thm 7.5 from P. Wagreich's paper "Elliptic Singularities of Surfaces" (page 452): Denote by $\mathcal{N}$ the kernel of $A \to H^0(O_Z)$ (sorry, I haven't ...
4
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0answers
281 views

Keep blowing up all $k$-rational points

In the construction of Drinfeld space, one uses the following construction of formal schemes (see appendix B of "Period mappings and differential equations. From $\mathbb C$ to $\mathbb C_p$" by Yves ...
6
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0answers
106 views

Classification of one dimensional (non-commutative) formal group laws over $k[\epsilon]/(\epsilon^n)$

It's well-known that any one dimensional formal group law over a $\mathbb Q$-algebra or a reduced ring is commutative, but there are one dimensioal non-commutative formal group laws over rings like $k[...
5
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1answer
372 views

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

$\let\opn=\operatorname$For my BA thesis I have to describe formal groups from the functorial point of view. I am hence reading Strickland - Formal Schemes and Formal Groups, which is apparently the ...
3
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0answers
162 views

Reference Request: Categorical/Functorial approach to Formal Schemes and Formal Groups

For my Bachelor thesis, I am looking for refernces about Formal Schemes and Formal Groups from the categorical point of view. So far I was only able to find Strickland's article Formal Schemes and ...
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0answers
75 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 ...
16
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1answer
694 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
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1answer
114 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 ...
3
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1answer
266 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 ...
6
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1answer
247 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
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3answers
726 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
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1answer
306 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 $A$-...
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0answers
196 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 ...
5
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1answer
700 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 ...
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0answers
104 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 ...
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0answers
178 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
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0answers
123 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 $\...
9
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1answer
314 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 ...
10
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0answers
236 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
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1answer
328 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
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1answer
243 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
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1answer
373 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
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1answer
625 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 ...
5
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0answers
557 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
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0answers
378 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
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0answers
97 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
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1answer
179 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
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1answer
540 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
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0answers
195 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
342 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 ...
15
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2answers
1k 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
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0answers
242 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
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1answer
224 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
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1answer
157 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
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0answers
224 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 $X$ ...
13
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1answer
617 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 ...
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0answers
58 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
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1answer
547 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 \...
22
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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 ...
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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
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0answers
476 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
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1answer
674 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
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1answer
545 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
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1answer
387 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 $\...
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0answers
246 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 ...