The formal-schemes tag has no usage guidance.

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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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275 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):
"...

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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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233 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 ...

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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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203 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 ...

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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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378 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 ...

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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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**1**answer

316 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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### 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 \...

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### Functorial point of view for formal schemes

Giving a scheme is the same as giving the corresponding functor from the category of rings to the category of set, and there are characterization of what functors arise in this way. This is explained ...

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### Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0

In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...

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### formal completion

When I study formal completion and formal schemes, on p.194 of Hartshorne's "Algebraic Geometry", he said "One sees easily that the stalks of the sheaf $\mathcal{O}_{\hat{X}}$ are local rings."
...

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### When do equivariant sheaves on a formal neighborhood extend?

Suppose that $X$ is a variety (in char 0) with an action of an affine algebraic group $G$. Let $Y \subset X$ be a subvariety fixed by $G$--the action map agrees with projection upon restriction to $Y$...

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### Can isomorphisms of schemes be constructed on formal neighborhoods?

Let (A,m) be a complete local Noetherian ring and let X and Y be two schemes of finite type over A (and flat over A). Let Xn and Yn be the reductions of X and Y mod mn+1.
Question: Suppose there ...