The formal-schemes tag has no wiki summary.

**1**

vote

**0**answers

43 views

### uniformization of non-algebraizable formal schemes

A construction of Mumford produces totally degenerate deformations of Abelian varieties by quotienting a certain formal scheme which is only locally of finite type by an action of a discrete group. ...

**1**

vote

**1**answer

118 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 ...

**7**

votes

**2**answers

301 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
$$
...

**4**

votes

**0**answers

178 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

**1**answer

170 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 ...

**1**

vote

**1**answer

86 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

**0**answers

159 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 $X$ ...

**9**

votes

**1**answer

364 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

**0**answers

51 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 ...

**2**

votes

**1**answer

266 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 ...

**11**

votes

**2**answers

947 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

**0**answers

101 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 ...

**3**

votes

**0**answers

185 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 ...

**1**

vote

**1**answer

258 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

**1**answer

463 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 ...

**2**

votes

**1**answer

291 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

**0**answers

186 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 ...

**15**

votes

**3**answers

1k 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

**0**answers

145 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

**2**answers

809 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

**2**answers

625 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 ...

**2**

votes

**1**answer

177 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

**1**answer

434 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 ...

**10**

votes

**3**answers

1k 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 ...

**5**

votes

**0**answers

447 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

**0**answers

555 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 ...

**15**

votes

**2**answers

1k views

### 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 ...

**7**

votes

**1**answer

469 views

### 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 ...

**7**

votes

**1**answer

2k views

### 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."
...

**4**

votes

**0**answers

204 views

### 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 ...

**5**

votes

**4**answers

413 views

### 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 ...