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1answer
74 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
126 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 $X$ ...
8
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0answers
267 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
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0answers
45 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
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1answer
182 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 ...
9
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2answers
527 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
93 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
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0answers
137 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
1answer
209 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 ...
2
votes
0answers
287 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
1answer
261 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
158 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 ...
14
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2answers
655 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
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0answers
135 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 ...
13
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2answers
638 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
559 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
1answer
167 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
399 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 ...
7
votes
3answers
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 ...
4
votes
0answers
412 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
511 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 ...
14
votes
2answers
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 ...
6
votes
1answer
452 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 ...
5
votes
1answer
1k 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
0answers
190 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
4answers
384 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 ...