Questions tagged [formal-schemes]

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Projective reduction of image of power series is algebraic?

Let $K$ be a non-archimedean field with closed unit disk $\mathcal{O}\subset K$, open unit disk $\mathfrak{m}\subset \mathcal{O}$ and residue field $k = \mathcal{O}/\mathfrak{m}$. Examples to keep in ...
Jef's user avatar
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1 answer
149 views

Meaning of dagger cohomology $H^{1 \dagger}(G^\dagger)$ in "Frobenius and Monodromy Operators" by Coleman and Iovita

Let $G$ be an abelian variety with good reduction over a finite extension $K$ of $\mathbb{Q}_p$. In equation (2.4) on page 179 of my edition of "The Frobenius and monodromy operators for curves ...
Vik78's user avatar
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Does there exist an upper triangular set of gens of $ k[[X]] $ for a unipotent formal group acting on $ \operatorname{Spec}(k[[X]]) $?

For this post I define a formal group to be a group object in the category of formal schemes. Let $ G $ be a linear algebraic group with affine coordinate ring $ k[G] $. If $ \mathfrak{m}_{e} $ is ...
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5 votes
1 answer
269 views

What's the relation between pseudo-compact and admissible rings?

We recall two definitions. Let $A$ be a linearly topologized ring which is complete and Hausdorff. We say that $A$ is pseudo-compact if, for every open ideal $I\subset A$, the ring $A/I$ is artinian. ...
Gabriel's user avatar
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3 votes
1 answer
162 views

On the stability of having a normal formal model under finite extensions of the base field

Let $K$ be a finite extension of the $p$-adic numbers with valuation ring $\mathcal{R}$ and uniformizer $\pi$. Consider a smooth and connected rigid $K$-variety $X=Sp(A)$ and assume that the affine ...
Fernando Peña Vázquez's user avatar
1 vote
0 answers
125 views

For what properties $ (\mathcal{P}) $ (if any) does $ (\mathcal{P}) $ + analytic isomorphism imply birationality?

In general if $ \mathfrak{p} \in \operatorname{Spec}(A) $ and $ \mathfrak{q} \in \operatorname{Spec}(B) $ are two (not necessarily closed) points such that $ (\mathfrak{p}, \operatorname{Spec}(A)) $ ...
Schemer1's user avatar
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2 votes
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Does there exists a "local slice" for an action $ \widehat{\mathbb{G}_{a}} $ on $ \operatorname{Spf}(\widehat{A}) $ (char zero)?

Every action $ \beta $ of $ \mathbb{G}_{a} $ on a variety $ \operatorname{Spec}(A) $ over a field of characteristic zero is obtained from a locally nilpotent derivation $ \delta $ via $ f(t_{0} \ast x)...
Schemer1's user avatar
  • 789
1 vote
0 answers
228 views

Does analytic isomorphism imply local isomorphism?

If $ \mathfrak{p} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(A) $, and $ \mathfrak{q} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(B) $ such ...
Schemer1's user avatar
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Contracting an effective Cartier divisor to a point

I am trying to understand contractibility of a subspace, in the case of contracting a divisor to a point. More precisely, what I mean by contraction is the following. Given an algebraic space(or ...
Sungwoo's user avatar
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Theorem on formal functions when the initial data is a proper map of formal schemes

Let $\pi: X \to S:=\mathrm{Spf}\text{ } A$ be a proper morphism of $\mathbb{Z}_p$-admissible formal schemes and $\mathcal{F}$ be a coherent sheaf on $X$. Set $S_0=\{x\}$ be a closed point of $S$ and $...
Adel BETINA's user avatar
  • 1,046
1 vote
0 answers
82 views

Global generation of twists of formal vector bundles

Let $X$ be a smooth projective variety (over $\mathbb C$) and $Y\subset X$ a complete intersection of ample divisors $H_1\cap H_2\cap \cdots\cap H_r$. In the proof of Grothendieck-Lefschetz, a crucial ...
pi_1's user avatar
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240 views

Are unramified simple Rapoport-Zink spaces smooth?

I have read on different occasions that unramified simple Rapoport-Zink spaces are formally smooth, eg. stated in Remark 4.13 of Rapoport and Viehman's survey article. These spaces are formal schemes $...
Suzet's user avatar
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562 views

Scholze and Weinstein's $\operatorname{Spa}\mathbf{Z}_p\times \operatorname{Spa}\mathbf{Z}_p$

In their Berkeley Lectures, to motivate the introduction of Diamonds, Scholze and Weinstein discuss what should be the definition of $\operatorname{Spa}\mathbf{Z}_p\times \operatorname{Spa}\mathbf{Z}...
Stabilo's user avatar
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Does $R\hat{f}_*\mathcal{F}=\hat{f}_*\mathcal{F}$ hold for affine adic morphisms?

Let $f:X\to Y$ be an affine morphism of locally Noetherian schemes. By this, we know that $Rf_*\mathcal{F}=f_*\mathcal{F}$ for any quasi-coherent sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules (the ...
Stabilo's user avatar
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12 votes
1 answer
960 views

Motivation for Henselian rings in algebraic geometry

In Andrew Kobin's script on Algebraic Geometry I found on page 355 a comment I would like better understand. It states Another way to view formal smoothness is as an abstraction of Hensel's Lemma. ...
user267839's user avatar
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2 votes
0 answers
147 views

Colimits of the infinitesimal neighborhoods of symmetric product in the category of schemes

This problem is highly related to this one and in fact it is the same question applied to a very specific situation. Given a smooth projective curve $C$, let $\text{Sym}^i(C)$ be the $i$-th symmetric ...
user127776's user avatar
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Algebraization of vector bundles over non-algebraically closed fields

I've asked this question here but never got an answer, a simplified version of the question is the following: Given an intersection of hypersurfaces $Y$ on a smooth projective variety $X$ over a ...
user127776's user avatar
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4 votes
1 answer
311 views

Symmetric powers of curves and completion along the diagonal

Given a smooth curve $C$, denote by $\text{Sym}^d(C)$ its $d$-th symmetric power. Let $\Delta$ be the diagonal subvariety which is defined as the codimension $1$ subvariety that at least two of the ...
user127776's user avatar
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4 votes
0 answers
258 views

How much does the formal completion know about the ambient variety?

How much information does the formal completion along an ample divisor hold about the ambient variety? Given two smooth projective varieties such that they have isomorphic ample divisors, where the ...
user127776's user avatar
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3 votes
0 answers
136 views

On formal completions of normal bundles in the non-affine case

According to this post. In the affine case the formal completion of $A$ along an ideal $I$ coincides with the formal is isomorphic to the completion of the normal bundle along its zero section. I was ...
user127776's user avatar
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11 votes
0 answers
333 views

Quasi-separated rigid-analytic space without a formal model?

Well, my question is slightly embarrassing. When learning rigid geometry (mostly from Bosch's book) I realized that I don't know the answer to the following basic question. Question. Is there an ...
Piotr Achinger's user avatar
2 votes
0 answers
243 views

Zariski descent of algebraic $K$-theory on formal schemes

This question is highly related to some other questions that I've previously asked, especially to this one. In this problem we have a scheme $X$ and a closed subscheme $Z$ the formal completion $X_Z$. ...
user127776's user avatar
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5 votes
0 answers
250 views

Colimit of nilpotent thickenings in the category of schemes

This question is highly related to this and this one. Given a ring $A$ and an ideal $I$, the direct system of schemes $\text{Spec}(A/I)\rightarrow \text{Spec}(A/I^2)\rightarrow \ldots$ has a colimit ...
user127776's user avatar
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12 votes
0 answers
562 views

Global version of Gabber's rigidity theorem

I had a question regarding Gabber's rigidity. Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), ...
user127776's user avatar
  • 5,831
4 votes
1 answer
365 views

Vector bundles on complete rings

Given a ring $A$ and an ideal $I$, consider the completion $\hat{A}$. What does usually mean by a vector bundle on $\hat{A}$? One way is to consider projective $\hat{A}$-modules. Another one is a ...
user127776's user avatar
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1 vote
0 answers
182 views

Moduli interpretation of normalization of moduli space

The question is about formal and rigid geometry, but I would be interested in an answer from an algebraic geometry point of view as well. Let $\mathfrak{X}$ be a formal moduli space (e.g., the formal ...
Jon Aycock's user avatar
1 vote
0 answers
139 views

Examples of effective Lefschetz condition

When a closed subvariety $Y$ of $X$ satisfy effective Lefschetz condition $Leff(X,Y)$, it implies there is an equivalence if categories between the vector bundles on the formal neighborhood of $Y$ in $...
user127776's user avatar
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3 votes
0 answers
197 views

A question about extending vector bundles from formal neighborhood to a coherent sheaf

I have a question which probably is very straightforward but because of my lack of knowledge of formal schemes I'm asking it here. Let's assume we have a vector bundle $E$ on the formal completion $...
user127776's user avatar
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3 votes
0 answers
124 views

how much information $O_K$ points of a formal scheme over $\mathbb{Z}_p$ contain

assume that $Spf\,A\to Spf\,\mathbb{Z}_p[[t_1,...,t_n]]$ is a closed immersion of flat integral formal schemes over $\mathbb{Z_p}$. I see Kisin several time use that if $Spf\,A(O_K)\subset SPf\,\...
ali's user avatar
  • 1,043
4 votes
0 answers
264 views

nearby cycles map for affine formal schemes

Assume that $X=Spf R$ is p-adic formal scheme over $O_{C_p}$ with generic fiber $X_{\eta}$. I want to know why the nearby cycles map $Ru^\star \mathbb{Z/p}$ is equal to $R\Gamma_{et}(spec R[1/p],\...
ali's user avatar
  • 1,043
4 votes
1 answer
456 views

When is a formal group smooth?

This is a question that I suspect is simply a matter of technical issues written down or clarified somewhere in the literature, but which I can't find. Suppose we're working over an arbitrary base ...
xir's user avatar
  • 1,964
2 votes
1 answer
547 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 ...
Federico Barbacovi's user avatar
11 votes
1 answer
727 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 ...
Dmitry Vaintrob's user avatar
3 votes
1 answer
512 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{...
sagirot's user avatar
  • 455
6 votes
1 answer
646 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 ...
user267839's user avatar
  • 5,948
9 votes
1 answer
1k 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}...
Jon Aycock's user avatar
0 votes
0 answers
179 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 ...
user267839's user avatar
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4 votes
0 answers
317 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 ...
sawdada's user avatar
  • 6,148
6 votes
0 answers
243 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[...
sawdada's user avatar
  • 6,148
6 votes
1 answer
626 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 ...
sagirot's user avatar
  • 455
3 votes
0 answers
179 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 ...
sagirot's user avatar
  • 455
2 votes
0 answers
92 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 ...
slin0's user avatar
  • 121
16 votes
1 answer
968 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 ...
user avatar
0 votes
1 answer
121 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 ...
user avatar
3 votes
1 answer
315 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 ...
Hang's user avatar
  • 2,719
6 votes
1 answer
316 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 ...
Hang's user avatar
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21 votes
3 answers
1k 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 ...
Qfwfq's user avatar
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4 votes
1 answer
443 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 $A$-...
user avatar
2 votes
0 answers
253 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 ...
user avatar
12 votes
1 answer
2k 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 ...
rime's user avatar
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