Questions tagged [formal-schemes]

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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 $...
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  • 1,016
1 vote
0 answers
77 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 ...
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  • 1,361
7 votes
0 answers
185 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 $...
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  • 425
7 votes
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524 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}...
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2 votes
1 answer
385 views

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 ...
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  • 1,325
11 votes
1 answer
701 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. ...
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2 votes
0 answers
137 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 ...
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2 votes
0 answers
142 views

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 ...
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  • 4,966
4 votes
1 answer
270 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 ...
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4 votes
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198 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 ...
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3 votes
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117 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 ...
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10 votes
0 answers
221 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 ...
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1 vote
0 answers
187 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 $X_Z$. ...
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5 votes
0 answers
231 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 ...
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11 votes
0 answers
468 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 $I$), ...
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4 votes
1 answer
303 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 ...
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1 vote
0 answers
147 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 ...
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1 vote
0 answers
93 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 $...
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2 votes
0 answers
142 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 $...
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3 votes
0 answers
118 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\,\...
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  • 1,126
4 votes
0 answers
195 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],\...
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  • 1,126
4 votes
1 answer
351 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 ...
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  • 1,399
2 votes
1 answer
343 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 ...
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10 votes
1 answer
629 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 ...
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3 votes
1 answer
362 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{...
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  • 425
5 votes
1 answer
373 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 ...
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8 votes
1 answer
897 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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0 votes
0 answers
171 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 ...
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4 votes
0 answers
310 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 ...
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6 votes
0 answers
154 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[...
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  • 6,040
5 votes
1 answer
472 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 ...
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  • 425
3 votes
0 answers
172 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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  • 425
2 votes
0 answers
85 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 ...
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  • 121
16 votes
1 answer
858 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 ...
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0 votes
1 answer
118 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 ...
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3 votes
1 answer
287 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 ...
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  • 2,611
6 votes
1 answer
267 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 ...
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20 votes
3 answers
880 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 ...
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4 votes
1 answer
380 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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2 votes
0 answers
222 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 ...
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10 votes
1 answer
1k 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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0 votes
0 answers
215 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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2 votes
0 answers
247 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 ...
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5 votes
0 answers
148 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 $\...
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9 votes
1 answer
357 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 ...
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10 votes
0 answers
287 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 ...
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  • 10k
3 votes
1 answer
366 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 ...
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4 votes
2 answers
399 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 ...
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3 votes
1 answer
456 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 ...
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5 votes
1 answer
769 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 ...
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