Questions tagged [formal-schemes]
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83
questions
2
votes
1answer
340 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 ...
8
votes
1answer
435 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.
...
2
votes
0answers
134 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 ...
2
votes
0answers
134 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 ...
4
votes
1answer
239 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 ...
4
votes
0answers
163 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 ...
3
votes
0answers
105 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 ...
9
votes
0answers
144 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 ...
1
vote
0answers
146 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 $X_Z$. ...
5
votes
0answers
213 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 ...
11
votes
0answers
392 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 $I$), ...
4
votes
1answer
259 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 ...
1
vote
0answers
128 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 ...
1
vote
0answers
79 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 $...
2
votes
0answers
120 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 $...
3
votes
0answers
110 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\,\...
4
votes
0answers
157 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],\...
1
vote
0answers
63 views
Formal relative tangent space
Let $F$ and $G$ be two functor from Artin local schemes over complex numbers to sets and suppose both satisfy all the Schlesinger's conditions and let $f: F\rightarrow G$ be a natural transformation ...
4
votes
1answer
277 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 ...
2
votes
1answer
231 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 ...
9
votes
1answer
548 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 ...
3
votes
1answer
336 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{...
4
votes
1answer
274 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 ...
8
votes
1answer
743 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}...
0
votes
0answers
169 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 ...
4
votes
0answers
296 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 ...
6
votes
0answers
128 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[...
5
votes
1answer
412 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 ...
3
votes
0answers
166 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 ...
2
votes
0answers
81 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 ...
16
votes
1answer
778 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 ...
0
votes
1answer
117 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 ...
3
votes
1answer
275 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 ...
6
votes
1answer
256 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 ...
20
votes
3answers
796 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 ...
4
votes
1answer
338 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 $A$-...
2
votes
0answers
211 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 ...
8
votes
1answer
928 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 ...
0
votes
0answers
192 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 ...
2
votes
0answers
212 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 ...
5
votes
0answers
129 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 $\...
9
votes
1answer
338 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 ...
10
votes
0answers
261 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 ...
3
votes
1answer
350 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 ...
3
votes
2answers
318 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 ...
3
votes
1answer
414 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 ...
5
votes
1answer
696 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 ...
5
votes
0answers
614 views
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). ...
5
votes
0answers
413 views
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 ...
2
votes
0answers
98 views
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 ...
