Questions tagged [ind-schemes]

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Set of smooth points in an ind-variety

There is a remark (Remark IV.4.3.2) in Shrawan Kumar's book* that says it is unknown to the author that the set of smooth points of an ind-variety is open. I was wondering if this has been answered ...
Amir's user avatar
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1 vote
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Generic properties of families of algebras over an infinite dimensional base space

Let $\mathbb{k}$ be an algebraically closed field and let $A$ be a $\mathbb{k}$-algebra which is a free module of rank $r$ over some central subalgebra $Z_0$. If $Z_0$ is affine and $r$ is a finite ...
Lewis Topley's user avatar
2 votes
0 answers
46 views

Special unitary group of an affine algebra is integral

Let $R$ be an affine $\mathbb C-$algebra with a linear involution $x\rightarrow \bar x=\iota(x)$, let $S=R/\iota$ and $\psi:R^n\times R^n\rightarrow R$ be an $R/S-$hermitian form. Finaly let $$SU_n(R)=...
7100note4's user avatar
4 votes
0 answers
139 views

Amalagamation of a sequence of closed immersions of schemes

Let $(X_n)_{n \geq 0}$ be a family of schemes. Let $$X_0 \to X_1 \to X_2 \to \dotsc$$ be a sequence of closed immersions (which therefore gives rise to an ind-scheme). Under which (necessarly and/or ...
HeinrichD's user avatar
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21 votes
1 answer
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Definition of ind-schemes

What is the correct definition of an ind-scheme? I ask this because there are (at least) two definitions in the literature, and they really differ. Definition 1. An ind-scheme is a directed colimit ...
Martin Brandenburg's user avatar
9 votes
1 answer
482 views

How algebraic is the holonomy map?

Let $G$ be either a compact, simple, simply-connected Lie group, or a simply-connected complex reductive group (so either $SU(n)$ or $SL(n,\mathbb{C})$, for instance), or even complex affine. I don't ...
David Roberts's user avatar
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2 votes
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277 views

infinite dimensional germs of schemes and tangent spaces

(The question of the type "how to define?") Let $(R,\mathfrak{m})$ be a local ring over a field $k$ of zero characteristic. Consider the matrices over this ring, $Mat(m,R)$. I think of $Mat(m,R)$ as ...
Dmitry Kerner's user avatar
3 votes
1 answer
262 views

Infinite Dimensional Weil Restriction and Ind-scheme

I'm studying the Weil Restriction $\mbox{Res}_{k/\textbf{F}_p}$ where $\textbf{F}_p$ is the field with $p$ elements and $k$ is a perfect field over $\textbf{F}_p$, not necessarily finite. In this ...
david's user avatar
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7 votes
1 answer
846 views

Is there an underlying topological space for ind-schemes?

An ind-scheme over a base scheme $S$ can be defined in several ways. For simplicity, lets assume that $S$ is the spectrum of an algebraically closed field $k$. We can define a $k$-ind-scheme as a ...
KotelKanim's user avatar
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1 vote
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ind scheme and Jacobson property

Let $G$ a semisimple group over $k$ and $k$ algebraically closed. Let $G(k((t)))$ the corresponding ind-scheme, does it satisfies the Jacobson property, say closed points are dense in it?
prochet's user avatar
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1 vote
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closed subscheme of ind scheme

Let $X$ a ind-scheme of ind-finite type and ind-affine. (e.g, take a k- smooth, affine scheme of finte type $T$, $C$ a smooth projective curve over $k$ and $x$ a closed point, then $X=T(C-x)$ verifies ...
prochet's user avatar
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1 vote
1 answer
334 views

quotient of ind scheme

Is I consider an ind scheme such as $G(k((t)))$ for a reductive connected group over $k=\bar{k}$ I have the conjugacy action of $G(k[[t]])$. In what category can I make the quotient $[G(k((t))/ad(G(...
prochet's user avatar
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11 votes
2 answers
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Do smooth ind schemes have Dualizing sheafs?

Say I have an ind scheme $X = \cup_i X_i$ over a field $k$. I have its tangent bundle $\hom_k(k[\epsilon], X)$ which I can think of as ind scheme via $\cup_i \hom_k(k[\epsilon],X_i)$. The problem is ...
solbap's user avatar
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3 votes
1 answer
556 views

Line bundles on Ind Schemes

I've learned when you have a integral smooth scheme line bundles are the same as Cartier divisors are the same as Weil divisors. My question is to what extent does this continue to hold (if at all) ...
solbap's user avatar
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4 votes
1 answer
438 views

categorical formulation for projective ind-scheme

It is well known that Serre [FAC] gave us a nice categorical description for quasi coherent sheaves on projective scheme, it is a proj-category.(graded modules category localized by Serre subcategory) ...
Shizhuo Zhang's user avatar
3 votes
1 answer
421 views

Principal bundle for contractible group is weak homotopy equivalence for ind schemes

This is may be obvious, but I am not comfortable with ind-schemes. I have an ind-scheme $X$ over $\mathbb{C}$. Every point has a neighborhood which can be written as an ascending union of regular ...
David E Speyer's user avatar