Questions tagged [ind-schemes]
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16 questions
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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 ...
- 111
1
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0
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75
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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 ...
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2
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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 $$SU_n(R)=...
- 51
4
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Amalagamation of a sequence of closed immersions of schemes
Let $(X_n)_{n \geq 0}$ be a family of schemes. Let $$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 ...
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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 ...
- 67.4k
9
votes
1
answer
572
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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 ...
- 37.3k
2
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0
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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 ...
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3
votes
1
answer
338
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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 ...
- 61
8
votes
1
answer
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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 ...
- 2,215
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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?
- 3,642
1
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182
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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 ...
- 3,642
1
vote
1
answer
387
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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(...
- 3,642
11
votes
2
answers
1k
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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 ...
- 4,008
3
votes
1
answer
664
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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) ...
- 4,008
4
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1
answer
469
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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)
...
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