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Suppose $G=GL(n) $ and let $LG$ be the loop group and $Gr_{G}$ be the affine grassmanian. It's a fact mentioned in books and notes on affine grassmanian that these are not schemes but rather ind-schemes. My question is that how do we see that $LG$ and $Gr_{G}$ are not schemes, which properties of a scheme do they not satisfy.

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I don't want to be too trite, but they lack the property of being schemes. There are no locally ringed spaces with open coverings by affine schemes representing $LG$ or $Gr_G$.

Given that $LG$ and $Gr_G$ are ind-schemes, there is a notion of open immersion, so we could rephrase the above to say: There is no surjective morphism $U \to LG$, resp. $U \to Gr_G$, such that $U = \coprod U_i$ is a disjoint union of schemes, and each $U_i \to LG$, resp. $U_i \to Gr_G$, is an open immersion.

If you're asking for a proof of non-representability, I'm still pretty new at this and I don't have one right now. But it should be immediately clear by construction, at least for $LG$, that it is not a subscheme of infinite-dimensional affine space.

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