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by representable functors. As such, although it is not a scheme, it may be thought of as a union of schemes, and this is enough to profitably apply geometric methods to study it.
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27:
For the variety of all k-dimensional affine subspaces of a finite-dimensional vector space (a smooth finite-dimensional variety over k), see
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is in fact ind-projective, i.e., an inductive limit of projective schemes.
251:-algebras to sets which is not itself representable, but which has a
158:-algebras and the category of sets respectively. Through the
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77:))) and which describes the representation theory of the
344:, it is also possible to specify this data by fixing an
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621:{\displaystyle G({\mathcal {K}})/G({\mathcal {O}})}
557:{\displaystyle \operatorname {Spec} {\mathcal {O}}}
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212:{\displaystyle X:k{\text{-Alg}}\to \mathrm {Set} }
211:
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118:
8:
247:. The affine Grassmannian is a functor from
641:Affine Flag Manifolds and Principal Bundles
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284:the set of isomorphism classes of pairs (
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93:Definition of Gr via functor of points
462:{\displaystyle {\mathcal {K}}=k((t))}
312:is an isomorphism, defined over Spec
7:
638:Alexander Schmitt (11 August 2010).
276:is the functor that associates to a
574:is identified with the coset space
239:. We then say that this functor is
530:. By choosing a trivialization of
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515:{\displaystyle {\mathcal {O}}=k]}
57:—a colimit of finite-dimensional
87:geometric Satake correspondence
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147:{\displaystyle \mathrm {Set} }
119:{\displaystyle k{\text{-Alg}}}
61:—which can be thought of as a
29:affine Grassmannian (manifold)
1:
85:through what is known as the
154:the category of commutative
420:Definition as a coset space
298:principal homogeneous space
262:be an algebraic group over
692:
644:. Springer. pp. 3–6.
101:be a field, and denote by
26:
342:Beauville–Laszlo theorem
174:, which is the functor
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558:
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471:formal Laurent series
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388:a trivialization on (
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170:is determined by its
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18:Infinite Grassmannian
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522:the ring of formal
268:affine Grassmannian
65:for the loop group
40:affine Grassmannian
676:Algebraic geometry
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116:
651:978-3-0346-0287-7
424:Let us denote by
324:with the trivial
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172:functor of points
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16:(Redirected from
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408:reductive group
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346:algebraic curve
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79:Langlands dual
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568:-points of Gr
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469:the field of
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49:over a field
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655:. Retrieved
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534:over all of
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524:power series
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220:
219:which takes
167:
163:
160:Yoneda lemma
155:
98:
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82:
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63:flag variety
50:
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39:
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375:-bundle on
235:-points of
223:to the set
162:, a scheme
36:mathematics
657:1 November
632:References
304:over Spec
253:filtration
55:ind-scheme
545:
477:, and by
292:), where
280:-algebra
196:→
670:Category
371:to be a
328:-bundle
402:. When
359:-point
332:× Spec
320:)), of
59:schemes
648:
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266:. The
81:group
53:is an
42:of an
38:, the
526:over
473:over
406:is a
351:over
296:is a
231:) of
659:2012
646:ISBN
542:Spec
410:, Gr
384:and
355:, a
300:for
258:Let
192:-Alg
126:and
113:-Alg
97:Let
363:on
34:In
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270:Gr
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434:K
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394:x
390:X
386:φ
381:A
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369:E
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330:G
326:G
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318:t
314:A
310:φ
306:A
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294:E
290:φ
286:E
282:A
278:k
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264:k
260:G
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233:A
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