By M. Spivak

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**Additional info for Comprehensive Intro to Differential Geometry [Vols 1, 2)**

**Sample text**

The next important topic we want to review brieﬂy is descent theory ([57], Expos´e VIII). This technology plays a key role in the theory of stacks as a substitute of the usual gluing process along an open covering. The key words are descent datum, cocycle condition, and eﬀectiveness. 22). Let ???? be a topological space and let ???? = (???????? ) be an open cover of ????, or a collection of open embeddings (???????? → ???? )???? . Suppose that we are given for each ???? a sheaf ℱ???? on ???????? , and for each ????, ???? an isomorphism ∼ ???????????? : ℱ???? ∣???????? ∩???????? −→ ℱ???? ∣???????? ∩???????? such that for each ???? we have ???????????? = ????????, and for each (????, ????, ????) we have ???????????? = ???????????? ∘ ???????????? on ???????? ∩ ???????? ∩ ???????? (this is called the cocycle condition).

The functor ℎ???? is a Zariski sheaf, thus we can easily reduce the checking to the case where ???? is aﬃne. Assuming this, since ???? is quasi-compact, ???? ′ is covered by ﬁnitely many aﬃne open sets, let ???? ′ = ∪????????=1 ????????′ , ????????′ = Spec(????′???? ) ˜ = be such a covering. This Zariski covering yields a faithfully ﬂat morphism ???? 13 This means that the inverse image of a quasi-compact subset is quasi-compact. 47). We have the following commutative diagram ???? ′ ×y???? ???? ′ ????1 ????2 ???? ˜ ˜ ×???? ???? ???? GG ????y ′ ???? ????1 ????2 GG ˜ ???? ????˜′ ×y ???? ???? G???? ????˜ G???? ????′ ????˜ the arrows being the obvious ones.

41) ???????? G???? ???????? × ???? ???? The hypothesis shows that ???????? = ???????? ×???? ???? → ???? is an open immersion, furthermore (???????? )???? is an open cover of ????. 42) factors through ????. This amounts to seeing the equality on each intersection ????????,???? = ???????? ∣???????? ∩???????? = ???????? ???????? ∩???????? = ????????,???? . 43) This is a simple consequence of the functoriality of the gluing morphism ????, leading to the commutative diagram (setting ????????,???? = ????????,???? = ???????? ∩ ???????? ): ????????,???? y ????????,???? ????????,???? ????????,???? G ????????,???? y ????????,???? ????????,???? .

### Comprehensive Intro to Differential Geometry [Vols 1, 2) by M. Spivak

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