Descent data

Theorem. Let R- S be faithfully flat. Then R-modules are naturally equivalent to S-modules with descent data. 

The condition 9l = 9°92 then says that localized to RiJk agrees with 9Jk9ij. Thus descent data are patching information , isomorphisms on overlaps which are compatible on multiple overlaps. Our theorem here says then that an R-module, R-algebra, etc. can be constructed by taking ones over the various R with compatible isomorphisms over Rj . ... 

The general result of course covers much more ground in the next chapter, for instance, R and S will usually be fields. But one can still think of descent data as patching information for a covering in the fpqc topology. 

Now different

basic theorem shows that two forms are isomorphic over R iff there is an isomorphism over S commuting with the descent data. 

Let N be a finitely generated free (or projective) R-module corresponding to descent data (M. 6). Show AkN corresponds to (AkM, Ak6). Compute the descent data on MD giving N°. 

Dense set 157 Deploye, see Split Derivation 83 Derived group 73 Descent data 131 Diagonalizable group scheme 14 Differential field 77 Differential operator 99 Differentials of an algebra 84 Dimension of an algebraic G 88 Direct limit 151... 

We must show that t is a morphism of objects with descent data. To see... 

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