Quasi-coherent modules

Almost all local theory involves modules as well as rings and ideals. The purpose of this section is to put modules in their convenient geometric setting. This is really just a technical digression before we get on to the more interesting geometry. 

Basic Example. Let X j= Spec R, and let M be an /2-module. We define an associated ox-module M in the same way in which we defined ox itself. To the distinguished open set Xd we assign the module M/, checking that when Xf C Xg we get a natural map Mh - Mf. We check that lim Mf = Mp, 

Proposition 1. Let M,N be R-modules. The natural map hom2x(M,N) homft(M, N) gotten by taking global sections is a bijection. 

Given a homomorphism / M - iV we get induced homomorphisms (ff Mf - Nf for every /. This gives us homomorphisms r(U,M) - r(U,N) on every distinguished open set U, compatible with restriction that necessarily induces homomorphismsJbr all open sets U. It is easily seen that this process gives an inverse to hom(M, N) hom(M, N).  

Corollary. The category of R-modules is equivalent to the category of ox-modules of the form M. 

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Assume that whenever M is a quasi-coherent module on an affine open... 

This tells us that for any quasi-coherent module M on U the map Ext1(A.G(n), M) -... 

Barsotti-Tate group on S and M a quasi-coherent module, Horn... 

Since S is affine and Y(G) is a quasi-coherent module all torseurs... 

Note that an ox-ideal Q is quasi-coherent in this definition if and only if it is quasi-coherent in our former sense (Ch. 2, 5). Moreover, kernels, cokernels, and images of maps between quasi-coherent modules are quasi-coherent. For by the theorem quasi-coherence is a local condition, and we know the result for affine schemes. Yet another way of combining quasi-coherent o -modules is this Let Q be a quasi-coherent o -ideal, T a quasi-coherent -module. Let Q T be the least submodule of T containing all elements a s (a G r(U, Q), s G rfU, )). Then Q T is also quasi-coherent, and if U is any open affine, Q T v = I M where I = r(C7, Q), M = r(U, T). [The first statement follows from the second, whoose proof is left as an exercise.]... 

As a less trivial example, we will define for every morphism X Y a very important quasi-coherent -module fix/y-, called the sheaf of relative differentials. First, we recall the commutative algebra involved. 

Definition 2. fix/y is the quasi-coherent -module obtained by carrying Q/Q2 back to X by the isomorphism A X Z. 

D. Let / X — Y be a morphism. Assume Y is irreducible and reduced with generic point y. Let T be a quasi-coherent -module flat over oy. Then for all x G X, Tx is a torsion-free y-module. If X is noetherian and T is a coherent ox-module, this means that all associated points of T lie over y. Conversely, this property implies that T is flat over oy if all stalks oy Y are valuation rings (e.g., Y a non-singular curve, or Spec (Z)). 

Lemme II 1.1. Soient f X — S up morphisme de schemas, quasi—compact et quasi-separe, P un Q -Module quasi-coherent. Alors, les conditions suivantes sont dquivalentes ... 

Pour tout morphisme de schemas g S —S, si l on d signe par X f, P les objets d duits de X,f,F par le changement de base g, alors il existe un 0g(-Module quasi-coh rent Q et un lsomorphisme, fondtoriel par rapport au 0g(-Module quasi-coherent M s... 

Le lemme resulte immediatement du fait que l on a des isomorphismes cano-niques, fonctoriels en le 0, -Module quasi-coherent M s... 

Lemme II 2.1. Soient S le spectre d un anneau local noetherien, s le point ferme de S, f X > S un S—schema localement noetherien, P un O -Module quasi—coherent et f-plat. Alors les conditions suivantes sont 6qui-valentes s... 

Definitional 2.2. Soient f t X —-> S un morphisme de schdmas localement noe-thdriens, F un O -Module quasi—coherent et f—plat. 

S-sch4mas localement de type fini, F un O -Module quasi-coherent (resp. G... 

Lemme II 2.7. Soient S un schema local) artinien, de point feme s, X un S—schema quasi-compact et quasi-separe, F un 0 —Module quasi-coherent et S-plat. Pour que P verifie W, il faut et il suffit que P(Xa)Pg) soit de dimension finie sur k(s). 

Module quasi-coherent P est ponctuellement libre de rang r (i.e. ses... 

Lemme 111 1.1. Soient f X —un morphisme de schemas, quasi-compact et quasi-s pard, P un O -Module quasi-coherent qui satisfait a V (II 1.3). Alors les conditions suivantes sont 4quivalentes ... 

Both a and uj give contravariant functors of the category of finite flat group schemes over S into the category of quasi-coherent Os-modules. The sheaf ug IS finitely generated as an Os-module. 

XIV) Flatness is preserved under base chanoe if f % - 5 is a morphism and F is a quasi-coherent sheaf of O -modules, flat over 5, then for every morphism g 5 —> 5 the sheaf g (F), where g DGxs5 X is the projection, is flat over 5 ... 

V et V cesmne des foncteurs de la cat gorie (CL-rMod. q.c.) des CU-modules quasi—coherents dans la cat gorie des O -modules... 

Theorems 5.3 3- Soient S un schema affine et G un S-groupe diago-nalisable. Pour tout G-Og-module quasi-coherent P, on a Hn(G, P) 0, n > 0. ... 

Soient maintenant M et N deux Og-modules quasi-coherents. Le diagrams commutatif... 

Examples de bons Og-modules s pour tout Og-Module quasi-coherent E, les Og-modules V(E) et W(e) d finis en I 4.6 sont bons. ... 

Rappel 0.13. L image directe d un module quasi-coherent par un morphiane de presentation finie est quasi-coherente. Sous les mimes conditions, la formation de l image directe commute au changement de base plat dans la situation... 

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