Characterization of quasi-coherent modules that are module schemes
Amelia Álvarez Sánchez, Carlos Sancho de Salas, Pedro Sancho de Salas
aa r X i v : . [ m a t h . A C ] S e p CHARACTERIZATION OF QUASI-COHERENT MODULES THATARE MODULE SCHEMES
AMELIA ´ALVAREZ, CARLOS SANCHO, AND PEDRO SANCHO
Introduction
Let R be a commutative ring with unit. All functors we consider are functorsover the category of commutative R -algebras. Given an R -module E , we denoteby E the functor of R -modules E ( B ) := E ⊗ R B . We will say that E is a quasi-coherent R -module. If E is an R -module of finite type we will say that E is acoherent R -module. Given F , H functors of R -modules, Hom R ( F, H ) will denotethe functor of R -modules Hom R ( F, H )( B ) = Hom B ( F | B , H | B )where F | B is the functor F restricted to the category of commutative B -algebras. Itholds that Hom R ( E , E ′ ) = Hom R ( E, E ′ ). The category of R -modules is equivalentto the category of quasi-coherent R -modules ([A, 1.12]).We denote F ∗ := Hom R ( F, R ). We will say that E ∗ is an R -module scheme.The R -module functors that are essential for the development of the theory ofthe linear representations of an affine R -group are the quasi-coherent R -modulesand the R -module schemes ([A]). The aim of this paper is to study when a quasi-coherent R -module is an R -module scheme. We will prove that it is equivalent togiving a characterization of projective R -modules of finite type.The main result we are going to use is the following proposition. Proposition 0.1. [A, 1.8]
Let E , E ′ be R -modules. Then: Hom R ( E ∗ , E ′ ) = E ⊗ R E ′ . Two immediate consequences are: (a) E ∗∗ = E ([A, 1.10]) . (b) If E is a projective module, the image of a morphism E ∗ → V is a coherentmodule ([A, 4.5]) . Characterization of quasi-coherent modules that are moduleschemes
Theorem 1.1.
Let E be an R -module. Then, E is an R -module scheme if andonly if E is a projective module of finite type.Proof. ⇒ ) Assume E = V ∗ . For every R -module E ′ ,Hom R ( E, E ′ ) = Hom R ( E , E ′ ) = Hom R ( V ∗ , E ′ ) = V ⊗ R E ′ . Date : June, 2007.2000
Mathematics Subject Classification.
Primary 13C10. Secondary 20C99. E is a projective module because Hom R ( E, − ) = V ⊗ R − , which is exact on theright.Since V = V ∗∗ = E ∗ , we have that V is a projective module. The image ofthe isomorphism V ∗ = E is coherent (0.1 (b)). Then, E is an R -module of finitetype. ⇐ ) Let us consider an epimorphism from a finitely generated free module L to E , L → E . Taking dual we have an injective morphism E ∗ ֒ → L ∗ . L ∗ is isomorphic toa coherent module. The image of the morphism E ∗ ֒ → L ∗ , which is E ∗ , is coherent(0.1(b)). Then, E = E ∗∗ is a module scheme. (cid:3) Corollary 1.2. An R -module E is projective of finite type if and only if Hom R ( E, B ) = Hom R ( E, R ) ⊗ R B for every commutative R -algebra B .Proof. Hom R ( E, B ) = Hom R ( E, R ) ⊗ R B for every commutative R -algebra if andonly if E ∗ is a quasi-coherent R -module. That is to say, if and only if E is a modulescheme, from the previous proposition, if and only if E is a projective module offinite type. (cid:3) Corollary 1.3. An R -module E is projective of finite type if and only if Hom R ( E, E ′ ) = Hom R ( E, R ) ⊗ R E ′ for every R -module E ′ . In [B, Ch. II, § P i w i ⊗ e i ∈ Hom R ( E, E ) = E ∗ ⊗ R E , then E = < e i > is a module of finite type. Corollary 1.4.
The quasi-coherent R -module corresponding to the R -module E ∗ is E ∗ if and only if E is a projective module of finite type. References [A] ´Alvarez, A., Sancho, C., Sancho,P.,
Algebra schemes and their representations , J. Algebra (2006) 110-144.[B]
Bourbaki, N., ´El´ements de Math´ematique. Alg`ebre I, Hermann, Paris, 1970.
Departamento de Matem´aticas, Universidad de Extremadura, Avenida de Elvas s/n,06071 Badajoz, Spain
E-mail address : [email protected] Departamento de Matem´aticas, Universidad de Salamanca, Plaza de la Merced 1-4,37008 Salamanca, Spain
E-mail address : [email protected] Departamento de Matem´aticas, Universidad de Extremadura, Avenida de Elvas s/n,06071 Badajoz, Spain
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