Full subcategory

Definition 1.3. Let Ag>s be the full subcategory of Ag,d consisting of those objects (5,X, A) of A9id such that there exists a faithfully flat T — S such that Ker(A)r zz K( )t-... 

Therefore we get our functor Ms C(l)s0 — M(l)s0 a pair ( >/s) as above exists. Suppose that So is the spectrum of a Noetherian complete local ring whose residue field has a finite p-basis. In this case the functor Ms induces an equivalence of the full subcategory of C(l)s0 consisting of group schemes which have multiplicative part equal to zero and a corresponding subcategory of At (1) s0 (Theorem 9.3). If the residue field is perfect then we can show under some additional hypotheses on (5,/s) that Ms C(l)s0 At(l)s0 is an equivalence (Theorem 10.2). 

Again we let K = Ker(FH). Further we let C F 0)so he full subcategory of C l)so consisting of objects whose Frobenius morphism is zero. Then in this case... 

Let be a locally noetherian, normal and connected formal scheme and ( j i6I locally finite set of regular divisors with normal crossings on. Put D= E D.. Consider in the category of formal u -schemes the following full subcategories ... 

Moreover [H, p. 33, Prop. 3.4] any morphism Q —> Q2 of such functors extends uniquely to a morphism Q[ —> Q. In other words, composition with Q gives, for any category E, an isomorphism of the functor category Hom(D, E) onto the full subcategory of Hom(K, E) whose objects are the functors K —> E which transform quasi-isomorphisms in K into isomorphisms in E. 

Now let KJ" be the full subcategory of K" " whose objects are the bounded-below I-complexes. Since the additive subcategory I C is closed under finite direct sums, one sees that Kj" is a A-subcategory of K. According to (1.7.2)°P, the derived category DJ" of KJ" can be identified with a A-subcategory of D , and the above family [Pg.34]

For example, if I is the full subcategory of A whose objects are all the injectives in A, then by [H, p.41, Lemma 4.5] every quasi-isomorphism in KJ" is an isomorphism, so that KJ" can be identified with its derived... 

For example, if 17 is a topological space, CMs a sheaf of rings on U, and Tlis the abelian category of (sheaves of) left O-modules, then we can take P to be the full subcategory of A whose objects are all the flat O-modules [H, p. 86, Prop. 1.2]. 

Let. A be a plump subcategory of the abelian category A, i.e., a full subcategory containing 0 and such that for every exact sequence in A... 

One checks that r> (respectively t< ) extends naturally to an additive functor of complexes which preserves homotopy and takes quasi-isomorphisms to quasi-isomorphisms, and hence induces an additive functor D D, see 1.2. In fact if D ) is the full subcategory of D whose objects are the complexes A such that H A ) = 0 for m > n (resp. m < n) then we have additive functors... 

For a subcategory E of D( ), E< (resp. E> ) will denote the full subcategory of E whose objects are those complexes A such that = 0... 

Proof We check first that C is actually a morphism of A-functors. Consider a map u X - Y m. J". Since Q px) i isomorphism, there is a unique map u Ax —> Ay in D" (and hence in the full subcategory D ) making the following D"-diagram commute ... 

This map plays a crucial role in Grothendieck duality theory on, say, the full subcategory of S whose objects are all the concentrated schemes, in which situation the right adjoints and exist, see (4.1.1) below. 

In the case where G is trivial, / is defined as follows. For a scheme Z, we denote the category of Gz-modules by Mod(Z). By deflnition, a plump suhcategory of an abelian category is a non-empty full subcategory which is closed under kernels, cokernels, and extensions [26, (1.9.1)]. We denote the plump subcategory of Mod(Z) consisting of quasi-coherent Gz-modules by Qch(Z). 

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