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254 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3.3.7 For a map U ! V we have an isomorphism of sites S=U Š .S=V /=.U ! V/.Several formulas below implicitly contain this identification. For example, we have acanonical isomorphismF jU Š .F jV / jU !Vfor F 2 ShS. ForG; H 2 ShS this induces isomorphismsHom Sh S .G; H / jV Š Hom Sh S=V .G jV ;H jV /: (14)3.3.8 We now consider a sheaf of abelian groups H 2 Sh Ab S. If 2 cov S .U / is acovering family of U and H 2 Sh Ab S, then we can define the Čech complex LC .I H/(see [BSS, 2.3.5]).Definition 3.10. The sheaf H is called flabby if H i . LC .I H// Š 0 for all i 1,U 2 S and 2 cov S .U /.For A 2 S we have the section functor.AI :::/W Sh Ab S ! Ab;.AI H/ WD H.A/:This functor is left exact and admits a right-derived functor R.AI :::/which can becalculated using injective resolutions. Note that .AI :::/is acyclic on flabby sheaves,i.e. R i .AI H/ Š 0 for i 1 if H is flabby. Hence, the derived functor R.AI :::/can also be calculated using flabby resolutions. Note that an injective sheaf is flabby.Lemma 3.11. The restriction functor Sh Ab S ! Sh Ab S=Y preserves flabby sheaves.Proof. Let H 2 Sh Ab S be flabby. For .U ! Y/ 2 S=Y and 2 cov S .U / welet jY 2 cov S=Y .U ! Y/ be the induced covering family as in 3.3.1. Note thatLC . jY I H jY / Š LC .I H/is acyclic. Since cov S=Y .U ! Y/is exhausted by familiesof the form jY , 2 cov S .U / it follows that H jY is flabby.3.3.9 We now consider sheaves of abelian groups F;G 2 Sh Ab S.Proposition 3.12. Assume that the site S has the property that for all U 2 S therestriction ShS ! ShS=U preserves representable sheaves. If Y 2 S, then inD C .Sh Ab S=Y / there is a natural isomorphismRHom ShAb S .F; G/ jY Š RHom ShAb S=Y .F jY ;G jY /:Proof. We choose an injective resolution G ! I and furthermore an injective resolutionI jY ! J . Note that by Lemma 3.11 restriction is exact and therefore G jY ! J isa resolution of G jY . Then we haveRHom ShAb S .F; G/ jY Š Hom ShAb S .F; I / jY(14)Š Hom ShAb S=Y .F jY ;I jY /:Furthermore,RHom ShAb S=Y .F jY ;G jY / Š Hom ShAb S=Y .F jY ;J/