# An introduction to the derived category, Edition: version 7 by Theo Bühler

By Theo Bühler

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Indeed, (f∗ YX )(Z) = HomX (Z ×X X , Y ) = HomX (Z, Y ) = YX (Z). The direct image functor f∗ : S(X ) → S(X) has a left adjoint functor (see [Mi], p. 68) which is denoted by f ∗ : S(X) → S(X ) and called the inverse image functor. Thus f ∗ is the unique functor which satisﬁes Hom(f ∗ L, L ) = Hom(L, f∗ L ) (L ∈ S(X), L ∈ S(X )). If YX is a locally constant sheaf on X, then its inverse image f ∗ YX is the locally constant sheaf YX on X deﬁned by the ﬁnite ´etale morphism Y = Y ×X X → X ; indeed, HomX (Z , Y ) = HomX (Z , Y ).

L|S0 ) of the restriction Rb! L|S0 = j ∗ Rb! L to S0 of the complex Rb! L of sheaves in the derived category D(X). 1. For any Z -sheaf L on X and morphism b : X → S, the sheaf Ri b! L is constructible. Consequently there is an open dense subscheme j : S0 → S such that the restriction Ri b! L|S0 is a smooth sheaf, for all i ≥ 0. 50 YUVAL Z. FLICKER Proof. The ﬁrst claim is the constructibility theorem of [SGA4], XVII, p. 364. 9, p. 236. The second follows from the deﬁnition of constructibility. Let S be an irreducible scheme.

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