Common use of Remark Clause in Contracts

Remark. If X(R) = ∅, then RΓc(Xét, Z(n)) is the same as RΓc(Xét, Z(n)) (see 0.9.2), so that in this case we have an isomorphism of distinguished tri- angles RHom(RΓ(Xét, Zc(n)), Q[−2]) id RHom(RΓ(Xét, Zc(n)), Q[−2]) RHom(RΓ(Xét, Zc(n)), Q/Z[−2]) ' RHom(RΓ(Xét, Zc(n)), Z[−1]) ' RΓc(Xét, Z(n)) RΓfg(X, Z(n)) RHom(RΓ(Xét, Zc(n)), Q[−1]) id RHom(RΓ(Xét, Zc(n)), Q[−1]) where the left column is the result of application of RHom(RΓ(Xét, Zc(n)), ) to an appropriate rotation of the triangle We conclude that RΓfg(X, Z(n)) ' RHom(RΓ(Xét, Zc(n)), Z[−1]). However, this holds only if X(R) = ∅. In what follows, we are not going to make such an assumption on X, even though it would save quite some technical work. It is still helpful to keep in mind the special case X(R) = ∅. The complex of sheaves Zc(n) is bounded from below, under the as- sumption that their cohomology groups are finitely generated (which is our conjecture Lc (Xét, n), stated in 1.1.1).