Common use of Remark Clause in Contracts

Remark. One can also obtain information about HomD(A )(A•, B• ) using the following hyperext spectral sequence: Epq = ∏ Extp (Hi(A•), Hq+i(B•)) =⇒ Extp+q (A•, B•) D(A ) (see e.g. [▇▇▇▇▇▇▇-thèse, Chapitre III, §4.6.10] and [Wei1994, §5.7.9]). For a hereditary category Extp = 0, unless p = 0, 1, and this spectral sequence consists of two columns and therefore gives us short exact sequences 0 → ∏ Ext1 (Hi(A•), Hi−1(B•)) → HomD(A )(A•, B•) → ∏ HomA (Hi(A•), Hi(B•)) → 0 i∈Z However, one should be careful with boundedness of A• and B• to make sure that the spectral sequence exists. Recall that a complex of abelian groups C• is called perfect if it is quasi- isomorphic to a bounded complex of finitely generated free (= projective) abelian groups. This is the same as asking Hi (C• ) to be finitely generated abelian groups, and Hi (C• ) = 0 for all but finitely many i. In §1.5 we are going to construct certain complexes RΓfg(X, Z(n)) that are almost perfect, in the sense that their cohomology groups Hi (X, Z(n)) are finitely generated, vanish for i 0, and for i 0 they are finite 2-torsion (that is, killed by multiplication by 2). Let us introduce the following notion.