Categories of Motives for Additive Categories. Ii

This is a continuation of the paper by the same author published in this journal, v. 19 (2007), no. 6. This paper is a continuation of the paper [4] and makes a whole with it. We retain the notation, the definitions, and the assumptions of [4], and we continue the numeration of sections, theorems and lemmas. The proofs of Theorems 2–4 in §6 of [4] were in fact missing; it was said only that one can argue as in [3]. Next, there were some inaccuracies in that section. For this reason, the first section of the present paper is again §6, and it replaces §6 of the first part. §6. Category Ď M In this section, the study of arbitrary objects of the category M will be reduced largely to the study of objects whose additive groups of the endomorphism rings are torsion-free. For this purpose we shall introduce a new category Ď M . We start with some comments on one of the notions introduced earlier, and we give a new definition. In §5 we defined homomorphisms that factor through the torsion, and we proved some important properties of such homomorphisms. But even the notion of a homomorphism of this sort depends on the structure of the groups of homomorphisms from arbitrary objects of M to infinite direct sums of l-periodic objects (such direct sums can exist in the category M). Nothing was said about these groups of homomorphisms, and they can be different for two categories, even if these categories are identical in all other respects. For example, it can happen that in our initial category no infinite direct sums of periodic objects are allowed; we can adjoin them to the category artificially, but the groups of homomorphisms in such infinite direct sums (which in general are not direct products!) can be defined in various ways. But not many properties of such homomorphisms were in fact used in our proofs; for this reason it is natural to consider not only homomorphisms that factor through the torsion, but also homomorphisms that could factor through the torsion for suitable groups of homomorphisms to periodic objects and could be described in terms of the initial category. Recall that any homomorphism that factors through the torsion is divisible modulo the periodic part of the group of homomorphisms (Lemma 5); moreover, if a homomorphism a : A → B factors through t(B), then, for any object C, in the group HomM (A,C) there exists the product of a with every homomorphism from t(B) to t(C) (the argument in §5 was based precisely on this property). Roughly speaking, we say that a homomorphism potentially factors through the torsion if it satisfies the above conditions. Before giving the precise definition, we recall some notation. For an object A of the category M , we denote by Al the l-periodic component of A and by i A l : Al → A, 2000 Mathematics Subject Classification. Primary 18E05.