The author describes the category of motives \(DM_{\text{gm}}^{\text{eff}}\) of V. Voevodsky [in: Cycles, transfers, and motivic homology theories. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 143, 188–238 (2000; Zbl 1019.14009)] as well as its differential graded enhancement. Let \(DM^{s}\) be the full triangulated subcategory of the category \(DM_{\text{gm}}^{\text{eff}}\) generated by motives of smooth varieties. It is proved that for any motivic complex \(M\) there exists a quasi-isomorphic complex \(M^{\prime}\), unique up to homotopy, constructed by means of the Suslin complexes of smooth projective varieties. The author introduces the category \(\mathfrak H\) of twisted complexes over a differential graded category whose objects are cubical Suslin complexes. The equivalence \(m: {\mathfrak H} \rightarrow DM^{s}\) is also constructed. The description of \(DM^{s}\) provides an enhancement of this category in the sense of A. I. Bondal and M. M. Kapranov [Math. USSR, Sb. 70, No. 1, 93–107 (1991); translation from Mat. Sb. 181, No. 5, 669–683 (1990; Zbl 0729.18008)]. The category \(\mathfrak H\) allows one to describe subcategories generated by fixed set of objects. The localizations of \(\mathfrak H\) can be explicitely described.
In the first section the author recalls some basic notation from [Zbl 1019.14009] and gives the description and properties of Suslin complexes. In section 2 the formalism of differential graded categories, twisted complexes and the construction of the category \(\mathfrak H\) as well as a functor \(h\) from \(\mathfrak H\) into the homotopy category of complexes of Nisnevich sheaves with transfers are given. In section 3 the author proves that \(m: {\mathfrak H} \rightarrow DM^{s}\) is an equivalence. Section 4 contains the proof that Voevodsky’s \(DM_{\text{gm}}{\mathbb Q}\) is anti-isomorphic to Hanamura’s \({\mathcal D}(k).\) In section 5 some properties of cubical Suslin complexes are verified. In section 6 the author defines and studies the truncation functors. In section 7 the realizations of the category of motives and their connections with weight filtrations are studied. Section 8 is devoted to the localizations of the category \(\mathfrak H\).