ABSTRACT
Symmetric monoidal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Closed monoidal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Closed monoidal categories on chain complexes over a ring . . . . . 65 Tensor product of chain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Bar tensor product of chain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Derived functors of ´b´ and Homp´,´q . . . . . . . . . . . . . . . . . . . . . . 77 Torsion functors and preservation of direct limits . . . . . . . . . . . . . . . 80 Suspension of chain complexes and extensions . . . . . . . . . . . . . . . . . . 81 Pontryagin complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Enriched adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.5 Flat chain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Flat chain complexes with respect to b . . . . . . . . . . . . . . . . . . . . . . . . . 92 Flat chain complexes with respect to b . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.6 Torsion functors and flat dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1 Introduction This chapter starts by presenting the notion of monoidal categories, as a
categorical generalizations of the notion of tensor products of modules over a ring. We give three examples of monoidal structures: one on the category of modules over a ring, and two on the category of chain complexes. We introduce the notion of flatness with respect to tensor products, and study certain properties of flat modules and flat complexes. As we will define two tensor products for chain complexes, we will deal with two types of flat complexes, the first one being the usual flat complexes, i.e., exact complexes with flat cycles, and the second one known as degreewise flat complexes, i.e., complexes with flat components. Last, we study the corresponding derived functors of the given examples of tensor products, known as torsion functors.
