Homotopy Theory of Differential Graded Modules and Adjoints of Restriction of Scalars

This study constructs two different model structures called projective and injective
model on the category of differential graded modules over a differential graded ring
and also provides an explicit description of fibrant and cofibrant objects for these
models. The constructions are based on the concept and properties of semi-projective
and semi-injective modules and other kinds of projectivity and injectivity in the
category of differential graded modules.
Also an analysis of behavior of functors; restriction, extension and co-extension of
scalars is given. Furthermore, some conditions under which an adjunction becomes
a Quillen pair and a Quillen pair becomes a Quillen equivalence are described. Additionally, a relationship between restriction and co-extension for compact Lie groups
is discovered.