Given and a module N we put The right

Given and a -module N, we put The right hand side makes sense because we regard g as an CHZ868 receptor of and meanwhile . It is also a skew group ring, identified with via the following equation It follows that becomes a -module, with Analogous to the situation of group representations, the underlying space of N admits a -module structure via the linear isomorphism . We shall denote it by .
In the proof, we actually showed that is a free right -module. It means that is exact. Let M be a -module and H be a subgroup of G. If the -module epimorphism is split, then M is said to be projective relative to , or relatively -projective. In [5], [4], M was called relatively H-projective, for the obvious reasons. We use the more cumbersome terminology because, again, we want to be consistent with the rest of the paper. Dade proved several equivalent conditions for M to be relatively -projective, which no doubt are parallel to the case of group representations. (Using a relative trace map of Miyashita, defined entirely analogous to the group case, he also obtained a Higman's criterion.)
Based on the previous theorem, in [5], [4], H was called a “vertex” of the indecomposable -module M. However, we only go down to the subalgebra , not RH. We shall see later on that this is a reason why Dade's construction may be improved for . To study the block theory of a fully group-graded algebra, Boisen introduced the concept of a diagonal subalgebra. We also recall it for skew group algebras. Given , we set , and . Note that may be identified with the product group , and there is a group isomorphism . The group acts on via , and the “diagonal subalgebra” of is defined to be As an example, regarding kG as a fully G-graded algebra, we have . Assume R is a field of characteristic . Boisen proved that an indecomposable summand B (a block) of the -module A is relatively -projective, for a minimal p-subgroup . In light of this, he called H a “defect group” of B. Analogous to the group case, he subsequently defined a generalized Brauer correspondence and established a generalized Brauer's First Main Theorem. As in Dade's treatment, his “defect” is also too big when .
Transporter categories and their algebras We shall study a special class of skew group algebras, namely the transporter category algebras , because has “local structure”. Like a group algebra or an incidence algebra, possesses a canonical base, which is the morphism set of the transporter category . This basis has an intrinsic structure and is crucial to our theory. Particularly it allows us to introduce transporter subcategories of , based on which we will be able to discuss the interactions between representations of and those of .
Vertices and sources In Section 2, we briefly explained, given a skew group algebra , how Dade conceived a theory of vertices and sources, through comparing it with various subalgebras , for . When with left G-multiplication, is Morita equivalent to kH. If we take the trivial module -mod (corresponding to -mod), a “vertex” of , in the sense of Dade, would be T, a Sylow p-subgroup of H. However, it actually means that is projective relative to , which is not Morita equivalent to kT, and thus does not match the classical theory for group algebras. See Example 4.24 for more details. We shall fix the problem and get a sharpened definition.
Block theory Boisen [4] studied the block theory of fully group-graded algebras. In particular, based on Dade's work, he defined “defect groups” of blocks, and established a generalized Brauer's First Main Theorem. These constructions and results certainly are valid for transporter category algebras. However, Boisen's “defects” are not truly subgroups, and they are too big in the case of transporter category algebras. We shall improve a few of the existing results, and then propose some entirely new constructions and theorems. Especially, for a transporter category algebra, we can talk about defect transporter categories of its blocks. Since there is a trivial representation , we also have a notion of the principal block of a transporter category algebra.