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  • The main results whose proofs depend on the

    2022-05-18

    The main results, whose proofs depend on the explicit calculation of , include To set our work into the historical context, we note that the transporter category algebras are skew group algebras, and thus are fully group-graded algebras. This work is partially motivated by the papers on fully group-graded algebras by Boisen [4], Dade [5], [6], [7], and Miyashita [8] (the latter in the context of G-Galois theory). Especially, Dade conceived a Sitagliptin phosphate monohydrate of vertices and sources (for fully group-graded algebras). However, his “vertices” seem to be too big, see Example 4.15, Example 4.24. Same problem occurs in Boisen's definition of a “defect” of a block, because a “defect” is a “vertex”, in the sense of Dade, of some module. We shall propose a sharpened definition of a vertex, incorporating our earlier work on general EI category algebras [10], and prove it is appropriate. For the reader's convenience, some key constructions and results, from the previously mentioned papers, are quoted here. The approach in this paper is mostly parallel to the standard one for group representations. However the extra G-poset structure does require more than mere technicality. The paper is organized as follows. In Section 2, we recall relevant results for fully group-graded algebras. Then we examine local structures of transporter categories in Section 3. Subsequently the Kan extensions for investigating representations will be thoroughly discussed from the beginning of Section 4. A generalized theory of vertices and sources will be given. Finally in Section 5, we study the block theory of transporter category algebras.
    Results from fully group-graded algebras In the present paper, we want to develop modular representation theory of transporter category algebras. Some known results on fully group-graded algebras of Boisen [4], Dade [5], [6], [7], and Miyashita [8], will specialize to our situation and they will pave the way towards our key constructions. We shall quote these results mainly for skew group algebras. Some proof are given if they are needed in our presentation. Let R Sitagliptin phosphate monohydrate be a commutative ring with identity. Suppose G is a group and A is a G-graded R-ring. It means that, as R-modules, we have satisfying . If A meets the extra condition that , then we say A is fully G-graded. Suppose H is a subgroup of G. We may define a subalgebra . Particularly becomes a subalgebra. Suppose S is an R-ring that admits a G-action. We say S has a G-action, if there exists a group homomorphism . Under the circumstance, we also call S a G-ring. We usually denote the G-action by for all and . Then we may continue to define the skew group ring. As an R-module, it is simply . For convenience, we write , instead of , for an element in the skew group ring. The multiplication is determined by , for and . This ring contains subrings and for some . We may wish to take a larger ring fixed by G so that . We assume S is free as an R-module. For the sake of simplicity, for each and , we shall write and as elements of , when there is no confusion. The skew group ring is fully G-graded, if we put for each . Here we shall mainly recall constructions and results by Dade [5], [6] and Boisen [4]. For future applications, we will only state known results from [5], [6], [4] in the special forms for skew group algebras. We also note that Reiten and Riedtmann [9] studied the representation theory of skew group algebras over , the complex numbers. See [3] for another presentation. If , we have an inclusion , and thus the induction and restriction For instance . In [5], [4], these two functors are denoted by symbols and since S is unchanged and it matches the special case of groups. We refrain from using the latter in order to be consistent throughout this paper. In general, we have a decomposition and the -modules structure is obtained by a “twisted permutation” of summands if for some . Parallel to this, if is a right -module, then the induced right -module may be written as It is a bit surprising, but the reasonable right -action is for all and . This difference attributes to the fact that usually .