Graded Algebra

Since our approach to structures over valued fields relies in a fundamental way on filtrations and associated graded structures, our arguments often require information that is specific to graded modules. We collect in this chapter the basic definitions and results on graded algebras and modules that will be of constant use in subsequent chapters. In §2.1 we define graded rings, modules, and the gradings on their homomorphism groups and tensor products. For finite-dimensional semisimple graded algebras A we prove in §2.2 graded analogues to the classical Wedderburn Theorems. When A is graded simple we also prove graded versions of the Double Centralizer Theorem and Skolem–Noether Theorem. For A graded simple, its degree-0 component A 0 is semisimple, though often not simple. In §2.3 we relate the grade set Γ A to the structure of A 0 via the map $\theta_{\mathsf {A}}\colon \Gamma^{\times}_{\mathsf {A}}\to \operatorname {\mathcal {G}}(Z(A_{0})/(Z(\mathsf {A}))_{0})$ induced by inner automorphisms of homogeneous units. We also describe inertial graded algebras A, which are completely determined by A 0.