Titlebook: Value Functions on Simple Algebras, and Associated Graded Rings; Jean-Pierre Tignol,Adrian R. Wadsworth Book 2015 Springer International P

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Graded and Valued Field Extensions,sion of graded fields. Henselian fields are shown to satisfy a tame lifting property from graded field extensions, generalizing the inertial lifting property. In §., we lay the groundwork for the subsequent developments by an independent study of graded fields, their algebraic extensions and their Galois theory.

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Valuations on Division Rings, we define valuations and describe the associated structures familiar from commutative valuation theory: the valuation ring ., its unique maximal left and maximal right ideal ., the residue division algebra ., and the value group Γ.. We also describe an important and distinctively noncommutative fea

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Graded Algebra,equire 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 §. we define graded rings, modules, and the gradings on their homomorphism groups and tensor prod

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Value Functions,ld associated graded structures that are graded vector spaces and graded algebras. We emphasize value functions whose associated graded structures have good properties: those called . on vector spaces and . on algebras. The use of gauges is a key aspect of our approach to valuation theory on divisio

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Graded and Valued Field Extensions,valued field is an enhanced version of the residue field, inasmuch as it encapsulates information about the value group in addition to the residue field. It thus captures much of the structure of the field, particularly in the Henselian case. This point is made clear in §., where we show that—when t