Descent Theory and Cohomology
Many constructions in the theory of hermitian and, specially, quadratic spaces over rings use the technique of faithfully flat descent. In particular the notion of twisted forms plays an important role. In Chapter III we develop this technique and the corresponding cohomological tools. Applications are given to discriminant modules, quadratic algebras, Azumaya algebras and projective modules of rank one over quaternion algebras. Finally we study involutions on Azumaya algebras and, in the last section, we introduce twisted forms of the pfaffian of an alternating matrix. These twisted forms are quite useful in the study of forms of rank 5 and 6.
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