Generalized quadratic modules (Q1929711)

Let \(K\) be a commutative associative unital ring. \textit{T. Kanzaki} [Nagoya Math. J. 49, 127--141 (1973; Zbl 0238.13018)] introduced two categories of generalized quadratic modules over \(K\). The first category is related to the generalized Clifford algebras and the second category is related to extended Witt rings. In this article, the authors study the structures of the two categories and in most sections \(K\) is a field of characteristic \(2\). Section 2, 3 and 4 is related to the first catergory \(Quad_1(K)\), the objects are \((M, f, q)\) where \(M\) is a \(k\)-module, \(f\) is a linear form on \(M\) and \(q\) is a quadratic form on \(M\), the morphisms are \((\phi, g):(M,f,q)\rightarrow (M',f',q;)\) where \(\phi:M\rightarrow M'\) and \(g:M\rightarrow K\) are linear and that \[ f'\circ \phi=f-2g, \quad q'\circ \phi=q+fg-g^2. \] The authors give a general description of \(Quad_1(K)\) in Section 2 and prove a list of isomorphy theorems for generalized Clifford algebras where \(K\) is a field of characteristic 2 in Section 3 and 4. Section 5, 6, 7, 8 and 9 is related to the second category \(Quad_2(K)\), the objects are again \((M, f, q)\) and the morphisms are \((\phi, g)\) satisfying a different condition \[ f'\circ \phi=f-2g, \quad q'\circ \phi=q+2fg-2g^2. \] The authors give a general description of \(Quad_2(K)\) in Section 6, and prove some theorems when \(K\) is a field of characteristic 2 in Section 6,7,8 and 9. In particular, they study the invariants of the extended Witt rings. When \(K\) is of cardinality \(2^k\), the extended Witt ring can be determined by three invariants and be applied to solve a problem of enumeration.