Quadratic forms I (Ex)

Latest revision as of 18:22, 29 May 2012

We follow [Lück2001, Definition 4.22] Let $<wikitex>; We follow {{citeD|Lück2001|Definition 4.22}} Let $P$ be a finitely generated free $\Lambda$ module, $\Lambda$ a ring with involution. Let $$ T \colon \textup{Hom}_{\Lambda} (P, P^*) \to \textup{Hom}_{\Lambda} (P, P^*), \quad f \mapsto f^* \circ e(P)$$ be the involution where $e(P) \colon P \to P^{**}$ is the canonical isomorphism given by evaluation. Define $$ Q_\epsilon(P) : = \textup{coker}(1 - \epsilon T) \colon \left( \textup{Hom}_{\Lambda} (P, P^*) \to \textup{Hom}_{\Lambda} (P, P^*) \right).$$ {{beginthm|Exercise}} Show that $\theta \in Q_\epsilon(P)$ defines a unique $\epsilon$-quadratic form on $P$ and that every such form arises in this way. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]P$ be a finitely generated free $\Lambda$ module, $\Lambda$ a ring with involution. Let

$$\displaystyle T \colon \textup{Hom}_{\Lambda} (P, P^*) \to \textup{Hom}_{\Lambda} (P, P^*), \quad f \mapsto f^* \circ e(P)$$

be the involution where $e(P) \colon P \to P^{**}$ is the canonical isomorphism given by evaluation. Define

$$\displaystyle Q_\epsilon(P) : = \textup{coker}(1 - \epsilon T) \colon \left( \textup{Hom}_{\Lambda} (P, P^*) \to \textup{Hom}_{\Lambda} (P, P^*) \right).$$