# Quadratic mappings and Clifford algebras by Jacques Helmstetter, Artibano Micali

By Jacques Helmstetter, Artibano Micali

After a classical presentation of quadratic mappings and Clifford algebras over arbitrary earrings (commutative, associative, with unit), different themes contain extra unique equipment: inside multiplications enable a good remedy of deformations of Clifford algebras; the family among automorphisms of quadratic types and Clifford algebras are in accordance with the concept that of the Lipschitz monoid, from which a number of teams are derived; and the Cartan-Chevalley concept of hyperbolic areas turns into even more common, distinctive and effective.

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K. Proof. 7) we derive (b)⇔(a)⇒(c), and it is obvious that (d)⇔(e), because for a subset of elements s of K the following three assertions are equivalent: – it is contained in no maximal ideal m; – it generates K as an ideal; – it contains a ﬁnite subset generating K as an ideal. Therefore it suﬃces to prove (c)⇒(d) and (e)⇒(a). Let us prove (c)⇒(d). Let m be a maximal ideal, r the rank of P at m, and (x1 , x2 , . . , xn ) a ﬁnite family of generators of P such that (x1 /1, x2 /1, . . , xr /1) is a minimal family of generators of Pm .

2) should be a suﬃcient model for all the others. Observe that the L-algebras TL (L ⊗K M ) and L ⊗K TK (M ) are generated by the elements which in each algebra are written 1L ⊗ x (with x ∈ M ). The mapping which maps every λ ⊗ x in L ⊗K M to the corresponding λ ⊗ x in L ⊗K TK (M ), is L-linear, and therefore extends to a morphism of L-algebras TL (L ⊗K M ) → L ⊗K TK (M ). 9. Changes of basic rings 19 of K-algebras TK (M ) → TL (L ⊗K M ), and by combining it with the canonical morphism L → TL (L⊗K M ), the image of which lies in the center of TL (L⊗K M ), we get a morphism of K-algebras L ⊗K TK (M ) → TL (L ⊗K M ).

Later it shall be necessary to know the prime ideals of S −1 K. As explained above for the submodules of any K-module M , to each ideal a of K corresponds an ideal S −1 a of S −1 K; it is the ideal generated by f (a). Conversely with every ideal b of S −1 K we associate the ideal f −1 (b) of K; it is clear that f −1 (b) ∩ S = ∅ whenever b = S −1 K. The proof of the following lemma is left to the reader. 9) Lemma. The mapping q −→ f −1 (q) is a bijection from the set of prime ideals q of S −1 K onto the set of prime ideals p of K such that p ∩ S is empty; the converse bijection is p −→ S −1 p.