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Automorphisms of products of Witt rings of local type. (English) Zbl 1043.11038
This article describes all Harrison automorphisms of Witt rings that are products of Witt rings of local type, by studying the $$Q$$-automorphisms on the associated quaternionic pairings. The Witt rings under consideration here are abstract Witt rings in the sense of Marshall, defined as a pair $$(R,G)$$ where $$R$$ is a commutative ring with $$1$$ and $$G$$ is a subgroup of the multiplicative group $$R^*$$ of exponent $$2$$ and containing $$-1$$, which generates $$R$$ additively, and satisfying certain additional axioms. A Harrison automorphism of a Witt ring $$W = (R,G)$$ is a ring automorphism of $$R$$ that sends $$G$$ to $$G$$. There is a natural equivalence of categories between the category of Witt rings and the category of quaternionic structures or $$Q$$-structures, which are triplets $$(G,Q,q)$$ where $$G$$ is an elementary $$2$$-group with distinguished $$-1$$, $$Q$$ is a pointed set, and $$q:G \times G \to Q$$ is a surjective mapping satisfying certain conditions.
If $$(G,Q,q)$$ is the $$Q$$-structure of a Witt ring $$W = (R,G)$$, then there is a canonical isomorphism between the group of $$Q$$-automorphisms of $$(G,Q,q)$$ and the group $$\text{Aut}_H(W)$$ of Harrison automorphisms of $$W$$. Moreover, when $$(G,Q,q)$$ is a $$Q$$-structure of local type, then it can be viewed as a bilinear space $$(G,q)$$ over the field $$F_2$$ of two elements, and its group of automorphisms is a group of automorphisms of the orthogonal space $$(G,q)$$ over $$F_2$$. The main result of the paper is the following:
Theorem: Let $$W$$ be a finite product of Witt rings of local type. Divide these Witt rings of local type into isomorphism classes $$C_i, 1 \leq i \leq m$$ of cardinality $$k_i$$, and choose one representative $$W_i$$ of each class. Then $\text{Aut}_H(W) \cong \prod_{i=1}^m (\text{Aut}_H(W_i))^{k_i} \ltimes S_{k_i}.$

MSC:
 1.1e+82 Algebraic theory of quadratic forms; Witt groups and rings
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References:
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