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Zbl 1043.11038
Stȩpień, Marcin
Automorphisms of products of Witt rings of local type. (English)
[J] Acta Math. Inform. Univ. Ostrav. 10, No. 1, 125-131 (2002). ISSN 1211-4774

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. \par 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: \par Theorem: Let $W$ be a finite product of Witt rings of local type. Divide these Witt rings of local type into isomorphism classes $C_i, 1 \le i \le m$ of cardinality $k_i$, and choose one representative $W_i$ of each class. Then $$\text {Aut}_H(W) \cong \prod_{i=1}^m (\text {Aut}_H(W_i))^{k_i} \ltimes S_{k_i}.$$
[Tara L. Smith (Cincinnati)]

MSC 2000:: *11E81 Algebraic theory of quadratic forms

Keywords: Witt ring; quaternionic structure; Harrison automorphisms

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