Automorphisms of products of Witt rings of local type.

*(English)*Zbl 1043.11038This 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 \[ \text{Aut}_H(W) \cong \prod_{i=1}^m (\text{Aut}_H(W_i))^{k_i} \ltimes S_{k_i}. \]

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 \[ \text{Aut}_H(W) \cong \prod_{i=1}^m (\text{Aut}_H(W_i))^{k_i} \ltimes S_{k_i}. \]

Reviewer: Tara L. Smith (Cincinnati)

##### MSC:

11E81 | Algebraic theory of quadratic forms; Witt groups and rings |

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\textit{M. Stȩpień}, Acta Math. Inform. Univ. Ostrav. 10, No. 1, 125--131 (2002; Zbl 1043.11038)

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##### References:

[1] | C. M. Cordes: The Witt Group and the Equivalence of Fields with Respect to Quadratic Forms. J. of Algebra 26 (1973), pp. 400-421. · Zbl 0288.12101 · doi:10.1016/0021-8693(73)90002-1 |

[2] | M. Marshall: Abstract With Rings. Queen’s Papers in Pure and Applied Math. 57, Queen’s University, Kingston, Ontario (1980). |

[3] | K. Szymiczek: Bilinear Algebra: An Introduction to the Algebraic Theory of Quadratic Forms. Algebra, Logic and Applications Series Vol. 7, Gordon and Breach Science Publishers 1997. · Zbl 0890.11011 |

[4] | A. Wesolowski: Automorphisms of forms of higher degrees. unpublished Ph. D. dissertation, Silesian University, Katowice (1998), in polish. |

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