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Grothendieck groups of sesquilinear forms over a ring with involution. (English) Zbl 0616.57017
For any ring with unit R equipped with an involution, we consider the sets FP(R) and F(R) of isomorphism classes of unimodular sesquilinear forms defined on finitely generated projective, respectively free R- modules. These are monoids with respect to the orthogonal sum operation. We also define a natural notion of exactness for triples of elements of these sets. In the case of F(R) it has the following form: the triple \((B_ 1,B_ 2,B_ 3)\) of elements of F(R) is exact if there is a matrix X such that \(B_ 2\) is congruent to \(\left[\begin{matrix} B_ 1&0\\ X&B_ 3\end{matrix} \right]\).
Our aim is to compute the corresponding Grothendieck groups KF(R) and KFP(R). We prove that there is an exact sequence connecting KF(R), KFP(R) and a subgroup of the projective class group of the ring R. We denote by KF(R) the kernel of the rank map \(\rho\) : KF(R)\(\to {\mathbb{Z}}\). We show that KF(R) is naturally isomorphic to \(K_ 1(R)/NK_ 1(R)\), where \(NK_ 1(R)\) denotes the subgroup of \(K_ 1(R)\) of elements of the form \(X\cdot X^ *\), where \(X^ *\) is the transpose-\(conjugate\) of X.
We prove that the set \(\Sigma\) (R) of related stable equivalence classes of matrices over R is an abelian group with respect to block sum. The group \(\Sigma\) (R) turns out to be also a quotient of \(K_ 1(R)\). In the commutative case, \(\Sigma\) (R) can be naturally identified with \(SK_ 1(R)/NSK_ 1(R)\). The group \(\Sigma\) (R) depends on the way the transpose-\(conjugation\) acts on \(K_ 1(R)\), and using topological K-theory we give different instances of this action.
The fact that \(\Sigma\) (\({\mathbb{Z}})\) is trivial has a geometric interpretation in high-dimensional knot theory.
MSC:
57R67 Surgery obstructions, Wall groups
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
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