On Brauer \(p\)-dimensions and index-exponent relations over finitely-generated field extensions.
Recall that if \(A\) is a central division algebra over a field \(E\), then its exponent (i.e. the order of its class \([A]\) in the Brauer group \(\text{Br}(E)\)) and its index form a Brauer pair \((\exp(A),\text{ind}(A))\), i.e. \(\exp(A)\) divides \(\text{ind}(A)\) and each prime dividing \(\text{ind}(A)\) also divides \(\exp(A)\). Let \(\mathbb P\) denote the set of all primes and let \(p\in\mathbb P\). The Brauer \(p\)-dimension \(\text{Brd}_p(E)\) of \(E\) is defined to be the least integer \(n\geq 0\) such that for each central simple algebra \(A\) over \(E\) of \(p\)-power exponent one has that \(\text{ind}(A)\leq\exp(A)^n\), provided such an integer exists. Otherwise, one defines \(\text{Brd}_p(E)=\infty\). The absolute Brauer \(p\)-dimension \(\text{abrd}_p(E)\) is the supremum of all \(\text{Brd}_p(F)\) where \(F\) runs over all finite separable extensions of \(E\). The (absolute) Brauer dimension of \(E\) is then defined to be \(\text{Brd}(E)=\sup\{\text{Brd}_p(E)\mid p\in\mathbb P\}\) (resp. \(\text{abrd}(E)=\sup\{\text{abrd}_p(E)\mid p\in\mathbb P\}\)). In the present paper, the author studies the behaviour of these invariants when passing from the field \(E\) to a finitely generated extension \(F\) over \(E\) of transcendence degree \(\kappa\geq 1\). The main results are as follows. If \(\text{abrd}_p(E)<\infty\) and if \(F/E\) is rational, then \(\text{Brd}_p(F)\geq\text{abrd}_p(E)+\kappa-1\). If \(\text{abrd}_p(E)=\infty\), then \(\text{Brd}_p(F)=\infty\), and to all integers \(n\geq m>0\) there exists a central division algebra \(D_{n,m}\) over \(F\) with \((\exp(D_{n,m}),\text{ind}(D_{n,m}))=(p^m,p^n)\). Suppose now that \(\text{char}(E)=p\). If \([E:E^p]=\infty\) then \(\text{Brd}_p(F)=\infty\), and if \([E:E^p]=p^\nu<\infty\) then \(\nu+\kappa-1\leq\text{Brd}_p(F)\leq\nu+\kappa\). As a consequence, it is shown that if \(\text{abrd}(E)=\infty\), then \(\text{Brd}(F)=\infty\) for every finitely generated purely transcendental extension. The proofs are heavily based on valuation theoretic arguments.
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