The essential dimension (abbreviated as es-dimension) of a functor \(\mathcal{F}:\text{Fields}\to\text{Sets}\) from the category of field extensions of a fixed base field \(F\) to the category of sets is defined as the least integer \(n\), such that every object \(\alpha\in\mathcal{F}(K)\) over a field extension \(K/F\) is defined over a subextension \(K _ 0/F\) of transcendence degree at most \(n.\) For an algebraic group \(G\), \(\mathrm{ed}(G)\) is defined as the es-dimension of the Galois cohomology functor \(H^1(-, G):\text{Fields} _ F\to\text{Sets}\), defined by the rule \(K\to H^1(K,G)\). It measures the complexity of \(G\)-torsors, up-to an isomorphism, hence, of isomorphism classes of central simple algebras (CSA-s (for projective linear groups), quadratic forms (for orthogonal groups), étale algebras (for symmetric groups), see, e.g., A. S. Merkurjev’s survey in [Transform. Groups 18, No. 2, 415–481 (2013; Zbl 1278.14066)].
The paper under review deals with the study of the es-dimensions of separable subalgebras of CSA-s. By a separable algebra over a field \(E\), we mean a finite-dimensional \(E\)-algebra \(T\) with \(T _ E':=T\otimes _ EE'\) a semisimple \(E'\)-algebra, for every field extension \(E'/E\). The class of these algebras contains both CSA-s and étale algebras. The author considers separable algebras over extensions \(K/F\), which embed in the \(K\)-algebra \(A _ K\), where \(A\) is a given CSA of prime power degree \(\deg(A)\) over a fixed arbitrary field \(F\). He studies the functor \(\mathbf{Forms}(B):\text{Fields}\to\text{Sets}\) that takes a field extension \(K/F\) to the set of isomorphism classes of \(K\)-algebras \(B'\) which become isomorphic over a separable closure \(F_ {\text{sep}}\) to a given separable \(K\)-algebra \(B\) included in \(A\) as an \(F\)-subalgebra. The author restricts to the case where \(B\) has type \(\theta_B\) in \(A\) equal to the multiset \([(d, r),\dots,(d,r)]\) (\(m\) times), for some \(r,d,m\geq 1\). This means that \(B\) and the centralizer \(C=C_A(B)\) satisfy the following over \(F_{\text{sep}}:B_{\text{sep}}\) and \(C_{\text{sep}}\) are \(F_{\text{sep}}\)-isomorphic to the direct product of \(m\) isomorphic copies of the full matrix rings \(M_d(F_{\text{sep}})\) and \(M _ r(F_{\text{sep}})\), respectively. The paper concentrates on the es-dimension \(\mathrm{ed}(\mathbf{Forms}_ A^{\theta}(B))\), where \(\mathbf{Forms}_ A^{\theta}(B)\) is the subfunctor of \(\mathbf{Forms}(B)\) formed by those isomorphism classes \(B'\) of forms of \(B\), which admit an embedding in \(A\) of type \(\theta _ B\). Its main result shows that if the Schur index \(\operatorname{ind}(A)\) is at most equal to \(r/d\), then \(\mathbf{Forms}(B)=\mathbf{Forms}_ A^{\theta}(B)\), and the functors \(\mathbf{Forms}(B)\), \(H^1(-, (\mathrm{PGL}_ d)^m\rtimes S _ m)\) and \(H^1(-,G)\) are isomorphic, where \(G\) is the normalizer of \(\mathrm{GL}_ 1(B)\) in \(\mathrm{GL}_ 1(A)\). When \(\operatorname{ind}(A) > r/d\), it determines the exact value of \(\mathrm{ed}(\mathbf{Forms}_ A^{\theta}(B))\) except possibly in the case where \(\operatorname{ind}(A)=2\) and \(d=r>1\). Information concerning the excluded case is derived as well.