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Embeddings of fields into simple algebras: generalizations and applications. (English) Zbl 1270.11120
The classical theory of Brauer groups yields the result that in a central simple algebra \(A\) of dimension \(n^2\) over a field \(F\), an extension field \(E\) of \(F\) of degree \(n\) embeds as an \(F\)-algebra if and only if \(E\) splits \(A\). In this beautiful paper, the author derives general necessary and sufficient conditions for an \(F\)-algebra homomorphism to exist between two semisimple \(F\)-algebras. Following that, the author obtains conditions for the existence of an \(F\)-algebra homomorphism of semisimple \(F\)-algebras which is also an embedding. In addition, towards a generalization of the Skolem-Noether theorem, the author investigates the orbit set \(B^{\ast}/\operatorname{Hom}_F(A,B)\). It is shown that this orbit may be infinite and, a necessary and sufficient condition is given for the orbit to be finite. In the last section, applications to endomorphisms of abelian varieties are given as well as the special case of global fields is discussed vis-a-vis the obstruction to a local-global principle holding good.
These results are really a welcome addition to the literature on this subject and should be of interest to many mathematicians. All the results are deduced from the following general result which we state now.
Let \(F\) be an arbitrary field. Let \(A,B\) be finite-dimensional semisimple \(F\)-algebras. Realize \(B\) as \(\prod_{i=1}^r \text{End}_{\Delta_j}(V_j)\), where each \(\Delta_j\) is a division \(F\)-algebra and \(V_j\) is a right \(\Delta_j\)-module. Write \(A = \prod_{i=1}^s A_i\), a product of simple \(F\)-algebras. Then,
(i) The set \(\operatorname{Hom}_{F\text{-alg}}(A,B)\) is non-empty if, and only if, for each \(j \leq r\), there is a decomposition \(V_j = \oplus_{k=1}^s V_{jk}\) into \(\Delta_j\)-subspaces such that each \(V_{jk}\) has the structure of a \(\Delta_j \otimes_F A_k^{op}\)-module;
(i)’ more precisely, for each \(j\), write the maximal semisimple quotient \[ (\Delta_j \otimes_F A^{op})^{ss} = \prod_{k=1}^{t_j} M_{m_{jk}}(D_{jk}), \] a product of its simple factors. Then, the set \(\operatorname{Hom}_{F\text{-alg}}(A,B)\) is non-empty if, and only if, there are non-negative integers \(x_{jk}\) (\(k \leq t_j, j \leq r\)) such that
\(\sum_{k=1}^{t_j} x_{jk} =\dim_{\Delta_j}(V_j)\) for all \(j \leq r\); and

\(\frac{m_{jk}[D_{jk}:F]}{[\Delta_j:F]}\) divides \(x_{jk}\) for all \(j,k\);

(ii) the subset of embeddings of \(\operatorname{Hom}_{F\text{-alg}}(A,B)\) is non-empty if, and only if, in addition, for each \(k \leq s\), each direct sum \(\oplus_{j=1}^r V_{jk}\) is non-zero;
(ii)’ more precisely, for each \(j,i\), write the maximal semisimple quotient \[ (\Delta_j \otimes_F A_i^{op})^{ss} = \prod_{k=1}^{t_{ji}} M_{m_{jik}}(D_{jik}), \] a product of its simple factors. Then, the subset of embeddings in \(\operatorname{Hom}_{F\text{-alg}}(A,B)\) is non-empty if, and only if, there are non-negative integers \(x_{jik}\) (\(k \leq t_j, j \leq r\)) such that
\(\sum_{k=1}^{t_j} \sum_{i=1}^s x_{jik} = \dim_{\Delta_j}(V_j)\) for all \(j \leq r\);

\(\frac{m_{jik}[D_{jik}:F]}{[\Delta_j:F]}\) divides \(x_{jik}\) for all \(j,i,k\); and

\(\sum_{j,k} x_{jik}\) is positive for each \(i \leq s\).

MSC:
11R52 Quaternion and other division algebras: arithmetic, zeta functions
16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings
16K20 Finite-dimensional division rings
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