On extending \(\mathcal A\) -modules through the coefficients (Q1943563)

Ambient space in this paper is the category of free \(\mathcal A\)-modules, where \(\mathcal A\) is an \(\mathbb R\)-algebra sheaf. The authors model their results on free \(\mathcal A\)-modules on the results in the classical case. Complexification \(\mathcal A_\mathbb C\) of free \(\mathcal A\)-modules is the process of obtaining new free \(\mathcal A\)-modules by enlarging the \(\mathbb R\)-algebra sheaf \(\mathcal A\) to a \(\mathbb C\)-algebra sheaf \(\mathcal A_\mathbb C=\mathbb C\otimes_\mathbb R \mathcal A\). For an \(\mathcal A\)-module \(\mathcal E\) , the complete presheaf of sections of \(\mathcal E\) (denoted by \(\Gamma(\mathcal E)\)) is a complete \(\Gamma(\mathcal A)\)-presheaf. Some of the results are as follows: Theorem 1.1: Let \(\mathcal A\) be an algebra sheaf, \(\mathcal B\) a subalgebra sheaf in \(\mathcal A\), \(\mathcal E\) a free \(\mathcal B\)-module and let \(\vartheta\) be the canonical \(\Gamma(\mathcal B)\)-morphism \(\vartheta=1\otimes id_{\Gamma(\mathcal E)}\). Then, the ordered pair \((\Gamma(\mathcal E_\mathcal A), \vartheta)\) has the following universal property: For every pair \((\Gamma(\mathcal F),\nu)\), where \(\mathcal F\) is an \(\mathcal A\)-module and \(\nu\) is a \(\Gamma(\mathcal B)\) morphism of the \(\Gamma(\mathcal B)\) presheaf \(\Gamma(\mathcal E)\) into the underlying \(\Gamma(\mathcal B)\)-presheaf of \(\Gamma(\mathcal F)\), there is a uniquely defined \(\Gamma(\mathcal A)\) morphism \(\tilde\nu\) sending \(\Gamma(\mathcal E_\mathcal A)\) into \(\Gamma(\mathcal F)\), such that \(\nu=\tilde\nu\circ\vartheta\). Theorem 2.1: Let \(\mathcal A\) be an ordered nonzero-nilpotent free \(\mathbb R\)-algebra sheaf, and \(\mathcal E\) a free \(\mathcal E\)-module. Then, \(\mathcal E \) carries an \(\mathcal A_\mathbb C\)-module structure if and only if, there exists a \(J\in\)End\(_\mathcal A\mathcal E\), such that \(J^2=-1\). Theorem 2.2: The sheaf \(\mathcal S\) of sets of almost complex structures on \((_\mathbb R\mathcal A)^{2n}\) is such that, for every open \(U\subseteq X\), \(S(U)\) consists of all \(A^{-1}JA\), where \(A\in \mathcal G\mathcal L(2n,_\mathbb R\mathcal A)(U)\) and \(J\) is the standard almost complex structure of \((_\mathbb R\mathcal A)^{2n}(U)\). That is, the same is determined by the quotient sheaf \(\mathcal G\mathcal L(2n,_\mathbb R\mathcal A)/\mathcal G\mathcal L(n,\mathcal A)\).