Witt's theorem in abstract geometric algebra

In an earlier paper of the author, a version of the Witt’s theorem was obtained within a specific subcategory of the category of A-modules: the full subcat-egory of convenient A-modules. A further investigation yields two more versions of the Witt’s theorem by revising the notion of convenient A-modules. For the first version, the A-bilinear form involved is either symmetric or antisymmetric, and the two isometric free sub-A-modules, the isometry between which may extend to an isom-etry of the non-isotropic convenient A-module concerned onto itself, are assumed pre-hyperbolic. On the other hand, for the second version, the A-bilinear form defined on the non-isotropic convenient A-module involved is set to be symmetric, and the two isometric free sub-A-modules, whose orthogonals are to be proved isometric, are assumed strongly non-isotropic and disjoint.http://www.springer.com/mathematics/journal/1158

Reflexivity of orthogonality in A-modules

In this paper, as part of a project initiated by A. Mallios consisting of exploring new horizons for Abstract Differential Geometry (`a la Mallios), [5, 6, 7, 8], such as those related to the classical symplectic geometry, we show that essential results pertaining to biorthogonality in pairings of vector spaces do hold for biorthogonality in pairings of A-modules. We single out that orthogonality is reflexive for orthogonally convenient pairings of free A-modules of finite rank, governed by non-degenerate A-morphisms, and where A is a PID (Corollary 3.8). For the rank formula (Corollary 3.3), the algebra sheaf A is assumed to be a PID. The rank formula relates the rank of an A-morphism and the rank of the kernel (sheaf) of the same A-morphism with the rank of the source free A-module of the A-morphism concerned.http://www.tandfonline.com/loi/tqma202015-12-30hb201

On the group sheaf of A-symplectomorphisms

This is a part of a further undertaking to affirm that most of classical module theory may be retrieved in the framework of Abstract Differential Geometry (`a la Mallios). More precisely, within this article, we study some defining basic concepts of symplectic geometry on free A-modules by focussing in particular on the group sheaf of A-symplectomorphisms, where A is assumed to be a torsion-free PID C-algebra sheaf. The main result arising hereby is that A-symplectomorphisms locally are products of symplectic transvections, which is a particularly well-behaved counterpart of the classical result.http://link.springer.com/journal/12175hb201

Clifford A-algebras of quadratic A-modules

A Clifford A-algebra of a quadratic A-module (E, q) is an associative and unital A-algebra (i.e. sheaf of A-algebras) associated with the quadratic ShSetX-morphism q, and satisfying a certain universal property. By introducing sheaves of sets of orthogonal bases (or simply sheaves of orthogonal bases), we show that with every Riemannian quadratic free A-module of finite rank, say, n, one can associate a Clifford free A-algebra of rank 2n. This “main” result is stated in Theorem 3.2.http://link.springer.com/journal/6hb2016Mathematics and Applied Mathematic

Universal problem for Kähler differentials in A-modules : non-commutative and commutative cases

Let A be an associative and unital K-algebra sheaf, where K is a commutative ring sheaf, and ε an (A, A)-bimodule, that is, a sheaf of (A, A)-bimodules. We construct an (A, A)-bimodulc which is K-isomorphic with the K-module D K (A, ε) of germs of K-derivations. A similar isomorphism is obtained, this time around with respect to A, between the K-module D K (A, ε) with the A-module Hom A (Ω K (A), ε). where A, in addition of being associative and unital, is assumed to be commutative, and Ω K (A) denotes the A-module of germs of Kähler differentials. Finally, we expound on functoriality of Kähler differentials.http://link.springer.com/journal/13226hb201

On the commutativity of the Clifford and "extension of scalars" functors

Please read abstract in the article.http://www.elsevier.com/locate/topolhb201

The symplectic Gram-Schmidt theorem and fundamental geometries for A\mathcal A-modules

summary:Like the classical Gram-Schmidt theorem for symplectic vector spaces, the sheaf-theoretic version (in which the coefficient algebra sheaf $\mathcal A$ is appropriately chosen) shows that symplectic $\mathcal A$-morphisms on free $\mathcal A$-modules of finite rank, defined on a topological space $X$, induce canonical bases (Theorem 1.1), called symplectic bases. Moreover (Theorem 2.1), if $(\mathcal {E}, \phi )$ is an $\mathcal A$-module (with respect to a $\mathbb C$-algebra sheaf $\mathcal A$ without zero divisors) equipped with an orthosymmetric $\mathcal A$-morphism, we show, like in the classical situation, that “componentwise” $\phi$ is either symmetric (the (local) geometry is orthogonal) or skew-symmetric (the (local) geometry is symplectic). Theorem 2.1 reduces to the classical case for any free $\mathcal A$-module of finite rank