Division, Adjoints, and Dualities of Bilinear Maps

Abstract

The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the adjoint category is not a module category but nevertheless it is suitably familiar. The universal properties have geometric perspectives. For example, products are orthogonal sums. The nonsingular bilinear maps are the simple bimaps with respect to nondegenerate adjoint-morphisms. That formalizes the understanding that the atoms of linear geometries are algebraic objects with no zero-divisors. Adjoint-isomorphism coincides with principal isotopism; hence, semifields can be studied within this framework.

Key Words:

• Adjoint
• Bilinear
• Duality
• Nonsingular

2000 Mathematics Subject Classification:

• Primary 15A63, 18A40
• Secondary 17A35

ACKNOWLEDGMENTS

Thanks to the referee for better reasoning, to D. Oury for helpful discussion, and to W. Kantor and D. Shapiro for remarks on semifields and nonsingular bimaps.

Notes

¹Here a form means a bimap B: U × V → W where W, is a cyclic (bi-)module; cf. Section 1.1.

Indeed, Adj(W) cannot be a Grothendieck category since then it would also be a co-Grothendieck; in such categories all objects are zero [9, p. 116].

We resist writing this product with ⊕ since the product in the category of homotopisms on bimaps has a superseding requirement that B: U × V → W and C: X × Y → Z have a product B ⊕ C: (U ⊕ X) × (V ⊕ Y) → (W ⊕ X).

Note that construction cannot be modified to show is injective in Mod _S .

Communicated by A. Elduque.