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Quaternionic algebra and sheaves on the Riemann sphere. (English) Zbl 0921.14003
The author gives a thorough sheaf-theoretic interpretation of Joyce’s quaternionic algebra [see D. Joyce, Hypercomplex algebraic geometry, Q. J. Math., Oxf., II. Ser. 49, 129-162 (1998)]. The main result is the construction of a contravariant equivalence between the category of positive augmented $$\mathbb{H}$$-modules and that of regular sheaves on the Severi-Brauer variety, which is the real form of $$\mathbb{C} \mathbb{P}^1$$ defined by the antipodal map. Under this equivalence regular vector bundles correspond to stable augmented $$\mathbb{H}$$-modules.

##### MSC:
 14A22 Noncommutative algebraic geometry 16K20 Finite-dimensional division rings
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