An invitation to dyslectic geometry (Q1210083)

``Dyslectic geometry is a ``\(G\)-commutative'' geometry generalizing the physicists' \(\mathbb{Z}/2\mathbb{Z}\)-commutative supergeometry to the case of an arbitrary group \(G\)\dots'' After defining the notion of \(G\)-commutative \(G\)-algebra and sketching the general theory of such objects, the author provides some examples (Connes' \(C^ \ast\)-algebra associated with the Kronecker flow on the torus, certain algebras constructed from non-Archimedean local fields, superalgebras, etc.). Then the author undertakes the construction of a theory of \(G\)-schemes: a \(\text{Spec}\) functor is introduced, together with the notion of locally ringed \(G\)-space. Such objects are shown to admit an infinitesimal theory of differentials and a global cohomology theory. This interesting paper proposes an avenue for the development of a noncommutative geometry which is closer in spirit to standard algebraic geometry than Connes' approach.