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By Richard Vale


Abstract. For every ring R, we construct a ringed space NCSpec(R) and for every ring homomorphism R → S, a morphism of ringed spaces NCSpec(S) → NCSpec(R). We show that this gives a fully faithful contravariant functor from the category of rings to a category of ringed spaces. If R is a commutative ring, we show that Spec(R) embeds naturally in NCSpec(R) as a dense subspace. We then explain how the spaces NCSpec(R) may be glued, and study quasicoherent sheaves on them. As an example, we compute the category of quasicoherent sheaves on a space constructed from a skew-polynomial ring R by an analogue of the Proj construction. 1

Topics: category of ringed spaces. More precisely, there is a contravariant functor from the category of commutative rings to the category of ringed sp, which associates to a ring R its prime spectrum Spec(R) equipped with the structure sheaf O. This fu
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