Sections of a differential spectrum and factorization-free computations

We construct sections of a differential spectrum using only localization and projective limits. For this purpose we introduce a special form of a multiplicative system generated by one differential polynomial and call it $D$-localization. Owing to this technique one can construct sections of a differential spectrum of a differential ring $\mathcal R$ without computation of $\operatorname{diffspec}\mathcal R$. We compare our construction with Kovacic's structure sheaf and with the results obtained by Keigher. We show how to compute sections of factor-rings of rings of differential polynomials. All computations in this paper are factorization-free.