Section of a differential spectrum and factorization-free computations. (Russian, English) Zbl 1073.12004

Ovchinnikov, A. I.

Fundam. Prikl. Mat. 9, No. 3, 133-144 (2003); translation in J. Math. Sci., New York 135, No. 5, 3355-3362 (2006).

The paper is concerned with the schemes over differential algebras. The notions of sub-sheafs and sheafs are introduced. Then the author constructs sections of a differential spectrum by using only localizations and projective limits. It is proved that for a factorial ring \(R\) the structural sub-sheaf is a sheaf of differential rings of \(\text{diffspec}(R)\) and proposed approach is equivalent to the approach of J. J. Kovacic [Trans. Am. Math. Soc. 335, No. 11, 4475–4522 (2003; Zbl 1036.12005)]. Also examples are provided which show that for non-factorial rings both aforesaid results in general do not hold. Results obtained give rise to the construction of sections of a differential spectrum of a differential ring \(R\) without computations of \(\text{diffspec}(R)\). The paper contains many interesting examples and open problems.

Reviewer: Alexander E. Guterman (Moskva)

MSC:

12H05

Differential algebra

Keywords:

differential algebras; differential schemes; spectrum

Citations:

Zbl 1036.12005