nLab germ-determined C-infinity ring

Definition

Definition

(germ-determined $C^\infty$-ideal)

For $M$ a smooth manifold, an ideal $I \,\subset\, C^\infty(M)$ in its C-infinity ring (smooth algebra) of smooth functions is called germ determined if for any $f \in C^\infty(M)$ such that the germ of $f$ at any point $x$ in the zero locus of $I$ in $M$ coincides with the germ of some element of $I$ at $x$, then $f$ itself belongs to $I$.

Equivalently, germ-determined ideals are precisely those ideals that are closed under (possibly infinite) sums of locally finite families. One can also take sums with coefficients in a partition of unity.

Definition

(germ-determined $C^\infty$-ring)

A finitely generated C-infinity ring of the form $C^\infty(M)/I$ for some smooth manifold $M$ and ideal $I$ is germ-determined if the ideal $I$ is a germ-determined ideal (Def. ).

Related concepts

• Dubuc topos

References

• Eduardo Dubuc, Sur les modèles de la géométrie différentielle synthétique, Cahiers de Topologie et Geometrie Differentielle Vol. XX-3

  (1979) (numdam:CTGDC_1979__20_3_231_0)

• Eduardo Dubuc, Schemas $C^\infty$, “Prepublications de la Universite de Montreal” 80-81 edited by G. Reyes (1980).

• Eduardo Dubuc, $C^\infty$ Schemes, American Journal of Mathematics, John Hopkins University, Vol. 103-4 (1981) (jstor:2374046)

• Eduardo Dubuc, Open Covers and Infinitary Operations in C-inf-rings, Cahiers de Topologie et Geometrie Differentielle Vol. XXII-3 (1981) (numdam:CTGDC_1981__22_3_287_0)

• Eduardo Dubuc, Integracion de campos vectoriales y Geometria Diferencial Sintetica, Proceedings of the VII Sem. Nac. de Mat., FAMAF, Univ. Nac. de Cordoba (1984). journal volume, pdf

• Eduardo Dubuc, Germ representability and Local integration of vector fields in a well adapted model of SDG, Aarhus Univ. Math. Inst. preprint series (1985/1986), published in Journal of Pure and Applied Algebra Vol. 64, (1990) (doi:10.1016/0022-4049(90)90152-8)

• Marta Bunge, Eduardo Dubuc, Archimedian Local C-inf-rings and Models of SDG, Cahiers de Topologie et Geometrie Differentielle Vol. XXVII-3 (1986). NUMDAM

• Marta Bunge, Eduardo Dubuc, Local concepts in SDG and germ representability, in D. Kueker et al. (ads), Mathematical Logic and Theoretical Computer Science, Lecture Notes in Pure and Applied Mathematics 106, Marcel Dekker (1987)