## These are the differentials of order $n$

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- by Dan Laksov and Anders Thorup PDF
- Trans. Amer. Math. Soc.
**351**(1999), 1293-1353 Request permission

## Abstract:

We answer P.-A. Meyer’s question “Qu’est ce qu’une différentielle d’ordre $n$?”. In fact, we present a general theory of higher order differentials based upon a construction of universal objects for higher order differentials. Applied to successive tangent spaces on a differentiable manifold, our theory gives the higher order differentials of Meyer as well as several new results on differentials on differentiable manifolds. In addition our approach gives a natural explanation of the quite mysterious multiplicative structure on higher order differentials observed by Meyer. Applied to iterations of the first order Kähler differentials our theory gives an algebra of higher order differentials for any smooth scheme. We also observe that much of the recent work on higher order osculation spaces of varieties fits well into the framework of our theory.## References

- A. Buium,
*Intersections in jet spaces and a conjecture of S. Lang*, Ann. of Math. (2)**136**(1992), no. 3, 557–567. MR**1189865**, DOI 10.2307/2946600 - Alexandru Buium,
*Geometry of differential polynomial functions. I. Algebraic groups*, Amer. J. Math.**115**(1993), no. 6, 1385–1444. MR**1254738**, DOI 10.2307/2374970 - Alexandru Buium,
*Geometry of differential polynomial functions. II. Algebraic curves*, Amer. J. Math.**116**(1994), no. 4, 785–818. MR**1287940**, DOI 10.2307/2375002 - Alexandru Buium,
*Geometry of differential polynomial functions. III. Moduli spaces*, Amer. J. Math.**117**(1995), no. 1, 1–73. MR**1314457**, DOI 10.2307/2375035 - Susan Jane Colley and Gary Kennedy,
*A higher-order contact formula for plane curves*, Comm. Algebra**19**(1991), no. 2, 479–508. MR**1100358**, DOI 10.1080/00927879108824150 - Susan Jane Colley and Gary Kennedy,
*The enumeration of simultaneous higher-order contacts between plane curves*, Compositio Math.**93**(1994), no. 2, 171–209. MR**1287696** - Alberto Collino,
*Evidence for a conjecture of Ellingsrud and Strømme on the Chow ring of $\textrm {Hilb}_d(\textbf {P}^2)$*, Illinois J. Math.**32**(1988), no. 2, 171–210. MR**945858** - J.-P. Demailly,
*Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials*, Lecture Notes, AMS Summer Research Institute Santa Cruz 1995, PSPM, vol. 62, part 2, 1997. - A. Grothendieck,
*Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV*, Inst. Hautes Études Sci. Publ. Math.**32**(1967), 361 (French). MR**238860** - Shigeru Iitaka,
*Symmetric forms and Weierstrass cycles*, Proc. Japan Acad. Ser. A Math. Sci.**54**(1978), no. 4, 101–103. MR**491688** - Shigeru Iitaka,
*Duality theorems for symmetric differential forms*, Proc. Japan Acad. Ser. A Math. Sci.**55**(1979), no. 2, 53–58. MR**528228** - Shigeru Iitaka,
*Symmetric forms and Weierstrass semigroups*, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978) Lecture Notes in Math., vol. 732, Springer, Berlin, 1979, pp. 157–170. MR**555697** - Shigeru Iitaka,
*Weierstrass forms associated with linear systems*, Adv. in Math.**33**(1979), no. 1, 14–30. MR**540635**, DOI 10.1016/S0001-8708(79)80008-0 - Joseph Johnson,
*Order for systems of differential equations and a generalization of the notion of differential ring*, J. Algebra**78**(1982), no. 1, 91–119. MR**677713**, DOI 10.1016/0021-8693(82)90103-X - Joseph Johnson,
*Prolongations of integral domains*, J. Algebra**94**(1985), no. 1, 173–210. MR**789546**, DOI 10.1016/0021-8693(85)90209-1 - E. R. Kolchin,
*Differential algebra and algebraic groups*, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR**0568864** - E. R. Kolchin,
*Differential algebraic groups*, Pure and Applied Mathematics, vol. 114, Academic Press, Inc., Orlando, FL, 1985. MR**776230** - Dan Laksov and Anders Thorup,
*The Brill-Segre formula for families of curves*, Enumerative algebraic geometry (Copenhagen, 1989) Contemp. Math., vol. 123, Amer. Math. Soc., Providence, RI, 1991, pp. 131–148. MR**1143551**, DOI 10.1090/conm/123/1143551 - Dan Laksov and Anders Thorup,
*Weierstrass points and gap sequences for families of curves*, Ark. Mat.**32**(1994), no. 2, 393–422. MR**1318539**, DOI 10.1007/BF02559578 - Dan Laksov and Anders Thorup,
*Weierstrass points on schemes*, J. Reine Angew. Math.**460**(1995), 127–164. MR**1316575** - P.-A. Meyer,
*Formes differentielles d’ordre $n>1$*, Publication IRMA, Université Louis Pasteur, Strasbourg, 1979/80. - Paul-André Meyer,
*Qu’est ce qu’une différentielle d’ordre $n$?*, Exposition. Math.**7**(1989), no. 3, 249–264 (French, with English summary). MR**1007886** - Charles Hopkins,
*Rings with minimal condition for left ideals*, Ann. of Math. (2)**40**(1939), 712–730. MR**12**, DOI 10.2307/1968951 - Robert Speiser,
*Derived triangles and differential systems*, Projective geometry with applications, Lecture Notes in Pure and Appl. Math., vol. 166, Dekker, New York, 1994, pp. 97–109. MR**1302944**

## Additional Information

**Dan Laksov**- Affiliation: KTH, S–100 44 Stockholm, Sweden
- Email: laksov@math.kth.se
**Anders Thorup**- Affiliation: Matematisk Afdeling, Københavns Universitet, Universitetsparken 5, DK–2100 København Ø, Denmark
- Email: thorup@math.ku.dk
- Received by editor(s): January 30, 1997
- Additional Notes: The first author was partially supported by The Göran Gustafsson Foundation for Research in Natural Sciences and Medicine.

The second author was supported in part by the Danish Natural Science Research Council, grant 11–7428. - © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 1293-1353 - MSC (1991): Primary 13N05, 14F10; Secondary 16Sxx
- DOI: https://doi.org/10.1090/S0002-9947-99-02120-0
- MathSciNet review: 1458328