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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

These are the differentials of order $n$


Authors: Dan Laksov and Anders Thorup
Journal: Trans. Amer. Math. Soc. 351 (1999), 1293-1353
MSC (1991): Primary 13N05, 14F10; Secondary 16Sxx
MathSciNet review: 1458328
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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.


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Additional Information

Dan Laksov
Affiliation: KTH, S–100\thinspace44 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

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02120-0
PII: S 0002-9947(99)02120-0
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.
Article copyright: © Copyright 1999 American Mathematical Society