no. 4

Algebraically constructible chains
[Chaî nes algébriquement constructibles]
Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 939-994.

On construit pour une variété algébrique réelle (plus généralement, pour un schéma essentiellement de type fini sur un corps de caractéristique 0) des complexes de chaînes algébriquement constructibles et k-algébriquement constructibles, dont on étudie la fonctorialité et dont on calcule l’homologie pour les espaces affines et projectifs. Puis on montre que les cycles lagrangiens algébriquement constructibles du fibré cotangent sont exactement les cycles caractéristiques des fonctions algébriquement constructibles.

We construct for a real algebraic variety (or more generally for a scheme essentially of finite type over a field of characteristic 0) complexes of algebraically and k- algebraically constructible chains. We study their functoriality and compute their homologies for affine and projective spaces. Then we show that the lagrangian algebraically constructible cycles of the cotangent bundle are exactly the characteristic cycles of the algebraically constructible functions.

DOI : https://doi.org/10.5802/aif.1841
Classification : 14P25,  14F43
Mots clés : algébriquement constructible, homologie, cycle caractéristique 
@article{AIF_2001__51_4_939_0,
     author = {Pennaneac'h, H\'el\`ene},
     title = {Algebraically constructible chains},
     journal = {Annales de l'Institut Fourier},
     pages = {939--994},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {51},
     number = {4},
     year = {2001},
     doi = {10.5802/aif.1841},
     zbl = {1036.14029},
     mrnumber = {1849211},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1841/}
}
Pennaneac'h, Hélène. Algebraically constructible chains. Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 939-994. doi : 10.5802/aif.1841. http://archive.numdam.org/articles/10.5802/aif.1841/

[B] I. Bonnard Un critère pour reconnaître les fonctions algébriquement constructibles, J. reine angew. Math., Volume 526 (2000), pp. 61-88 | Article | MR 1778301 | Zbl 0959.14035

[BB] E. Becker; L. Bröcker On the description of the reduced Witt ring, J. Alg., Volume 52 (1978), pp. 328-346 | Article | MR 506029 | Zbl 0396.10012

[BCR] J. Bochnak; M. Coste; M.-F. Roy Géométrie algébrique réelle, Ergebnisse der Math., 3 Folge, Vol. 12, Springer, 1987 | MR 949442 | Zbl 0633.14016

[DM] M.A. Dickmann; F. Miraglia On quadratic forms whose total signature is zero mod 2 n , Invent. Math., Volume 133 (1998) no. 2, pp. 243-278 | Article | MR 1632786 | Zbl 0908.11022

[G] C.G. Gibson; K. Wirthmüller; A.A. du Plessis; E. Looijenga Topological stability of smooth mappings, Lecture Notes in Mathematics, Vol. 552, Springer-Verlag, Berlin-New York, 1976 | MR 436203 | Zbl 0377.58006

[Ha] R. Hartshorne Algebraic Geometry, Graduate Texts in Math., Springer Verlag, 1977 | MR 463157 | Zbl 0367.14001

[KnS] M. Knebusch; C. Scheiderer Einführung in die reelle Algebra, Vieweg Studium: Aufbaukurs Mathematik, 63, Vieweg und Sohn, Braunschweig, 1989 | MR 1029278 | Zbl 0732.12001

[KS] M. Kashiwara; P. Schapira Sheaves on manifolds, Springer-Verlag, Berlin, 1990 | MR 1074006 | Zbl 0709.18001

[Ku] E. Kunz Kähler Differentials, Advanced Lectures in Math., Friedr. Vieweg and Sohn, Braunschweig, 1986 | MR 864975 | Zbl 0587.13014

[MP] C. McCrory; A. Parusiński Algebraically constructible functions, Ann. Scient. École Norm. Sup. (4), Volume 30 (1997), pp. 527-552 | Numdam | MR 1456244 | Zbl 0913.14018

[R] M. Rost Chow groups with coefficients, Docu. Math., Volume 1 (1996), pp. 319-383 | MR 1418952 | Zbl 0864.14002

[S] M. Schmid Wittringhomologie Thesis University of Regensburg (available on the web page of M. Rost), http://www.math.ohio-state.edu/~rost/papers.html

[Sch] C. Scheiderer Purity theorems for real spectra and applications, Real analytic and algebraic geometry (Trento, 1992) (1995), pp. 229-250 | Zbl 0840.14035

[SV] W. Schmid; K. Vilonen Characteristic cycles of constructible sheaves, Invent. Math., Volume 124 (1996), pp. 451-502 | Article | MR 1369425 | Zbl 0851.32011

[ZS] O. Zariski; P. Samuel Commutative Algebra, van Nostrand, Princeton-London-Toronto, 1958 | MR 90581 | Zbl 0081.26501