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Sheaves on subanalytic sites. (English) Zbl 1171.32002
Let \(k\) be a field and \(X\) be a real analytic manifold. The spaces of functions which are not defined by local properties do not define sheaves on \(X\), they define sheaves on a site associated to \(X\), the subanalytic site \(X_{sa}\).
M. Kashiwara and P. Schapira [Ind-sheaves. Astérisque. 271. Paris: Société Mathématique de France (2001; Zbl 0993.32009)] introduced the notion of ind-sheaves and defined the so-called six Grothendieck operations in this framework. Restriction to \(\mathbb{R}\)-constructible sheaves gives an equivalence between the category of ind \(\mathbb{R}\)-constructible sheaves on \(X\) and the categroy \(\text{Mod}(k_{X_{sa}})\) of sheaves on the subanalytic site associated to \(X\).
The paper gives a direct, self-contained construction of the six Grothendieck operations on \(\text{Mod}(k_{X_{sa}})\) without using the theory of ind-sheaves.

32B99 Local analytic geometry
14P05 Real algebraic sets
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
35C05 Solutions to PDEs in closed form
32S99 Complex singularities
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[1] E. BIERSTORNE - D. MILMANN, Semianalytic and subanalytic sets, Publ. I.H.E.S., 67 (1988), pp. 5-42. Zbl0674.32002 MR972342 · Zbl 0674.32002 · doi:10.1007/BF02699126 · numdam:PMIHES_1988__67__5_0 · eudml:104032
[2] S. I. GELFAND - YU. I. MANIN, Methods of homological algebra, SpringerVerlag, Berlin (1996). Zbl0855.18001 MR1438306 · Zbl 0855.18001
[3] R. GODEMENT, Topologie algébrique et théorie des faisceaux, Hermann, Paris (1958). Zbl0080.16201 MR102797 · Zbl 0080.16201
[4] M. KASHIWARA, The Riemann-Hilbert problem for holonomic systems, Publ. RIMS, Kyoto Univ., 20 (1984), pp. 319-365. Zbl0566.32023 MR743382 · Zbl 0566.32023 · doi:10.2977/prims/1195181610
[5] M. KASHIWARA - P. SCHAPIRA, Sheaves on manifolds, Grundlehren der Math. 292, Springer-Verlag, Berlin (1990). Zbl0709.18001 MR1074006 · Zbl 0709.18001
[6] M. KASHIWARA - P. SCHAPIRA, Moderate and formal cohomology associated with constructible sheaves, Mémoires Soc. Math. France, 64 (1996). Zbl0881.58060 MR1421293 · Zbl 0881.58060 · numdam:MSMF_1996_2_64__1_0 · eudml:94916
[7] M. KASHIWARA - P. SCHAPIRA, Ind-sheaves, Astérisque, 271 (2001). Zbl0993.32009 MR1827714 · Zbl 0993.32009
[8] M. KASHIWARA - P. SCHAPIRA, Categories and sheaves, Grundlehren der Math., 332, Springer-Verlag, Berlin (2005). Zbl1118.18001 MR2182076 · Zbl 1118.18001 · doi:10.1007/3-540-27950-4
[9] B. KELLER, Derived categories and their uses, Handbook of algebra vol. 1 pp. 671-701 North Holland, Amsterdam (1996). Zbl0862.18001 MR1421815 · Zbl 0862.18001
[10] S. LOJACIEWICZ, Sur la géométrie semi- et sous-analytique, Ann. Inst. Fourier, 43 (1993), pp. 1575-1595. Zbl0803.32002 MR1275210 · Zbl 0803.32002 · doi:10.5802/aif.1384 · numdam:AIF_1993__43_5_1575_0 · eudml:75048
[11] S. LOJACIEWICZ, Sur le problème de la division, Studia Mathematica, 8 (1959) pp. 87-136. Zbl0115.10203 MR107168 · Zbl 0115.10203
[12] B. MALGRANGE, Ideals of differentiable functions, Tata Institute, Oxford University Press (1967). Zbl0177.17902 MR212575 · Zbl 0177.17902
[13] L. PRELLI, Sheaves on subanalytic sites, Phd Thesis, Universities of Padova and Paris 6 (2006). · Zbl 1171.32002
[14] G. TAMME, Introduction to étale cohomology, Universitext Springer-Verlag, Berlin (1994). Zbl0815.14012 MR1317816 · Zbl 0815.14012
[15] J. L. VERDIER, Catégories dérivées, état 0, SGA 41 2, Lecture Notes in Math., 569 Springer-Verlag, Berlin (1977). Zbl0407.18008 · Zbl 0407.18008
[16] SGA4: Sém. Géom. Algébrique du Bois-Marie by M. Artin, A. Grothendieck, J. L. Verdier, Lecture Notes in Math. 269 Springer-Verlag, Berlin (1972). MR354653
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