Prelli, Luca Sheaves on subanalytic sites. (English) Zbl 1171.32002 Rend. Semin. Mat. Univ. Padova 120, 167-216 (2008). 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. Reviewer: Gerhard Pfister (Kaiserslautern) Cited in 19 Documents MSC: 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 Keywords:subanalytic site; ind-sheaf; Grothendieck operation PDF BibTeX XML Cite \textit{L. Prelli}, Rend. Semin. Mat. Univ. Padova 120, 167--216 (2008; Zbl 1171.32002) Full Text: DOI Link EuDML arXiv References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.