the authors introduced the notion of ind-sheaves and defined the six Grothendieck operations in this framework. As a byproduct, they obtained subanalytic sheaves and the six Grothendieck operations on them. The aim of this paper is to give a direct construction of the six Grothendieck operations in the framework of subanalytic sites avoiding the heavy machinery of ind-sheaves. As an application, we show how to recover the subanalytic sheaves y t and y w of temperate and Whitney holomorphic functions respectively.
Introduction.Let X be a real analytic manifold and k a field. The spaces of functions which are not defined by local properties, such as tempered distributions, tempered and Whitney g I functions, etc., are very useful in the study of systems of linear partial differential equations (Laplace transform, tempered holomorphic solutions of h-modules etc.). Although these spaces do not define sheaves on X, they define sheaves on a site associated to X, the subanalytic site X sa , where one just considers open subanalytic sets and locally finite coverings.In [7], Kashiwara and Schapira, motivated by the construction of the microlocalization functor, treated a more general theory, namely that of ind-sheaves. They defined the category I(k X ) of ind-sheaves on X as the category of ind-objects of the category Mod c (k X ) of sheaves with compact support and they developped the six Grothendieck operations in this framework. When restricting to R-constructible sheaves, they showed the equivalence between the category I R-c (k X ) Ind(Mod c R-c (k X )) of ind-Rconstructible sheaves on X and the category Mod(k X sa ) of sheaves on the subanalytic site associated to X. In this way, tempered distributions, tempered and Whitney g I functions, etc., are obtained as a byproduct of the whole theory of ind-sheaves. It turns out to be useful to have a more straightforward introduction of these sheaves.Our aim in this paper is to give a direct, self-contained and elementary construction of the six Grothendieck operations on Mod(k X sa ), without using the more sophisticated and much more difficult theory of indsheaves. Indeed, contrary to the category I(k X ), the category Mod(k X sa ) is a Grothendieck category.We will start by recalling some results of [7], the definition of a subanalytic site, the natural functor of sites r : X 3 X sa , and the functors r à , r À1 and r ! relating the categories of``classical'' and subanalytic sheaves. We also recall a very useful description of subanalytic sheaves as inductive limits of R-constructible sheaves.Then we go into the study of subanalytic sheaves, without using the notion of ind-sheaf. Let f : X 3 Y be a morphism of real analytic manifolds. The functors rom, , f à and f À1 are well defined since X sa is a site.We introduce the proper direct image functor f !! and we study the relations between the above operations and the functors r à , r À1 and r ! . In the derived category we obtain an exceptional inverse image, denoted by f ! , right adjoint to Rf !! . This is obtained vi...
