Ind-sheaves.

*(English)*Zbl 0993.32009
Astérisque. 271. Paris: Société Mathématique de France, vi, 136 p. (2001).

Sheaf theory works well in situations where compatible local information can be glued together to produce global information. Since the 1960s, thanks to Grothendieck and his school [see, for example, “Sémin. Géom. algébrique 1963-64” (SGA4), M. Artin, A. Grothendieck, and J.-L. Verdier, Lect. Notes Math. 269 (1972; Zbl 0234.00007)], more and more mathematicians have viewed the category of sheaves on a space as a valuable generalization of that space.

The authors of this monograph are interested in analysis on manifolds, and using sheaf theory in that study. But on an analytic manifold there are problems with tempered functions and distributions, for example. Their approach to meeting some of these problems is to replace the “space” of sheaves on the manifold with the category Mod\(^{c}(k_{X})\) of sheaves of \(k\)-modules on \(X\) with compact support, and then to work in the category of ind-objects (systems of objects defined on a filtered category, again an idea that goes back at least to Grothendieck; see the above reference) in that category of sheaves – hence the title for the monograph. This serves as a kind of completion for the generalized manifold, and in it they find much of what they are looking for. Passing then to the derived category of this completion [see, for example, “Catégories derivées (Etat O)”, J.-L. Verdier, Lect. Notes Math. 569, 262-311 (1977; Zbl 0407.18008)], which is something like the homotopy category of the generalized space, they produce the familiar external and internal operations (\(\operatorname{Hom}\), \(R\operatorname{Hom},\) \(Rf_{\ast},\) composition, etc.) and a few new ones useful for their purposes. As they say: “…ind-sheaves allow us to treat functions with growth conditions in the formalism of sheaves. On a complex manifold \(X\), we can define the ind-sheaf of “tempered holomorphic functions” \(\mathcal{O}_{X}^{t}\), or the ind-sheaf of “Whitney holomorphic functions” \(\mathcal{O}_{X}^{w}\), and obtain for example the sheaves of distributions or of \(C^{\infty}\)-functions using Sato’s construction of hyperfunctions, simply replacing \(\mathcal{O}_{X}\) with \(\mathcal{O}_{X}^{t}\) or \(\mathcal{O}_{X}^{w}\). We also prove an adjunction formula for integral transforms in this framework”.

The authors of this monograph are interested in analysis on manifolds, and using sheaf theory in that study. But on an analytic manifold there are problems with tempered functions and distributions, for example. Their approach to meeting some of these problems is to replace the “space” of sheaves on the manifold with the category Mod\(^{c}(k_{X})\) of sheaves of \(k\)-modules on \(X\) with compact support, and then to work in the category of ind-objects (systems of objects defined on a filtered category, again an idea that goes back at least to Grothendieck; see the above reference) in that category of sheaves – hence the title for the monograph. This serves as a kind of completion for the generalized manifold, and in it they find much of what they are looking for. Passing then to the derived category of this completion [see, for example, “Catégories derivées (Etat O)”, J.-L. Verdier, Lect. Notes Math. 569, 262-311 (1977; Zbl 0407.18008)], which is something like the homotopy category of the generalized space, they produce the familiar external and internal operations (\(\operatorname{Hom}\), \(R\operatorname{Hom},\) \(Rf_{\ast},\) composition, etc.) and a few new ones useful for their purposes. As they say: “…ind-sheaves allow us to treat functions with growth conditions in the formalism of sheaves. On a complex manifold \(X\), we can define the ind-sheaf of “tempered holomorphic functions” \(\mathcal{O}_{X}^{t}\), or the ind-sheaf of “Whitney holomorphic functions” \(\mathcal{O}_{X}^{w}\), and obtain for example the sheaves of distributions or of \(C^{\infty}\)-functions using Sato’s construction of hyperfunctions, simply replacing \(\mathcal{O}_{X}\) with \(\mathcal{O}_{X}^{t}\) or \(\mathcal{O}_{X}^{w}\). We also prove an adjunction formula for integral transforms in this framework”.

Reviewer: D.H.Van Osdol (Durham)

##### MSC:

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

32S60 | Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) |

18E30 | Derived categories, triangulated categories (MSC2010) |