Abstract. This a first step to develop a theory of smooth,étale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of pseudo finite type. Among our results we show that these infinitesimal properties of a map of usual schemes carry over into the completion with respect to suitable closed subsets. We characterize unramifiedness by the vanishing of the module of differentials. Also we see that a smooth morphism of noetherian formal schemes is flat and its module of differentials is locally free. The paper closes with a version of Zariski's Jacobian criterion.

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ABSTRACT. A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category.We also associate, in a 2-functorial way, to a 1-morphism of geometric stacks f : X → Y, an adjunction f * ⊣ f * for the corresponding categories of quasi-coherent sheaves that agrees with the classical one defined for schemes. This construction is described both geometrically in terms of the small flat site and algebraically in terms of comodules over the Hopf algebroid. CONTENTS

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a b s t r a c tWe provide the main results of a deformation theory of smooth formal schemes as defined in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Algebra 35 (2007) 1341-1367].Smoothness is defined by the local existence of infinitesimal liftings. Our first result is the existence of an obstruction in a certain Ext 1 group whose vanishing guarantees the existence of global liftings of morphisms. Next, given a smooth morphism f 0 : X 0 → Y 0 of noetherian formal schemes and a closed immersion Y 0 → Y given by a square zero ideal I, we prove that the set of isomorphism classes of smooth formal schemes lifting X 0 over Y is classified by Ext 1 ( 1 X0/Y0 , f * 0 I) and that there exists an element inwhich vanishes if and only if there exists a smooth formal scheme lifting X 0 over Y.

Communicated by E.M. Friedlander MSC:Primary: 14B10 secondary: 14B20 14B25 a b s t r a c tWe continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Alg. 35 (2007Alg. 35 ( ) 1341Alg. 35 ( -1367. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersions that are flat. Also, the local structure of smooth and étale morphisms between locally noetherian formal schemes is described: the former factors locally as a completion morphism followed by a smooth adic morphism and the latter as a completion morphism followed by an étale adic morphism. IntroductionFormal schemes have always been present in the backstage of algebraic geometry but they were rarely studied in a systematic way after the foundational [5, Section 10]. It has become more and more clear that the wide applicability of formal schemes in several areas of mathematics require such study. Let us cite a few of this applications. The construction of De Rham cohomology for a scheme X of zero characteristic embeddable in a smooth scheme P, studied by Hartshorne [9] (and, independently, by Deligne), is defined as the hypercohomology of the completion of the De Rham complex of the formal completion of P along X . Formal schemes play a key role in p-adic cohomologies (crystalline, rigid . . . ) and are also algebraic models of rigid analytic spaces. These developments go back to Grothendieck with further elaborations by Raynaud, in collaboration with Bosch and Lütkebohmert, and later work by Berthelot and de Jong. In a different vein, Strickland [15] has pointed out the importance of formal schemes in the context of (stable) homotopy theory.A particular assumption that it is almost always present in most earlier works on formal schemes is that morphisms are adic, i.e. that the topology of the sheaf of rings of the initial scheme is induced by the topology of the base formal scheme. This hypothesis on a morphism of formal schemes guarantees that its fibers are usual schemes, therefore an adic morphism between formal schemes is, in the terminology of Grothendieck's school, a relative scheme over a base that is a formal scheme. But there are important examples of maps of formal schemes that do not correspond to this situation. The first example that comes into mind is the natural map Spf(A[[X ]]) → Spf(A) for an adic ring A. This morphism has a finiteness $ This work was partially supported by Spain's MCyT and E.U.'s FEDER research project MTM2005-05754.

Abstract. In this paper, we prove that for a noetherian formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies Dqct(X) is generated by a single compact object. In an appendix we prove that the category of compact objects in Dqct(X) is skeletally small.