Infinitesimal Lifting and Jacobi Criterion for Smoothness on Formal Schemes

Tarrío, Leovigildo Alonso; López, Ana Jeremías; Rodríguez, Marta Pérez

doi:10.1080/00927870601115823

Cited by 14 publications

(42 citation statements)

References 7 publications

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“…The tube F x := ]x[ of x in X is canonically isomorphic to the generic fiber of the flat special formal R-scheme Spf O X,x (the R-structure being given by f ), by [8, 0.2.7]. In [31], we called F x the analytic Milnor fiber of f at x, based on a topological intuition explained in [33,4.1] and a cohomological comparison result: if k = C and X is smooth at x, then theétale ℓ-adic cohomology of F x corresponds to the singular cohomology of the classical topological Milnor fiber of f at x, by [31, 9.2]. If f has smooth generic fiber (e.g.…”

Section: The Analytic Milnor Fibermentioning

confidence: 99%

A trace formula for rigid varieties, and motivic Weil generating series for formal schemes

Nicaise

2008

Math. Ann.

117

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Abstract. We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on itsétale cohomology. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal R-scheme X of pseudo-finite type, via the construction of a GelfandLeray form on its generic fiber. Our trace formula yields a cohomological interpretation of this Weil generating series.When X is the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f . When X is the formal completion of f at a closed point x of the special fiber f −1 (0), we obtain the local motivic zeta function of f at x. In the latter case, the generic fiber of X is the so-called analytic Milnor fiber of f at x; we show that it completely determines the formal germ of f at x.

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Section: The Analytic Milnor Fibermentioning

confidence: 99%

A trace formula for rigid varieties, and motivic Weil generating series for formal schemes

Nicaise

2008

Math. Ann.

117

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“…• In Corollary 4.6 assertion (2) may be written: (2 ) For all x ∈ X, y = f (x), f −1 (y) is an unramified k(y)-scheme at x. From Proposition 4.5 we obtain the following result, in which we provide a description of pseudo-closed immersions that will be used in the characterization of completion morphisms (Theorem 7.5).…”

Section: 12mentioning

confidence: 97%

“…By hypothesis Ω 1 X 0 /Y 0 = 0 and thus, since f is adic it holds that From Nakayama's lemma we deduce that Ω 1 A/B = 0 and therefore, Ω 1 X/ = ( Ω 1 A/B ) = 0. Applying [2,Proposition 4.6] it follows that f is unramified.…”

Section: The Morphism F Is Unramified If and Only If The Induced Morpmentioning

confidence: 99%

“…1 The fact that pseudo-finite type morphisms need not be adic allows fibers that are not usual schemes, and the structure of these maps is, therefore, more complex than the structure of adic maps. The study of smoothness and, more generally, infinitesimal lifting properties in the context of noetherian formal schemes together with this hypothesis of finiteness was embraced in general in our previous work [2]. We should mention a preceding study of smooth morphisms under the restriction that the base is a usual scheme in [16] and also the overlap of several results in [2] and a set of results in [10,Section 2], based on Nayak's 1998 thesis.…”

mentioning

confidence: 99%

“…In [2] we studied the good properties of these definitions and the agreement of their properties with the corresponding behavior for usual noetherian schemes, obtaining the corresponding statement of Zariski's Jacobian criterion for smoothness. Now we concentrate on studying properties which make sense specifically in the formal context getting information about the infinitesimal lifting properties from information present in the structure of a formal scheme.…”

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confidence: 99%

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Local structure theorems for smooth maps of formal schemes

Tarrío

López

Rodríguez

2009

Journal of Pure and Applied Algebra

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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.

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