The logarithmic cotangent complex.

*(English)*Zbl 1095.14016More than thirty years ago, L. Illusie developed his highly abstract and sophisticated theory of cotangent complexes for morphisms of schemes [Complexe cotangent et deformations. I. Lect. Notes Math. 239 (1971; Zbl 0224.13014); Complexe cotangent et deformations. II, Lect. Notes Math. 283 (1972; Zbl 0224.13014)], thereby establishing an utmost general and powerful cohomological framework in algebraic deformation theory, which has found far-reaching applications ever since.

In the meantime, the appearance of logarithmic structures in algebraic geometry, together with their most recent applications to both Gromov-Witten theory and intersection theory on degenerations of algebraic varieties, has brought about the natural question of to what extent L. Illusie’s theory of cotangent complexes can be generalized to logarithmic geometry, and how this logarithmic version can be used to understand more complicated deformation-theoretic problems arising there. The paper under review is devoted to exactly this important question. As the author points out, his interest in the development of a logarithmic version of the theory of cotangent complexes for morphisms of fine log-schemes comes from two sources. First, there is some strong demand to generalize F. Kato’s so-called “log smooth deformation theory” [Tôhoku Math. J., II. Ser. 48, No. 3, 317–354 (1996; Zbl 0876.14007)] in order to tackle specific deformation problems in the logarithmic category. Secondly, the first steps into such a theory of the log cotangent complex have recently been made by K. Kato and T. Saito [Publ. Math., Inst. Hautes Étud. Sci. 100, 5–151 (2004; Zbl 1099.14009)], and a systematic elaboration of this approach seems to be just as worthwile as promising.

In this vein, the author provides a construction that associates to every morphism of fine log schemes \(f:X\to Y\) a projective system \(L_{X/Y}=(\dots\to L_{X/Y}^{\geq-n-1}\to L_{X/Y}^{\geq -n}\to\cdots\to L_{X/Y}^{\geq 0})\), where \(L_{X/Y}^{-n}\) is an essentially constant ind-object in the derived category of sheaves of \({\mathcal O}_X\)-modules with support in \([-n,0]\). The system \(L_{X/Y}\) is then called the log cotangent complex of the morphism \(f\), and the author shows that this object admits a number of nice functorial properties analoguous to those of L. Illusie’s classical cotangent complex for morphisms of ordinary schemes. The construction is based upon the author’s stack-theoretic approach to logarithmic geometry [M. C. Olsson, Ann. Sci. Ec. Norm. Supér., IV. Sér. 36, No. 5, 747–791 (2003; Zbl 1069.14022)], together with some suitable generalizations of the log stacks introduced there. As an application of his log cotangent complex, the author explains how to compute crystalline cohomology of so-called log complete intersections in terms this log cotangent complex. Modelled on L. Illusie’s original approach, the author’s logarithmic method leads to a description of the relationship between logarithmic cotangent complexes and deformations of fine log schemes, just as desired. In the sequel, the problem of the existence of a reasonable theory of cotangent complexes for log schemes admitting a distinguished triangle is analyzed. Finally, the author discusses an alternate approach to defining a log cotangent complex for morphisms of log schemes due to O. Gabber (unpublished), including a careful comparison between the two constructions (and their properties) delivered in the present paper. As the author points out in the acknowledgements, this extensive paper grew out of his Ph.D. thesis written under A. Ogus as academic supervisor.

In the meantime, the appearance of logarithmic structures in algebraic geometry, together with their most recent applications to both Gromov-Witten theory and intersection theory on degenerations of algebraic varieties, has brought about the natural question of to what extent L. Illusie’s theory of cotangent complexes can be generalized to logarithmic geometry, and how this logarithmic version can be used to understand more complicated deformation-theoretic problems arising there. The paper under review is devoted to exactly this important question. As the author points out, his interest in the development of a logarithmic version of the theory of cotangent complexes for morphisms of fine log-schemes comes from two sources. First, there is some strong demand to generalize F. Kato’s so-called “log smooth deformation theory” [Tôhoku Math. J., II. Ser. 48, No. 3, 317–354 (1996; Zbl 0876.14007)] in order to tackle specific deformation problems in the logarithmic category. Secondly, the first steps into such a theory of the log cotangent complex have recently been made by K. Kato and T. Saito [Publ. Math., Inst. Hautes Étud. Sci. 100, 5–151 (2004; Zbl 1099.14009)], and a systematic elaboration of this approach seems to be just as worthwile as promising.

In this vein, the author provides a construction that associates to every morphism of fine log schemes \(f:X\to Y\) a projective system \(L_{X/Y}=(\dots\to L_{X/Y}^{\geq-n-1}\to L_{X/Y}^{\geq -n}\to\cdots\to L_{X/Y}^{\geq 0})\), where \(L_{X/Y}^{-n}\) is an essentially constant ind-object in the derived category of sheaves of \({\mathcal O}_X\)-modules with support in \([-n,0]\). The system \(L_{X/Y}\) is then called the log cotangent complex of the morphism \(f\), and the author shows that this object admits a number of nice functorial properties analoguous to those of L. Illusie’s classical cotangent complex for morphisms of ordinary schemes. The construction is based upon the author’s stack-theoretic approach to logarithmic geometry [M. C. Olsson, Ann. Sci. Ec. Norm. Supér., IV. Sér. 36, No. 5, 747–791 (2003; Zbl 1069.14022)], together with some suitable generalizations of the log stacks introduced there. As an application of his log cotangent complex, the author explains how to compute crystalline cohomology of so-called log complete intersections in terms this log cotangent complex. Modelled on L. Illusie’s original approach, the author’s logarithmic method leads to a description of the relationship between logarithmic cotangent complexes and deformations of fine log schemes, just as desired. In the sequel, the problem of the existence of a reasonable theory of cotangent complexes for log schemes admitting a distinguished triangle is analyzed. Finally, the author discusses an alternate approach to defining a log cotangent complex for morphisms of log schemes due to O. Gabber (unpublished), including a careful comparison between the two constructions (and their properties) delivered in the present paper. As the author points out in the acknowledgements, this extensive paper grew out of his Ph.D. thesis written under A. Ogus as academic supervisor.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14A20 | Generalizations (algebraic spaces, stacks) |

14D15 | Formal methods and deformations in algebraic geometry |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

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

##### Keywords:

simplicial monoids; deformation theory; logarithmic geometry; derived category of sheaves; algebraic stacks; crystalline cohomology
Full Text:
DOI

##### References:

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