The logarithmic cotangent complex

Abstract

We define the cotangent complex of a morphism of fine log schemes, prove that it is functorial, and construct under certain restrictions a transitivity triangle. We also discuss its relationship with deformation theory.

$p$-adic periods and derived de Rham cohomology

“…There are two (in general, nonequivalent) ways to define the cotangent complex for log schemes due, respectively, to Gabber and Olsson; see [Ol]. 12 Gabber's approach ( [Ol,§8]) is more direct and precise; 13 we recall it briefly. For a commutative ring A, a prelog structure on A is a homomorphism of monoids α : L → A, where L is a commutative integral monoid (written multiplicatively) and A is viewed as a monoid with respect to the product.…”

“…For a commutative ring A, a prelog structure on A is a homomorphism of monoids α : L → A, where L is a commutative integral monoid (written multiplicatively) and A is viewed as a monoid with respect to the product. Rings equipped with prelog structures form a category in an evident way; denote its objects simply by In all situations that we will consider, the two versions coincide by [Ol,8.34]. 13 It produces a true complex, while Olsson's construction yields a mere compatible datum of the canonical filtration truncations.…”

“…It is a log complete intersection over O K (see [Ol], 6.8). If π is a generator of O K /O K , f (t) its minimal polynomial, then, by [Ol,6.9], L (K ,O K ) is quasi-isomorphic to the cone of the multiplication by…”

Logarithmic interpretation of the main component in toric Hilbert schemes

“…For an fs-log scheme and object 1.5.1 of H log (S, M S ), let L i denote the logarithmic cotangent complex (in the sense of [Ol1]) of the morphism i. The distinguished triangle…”

Deformation theory of representable morphisms of algebraic stacks

“…Throughout this paper we follow the conventions of ( [13]) except that we do not assume that our stacks are quasi-separated (this is important in the application to log geometry ( [14])). More precisely, by an algebraic stack we mean a stack X in the sense of ([13], 3.1) satisfying the following:…”