On Deformation Theory in Higher Logarithmic Geometry

Abstract: We initiate the study of deformation theory in the context of derived and higher log geometry. After reconceptualizing the "exactification"-procedures in ordinary log geometry in terms of Quillen's approach to the cotangent complex, we construct an "exactified tangent bundle" over the category of log ring spectra. The fibers recover the categories of modules over the underlying ring spectra, and the resulting cotangent complex functor specializes to log topological Andr\'e--Quillen homology on each fiber. As applications, we characterize log square-zero extensions and derive a log variant of \'etale rigidity, applicable to some tamely ramified extensions of ring spectra.