Thom spectra, higher THH and tensors in ∞–categories

Let $f:G\to \mathrm{Pic}(R)$ be a map of $E_\infty$-groups, where $\mathrm{Pic}(R)$ denotes the Picard space an $E_\infty$-ring spectrum $R$. We determine tensor $X\otimes_R Mf$ Thom $E_\infty$-$R$-algebra $Mf$ with $X$; when $X$ is circle, topological Hochschild homology over use theory localizations $\infty$-categories as technical tool: we contribute to this $\infty$-categorical analogue Day's reflection theorem about closed symmetric monoidal structures on localizations, and prove that for smashing localization $L$ $\infty$-category presentable $\infty$-categories, free $L$-local small simplicial set $K$ given by presheaves valued $L$-localization spaces. If pointed space, $g: A\to B$ spectra satisfies $X$-base change if $X\otimes pushout $A\to X\otimes A$ along $g$. Building result Mathew, $g$ \'etale then it provided connected. also multiplication $B$ equivalence. Finally, that, under some hypotheses, isomorphism Mahowald cannot instance $S^0$-base change.