Motivic and $$\ell $$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

Abstract We study the motivic and $$\ell $$ ℓ -adic realizations of dg category singularities zero locus a global section line bundle over regular scheme. will then use formula obtained in this way together with theorem due to D. Orlov J. Burke–M. Walker give for realization vector bundle. In particular, we obtain special fiber scheme local ring dimension n .