We study the motivic and $$\ell $$
ℓ
-adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the $$\ell $$
ℓ
-adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the $$\ell $$
ℓ
-adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.
Let p : X → S be a flat, proper and regular scheme over a strictly henselian discrete valuation ring. We prove that the singularity category of the special fiber with its natural two-periodic structure allows to recover the ℓ-adic vanishing cohomology of p.Along the way, we compute homotopy-invariant non-connective algebraic K-theory with compact support of certain embeddings X t ֒→ X T in terms of the motivic realization of the dg category of relatively perfect complexes.
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