On The Compatibility Of Derived Structures On Critical Loci

We study the problem of compatibility of derived structures on a scheme which can be presented as a critical locus in more than one way. We consider the situation when a scheme can be presented as the critical locus of a function w∈𝓞(S) and as the critical locus of the restriction w|ₓ∈𝓞(X) for some smooth subscheme X⊂S. In the case when S is the total space of a vector bundle over X, we prove that, under natural assumptions, the two derived structures coincide. We generalize the result to the case when X is a quantized cycle in S and also give indications how to proceed when X⊂S is a general closed embedding.