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We introduce a generalization of the notion of a negligible morphism andstudy the associated tensor ideals and thick ideals. These ideals are definedby considering deformations of a given monoidal category $\mathcal{C}$ over alocal ring $R$. If the maximal ideal of $R$ is generated by a single element,we show that any thick ideal of $\mathcal{C}$ admits an explicitely givenmodified trace function. As examples we consider various Deligne categories andthe categories of tilting modules for a quantum group at a root of unity andfor a semisimple, simply connected algebraic group in prime characteristic. Weprove an elementary geometric description of the thick ideals in quantum type Aand propose a similar one in the modular case.

Generalized negligible morphisms and their tensor ideals