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Using cyclotomic specializations of the equivariant $K$-theory with respectto a torus action we derive congruences for discrete invariants of exceptionalobjects in derived categories of coherent sheaves on a class of varieties thatincludes Grassmannians and smooth quadrics. For example, we prove that if$X={\Bbb P}^{n_1-1}\times...\times{\Bbb P}^{n_k-1}$, where $n_i$'s are powersof a fixed prime number $p$, then the rank of an exceptional object on $X$ iscongruent to $\pm 1$ modulo $p$.

K-theoretic exceptional collections at roots of unity