K-theoretic exceptional collections at roots of unity

Using cyclotomic specializations of the equivariant $K$-theory with respect to a torus action we derive congruences for discrete invariants of exceptional objects in derived categories of coherent sheaves on a class of varieties that includes 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 powers of a fixed prime number $p$, then the rank of an exceptional object on $X$ is congruent to $\pm 1$ modulo $p$.

💡 Research Summary

The paper investigates how equivariant K‑theory, when specialized at roots of unity, imposes strong arithmetic constraints on exceptional objects in the bounded derived category of coherent sheaves on a broad class of algebraic varieties. The authors begin by recalling that for a smooth projective variety (X) equipped with an action of an algebraic torus (T), the equivariant K‑group (K_T(X)) is a free module over the representation ring (R(T)=\mathbb Z