  We prove K-theoretic generalizations of the component formulas of Knutson,
Miller, and Shimozono, and deduce that K-theoretic quiver coefficients have
alternating signs. We also prove new variants of the factor sequences
conjecture, and a conjecture of Knutson, Miller, and Shimozono stating that the
double ratio formula agrees with the original quiver formulas. For completeness
we include a short proof of the ratio formula for quiver varieties.
