  We prove a formula for the structure sheaf of a quiver variety in the
Grothendieck ring of its embedding variety. This formula generalizes and gives
new expressions for Grothendieck polynomials. We furthermore conjecture that
the coefficients in our formula have signs which alternate with degree. The
proof of our formula involves $K$-theoretic generalizations of several useful
cohomological tools, including the Thom-Porteous formula, the Jacobi-Trudi
formula, and a Gysin formula of Pragacz.
