We produce a Grothendieck transformation from bivariant operational $K$-theory to Chow, with a Riemann–Roch formula that generalizes classical Grothendieck–Verdier–Riemann–Roch. We also produce Grothendieck transformations and Riemann–Roch formulas that generalize the classical Adams–Riemann–Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety $X$ whose equivariant $K$-theory of vector bundles does not surject onto its ordinary $K$-theory, and describe the operational $K$-theory of spherical varieties in terms of fixed-point data.

In an appendix, Vezzosi studies operational $K$-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic $K$-theory of relatively perfect complexes to bivariant operational $K$-theory.

我们从双变量运算中产生了Grothendieck变换 $ķ$-Chow的理论，其中使用了黎曼–罗赫公式，将经典的Grothendieck–Verdier–Riemann–Roch进行了概括。我们还产生了Grothendieck变换和Riemann-Roch公式，这些公式推广了经典的Adams-Riemann-Roch和等变局部定理。作为应用，我们展示了投射复曲面$X$ 其等变 $ķ$向量束的理论不会超越它的常态 $ķ$-理论，并描述操作 $ķ$定点数据方面的球形变体理论。

在附录中，Vezzosi研究了可操作性 $ķ$方案的理论，并从双变量代数构造Grothendieck变换 $ķ$-相对完美的复合体到双变量运算的理论 $ķ$-理论。