Wiki. “K-理论” [K-理论]

K-理论由 Grothendieck 在 Riemann–Roch 定理的推广工作中提出. Atiyah 和 Hirzebruch 发展了与之类似的拓扑 K-理论.

设 $\mathcal C$ 为稳定无穷范畴, 其 K-理论是一个谱 $K(\mathcal C)$, $\pi_0K(\mathcal C)$ 是 $\mathcal C$ 的 Grothendieck 群.

例

代数 K-理论

Historically, the algebraic K-theory of a commutative ring (what today is the “0th” algebraic K-theory group) was originally defined to be the Grothendieck group of its symmetric monoidal category of projective modules. Under the relation between modules and vector bundles, this is directly analogous to the basic definition of topological K-theory, whence the common term. (In fact when applied to the stack of vector bundles then algebraic K-theory subsumes topological K-theory and also differential K-theory.

Quillen 定义了环 $R$ 的高阶 K-群 (代数), 它是某个空间 $BGL(R)^+$ 的同伦群.