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  "body": "\n## Summary\n\nThe algebraic K-theory of a scheme $X$ is a sequence of abelian groups $K_0(X), K_1(X), \\ldots$ built from the category of [[vector bundles|P4DZM975BV]] on $X$.\nThe group $K_0(X)$ is the Grothendieck group of vector bundles, and the higher groups are defined by Quillen's Q-construction.\nK-theory is the setting of the [[Grothendieck–Riemann–Roch theorem|3HKREN3Y3F]].\nIt is linked to [[Chow groups|EBEF6PEK4N]] by the Chern character and to motivic cohomology by spectral sequences.\n\n## Explanations\n\n### L1 human\n\nThe group $K_0(X)$ treats vector bundles like numbers: one can add them by direct sum, and a bundle built from two others by an extension counts as their sum.\nFor a point, vector bundles are vector spaces, and $K_0$ is $\\Z$, measured by the dimension.\nOn the projective line, the bundles $\\mathcal{O}(n)$ have both a rank and a degree, and $K_0(\\mathbb{P}^1) \\cong \\Z^2$ records exactly these two numbers.\nThe higher groups $K_n$ capture finer information, such as units of rings for $K_1$.\n",
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