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  "body": "\n## Definition\n\nLet $X$ be a [[scheme|5A35TYZD02]], and let $\\mathcal{P}(X)$ be the category of [[vector bundles|P4DZM975BV]] on $X$, that is, locally free $\\mathcal{O}_X$-modules of finite rank.\nWith the short exact sequences of $\\mathcal{O}_X$-modules, $\\mathcal{P}(X)$ is an [[exact category|TFADSC9YPZ]].\nFor $n \\ge 0$, the *$n$-th algebraic K-group* of $X$ is $K_n(X) = \\pi_{n+1}\\big(B Q \\mathcal{P}(X), 0\\big)$.\nThis is a homotopy group of the classifying space of Quillen's Q-construction on $\\mathcal{P}(X)$, at the base point given by the zero bundle.\n\n## Remarks\n\nThe group $K_0(X)$ is the Grothendieck group of vector bundles.\nIt is the quotient of the [[free abelian group|0YMSEZE1VH]] on isomorphism classes $[E]$ of vector bundles by the following relations.\nFor every short exact sequence $0 \\to E' \\to E \\to E'' \\to 0$, one sets $[E] = [E'] + [E'']$.\nThe tensor product makes $K_0(X)$ a commutative ring.\nExample: for $X = \\operatorname{Spec} k$ with $k$ a field, every exact sequence of finite-dimensional vector spaces splits, and $K_0(X) \\cong \\Z$ through the dimension; also $K_1(X) \\cong k^{\\times}$.\nFor $X = \\mathbb{P}^n_k$, $K_0(X)$ is free of rank $n + 1$ with basis $[\\mathcal{O}], [\\mathcal{O}(-1)], \\ldots, [\\mathcal{O}(-n)]$.\nNon-example: using all coherent sheaves instead of vector bundles gives the groups $G_n(X)$, which differ in general.\nFor a regular separated Noetherian scheme the natural maps $K_n(X) \\to G_n(X)$ are isomorphisms, but for a singular one they need not be.\nFor general schemes, Thomason and Trobaugh gave a better-behaved version using perfect complexes, which agrees with this one when $X$ has an ample family of line bundles.\n",
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