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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity.\n\n1. $K_0(R)$ is the [[Grothendieck group|5FDMY7PDYY]] of the commutative monoid of isomorphism classes of finitely generated [[projective|BFFA03GE3V]] left $R$-modules under direct sum.\n2. Let $GL(R) = \\bigcup_{n \\ge 1} GL_n(R)$, where $GL_n(R)$ is the [[general linear group|0FCHE902D7]] and $GL_n(R) \\subseteq GL_{n+1}(R)$ through $g \\mapsto \\begin{pmatrix} g \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$.\nThen $K_1(R) = GL(R)/[GL(R), GL(R)]$.\n3. For $n \\ge 1$, $K_n(R) = \\pi_n(BGL(R)^+)$, the $n$-th [[homotopy group|P976WYEFMB]] of the plus construction of the [[classifying space|WA796ZCNRX]] $BGL(R)$ with respect to the commutator subgroup $[GL(R), GL(R)]$.\n\nThe *algebraic K-theory* of $R$ is the sequence of abelian groups $K_n(R)$, $n \\ge 0$.\n\n## Remarks\n\nFor $n = 1$ clause 3 gives back clause 2, since $\\pi_1(BGL(R)^+) = GL(R)/[GL(R), GL(R)]$.\nBy Whitehead's lemma, $[GL(R), GL(R)]$ is the subgroup generated by elementary matrices.\nExample: for a field $F$, every finitely generated module is free, so $K_0(F) \\cong \\Z$ by the dimension.\nThe [[determinant|00KDZRW4C0]] gives $K_1(F) \\cong F^\\times$, because the elementary matrices generate $SL(F)$.\nCaution: replacing projective modules by all finitely generated modules gives a different group, $G_0(R)$.\nFor $R = k[\\varepsilon]/(\\varepsilon^2)$ over a field $k$, $K_0(R) \\cong \\Z$ is generated by $[R]$, $G_0(R) \\cong \\Z$ is generated by $[k]$, and $[R] = 2[k]$.\n",
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