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  "body": "\n## Summary\n\nAlgebraic K-theory attaches to a [[ring|0B1261B5ED]] $R$ a sequence of abelian groups $K_n(R)$.\nThe group $K_0(R)$ is the [[Grothendieck group|5FDMY7PDYY]] of finitely generated [[projective modules|BFFA03GE3V]], and $K_1(R)$ is the abelianization of the infinite general linear group.\nDaniel Quillen defined the higher groups by homotopy theory, for instance through the plus construction.\nThe groups encode deep arithmetic information, such as class groups, units and special values of zeta functions.\n\n## Explanations\n\n### L1 human\n\nAlgebraic K-theory measures a ring through its modules and its invertible matrices.\nThe group $K_0$ counts projective modules, with the rule that a direct sum becomes a sum.\nFor a field, projective modules are vector spaces, and $K_0$ just records the dimension, so it is $\\Z$.\nThe group $K_1$ remembers invertible matrices up to elementary row operations; for a field this is the determinant, so $K_1$ is the group of nonzero elements.\nThe higher groups are harder to compute, and they connect algebra with topology and number theory.\n",
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