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  "body": "\n## Summary\n\nA Hopf module over a [[Hopf algebra|7XFYP0N84H]] $H$ is a vector space that is both an $H$-[[module|0NZ8JA25WK]] and an $H$-[[comodule|7WEM2E6RTV]].\nThe coaction must be compatible with the action.\nThe fundamental theorem of Hopf modules, due to Larson and Sweedler, says that every Hopf module is free.\nIt has the form $V \\otimes H$, where $V$ is its space of coinvariants.\nThis theorem is the main tool behind the uniqueness of [[integrals|RJA5026RGX]] and the Frobenius property of finite-dimensional Hopf algebras.\n\n## Explanations\n\n### L1 human\n\nThe Hopf algebra $H$ itself can be multiplied by elements of $H$ and also split by its comultiplication, and the two operations fit together.\nA Hopf module is any vector space with these two structures fitting together in the same way.\nThe simplest examples are copies of $H$ with a multiplicity space attached, $V \\otimes H$.\nThe surprising theorem is that there are no others: every Hopf module looks like $V \\otimes H$.\n",
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