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  "body": "\n## Definition\n\nLet $K$ be a field and let $H$ be a [[Hopf algebra|7XFYP0N84H]] over $K$ with comultiplication $\\Delta$.\nA *right Hopf module* over $H$ is a vector space $M$ over $K$ such that:\n\n1. $M$ is a right $H$-[[module|0NZ8JA25WK]], with action $m \\otimes h \\mapsto m \\cdot h$.\n2. $M$ is a right $H$-[[comodule|7WEM2E6RTV]], with coaction $\\rho \\colon M \\to M \\otimes H$.\n3. For all $m \\in M$ and $h \\in H$, in [[Sweedler notation|NC26CE0Y8B]],\n$$\\rho(m \\cdot h) = \\sum m_{(0)} \\cdot h_{(1)} \\otimes m_{(1)} h_{(2)}.$$\n\nThe *coinvariants* of $M$ are the elements of $M^{\\mathrm{co}H} = \\set{m \\in M : \\rho(m) = m \\otimes 1}$.\n\n## Remarks\n\nClause 3 says that $\\rho$ is a module map when $M \\otimes H$ carries the action $(m \\otimes x) \\cdot h = \\sum m \\cdot h_{(1)} \\otimes x h_{(2)}$.\nExample: for a vector space $V$, the space $M = V \\otimes H$ with $(v \\otimes x) \\cdot h = v \\otimes xh$ and $\\rho(v \\otimes x) = \\sum v \\otimes x_{(1)} \\otimes x_{(2)}$.\nClause 3 holds because $\\Delta(xh) = \\Delta(x)\\Delta(h)$.\nNon-example: $H = KG$ for a group $G \\ne 1$, acting on itself by multiplication, with the coaction $\\rho(x) = x \\otimes 1$.\nThis is a module and a comodule, but for $g \\ne 1$ in $G$, $\\rho(1 \\cdot g) = g \\otimes 1$ while the right side of clause 3 is $g \\otimes g$.\nThe fundamental theorem of Hopf modules states that the map $M^{\\mathrm{co}H} \\otimes H \\to M$, $m \\otimes h \\mapsto m \\cdot h$, is an isomorphism of Hopf modules.\n",
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