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  "body": "\n## Summary\n\nA comodule over a [[coalgebra|A06RA8KB0T]] $C$ is a vector space $M$ with a coaction $\\rho \\colon M \\to M \\otimes C$, the dual notion of a [[module|0NZ8JA25WK]] over an algebra.\nComodules over the coordinate ring of an affine algebraic group are the same as its rational representations.\nComodules over the group algebra $KG$ with its standard coalgebra structure are the same as $G$-graded vector spaces.\nComodules are as basic for coalgebras and [[Hopf algebras|7XFYP0N84H]] as modules are for rings.\n\n## Explanations\n\n### L1 human\n\nA module lets algebra elements act on vectors.\nA comodule goes the other way: each vector is expanded as a sum of vectors tagged by coalgebra elements.\nTake the coalgebra with basis a set $S$ and $\\Delta(s) = s \\otimes s$.\nA comodule over it is just a vector space split into pieces $M_s$, one for each $s$, and the coaction tags each vector of $M_s$ with $s$.\nThis is how gradings, such as the degree of polynomials, appear as comodule structures.\n",
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