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Consider the symmetric algebra $S(V)$, with its coalgebra structure: $\Delta(x)=1\otimes x+x\otimes1$ on $V$, extended multiplicatively. What are its subcoalgebras?

In some vague sense, they seem to be of the form "all polynomials of degree at most $n_i$ on some subvarieties $V_i$ of $V$".

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  • $\begingroup$ All the examples I can imagine have a basis consisting in homogeneous polynomials. Is this always true? $\endgroup$
    – grok
    Commented Jun 12, 2023 at 9:22
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    $\begingroup$ As category of coalgebras is contravariantly equivalent to the category of linearly compact (i. e. pro-finite dimensional) algebras, the answer is: coalgebras topologically dual to quotients of formal power series algebra. In other words, all subschemes of formal dim V-dimensional disk. $\endgroup$
    – Denis T
    Commented Jun 12, 2023 at 20:21
  • $\begingroup$ Thanks alot @DenisT! I now see that they don't have to be homogeneous. Does your answer require $V$ to be finite dimensional? Mine isn't. $\endgroup$
    – grok
    Commented Jun 14, 2023 at 6:07

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