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sorry for my bad English.

I am reading Mumford's "Abelian Variety", and in this book he defined hyperalgebra $\mathbb{H}$ in p98, as follows.

Let $k$ be algebraic closed field, and $G$ be group scheme of finite type over $k$.

And for closed point $x\in G$, $\mathbb{H}_x:=Hom_{cont}(\mathscr{O}_{x,G}, k)$ where cont means the maps $L:\mathscr{O}_{x,G}\to k$ which are continuous in the sense that $L({\frak m}^{N+1})=0$ for some $N$.

Then, $\mathbb{H}:=\oplus_{x\in G} \mathbb{H}_x$.

On the other hand, he said $\mathbb{H}$ is equal to $\varinjlim Hom_k(\Gamma(Z,\mathscr{O}_Z),k)$ where $Z$ runs 0-dim subschema of $G$.

But I don't understand this correspond.

I think if $L\in \mathbb{H}_x$, we take $Z:=Spec(\mathscr{O}_x/{\frak m}^{N+1})$. But I don't know this $Z$ is truely subscheme (i.e. locally closed as topological space).

Please tell me concrete correspond, thanks.

Yos
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1 Answers1

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Question: "Please tell me concrete correspond, thanks."

Answer: If $k$ is algebraically closed and $x \in U:=Spec(A) \subseteq G$ is an open affine subscheme containing $x$, it follows

$$\mathcal{O}_{G,x}/\mathfrak{m}_x^{l+1} \cong J^l_A \otimes \kappa(x)$$

where $J^l_A:=A\otimes_k A/I^{l+1}$ where $I \subseteq A\otimes_k A$ is the "ideal of the diagonal". The dual of $J^l_A$ is the module of $l$'th order differential operators

$$Hom_A(J^l_A,A) \cong Diff^l_{\kappa(x)}(A,A),$$

hence the dual of $\mathcal{O}_{G,x}/\mathfrak{m}_x^{l+1}$ is the fiber of the module of $l$'th order differential operators.

On the following threads you find an "elementary" construction of all cofinite ideals in any finitely generated $k$-algebra $A$, and a possible relation with "differential operators" that may be of help. The idea is that for any maximal ideal $\mathfrak{m}:=(x-a,y-b) \subseteq k[x,y]$, the canonical quotient map

$$p: k[x,y] \rightarrow k[x,y]/(x-a,y-b)$$

may be seen as "Taylor expansion" with respect to the point $t:=(x-a,y-b)$.

Applications of the Chinese remainder Theorem to the study of the Hilbert scheme of points and $(\mathfrak{m},l)$-squeezed ideals.

At the following link is a note on the relation between Hilbert schemes of points and "Taylor expansions". Given any cofinite ideal $I \subseteq A$ where $A$ is a finitely generated $k$-algebra, using the Chinese Remainder Theorem you may realize the canonical map $p: A \rightarrow A/I$ as a product of "Talylor expansions" and projection maps:

https://mathoverflow.net/questions/10014/applications-of-the-chinese-remainder-theorem/394852#394852

hm2020
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