What is the notation $\mathcal{A}_\infty$ for a sheaf of graded $\mathcal{O}_X$-algebras mean?

Question

Today I encountered Hartshorne's condition $(\dagger)$ for quasi-coherent sheaves of algebras. This states that gives a graded $\mathcal{O}_X$-module $\mathcal{A}$ which has the structure of a graded algebra

1. $\mathcal{A} = \oplus_{n\geq 0 }\mathcal{A}_n$ and $\mathcal{A}_0 = \mathcal{O}_X$
2. $\mathcal{A}_1$ is coherent as an $\mathcal{O}_X$-module
3. $\mathcal{A}_\infty$ generates $\mathcal{A}$ as an $\mathcal{O}_X$-module

What is $\mathcal{A}_\infty$?

Answer 1

The definition listed in the post does not agree with Hartshorne's dagger condition in his book "Algebraic Geometry". Here's what his is written as:

$X$ is a noetherian scheme, $\mathcal{J}$ is a quasi-coherent sheaf of $\mathcal{O}_X$-modules, which has the structure of a sheaf of graded $\mathcal{O}_X$-algebras. Thus $\mathcal{J} \cong \bigoplus_{d\geq0} \mathcal{J}_d$ where $\mathcal{J}_d$ is the homogeneous part of degree $d$. We assume furthermore that $\mathcal{J}_0=\mathcal{O}_X$, that $\mathcal{J}_1$ is a coherent $\mathcal{O}_X$-module, and that $\mathcal{J}$ is locally generated by $\mathcal{J}_1$ as an $\mathcal{O}_X$-algebra. (It follows that $\mathcal{J}_d$ is coherent for all $d\geq 0$.)

This is verbatim from the text (modulo declaring a curly script letter I can't figure out to be $\mathcal{J}$), see 2 paragraphs after example II.7.8.6, in the section entitled "Proj, $\Bbb P(\mathcal{E})$, and Blowing Up" (page 160 in my version).

Unfortunately, this seems to suggest either a typo (although how close $1$ and $\infty$ are on one's keyboard, I'm not sure) or a slight change in definition that was glossed over. Maybe you can try emailing the person who's hosting the notes?