Fix n and let Pn be projective n-space. Let S=k[x0,…,xn]. Set A0=⨁d≥0H0(Pn,O(d)) and An=⨁d<−nHn(Pn,O(d)).

I have been told that it is a "well-known" result that A0⊕An has an A∞-algebra structure that extends the usual algebra structure on A0 and so that using the higher multiplication maps the algebra is generated by H0(Pn,O(1)) and Hn(Pn,O(−n−1)).

Question: Is there a reference or can someone sketch a proof of the statement above?There is probably a more general statement involving a scheme X and an ample line bundle L on X, though I'd already be interested in this special case.

Some thoughts: I have tried to realize this by putting a dga structure on the standard Cech complex (as in the proof of Theorem III.5.1 of Hartshorne's Algebraic Geometry book) and it looks like it works, but there is a problem with grading shift. To be precise, we have a complex C∙ so that Cd=⨁I⊂{0,…,n}S[x−1I] where xI=xi0⋯xid and I ranges over all subsets of size d+1. Then there is an obvious multiplication structure on C∙ but with a shift: Cd⊗Ce→Cd+e+1.So it doesn't quite work (and also shifting the indexing does not resolve the problem because then I would be removing the algebra structure on A0) but it seems that this shift by 1 is a common occurrence (like the Whitehead product on the homotopy groups of a space).

**Answer**

**Attribution***Source : Link , Question Author : Steven Sam , Answer Author : Community*