A∞A_\infty structure on sum of twists of structure sheaf

Fix n and let Pn be projective n-space. Let S=k[x0,,xn]. Set A0=d0H0(Pn,O(d)) and An=d<nHn(Pn,O(d)).

I have been told that it is a "well-known" result that A0An 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(n1)).

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[x1I] where xI=xi0xid and I ranges over all subsets of size d+1. Then there is an obvious multiplication structure on C but with a shift: CdCeCd+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

Leave a Comment