## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 5, Number 1 (2003), 407-421.

### Extensions of homogeneous coordinate rings to $A_ \infty$-algebras

#### Abstract

We study $A_\infty$-structures extending the natural algebra structure on the
cohomology of $\oplus_{n\in\mathbb{Z}} L^n$, where $L$ is a very ample line bundle on a
projective $d$-dimensional variety $X$ such that $H^i(X,L^n)=0$ for 0 > i
> d and all $ n \in \mathbb{Z}$. We prove that there exists a unique such
*nontrivial*$A_{\infty}$-structure up to a strict
$A_{\infty}$-isomorphism (i.e., an $A_{\infty}$-isomorphism with the identity as
the first structure map) and rescaling.

In the case when $X$ is a curve we also compute the group of strict $A_{\infty}$-automorphisms of this $A_{\infty}$-structure.

#### Article information

**Source**

Homology Homotopy Appl., Volume 5, Number 1 (2003), 407-421.

**Dates**

First available in Project Euclid: 13 February 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1139839940

**Mathematical Reviews number (MathSciNet)**

MR2072342

**Zentralblatt MATH identifier**

1121.55005

**Subjects**

Primary: 18E30: Derived categories, triangulated categories

Secondary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)

#### Citation

Polishchuk, A. Extensions of homogeneous coordinate rings to $A_ \infty$-algebras. Homology Homotopy Appl. 5 (2003), no. 1, 407--421. https://projecteuclid.org/euclid.hha/1139839940