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Extensions of homogeneous coordinate rings to {\(A_\infty\)}-algebras. (English) Zbl 1121.55005

For a projective \(d\)-dimensional variety \(X\) over a field \(k\) with very ample line bundle \(L\), the extended homogeneous coordinate ring \(A_L=\bigoplus_{p,q}H^q(X,L^{\otimes p})\) comes equipped with the structure of an \(A_\infty\)-algebra for which \(m_1\) is zero and all other multiplications \(m_n\) are homogeneous with respect to the internal degree \(p\). The author shows that this \(A_\infty\)-structure is unique whenever \(H^p(X,L^{\otimes q})\) vanishes for \(q\not =0,d\). In particular, the \(A_\infty\)-structure does not depend on the affine covering used for its construction.

MSC:

55N30 Sheaf cohomology in algebraic topology
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18D50 Operads (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
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