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Homogeneous complete intersection Hodge algebras on simplicial complexes

Since De Concini-Eisenbυd-Procesi [1] defined Hodge algebra, two special classes have been studied, one of which is an ordinal Hodge algebra and the other is a square-free Hodge algebra. An ordinal Hodge algebra ( = algebra with straightening laws, ASL, for short) have been investigated in detail and we know that an ASL reflects strongly a nature of a poset. On the other hand, let A be a square-free Hodge algebra. By [1], we can associate to A a unique simplicial complex Δ. Then A should accordingly reflect a nature of Δ. We call A a Hodge algebra on the simplicial complex Δ. The purpose of the present article is to classify the simplicial complex on which there exists a homogeneous complete intersection Hodge algebra of dimension<3. We often employ the arguments in [5]. In §1, we recall the definition of Hodge algebra and elementary definitions in topology. In §2, we give a classification of simplicial complexes Δ when there exists a homogeneous Hodge iC-algebra on Δ which is a complete intersection. Its proof is given in §3. The author would like to express his sincere gratitude to Professors M. Miyanίshi and S. Tsunoda who gave him useful comments and valuable suggestions and to Professor T. Hibi for precious advice.