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Simplicial types and polynomial algebras

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*Francisco Gomez*

**Address.** Department of Algebra, Geometry and Topology, University
of Malaga, AP 59 29080-Malaga, SPAIN
**E-mail:** fgomez@agt.cie.uma.es

**Abstract.** This paper shows that the simplicial type of a finite
simplicial complex K is determined by its algebra A of polynomial functions
on the baricentric coordinates with coefficients in any integral domain.
The link between K and A is done through certain admissible matrix associated
to K in a natural way. This result was obtained for the real numbers by
I.V. Savel'ev [5], using methods of real algebraic geometry. D. Kan and
E. Miller had shown in [2] that A determines the homotopy type of the polyhedron
associated to K and not only its rational homotopy type as it was previously
proved by D. Sullivan in [6].

**AMSclassification.** 58A10, 55U10, 14F40.

**Keywords.** Simplicial complex, algebraic de Rham complex, Sullivan's
de Rham complex.