Let M and N be two monomials of the same degree, and let I be the smallest Borel ideal containing M and N . We show that the toric ring of I is Koszul by constructing a quadratic Gröbner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul.

This paper explores the relation between multi-Rees algebras and ideals that arise in the study of toric dynamical systems from the theory of chemical reaction networks.Date: February 7, 2019. 2010 Mathematics Subject Classification. 13A30, 92C42.

Let k be a field and R a standard graded k-algebra. We denote by H R the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of R. We discuss the relationship between the multiplicative structure of H R and the property that R is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen-Macaulay local ring are rational, sharing a common denominator.

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