Universally Koszul algebras defined by monomials. (Q2568690)

The author characterizes the universally Koszul algebras defined by monomials [see \textit{R. Fröberg} in: Adv. Commutative Ring Theory, Proc. 3rd Int. Conf., Lect. Notes Pure Appl. Math. 205, 337--350 (1999; Zbl 0962.13009)]. Main results: (a) (Theorem 5) Let \(R\) be an algebra defined over a field \(K\) of characteristic different from 2 by an ideal \(I\) generated by monomials of degree 2 in a set of variables \(X\). The following are equivalent: (1) \(R\) is universally Koszul. (2) \(R\) is obtained from the algebra \(H(n)\) by iterated polynomial extensions and fiber products, where \(H(n)= K[x_1,\dots,x_n]/I\), \(I= (x_1,\dots, x_{n-1})^2+ (x_n^2)\), \(n\geq 0\). (3) The restriction of \(I\) to any subset of variables of \(X\) does not give an ideal of types listed in lemma 4 (namely, generated by: (i) \((xy,z^2)\); (ii) \((x^2,xy,z^2)\); (iii) a monomial ideal whose squarefree generators are \(xy\), \(yz\), \(zt\); (iv) a monomial ideal whose squarefree generators are \(xy\), \(zt\); (v) \((x^2,y^2,z^2)\). (b) (Theorem 6) Under the same assumptions for \(R\) except the characteristic is 2, the following statements are equivalent: (1) \(R\) is universally Koszul. (2) \(R\) is obtained from the field \(K\) by iterated polynomial extensions, fiber products and extensions of the form \(R=K[x_1,\dots,x_n]/I\), with \((x_i)^2\) in \(I\) for all \(i\). (3) The restriction of \(I\) to any subset of variables \(X\) does not give an ideal of type (i)--(iv) from the above list in theorem 5, (3).