×

Universally Koszul algebras defined by monomials. (English) Zbl 1081.13505

The author characterizes the universally Koszul algebras defined by monomials [see 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).

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13D02 Syzygies, resolutions, complexes and commutative rings
13B25 Polynomials over commutative rings

Citations:

Zbl 0962.13009

Software:

CoCoA
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] A. CONCA, Universally Koszul algebras, Math. Ann., 317, no. 2 (2000), pp. 329-346. Zbl0960.13003 MR1764242 · Zbl 0960.13003 · doi:10.1007/s002080000100
[2] R. FRÖBERG, Koszul algebras, Advances in commutative ring theory (Fez, 1997), 337-350, Lecture Notes in Pure and Appl. Math., 205, Dekker, New York, 1999. Zbl0962.13009 MR1767430 · Zbl 0962.13009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.