The Rees algebra of a two-Borel ideal is Koszul

Abstract

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.

Koszul blowup algebras associated to three-dimensional Ferrers diagrams

“…One can verify directly that (14) minus (13) equals one, which corresponds to the 1 in (12). Thus, we have established (12).…”

“…Next, we consider the Rees algebra R(I D ) of I D . The strategy is similar to that in [12,Section 6]. Then there exists an integer k, and an integer q < j ≤ n such that x q f k x j ∈ I.…”

Rees algebras of ideals of star configurations

“…On the other hand, only a few classes of ideals have well-understood free resolutions, and usually a free resolution for I does not provide information on the free resolutions of its powers I m . Nevertheless, the problem becomes manageable if one imposes algebraic conditions on a presentation matrix of I (see for instance [29,25,21]), especially when one can exploit methods from algebraic combinatorics (see for instance [31,18,10,8,12,15,1]).…”

Multi-Rees Algebras of Strongly Stable Ideals

“…Note that when the first parameter r is equal to one, the objects are the Rees algebras of strongly stable ideals and the Koszulness of these objects are studied by controlling the remaining two parameters. Authors of [4], [10], and [11] investigate the Koszulness of the Rees algebras of strongly stable ideals (see Section 4 for details).…”

“…In [11,Question 6.4], the authors conclude their paper with a question asking whether the multi-Rees algebra of strongly stable ideals I 1 , . .…”