It gives a class of p-Borel principal ideals of a polynomial algebra over a field K for which the graded Betti numbers do not depend on the characteristic of K and the Koszul homology modules have monomial cyclic basis. Also it shows that all principal p-Borel ideals have binomial cycle basis on Koszul homology modules.

For symplectic Lie algebras sp(2n,C), denote by b and n its Borel subalgebra and maximal nilpotent subalgebra, respectively. We construct a relationship between the abelian ideals of b and the cohomology of n with trivial coefficients. By this relationship, we can enumerate the number of abelian ideals of b with certain dimension via the Poincaré polynomials of Weyl groups of type An−1 and Cn.

A family of 2א0 Borel ideals on a compact lattice are constructed such that no two quotient lattices by ideals from this family are isomorphic.

We obtain some results about Borel maps with meager fibers on completely metrizable separable spaces. The results are related to a recent dichotomy by Sabok and Zapletal, concerning Borel maps and σ-ideals generated by closed sets. In particular, we give a “classical” proof of this dichotomy. We shall also show that for certain natural σ-ideals I generated by closed sets in compact metrizable s...

In this paper, it is shown that the Hilbert series of a Borel type ideal may be computed within a complexity which is polynomial in Dn where n + 1 is the number of unknowns and D is the highest degree of a minimal generator of input (monomial) ideal.