In recent years, the RFRS condition has been used to analyze virtual fibering in 3-manifold topology. Agol’s work shows that any 3-manifold with zero Euler characteristic satisfying the RFRS condition on its fundamental group virtually fibers over the circle. In this note we will show that a finitely generated nilpotent group is either virtually abelian or is not virtually RFRS. As a corollary,...

Fix integers r, k, g with r > k > 0 and g ≥ 2. Let X be an integral projective curve with g := pa(X) and E a rank r torsion free sheaf on X which is a flat limit of a family of locally free sheaves on X. Here we prove the existence of a rank k subsheaf A of E such that r(deg(A)) ≥ k(deg(E)) − k(r − k)g. We show that for every g ≥ 9 there is an integral projective curve X,X not Gorenstein, and a...

An ideal I in a Noetherian ringR is normally torsion-free if Ass(R/I) = Ass(R/I) for all t ≥ 1. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a squarefree monomial ideal I is minimally not normally torsion-free then the least power t such that I has embedded primes is bigger than β1, where β1 is the monomial grade o...

We prove a Freiheitssatz for one-relator products of torsion-free groups, where the relator has syllable length at most [Formula: see text]. This result applications to equations over groups: in particular singular equation text] group solution some overgroup.