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Let K be a field of characteristic 0 and consider exterior algebras of finitedimensional K-vector spaces. In this short paper we exhibit principal quadricideals in a family whose Castelnuovo-Mumford regularity is unbounded. Thisnegatively answers the analogue of Stillman's Question for exterior algebrasposed by I. Peeva. We show that these examples are dual to modules overpolynomial rings that yield counterexamples to a conjecture of J. Herzog on theBetti numbers in the linear strand of syzygy modules.

Stillman's Question for Exterior Algebras and Herzog's Conjecture on Betti Numbers of Syzygy Modules