Upper bounds for the Betti numbers of graded ideals of a given length in the exterior algebra

Let E be the exterior algebra of a finite dimensional vector space over a field It. Hilbert functions and Betti numbers of graded ideals I subset of E are studied. It is shown that there exists a unique graded ideal J subset of E such that for all graded ideals I subset of E generated in degree greater than or equal to 2 such that dim(K)E/J = dim(K) E/I one has beta(i)(E/J) greater than or equal to beta(i)(E/I) for all i. Here beta(i)(M) = dim(K)Tor(i)(E)(M, K) is the i-th Betti number of a graded E-module M. The unique ideal J with the above property has "maximal" Hilbert function. Our result is the analogue of a theorem of Valla [10] which he proved for graded ideals in the polynomial ring. We further prove a similar inequality for the Bass numbers of E/I, and give a combinatorial application.