  This work concerns finite free complexes with finite length homology over a
commutative noetherian local ring $R$. The focus is on complexes that have
length $\mathrm{dim}\, R$, which is the smallest possible value, and in
particular on free resolutions of modules of finite length and finite
projective dimension. Lower bounds are obtained on the Euler characteristic of
such short complexes when $R$ is a strict complete intersection, and also on
the Dutta multiplicity, when $R$ is the localization at its maximal ideal of a
standard graded algebra over a field of positive prime characteristic. The key
idea in the proof is the construction of a suitable Ulrich module, or, in the
latter case, a sequence of modules that have the Ulrich property
asymptotically, and with good convergence properties in the rational
Grothendieck group of $R$. Such a sequence is obtained by constructing an
appropriate sequence of sheaves on the associated projective variety.
