This work concerns finite free complexes with finite-length homology over a commutative noetherian local
ring
. The focus is on
complexes that have length
,
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
is a strict complete intersection, and also on the Dutta multiplicity, when
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
.
Such a sequence is obtained by constructing an appropriate sequence of
sheaves on the associated projective variety.
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