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 $\dim <!--FUNCTION\; APPLICATION-->R$, 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.

这项工作涉及在可交换的诺特局部环上具有有限长度同调的有限自由复合体$R$. 重点是有长度的复合体$\mathrm{暗淡}<!--FUNCTION\; APPLICATION-->R$，这是最小的可能值，特别是在有限长度和有限射影维度的模块的自由分辨率上。在这种短配合物的欧拉特性上获得下界，当$R$是一个严格的完全交集，并且在 Dutta 多重性上，当$R$是标准分级代数在正素数特征域上的最大理想局部化。证明中的关键思想是构造一个合适的 Ulrich 模块，或者在后一种情况下，是一系列渐近具有 Ulrich 属性的模块，并且在有理格洛腾迪克群中具有良好的收敛性$R$. 这样的序列是通过在相关的投射变体上构建适当的滑轮序列来获得的。