Lower bounds on projective levels of complexes

Abstract

For an associative ring $R$, the projective level of a complex $F$ is the smallest number of mapping cones needed to build $F$ from projective $R$-modules. We establish lower bounds for the projective level of $F$ in terms of the vanishing of homology of $F$. We then use these bounds to derive a new version of The New Intersection Theorem for level when $R$ is a commutative Noetherian local ring.Comment: To appear in the Journal of Algebra. In this new version, the paper has been rewritten to study projective levels, and to account for the existence of balanced big Cohen-Macaulay algebra

Level and Gorenstein Projective Dimension

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We investigate the relationship between the level of a bounded complex over a commutative ring with respect to the class of Gorenstein projective modules and other invariants of the complex or ring, such as projective dimension, Gorenstein projective dimension, and Krull dimension. The results build upon work done by J. Christensen [6], H. Altmann et al [1], and Avramov et al [3] for levels with respect to the class of finitely generated projective modules.

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“…In this context, it is interesting to compare the level of an object with other more familiar homological invariants. For instance, the level of a finitely generated module (considered as a complex concentrated in degree zero) with respect to the ring is one more than the projective dimension of the module ( [6]; see also [1,Cor. 2.2]).…”

Big Cohen-Macaulay modules, morphisms of perfect complexes, and intersection theorems in local algebra

“…Equality holds if R is local and x is a system of parameters; see Theorem 4.2 below. However, span K(x) − level R K(x) can be arbitrarily large; see [1,Section 3].…”