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arXiv · 2112.09680

Relative perfect complexes

Leovigildo Alonso·Ana Jeremias·Fernando Sancho
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Abstract

Let $f \colon X \to Y$ be a morphism of concentrated schemes. We characterize $f$-perfect complexes $\mathcal{E}$ as those such that the functor $\mathcal{E} \otimes^{\mathbf{L}}_X \mathbf{L} f^*-$ preserves bounded complexes. We prove, as a consequence, that a quasi-proper morphism takes relative perfect complexes into perfect ones. We obtain a generalized version of the semicontinuity theorem of dimension of cohomology and Grauert's base change of the fibers. Finally, a bivariant theory of the Grothendieck group of perfect complexes is developed.

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Leovigildo Alonso, Ana Jeremias, Fernando Sancho. 2021-12-17. Relative perfect complexes. https://doi.org/10.1007/s00209-023-03294-7

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