Finiteness of homological dimensions of RHom complexes

Abstract

This paper studies derived Hom complexes with finite projective, Gorenstein projective, or complete intersection dimension over a Noetherian local ring $R$. We establish several stability results concerning finite complete intersection homological dimension of derived Hom complexes. In addition, for $R$-complexes $M$ and $N$ we study when ${\mathrm{RHom}}_R(M,N)$ has finite projective, complete intersection dimension or Gorenstein projective dimension under either of the assumptions that ${\mathrm{Gpd}}_R(N)<\infty$ or ${\mathrm{Gid}}_R(M)<\infty$. We derive isomorphisms that mimic derived Hom evaluation and derived tensor evaluation when the derived Hom complex has finite Gorenstein projective dimension; while finite projective or complete intersection dimension of the derived Hom complex yields new stability results.

arXiv comment15 pages. The previous version of the paper has been split in half in order to improve readability. This version is the first half of the original version, and the second half will be posted soon