Let R be a commutative ring, and let L and L' be R-modules. We investigate finiteness conditions (e.g., noetherian, artinian, mini-max, Matlis reflexive) of the modules ExtRi(L,L') and ToriR(L,L') when L and L' satisfy combinations of these finiteness conditions. For instance, if R is noetherian, then given R-modules M and M' such that M is Matlis reflexive and M' is mini-max (e.g., noetherian or artinian), we prove that ExtRi(M,M'), ExtRi(M',M), and ToriR(M,M') are Matlis reflexive over R for all i≥0 and that ExtRi(M,M')∨≅ToriR(M,M'∨) and ExtRi(M',M)∨≅ToriR(M',M∨).
Bibliographical noteFunding Information:
This material is based on work supported by North Dakota EPSCoR and National Science Foundation Grant EPS-0814442 . Micah Leamer was supported by a GAANN grant from the Department of Education. Sean Sather-Wagstaff was supported by a grant from the NSA .
- Bass number
- Betti number
- Matlis duality
- Tensor product