Ag.algic geometry – Tension with finite plane dimension complex in the derived category


Let $ (R, m) $ to be a local Noetherian ring, and $ X $, $ Y $ to be finite generation complexes $ R $ modules. assume $ X $ is delimited above and $ Y $ is delimited below. Let $ S $ bean $ R $-Algebra of dimension to finished flat.

Q. 1) Prove that $$ { bf R} Hom_R (X, Y) otimes_ {R} ^ { bf L} S cong { bf R} Hom_S (X otimes_ {R} ^ { bf L} S, Y otimes_ {R} ^ { bf L} S) $$.

Q. 2) If $ X $ and $ Y $ are $ R $ modules such as $ Tor_n ^ R (X, S) = 0 = Tor_n ^ R (Y, S) $ for everyone $ n geq 1 $ then prove that $$ { bf R} Hom_R (X, Y) otimes_R ^ { bf L} S cong { bf R} Hom_S (X otimes_RS, Y otimes_RS) $$.

PS: The answers could be simple, but it would be very useful for you to explain them with kindness. Thanks in advance.