# 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.