Remark 15.63.20. Let $R$ be ring map. Let $L$, $M$, $N$ be $R$-modules. Consider the canonical map

$\mathop{\mathrm{Hom}}\nolimits _ R(M, N) \otimes _ R L \to \mathop{\mathrm{Hom}}\nolimits _ R(M, N \otimes _ R L)$

Choose a two term free resolution $F_1 \to F_0 \to M \to 0$. Assuming $L$ flat over $R$ we obtain a commutative diagram

$\xymatrix{ 0 \ar[r] & \mathop{\mathrm{Hom}}\nolimits _ R(M, N) \otimes _ R L \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits _ R(F_0, N) \otimes _ R L \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits _ R(F_1, N) \otimes _ R L \ar[d] \\ 0 \ar[r] & \mathop{\mathrm{Hom}}\nolimits _ R(M, N \otimes _ R L) \ar[r] & \mathop{\mathrm{Hom}}\nolimits _ R(F_0, N \otimes _ R L) \ar[r] & \mathop{\mathrm{Hom}}\nolimits _ R(F_1, N \otimes _ R L) }$

with exact rows. We conclude that if $F_0$ and $F_1$ are finite free, i.e., if $M$ is finitely presented, then the first displayed map is an isomorphism. Similarly, if $M$ is $(-m)$-pseudo-coherent and still assuming $L$ is flat over $R$, then the map

$\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N) \otimes _ R L \to \mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N \otimes _ R L)$

is an isomorphism for $i < m$.

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