Lemma 15.83.7. Let R = \mathop{\mathrm{colim}}\nolimits _{i \in I} R_ i be a filtered colimit of rings. Let 0 \in I and R_0 \to A_0 be a flat ring map of finite presentation. For i \geq 0 set A_ i = R_ i \otimes _{R_0} A_0 and set A = R \otimes _{R_0} A_0.
Given an R-perfect K in D(A) there exists an i \in I and an R_ i-perfect K_ i in D(A_ i) such that K \cong K_ i \otimes _{A_ i}^\mathbf {L} A in D(A).
Given K_0, L_0 \in D(A_0) with K_0 pseudo-coherent and L_0 finite tor dimension over R_0, then we have
\mathop{\mathrm{Hom}}\nolimits _{D(A)}(K_0 \otimes _{A_0}^\mathbf {L} A, L_0 \otimes _{A_0}^\mathbf {L} A) = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(A_ i)}(K_0 \otimes _{A_0}^\mathbf {L} A_ i, L_0 \otimes _{A_0}^\mathbf {L} A_ i)
In particular, the triangulated category of R-perfect complexes over A is the colimit of the triangulated categories of R_ i-perfect complexes over A_ i.
Proof.
By Algebra, Lemma 10.127.6 the category of finitely presented A-modules is the colimit of the categories of finitely presented A_ i-modules. Given this, Algebra, Lemma 10.168.1 tells us that category of R-flat, finitely presented A-modules is the colimit of the categories of R_ i-flat, finitely presented A_ i-modules. Thus the characterization in Lemma 15.83.4 proves that (1) is true.
To prove (2) we choose P_0^\bullet representing K_0 and F_0^\bullet representing L_0 as in Lemma 15.83.6. Then E_0^\bullet = \mathop{\mathrm{Hom}}\nolimits ^\bullet (P_0^\bullet , F_0^\bullet ) satisfies
H^0(E_0^\bullet \otimes _{R_0} R_ i) = \mathop{\mathrm{Hom}}\nolimits _{D(A_ i)}(K_0 \otimes _{A_0}^\mathbf {L} A_ i, L_0 \otimes _{A_0}^\mathbf {L} A_ i)
and
H^0(E_0^\bullet \otimes _{R_0} R) = \mathop{\mathrm{Hom}}\nolimits _{D(A)}(K_0 \otimes _{A_0}^\mathbf {L} A, L_0 \otimes _{A_0}^\mathbf {L} A)
by the lemma. Thus the result because tensor product commutes with colimits and filtered colimits are exact (Algebra, Lemma 10.8.8).
\square
Comments (0)