hcju/stacksqa huggingface dataset:
Let $U$ be a scheme and let $U \to X$ be a surjective \'etale morphism. As blowing up commutes with flat base change (Lemma \ref{lemma-flat-base-change-blowing-up}) we can prove each of these statements after base change to $U$. This reduces us to the case of schemes. In this case the result is Divisors, Lemma \ref{divisors-lemma-blow-up-reduced-scheme}.
Let $U$ be a scheme and let $U \to X$ be a surjective \'etale morphism. As blowing up commutes with flat base change (Lemma \ref{lemma-flat-base-change-blowing-up}) we can prove each of these statements after base change to $U$ (see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-proper-local}). This reduces us to the case of schemes. In this case the morphism $b$ is projective by Divisors, Lemma \ref{divisors-lemma-blowing-up-projective} hence proper by Morphisms, Lemma \ref{morphisms-lemma-locally-projective-proper}.
Part (1) follows from the fact that blowup commutes with flat base change (Lemma \ref{lemma-flat-base-change-blowing-up}), the fact that \'etale morphisms are flat, and that the base change of an \'etale morphism is \'etale. Part (3) then follows from the fact that taking the sheaf of sections supported on a closed commutes with pullback by \'etale morphisms, see Limits of Spaces, Lemma \ref{spaces-limits-lemma-sections-supported-on-closed-subset}. Part (2) follows from (3) applied to $\mathcal{F} = \mathcal{O}_X$.
Let $X'' \to X$ be the blowup of $X$ in $f^{-1}Z$. By the universal property of blowing up (Lemma \ref{lemma-universal-property-blowing-up}) there exists a commutative diagram $$ \xymatrix{ X'' \ar[r] \ar[d] & X \ar[d] \\ B' \ar[r] & B } $$ whence a morphism $i : X'' \to X \times_B B'$. The first assertion of the lemma is that $i$ is a closed immersion with image $X'$. The second assertion of the lemma is that $\mathcal{F}' = i_*\mathcal{F}''$ where $\mathcal{F}''$ is the strict transform of $\mathcal{F}$ with respect to the blowing up $X'' \to X$. We can check these assertions \'etale locally on $X$, hence we reduce to the case of schemes (Divisors, Lemma \ref{divisors-lemma-strict-transform}). Some details omitted.
Let $\mathcal{I} \subset \mathcal{O}_V$ be the finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I})$ is disjoint from $U \cap V$ and such that $V'$ is isomorphic to the blowup of $V$ in $\mathcal{I}$. Let $\mathcal{I}' \subset \mathcal{O}_{U \cup V}$ be the quasi-coherent sheaf of ideals whose restriction to $U$ is $\mathcal{O}_U$ and whose restriction to $V$ is $\mathcal{I}$. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-extend} there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_X$ whose restriction to $U \cup V$ is $\mathcal{I}'$. The lemma follows.
Let $\mathcal{I}_i \subset \mathcal{O}_X$ be the finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I}_i)$ is disjoint from $U$ and such that $X_i$ is isomorphic to the blowup of $X$ in $\mathcal{I}_i$. Set $\mathcal{I} = \mathcal{I}_1 \cdot \ldots \cdot \mathcal{I}_n$ and let $X'$ be the blowup of $X$ in $\mathcal{I}$. Then $X' \to X$ factors through $b_i$ by Lemma \ref{lemma-blowing-up-two-ideals}.
Choose a finite type quasi-coherent sheaf of ideals $\mathcal{I}$, resp.\ $\mathcal{J}$ such that $X \setminus U = V(\mathcal{I})$, resp.\ $X \setminus V = V(\mathcal{J})$, see Limits of Spaces, Lemma \ref{spaces-limits-lemma-quasi-coherent-finite-type-ideals}. Then $|V(\mathcal{I}\mathcal{J})| = |X|$. Hence $\mathcal{I}\mathcal{J}$ is a locally nilpotent sheaf of ideals. Since $\mathcal{I}$ and $\mathcal{J}$ are of finite type and $X$ is quasi-compact there exists an $n > 0$ such that $\mathcal{I}^n \mathcal{J}^n = 0$. We may and do replace $\mathcal{I}$ by $\mathcal{I}^n$ and $\mathcal{J}$ by $\mathcal{J}^n$. Whence $\mathcal{I} \mathcal{J} = 0$. Let $b : X' \to X$ be the blowing up in $\mathcal{I} + \mathcal{J}$. This is $U \cup V$-admissible as $|V(\mathcal{I} + \mathcal{J})| = |X| \setminus |U| \cup |V|$. We will show that $X'$ is a disjoint union of open subspaces $X' = X'_1 \amalg X'_2$ as in the statement of the lemma. \medskip\noindent Since $|V(\mathcal{I} + \mathcal{J})|$ is the complement of $|U \cup V|$ we conclude that $V \cup U$ is scheme theoretically dense in $X'$, see Lemmas \ref{lemma-blowing-up-gives-effective-Cartier-divisor} and \ref{lemma-complement-effective-Cartier-divisor}. Thus if such a decomposition $X' = X'_1 \amalg X'_2$ into open and closed subspaces exists, then $X'_1$ is the scheme theoretic closure of $U$ in $X'$ and similarly $X'_2$ is the scheme theoretic closure of $V$ in $X'$. Since $U \to X'$ and $V \to X'$ are quasi-compact taking scheme theoretic closures commutes with \'etale localization (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image}). Hence to verify the existence of $X'_1$ and $X'_2$ we may work \'etale locally on $X$. This reduces us to the case of schemes which is treated in the proof of Divisors, Lemma \ref{divisors-lemma-separate-disjoint-opens-by-blowing-up}.
Kernels and cokernels commute with taking underlying $A$-modules. Similarly for direct sums and colimits. In other words, these operations in $\text{Mod}_{(A, \text{d})}$ commute with the forgetful functor to the category of $A$-modules. This is not the case for products and limits. Namely, if $N_i$, $i \in I$ is a family of differential graded $A$-modules, then the product $\prod N_i$ in $\text{Mod}_{(A, \text{d})}$ is given by setting $(\prod N_i)^n = \prod N_i^n$ and $\prod N_i = \bigoplus_n (\prod N_i)^n$. Thus we see that the product does commute with the forgetful functor to the category of graded $A$-modules. A category with products and equalizers has limits, see Categories, Lemma \ref{categories-lemma-limits-products-equalizers}.
The map $\pi \circ \text{d}_L \circ s$ is compatible with the $A$-module structure and the gradings by construction. It is compatible with differentials by Homology, Lemmas \ref{homology-lemma-ses-termwise-split-cochain}. Let $R$ be the ring that $A$ is a differential graded algebra over. The equality of maps is a statement about $R$-modules. Hence this follows from Homology, Lemmas \ref{homology-lemma-ses-termwise-split-cochain} and \ref{homology-lemma-ses-termwise-split-long-cochain}.
The case $n = 1$ is without content. Lemma \ref{lemma-make-injective} is the case $n = 2$. Suppose we have constructed the diagram except for $M_n$. Apply Lemma \ref{lemma-make-injective} to the composition $M_{n - 1} \to L_{n - 1} \to L_n$. The result is a factorization $M_{n - 1} \to M_n \to L_n$ as desired.
By Lemma \ref{lemma-make-commute-map} we can replace $b$ and $b'$ by homotopic maps such that the right square of the left diagram commutes and the left square of the right diagram commutes. In other words, we have $\Im(b) \subset \Im(K_2 \to L_2)$ and $\Ker((b')^n) \supset \Im(K_2 \to L_2)$. Then $b \circ b' = 0$ as a map of modules.
Namely, let $(s', \pi')$ be a second choice of splittings as in Lemma \ref{lemma-admissible-ses}. Then we claim that $\delta$ and $\delta'$ are homotopic. Namely, write $s' = s + \alpha \circ h$ and $\pi' = \pi + g \circ \beta$ for some unique homomorphisms of $A$-modules $h : M \to K$ and $g : M \to K$ of degree $-1$. Then $g = -h$ and $g$ is a homotopy between $\delta$ and $\delta'$. The computations are done in the proof of Homology, Lemma \ref{homology-lemma-ses-termwise-split-homotopy-cochain}.
Let $a^{-1} : K_2 \to K_1$ be a homomorphism of differential graded $A$-modules which is inverse to $a$ in $K(\text{Mod}_{(A, \text{d})})$. Let $b^{-1} : L_2 \to L_1$ be a homomorphism of differential graded $A$-modules which is inverse to $b$ in $K(\text{Mod}_{(A, \text{d})})$. Let $c' : C(f_2) \to C(f_1)$ be the morphism from Lemma \ref{lemma-functorial-cone} applied to $f_1 \circ a^{-1} = b^{-1} \circ f_2$. If we can show that $c \circ c'$ and $c' \circ c$ are isomorphisms in $K(\text{Mod}_{(A, \text{d})})$ then we win. Hence it suffices to prove the following: Given a morphism of triangles $(1, 1, c) : (K, L, C(f), f, i, p)$ in $K(\text{Mod}_{(A, \text{d})})$ the morphism $c$ is an isomorphism in $K(\text{Mod}_{(A, \text{d})})$. By assumption the two squares in the diagram $$ \xymatrix{ L \ar[r] \ar[d]_1 & C(f) \ar[r] \ar[d]_c & K[1] \ar[d]_1 \\ L \ar[r] & C(f) \ar[r] & K[1] } $$ commute up to homotopy. By construction of $C(f)$ the rows form admissible short exact sequences. Thus we see that $(c - 1)^2 = 0$ in $K(\text{Mod}_{(A, \text{d})})$ by Lemma \ref{lemma-nilpotent}. Hence $c$ is an isomorphism in $K(\text{Mod}_{(A, \text{d})})$ with inverse $2 - c$.
Proof of (1). We have $C(\alpha) = L \oplus K$ and we simply define $C(\alpha) \to M$ via the projection onto $L$ followed by $\beta$. This defines a morphism of differential graded modules because the compositions $K^{n + 1} \to L^{n + 1} \to M^{n + 1}$ are zero. Choose splittings $s : M \to L$ and $\pi : L \to K$ with $\Ker(\pi) = \Im(s)$ and set $\delta = \pi \circ \text{d}_L \circ s$ as usual. To get a homotopy inverse we take $M \to C(\alpha)$ given by $(s , -\delta)$. This is compatible with differentials because $\delta^n$ can be characterized as the unique map $M^n \to K^{n + 1}$ such that $\text{d} \circ s^n - s^{n + 1} \circ \text{d} = \alpha \circ \delta^n$, see proof of Homology, Lemma \ref{homology-lemma-ses-termwise-split-cochain}. The composition $M \to C(f) \to M$ is the identity. The composition $C(f) \to M \to C(f)$ is equal to the morphism $$ \left( \begin{matrix} s \circ \beta & 0 \\ -\delta \circ \beta & 0 \end{matrix} \right) $$ To see that this is homotopic to the identity map use the homotopy $h : C(\alpha) \to C(\alpha)$ given by the matrix $$ \left( \begin{matrix} 0 & 0 \\ \pi & 0 \end{matrix} \right) : C(\alpha) = L \oplus K \to L \oplus K = C(\alpha) $$ It is trivial to verify that $$ \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) - \left( \begin{matrix} s \\ -\delta \end{matrix} \right) \left( \begin{matrix} \beta & 0 \end{matrix} \right) = \left( \begin{matrix} \text{d} & \alpha \\ 0 & -\text{d} \end{matrix} \right) \left( \begin{matrix} 0 & 0 \\ \pi & 0 \end{matrix} \right) + \left( \begin{matrix} 0 & 0 \\ \pi & 0 \end{matrix} \right) \left( \begin{matrix} \text{d} & \alpha \\ 0 & -\text{d} \end{matrix} \right) $$ To finish the proof of (1) we have to show that the morphisms $-p : C(\alpha) \to K[1]$ (see Definition \ref{definition-cone}) and $C(\alpha) \to M \to K[1]$ agree up to homotopy. This is clear from the above. Namely, we can use the homotopy inverse $(s, -\delta) : M \to C(\alpha)$ and check instead that the two maps $M \to K[1]$ agree. And note that $p \circ (s, -\delta) = -\delta$ as desired. \medskip\noindent Proof of (2). We let $\tilde f : K \to \tilde L$, $s : L \to \tilde L$ and $\pi : L \to L$ be as in Lemma \ref{lemma-make-injective}. By Lemmas \ref{lemma-functorial-cone} and \ref{lemma-third-isomorphism} the triangles $(K, L, C(f), i, p)$ and $(K, \tilde L, C(\tilde f), \tilde i, \tilde p)$ are isomorphic. Note that we can compose isomorphisms of triangles. Thus we may replace $L$ by $\tilde L$ and $f$ by $\tilde f$. In other words we may assume that $f$ is an admissible monomorphism. In this case the result follows from part (1).
Proof of TR1. By definition every triangle isomorphic to a distinguished one is distinguished. Also, any triangle $(K, K, 0, 1, 0, 0)$ is distinguished since $0 \to K \to K \to 0 \to 0$ is an admissible short exact sequence. Finally, given any homomorphism $f : K \to L$ of differential graded $A$-modules the triangle $(K, L, C(f), f, i, -p)$ is distinguished by Lemma \ref{lemma-the-same-up-to-isomorphisms}. \medskip\noindent Proof of TR2. Let $(X, Y, Z, f, g, h)$ be a triangle. Assume $(Y, Z, X[1], g, h, -f[1])$ is distinguished. Then there exists an admissible short exact sequence $0 \to K \to L \to M \to 0$ such that the associated triangle $(K, L, M, \alpha, \beta, \delta)$ is isomorphic to $(Y, Z, X[1], g, h, -f[1])$. Rotating back we see that $(X, Y, Z, f, g, h)$ is isomorphic to $(M[-1], K, L, -\delta[-1], \alpha, \beta)$. It follows from Lemma \ref{lemma-rotate-triangle} that the triangle $(M[-1], K, L, \delta[-1], \alpha, \beta)$ is isomorphic to $(M[-1], K, C(\delta[-1]), \delta[-1], i, p)$. Precomposing the previous isomorphism of triangles with $-1$ on $Y$ it follows that $(X, Y, Z, f, g, h)$ is isomorphic to $(M[-1], K, C(\delta[-1]), \delta[-1], i, -p)$. Hence it is distinguished by Lemma \ref{lemma-the-same-up-to-isomorphisms}. On the other hand, suppose that $(X, Y, Z, f, g, h)$ is distinguished. By Lemma \ref{lemma-the-same-up-to-isomorphisms} this means that it is isomorphic to a triangle of the form $(K, L, C(f), f, i, -p)$ for some morphism $f$ of $\text{Mod}_{(A, \text{d})}$. Then the rotated triangle $(Y, Z, X[1], g, h, -f[1])$ is isomorphic to $(L, C(f), K[1], i, -p, -f[1])$ which is isomorphic to the triangle $(L, C(f), K[1], i, p, f[1])$. By Lemma \ref{lemma-rotate-cone} this triangle is distinguished. Hence $(Y, Z, X[1], g, h, -f[1])$ is distinguished as desired. \medskip\noindent Proof of TR3. Let $(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$ be distinguished triangles of $K(\mathcal{A})$ and let $a : X \to X'$ and $b : Y \to Y'$ be morphisms such that $f' \circ a = b \circ f$. By Lemma \ref{lemma-the-same-up-to-isomorphisms} we may assume that $(X, Y, Z, f, g, h) = (X, Y, C(f), f, i, -p)$ and $(X', Y', Z', f', g', h') = (X', Y', C(f'), f', i', -p')$. At this point we simply apply Lemma \ref{lemma-functorial-cone} to the commutative diagram given by $f, f', a, b$.
Let us show that both sides correspond to graded $A$-bilinear maps $$ M' \times M \longrightarrow N^\bullet $$ compatible with differentials. We have seen this is true for the right hand side in Section \ref{section-tensor-product}. Given an element $g$ of the left hand side, the equality of More on Algebra, Lemma \ref{more-algebra-lemma-compose} determines a map of complexes of $R$-modules $g' : \text{Tot}(M' \otimes_R M) \to N^\bullet$. In other words, we obtain a graded $R$-bilinear map $g'' : M' \times M \to N^\bullet$ compatible with differentials. The $A$-linearity of $g$ translates immediately into $A$-bilinarity of $g''$.
Let us show that both sides correspond to graded $A$-bilinear maps $$ M \times M' \longrightarrow N^\bullet $$ compatible with differentials. We have seen this is true for the right hand side in Section \ref{section-tensor-product}. Given an element $g$ of the left hand side, the equality of More on Algebra, Lemma \ref{more-algebra-lemma-compose} determines a map of complexes $g' : \text{Tot}(M' \otimes_R M) \to N^\bullet$. We precompose with the commutativity constraint to get $$ \text{Tot}(M \otimes_R M') \xrightarrow{\psi} \text{Tot}(M' \otimes_R M) \xrightarrow{g'} N^\bullet $$ which corresponds to a graded $R$-bilinear map $g'' : M \times M' \to N^\bullet$ compatible with differentials. The $A$-linearity of $g$ translates immediately into $A$-bilinarity of $g''$. Namely, say $x \in M^e$ and $x' \in (M')^{e'}$ and $a \in A^n$. Then on the one hand we have \begin{align*} g''(x, ax') & = (-1)^{e(n + e')} g'(ax' \otimes x) \\ & = (-1)^{e(n + e')} g(ax')(x) \\ & = (-1)^{e(n + e')} (a \cdot g(x'))(x) \\ & = (-1)^{e(n + e') + n(n + e + e') + n} g(x')(xa) \end{align*} and on the other hand we have $$ g''(xa, x') = (-1)^{(e + n)e'} g'(x' \otimes xa) = (-1)^{(e + n)e'} g(x')(xa)  $$ which is the same thing by a trivial mod $2$ calculation of the exponents.
We will use that $K(\text{Mod}_{(A, \text{d})})$ is a triangulated category (Proposition \ref{proposition-homotopy-category-triangulated}). Let $F_\bullet$ be a filtration on $P$ as in property (P). The short exact sequence of Lemma \ref{lemma-property-P-sequence} produces a distinguished triangle. Hence by Derived Categories, Lemma \ref{derived-lemma-representable-homological} it suffices to show that $$ \Hom_{K(\text{Mod}_{(A, \text{d})})}(F_iP, N) = 0 $$ for all acyclic differential graded $A$-modules $N$ and all $i$. Each of the differential graded modules $F_iP$ has a finite filtration by admissible monomorphisms, whose graded pieces are direct sums of shifts $A[k]$. Thus it suffices to prove that $$ \Hom_{K(\text{Mod}_{(A, \text{d})})}(A[k], N) = 0 $$ for all acyclic differential graded $A$-modules $N$ and all $k$. This follows from Lemma \ref{lemma-hom-from-shift-free}.
Set $M = M_0$. We inductively choose short exact sequences $$ 0 \to M_{i + 1} \to P_i \to M_i \to 0 $$ where the maps $P_i \to M_i$ are chosen as in Lemma \ref{lemma-good-quotient}. This gives a ``resolution'' $$ \ldots \to P_2 \xrightarrow{f_2} P_1 \xrightarrow{f_1} P_0 \to M \to 0 $$ Then we set $$ P = \bigoplus\nolimits_{i \geq 0} P_i $$ as an $A$-module with grading given by $P^n = \bigoplus_{a + b = n} P_{-a}^b$ and differential (as in the construction of the total complex associated to a double complex) by $$ \text{d}_P(x) = f_{-a}(x) + (-1)^a \text{d}_{P_{-a}}(x) $$ for $x \in P_{-a}^b$. With these conventions $P$ is indeed a differential graded $A$-module. Recalling that each $P_i$ has a two step filtration $0 \to P_i' \to P_i \to P_i'' \to 0$ we set $$ F_{2i}P = \bigoplus\nolimits_{i \geq j \geq 0} P_j \subset \bigoplus\nolimits_{i \geq 0} P_i = P $$ and we add $P'_{i + 1}$ to $F_{2i}P$ to get $F_{2i + 1}$. These are differential graded submodules and the successive quotients are direct sums of shifts of $A$. By Lemma \ref{lemma-target-graded-projective} we see that the inclusions $F_iP \to F_{i + 1}P$ are admissible monomorphisms. Finally, we have to show that the map $P \to M$ (given by the augmentation $P_0 \to M$) is a quasi-isomorphism. This follows from Homology, Lemma \ref{homology-lemma-good-resolution-gives-qis}.
We will use that $K(\text{Mod}_{(A, \text{d})})$ is a triangulated category (Proposition \ref{proposition-homotopy-category-triangulated}). Let $F_\bullet$ be a filtration on $I$ as in property (I). The short exact sequence of Lemma \ref{lemma-property-I-sequence} produces a distinguished triangle. Hence by Derived Categories, Lemma \ref{derived-lemma-representable-homological} it suffices to show that $$ \Hom_{K(\text{Mod}_{(A, \text{d})})}(N, I/F_iI) = 0 $$ for all acyclic differential graded $A$-modules $N$ and all $i$. Each of the differential graded modules $I/F_iI$ has a finite filtration by admissible monomorphisms, whose graded pieces are products of $A^\vee[k]$. Thus it suffices to prove that $$ \Hom_{K(\text{Mod}_{(A, \text{d})})}(N, A^\vee[k]) = 0 $$ for all acyclic differential graded $A$-modules $N$ and all $k$. This follows from Lemma \ref{lemma-hom-into-shift-dual-free} and the fact that $(-)^\vee$ is an exact functor.
We will use the functors $N \mapsto N^\vee$ (from left to right differential graded modules and from right to left differential graded modules) constructed in Section \ref{section-modules-noncommutative-differential-graded} and all of their properties. For every $k \in \mathbf{Z}$ let $Q_k$ be the free left $A$-module with generators $x, y$ in degrees $k$ and $k + 1$. Define the structure of a left differential graded $A$-module on $Q_k$ by setting $\text{d}(x) = y$ and $\text{d}(y) = 0$. Arguing exactly as in the proof of Lemma \ref{lemma-good-quotient} we find a surjection $$ \bigoplus\nolimits_{i \in I} Q_{k_i} \longrightarrow M^\vee $$ of left differential graded $A$-modules. Then we can consider the injection $$ M \to (M^\vee)^\vee \to (\bigoplus\nolimits_{i \in I} Q_{k_i})^\vee = \prod\nolimits_{i \in I} I_{k_i} $$ where $I_k = Q_{-k}^\vee$ is the ``dual'' right differential graded $A$-module. Further, the short exact sequence $0 \to A[-k - 1] \to Q_k \to A[-k] \to 0$ produces a short exact sequence $0 \to A^\vee[k] \to I_k \to A^\vee[k + 1] \to 0$. \medskip\noindent The result of the previous paragraph produces $M \to I$ having properties (1) and (3). To obtain property (2), suppose $\overline{m} \in \Coker(\text{d}_M)$ is a nonzero element of degree $k$. Pick a map $\lambda : M^k \to \mathbf{Q}/\mathbf{Z}$ which vanishes on $\Im(M^{k - 1} \to M^k)$ but not on $m$. By Lemma \ref{lemma-hom-into-shift-dual-free} this corresponds to a homomorphism $M \to A^\vee[k]$ of differential graded $A$-modules which does not vanish on $m$. Hence we can achieve (2) by adding a product of copies of shifts of $A^\vee$.
Set $M = M_0$. We inductively choose short exact sequences $$ 0 \to M_i \to I_i \to M_{i + 1} \to 0 $$ where the maps $M_i \to I_i$ are chosen as in Lemma \ref{lemma-good-sub}. This gives a ``resolution'' $$ 0 \to M \to I_0 \xrightarrow{f_0} I_1 \xrightarrow{f_1} I_1 \to \ldots $$ Denote $I$ the differential graded $A$-module with graded parts $$ I^n = \prod\nolimits_{i \geq 0} I^{n - i}_i $$ and differential defined by $$ \text{d}_I(x) = f_i(x) + (-1)^i \text{d}_{I_i}(x) $$ for $x \in I_i^{n - i}$. With these conventions $I$ is indeed a differential graded $A$-module. Recalling that each $I_i$ has a two step filtration $0 \to I_i' \to I_i \to I_i'' \to 0$ we set $$ F_{2i}I^n = \prod\nolimits_{j \geq i} I^{n - j}_j \subset \prod\nolimits_{i \geq 0} I^{n - i}_i = I^n $$ and we add a factor $I'_{i + 1}$ to $F_{2i}I$ to get $F_{2i + 1}I$. These are differential graded submodules and the successive quotients are products of shifts of $A^\vee$. By Lemma \ref{lemma-source-graded-injective} we see that the inclusions $F_{i + 1}I \to F_iI$ are admissible monomorphisms. Finally, we have to show that the map $M \to I$ (given by the augmentation $M \to I_0$) is a quasi-isomorphism. This follows from Homology, Lemma \ref{homology-lemma-good-right-resolution-gives-qis}.
We know that $H^0$ is a homological functor by the long exact sequence of homology (\ref{equation-les}). The kernel of $H^0$ is the subcategory of acyclic objects and the arrows with induce isomorphisms on all $H^i$ are the quasi-isomorphisms. Thus this lemma is a special case of Derived Categories, Lemma \ref{derived-lemma-acyclic-general}. \medskip\noindent Set theoretical remark. The construction of the localization in Derived Categories, Proposition \ref{derived-proposition-construct-localization} assumes the given triangulated category is ``small'', i.e., that the underlying collection of objects forms a set. Let $V_\alpha$ be a partial universe (as in Sets, Section \ref{sets-section-sets-hierarchy}) containing $(A, \text{d})$ and where the cofinality of $\alpha$ is bigger than $\aleph_0$ (see Sets, Proposition \ref{sets-proposition-exist-ordinals-large-cofinality}). Then we can consider the category $\text{Mod}_{(A, \text{d}), \alpha}$ of differential graded $A$-modules contained in $V_\alpha$. A straightforward check shows that all the constructions used in the proof of Proposition \ref{proposition-homotopy-category-triangulated} work inside of $\text{Mod}_{(A, \text{d}), \alpha}$ (because at worst we take finite direct sums of differential graded modules). Thus we obtain a triangulated category $\text{Qis}_\alpha^{-1}K(\text{Mod}_{(A, \text{d}), \alpha})$. We will see below that if $\beta > \alpha$, then the transition functors $$ \text{Qis}_\alpha^{-1}K(\text{Mod}_{(A, \text{d}), \alpha}) \longrightarrow \text{Qis}_\beta^{-1}K(\text{Mod}_{(A, \text{d}), \beta}) $$ are fully faithful as the morphism sets in the quotient categories are computed by maps in the homotopy categories from P-resolutions (the construction of a P-resolution in the proof of Lemma \ref{lemma-resolve} takes countable direct sums as well as direct sums indexed over subsets of the given module). The reader should therefore think of the category of the lemma as the union of these subcategories.