Lemma 23.12.1. In the situation above, if $R$ is Noetherian, then for every $n$ there exists an $N \geq n$ and maps

\[ K_ N \to A \to R/(f_1^ N, \ldots , f_ r^ N)\quad \text{and}\quad A \to K_ n \]

with the following properties

$(A, \text{d}, \gamma )$ is as in Definition 23.6.5,

$A \to R/(f_1^ N, \ldots , f_ r^ N)$ is a quasi-isomorphism,

the composition $K_ N \to A \to R/(f_1^ N, \ldots , f_ r^ N)$ is the canonical map,

the composition $K_ N \to A \to K_ n$ is the transition map,

$A_0 = R \to R/(f_1^ N, \ldots , f_ r^ N)$ is the canonical surjection,

$A$ is a graded divided power polynomial algebra over $R$ with finitely many generators in each degree, and

$A \to K_ n$ is a homomorphism of differential graded $R$-algebras compatible with divided powers which induces the canonical map $R/(f_1^ N, \ldots , f_ r^ N) \to R/(f_1^ n, \ldots , f_ r^ n)$ on homology in degree $0$.

Condition (4) means that $A$ is constructed out of $A_0$ by successively adjoining a finite set of variables $T$ in each degree $> 0$ as in Example 23.6.2 or 23.6.3.

**Proof.**
Fix $n$. If $r = 0$, then we can just pick $A = R$. Assume $r > 0$. By More on Algebra, Lemma 15.94.1 (translated into the language of chain complexes) we can choose

\[ n_{r} > n_{r - 1} > \ldots > n_1 > n_0 = n \]

such that the transition maps $K_{n_{i + 1}} \to K_{n_ i}$ on Koszul algebras (see above) induce the zero map on homology in degrees $> 0$. We will prove the lemma with $N = n_ r$.

We will construct $A$ exactly as in the statement and proof of Lemma 23.6.9. Thus we will have

\[ A = \mathop{\mathrm{colim}}\nolimits A(m),\quad \text{and}\quad A(0) \to A(1) \to A(2) \to \ldots \to R/(f_1^ N, \ldots , f_ r^ N) \]

This will immediately give us properties (1), (2), (5), and (6). To finish the proof we will construct the $R$-algebra maps $K_ N \to A \to K_ n$. To do this we will construct

an isomorphism $A(1) \to K_ N = K_{n_ r}$,

a map $A(2) \to K_{n_{r - 1}}$,

$\ldots $

a map $A(r) \to K_{n_1}$,

a map $A(r + 1) \to K_{n_0} = K_ n$, and

a map $A \to K_ n$.

In each of these steps the map constructed will be between differential graded algebras compatibly endowed with divided powers and each of the maps will be compatible with the previous one via the transition maps between the Koszul algebras and each of the maps will induce the obvious canonical map on homology in degree $0$.

Recall that $A(0) = R$. For $m = 1$, the proof of Lemma 23.6.9 chooses $A(1) = R\langle T_1, \ldots , T_ r\rangle $ with $T_ i$ of degree $1$ and with $\text{d}(T_ i) = f_ i^ N$. Namely, the $f_ i^ N$ are generators of the kernel of $A(0) \to R/(f_1^ N, \ldots , f_ r^ N)$. Thus for $A(1) \to K_ N = K_{n_ r}$ we use the map

\[ \varphi _1 : A(1) \longrightarrow K_{n_ r},\quad T_ i \longmapsto \xi _ i \]

which is an isomorphism.

For $m = 2$, the construction in the proof of Lemma 23.6.9 chooses generators $e_1, \ldots , e_ t \in \mathop{\mathrm{Ker}}(\text{d} : A(1)_1 \to A(1)_0)$. The construction proceeds by taking $A(2) = A(1)\langle T_1, \ldots , T_ t\rangle $ as a divided power polynomial algebra with $T_ i$ of degree $2$ and with $\text{d}(T_ i) = e_ i$. Since $\varphi _1(e_ i)$ is a cocycle in $K_{n_ r}$ we see that its image in $K_{n_{r - 1}}$ is a coboundary by our choice of $n_ r$ and $n_{r - 1}$ above. Hence we can construct the following commutative diagram

\[ \xymatrix{ A(1) \ar[d] \ar[r]_{\varphi _1} & K_{n_ r} \ar[d] \\ A(2) \ar[r]^{\varphi _2} & K_{n_{r - 1}} } \]

by sending $T_ i$ to an element in degree $2$ whose boundary is the image of $\varphi _1(e_ i)$. The map $\varphi _2$ exists and is compatible with the differential and the divided powers by the universal of the divided power polynomial algebra.

The algebra $A(m)$ and the map $\varphi _ m : A(m) \to K_{n_{r + 1 - m}}$ are constructed in exactly the same manner for $m = 2, \ldots , r$.

Given the map $A(r) \to K_{n_1}$ we see that the composition $H_ r(A(r)) \to H_ r(K_{n_1}) \to H_ r(K_{n_0}) \subset (K_{n_0})_ r$ is zero, hence we can extend this to $A(r + 1) \to K_{n_0} = K_ n$ by sending the new polynomial generators of $A(r + 1)$ to zero.

Having constructed $A(r + 1) \to K_{n_0} = K_ n$ we can simply extend to $A(r + 2), A(r + 3), \ldots $ in the only possible way by sending the new polynomial generators to zero. This finishes the proof.
$\square$

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