The Stacks project

Proposition 89.5.2. Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map. There exists a simplicial $A$-algebra $P_\bullet $ with an augmentation $\epsilon : P_\bullet \to B$ such that each $P_ n$ is a polynomial algebra of finite type over $A$ and such that $\epsilon $ is a trivial Kan fibration of simplicial sets.

Proof. Let $\mathcal{A}$ be the category of $A$-algebra maps $C \to B$. In this proof our simplicial objects and skeleton and coskeleton functors will be taken in this category.

Choose a polynomial algebra $P_0$ of finite type over $A$ and a surjection $P_0 \to B$. As a first approximation we take $P_\bullet = \text{cosk}_0(P_0)$. In other words, $P_\bullet $ is the simplicial $A$-algebra with terms $P_ n = P_0 \times _ A \ldots \times _ A P_0$. (In the final paragraph of the proof this simplicial object will be denoted $P^0_\bullet $.) By Simplicial, Lemma 14.32.3 the map $P_\bullet \to B$ is a trivial Kan fibration of simplicial sets. Also, observe that $P_\bullet = \text{cosk}_0 \text{sk}_0 P_\bullet $.

Suppose for some $n \geq 0$ we have constructed $P_\bullet $ (in the final paragraph of the proof this will be $P^ n_\bullet $) such that

  1. $P_\bullet \to B$ is a trivial Kan fibration of simplicial sets,

  2. $P_ k$ is a finitely generated polynomial algebra for $0 \leq k \leq n$, and

  3. $P_\bullet = \text{cosk}_ n \text{sk}_ n P_\bullet $

By Lemma 89.5.1 we can find a finitely generated polynomial algebra $Q$ over $A$ and a surjection $Q \to P_{n + 1}$. Since $P_ n$ is a polynomial algebra the $A$-algebra maps $s_ i : P_ n \to P_{n + 1}$ lift to maps $s'_ i : P_ n \to Q$. Set $d'_ j : Q \to P_ n$ equal to the composition of $Q \to P_{n + 1}$ and $d_ j : P_{n + 1} \to P_ n$. We obtain a truncated simplicial object $P'_\bullet $ of $\mathcal{A}$ by setting $P'_ k = P_ k$ for $k \leq n$ and $P'_{n + 1} = Q$ and morphisms $d'_ i = d_ i$ and $s'_ i = s_ i$ in degrees $k \leq n - 1$ and using the morphisms $d'_ j$ and $s'_ i$ in degree $n$. Extend this to a full simplicial object $P'_\bullet $ of $\mathcal{A}$ using $\text{cosk}_{n + 1}$. By functoriality of the coskeleton functors there is a morphism $P'_\bullet \to P_\bullet $ of simplicial objects extending the given morphism of $(n + 1)$-truncated simplicial objects. (This morphism will be denoted $P^{n + 1}_\bullet \to P^ n_\bullet $ in the final paragraph of the proof.)

Note that conditions (b) and (c) are satisfied for $P'_\bullet $ with $n$ replaced by $n + 1$. We claim the map $P'_\bullet \to P_\bullet $ satisfies assumptions (1), (2), (3), and (4) of Simplicial, Lemmas 14.32.1 with $n + 1$ instead of $n$. Conditions (1) and (2) hold by construction. By Simplicial, Lemma 14.19.14 we see that we have $P_\bullet = \text{cosk}_{n + 1}\text{sk}_{n + 1}P_\bullet $ and $P'_\bullet = \text{cosk}_{n + 1}\text{sk}_{n + 1}P'_\bullet $ not only in $\mathcal{A}$ but also in the category of $A$-algebras, whence in the category of sets (as the forgetful functor from $A$-algebras to sets commutes with all limits). This proves (3) and (4). Thus the lemma applies and $P'_\bullet \to P_\bullet $ is a trivial Kan fibration. By Simplicial, Lemma 14.30.4 we conclude that $P'_\bullet \to B$ is a trivial Kan fibration and (a) holds as well.

To finish the proof we take the inverse limit $P_\bullet = \mathop{\mathrm{lim}}\nolimits P^ n_\bullet $ of the sequence of simplicial algebras

\[ \ldots \to P^2_\bullet \to P^1_\bullet \to P^0_\bullet \]

constructed above. The map $P_\bullet \to B$ is a trivial Kan fibration by Simplicial, Lemma 14.30.5. However, the construction above stabilizes in each degree to a fixed finitely generated polynomial algebra as desired. $\square$


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