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cotangent.tex
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\input{preamble}
% OK, start here.
%
\begin{document}
\title{The Cotangent Complex}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
The goal of this chapter is to construct the cotangent complex of a
ring map, of a morphism of schemes, and of a morphism of algebraic spaces.
Some references are the notes \cite{quillenhomology}, the paper
\cite{quillencohomology}, and the books
\cite{Andre} and \cite{cotangent}.
\section{Advice for the reader}
\label{section-advice-reader}
\noindent
In writing this chapter we have tried to minimize
the use of simplicial techniques. We view the choice of a {\it resolution}
$P_\bullet$ of a ring $B$ over a ring $A$ as a tool to calculating the
{\it homology} of abelian sheaves on the category $\mathcal{C}_{B/A}$, see
Remark \ref{remark-resolution}. This is similar to the role played
by a ``good cover'' to compute cohomology using the {\v C}ech complex.
To read a bit on homology on categories, please visit
Cohomology on Sites, Section \ref{sites-cohomology-section-homology}.
The derived lower shriek functor $L\pi_!$ is to homology what
$R\Gamma(\mathcal{C}_{B/A}, -)$ is to cohomology. The category
$\mathcal{C}_{B/A}$, studied in Section \ref{section-compute-L-pi-shriek},
is the opposite of the category of factorizations $A \to P \to B$ where $P$
is a polynomial algebra over $A$. This category comes with maps of sheaves
of rings
$$
\underline{A} \longrightarrow \mathcal{O} \longrightarrow \underline{B}
$$
where over the object $U = (P \to B)$ we have $\mathcal{O}(U) = P$.
It turns out that we obtain the cotangent complex of $B$ over $A$ as
$$
L_{B/A} =
L\pi_!(\Omega_{\mathcal{O}/\underline{A}} \otimes_\mathcal{O} \underline{B})
$$
see Lemma \ref{lemma-compute-cotangent-complex}. We have consistently tried
to use this point of view to prove the basic properties of cotangent
complexes of ring maps. In particular, all of the results can be proven
without relying on the existence of standard resolutions, although we have
not done so. The theory is quite satisfactory, except that
perhaps the proof of the fundamental triangle
(Proposition \ref{proposition-triangle}) uses just a little
bit more theory on derived lower shriek functors.
To provide the reader with an alternative,
we give a rather complete sketch of an approach to this result
based on simple properties of standard resolutions in
Remarks \ref{remark-triangle} and \ref{remark-explicit-map}.
\medskip\noindent
Our approach to the cotangent complex for morphisms of ringed topoi,
morphisms of schemes, morphisms of algebraic spaces, etc
is to deduce as much as possible from the case of ``plain ring maps''
discussed above.
\section{The cotangent complex of a ring map}
\label{section-cotangent-ring-map}
\noindent
Let $A$ be a ring. Let $\textit{Alg}_A$ be the category of $A$-algebras.
Consider the pair of adjoint functors $(F, i)$ where
$i : \textit{Alg}_A \to \textit{Sets}$ is the forgetful functor and
$F : \textit{Sets} \to \textit{Alg}_A$ assigns to a set $E$ the polynomial
algebra $A[E]$ on $E$ over $A$. Let $X_\bullet$ be the simplicial object of
$\text{Fun}(\textit{Alg}_A, \textit{Alg}_A)$ constructed in
Simplicial, Section \ref{simplicial-section-standard}.
\medskip\noindent
Consider an $A$-algebra $B$. Denote $P_\bullet = X_\bullet(B)$ the resulting
simplicial $A$-algebra. Recall that $P_0 = A[B]$, $P_1 = A[A[B]]$, and so on.
In particular each term $P_n$ is a polynomial $A$-algebra.
Recall also that there is an augmentation
$$
\epsilon : P_\bullet \longrightarrow B
$$
where we view $B$ as a constant simplicial $A$-algebra.
\begin{definition}
\label{definition-standard-resolution}
Let $A \to B$ be a ring map. The {\it standard resolution of $B$ over $A$}
is the augmentation $\epsilon : P_\bullet \to B$ with terms
$$
P_0 = A[B],\quad P_1 = A[A[B]],\quad \ldots
$$
and maps as constructed above.
\end{definition}
\noindent
It will turn out that we can use the standard resolution
to compute left derived functors in certain settings.
\begin{definition}
\label{definition-cotangent-complex-ring-map}
The {\it cotangent complex} $L_{B/A}$ of a ring map $A \to B$
is the complex of $B$-modules associated to the simplicial $B$-module
$$
\Omega_{P_\bullet/A} \otimes_{P_\bullet, \epsilon} B
$$
where $\epsilon : P_\bullet \to B$ is the standard resolution
of $B$ over $A$.
\end{definition}
\noindent
In Simplicial, Section \ref{simplicial-section-complexes} we associate a
chain complex to a simplicial module, but here we work with cochain complexes.
Thus the term $L_{B/A}^{-n}$ in degree $-n$ is the $B$-module
$\Omega_{P_n/A} \otimes_{P_n, \epsilon_n} B$ and $L_{B/A}^m = 0$
for $m > 0$.
\begin{remark}
\label{remark-variant-cotangent-complex}
Let $A \to B$ be a ring map. Let $\mathcal{A}$ be the category of
arrows $\psi : C \to B$ of $A$-algebras and let $\mathcal{S}$ be
the category of maps $E \to B$ where $E$ is a set. There are adjoint
functors $i : \mathcal{A} \to \mathcal{S}$ (the forgetful functor)
and $F : \mathcal{S} \to \mathcal{A}$ which sends $E \to B$ to
$A[E] \to B$. Let $X_\bullet$ be the simplicial object of
$\text{Fun}(\mathcal{A}, \mathcal{A})$ constructed in
Simplicial, Section \ref{simplicial-section-standard}.
The diagram
$$
\xymatrix{
\mathcal{A} \ar[d] \ar[r] & \mathcal{S} \ar@<1ex>[l] \ar[d] \\
\textit{Alg}_A \ar[r] & \textit{Sets} \ar@<1ex>[l]
}
$$
commutes. It follows that $X_\bullet(\text{id}_B : B \to B)$
is equal to the standard resolution of $B$ over $A$.
\end{remark}
\begin{lemma}
\label{lemma-colimit-cotangent-complex}
Let $A_i \to B_i$ be a system of ring maps over a directed index
set $I$. Then $\colim L_{A_i/B_i} = L_{\colim A_i/\colim B_i}$.
\end{lemma}
\begin{proof}
This is true because the forgetful functor
$i : A\textit{-Alg} \to \textit{Sets}$ and its adjoint
$F : \textit{Sets} \to A\textit{-Alg}$ commute with filtered colimits.
Moreover, the functor $B/A \mapsto \Omega_{B/A}$ does as well
(Algebra, Lemma \ref{algebra-lemma-colimit-differentials}).
\end{proof}
\section{Simplicial resolutions and derived lower shriek}
\label{section-compute-L-pi-shriek}
\noindent
Let $A \to B$ be a ring map. Consider the category of $A$-algebra maps
$\alpha : P \to B$ where $P$ is a polynomial algebra over $A$
(in some set\footnote{It suffices to consider sets of cardinality
at most the cardinality of $B$.} of variables).
Let $\mathcal{C} = \mathcal{C}_{B/A}$ denote the {\bf opposite}
of this category. The reason for
taking the opposite is that we want to think of objects
$(P, \alpha)$ as corresponding to the diagram of affine schemes
$$
\xymatrix{
\Spec(B) \ar[d] \ar[r] & \Spec(P) \ar[ld] \\
\Spec(A)
}
$$
We endow $\mathcal{C}$ with the chaotic topology
(Sites, Example \ref{sites-example-indiscrete}), i.e., we endow
$\mathcal{C}$ with the structure of a site where coverings are given by
identities so that all presheaves are sheaves.
Moreover, we endow $\mathcal{C}$ with two sheaves of rings. The first
is the sheaf $\mathcal{O}$ which sends to object $(P, \alpha)$ to $P$.
Then second is the constant sheaf $B$, which we will denote
$\underline{B}$. We obtain the following diagram of morphisms of
ringed topoi
\begin{equation}
\label{equation-pi}
\vcenter{
\xymatrix{
(\Sh(\mathcal{C}), \underline{B}) \ar[r]_i \ar[d]_\pi &
(\Sh(\mathcal{C}), \mathcal{O}) \\
(\Sh(*), B)
}
}
\end{equation}
The morphism $i$ is the identity on underlying topoi and
$i^\sharp : \mathcal{O} \to \underline{B}$ is the obvious map.
The map $\pi$ is as in Cohomology on Sites, Example
\ref{sites-cohomology-example-category-to-point}.
An important role will be played in the following
by the derived functors
$
Li^* : D(\mathcal{O}) \longrightarrow D(\underline{B})
$
left adjoint to $Ri_* = i_* : D(\underline{B}) \to D(\mathcal{O})$ and
$
L\pi_! : D(\underline{B}) \longrightarrow D(B)
$
left adjoint to $\pi^* = \pi^{-1} : D(B) \to D(\underline{B})$.
\begin{lemma}
\label{lemma-identify-pi-shriek}
With notation as above let $P_\bullet$ be a simplicial $A$-algebra
endowed with an augmentation $\epsilon : P_\bullet \to B$.
Assume each $P_n$ is a polynomial algebra over $A$ and $\epsilon$
is a trivial Kan fibration on underlying simplicial sets. Then
$$
L\pi_!(\mathcal{F}) = \mathcal{F}(P_\bullet, \epsilon)
$$
in $D(\textit{Ab})$, resp.\ $D(B)$ functorially in $\mathcal{F}$ in
$\textit{Ab}(\mathcal{C})$, resp.\ $\textit{Mod}(\underline{B})$.
\end{lemma}
\begin{proof}
We will use the criterion of Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution} to prove this.
Given an object $U = (Q, \beta)$ of $\mathcal{C}$ we have to show that
$$
S_\bullet = \Mor_\mathcal{C}((Q, \beta), (P_\bullet, \epsilon))
$$
is homotopy equivalent to a singleton.
Write $Q = A[E]$ for some set $E$ (this is possible by our choice of
the category $\mathcal{C}$). We see that
$$
S_\bullet = \Mor_{\textit{Sets}}((E, \beta|_E), (P_\bullet, \epsilon))
$$
Let $*$ be the constant simplicial set on a singleton. For $b \in B$
let $F_{b, \bullet}$ be the simplicial set defined by the cartesian
diagram
$$
\xymatrix{
F_{b, \bullet} \ar[r] \ar[d] & P_\bullet \ar[d]_\epsilon \\
{*} \ar[r]^b & B
}
$$
With this notation $S_\bullet = \prod_{e \in E} F_{\beta(e), \bullet}$.
Since we assumed $\epsilon$ is a trivial Kan fibration we see that
$F_{b, \bullet} \to *$ is a trivial Kan fibration
(Simplicial, Lemma \ref{simplicial-lemma-trivial-kan-base-change}).
Thus $S_\bullet \to *$ is a trivial Kan fibration
(Simplicial, Lemma \ref{simplicial-lemma-product-trivial-kan}).
Therefore $S_\bullet$ is homotopy equivalent to $*$
(Simplicial, Lemma \ref{simplicial-lemma-trivial-kan-homotopy}).
\end{proof}
\noindent
In particular, we can use the standard resolution of $B$ over $A$
to compute derived lower shriek.
\begin{lemma}
\label{lemma-pi-shriek-standard}
Let $A \to B$ be a ring map. Let $\epsilon : P_\bullet \to B$
be the standard resolution of $B$ over $A$. Let $\pi$ be as in
(\ref{equation-pi}). Then
$$
L\pi_!(\mathcal{F}) = \mathcal{F}(P_\bullet, \epsilon)
$$
in $D(\textit{Ab})$, resp.\ $D(B)$ functorially in $\mathcal{F}$ in
$\textit{Ab}(\mathcal{C})$, resp.\ $\textit{Mod}(\underline{B})$.
\end{lemma}
\begin{proof}[First proof]
We will apply Lemma \ref{lemma-identify-pi-shriek}.
Since the terms $P_n$ are polynomial algebras we see the first
assumption of that lemma is satisfied. The second assumption is proved
as follows. By
Simplicial, Lemma \ref{simplicial-lemma-standard-simplicial-homotopy}
the map $\epsilon$ is a homotopy equivalence of underlying
simplicial sets. By
Simplicial, Lemma \ref{simplicial-lemma-homotopy-equivalence}
this implies $\epsilon$ induces a quasi-isomorphism of associated
complexes of abelian groups. By
Simplicial, Lemma \ref{simplicial-lemma-qis-simplicial-abelian-groups}
this implies that $\epsilon$ is a trivial Kan fibration of underlying
simplicial sets.
\end{proof}
\begin{proof}[Second proof]
We will use the criterion of Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}.
Let $U = (Q, \beta)$ be an object of $\mathcal{C}$.
We have to show that
$$
S_\bullet = \Mor_\mathcal{C}((Q, \beta), (P_\bullet, \epsilon))
$$
is homotopy equivalent to a singleton. Write $Q = A[E]$ for some set $E$
(this is possible by our choice of the category $\mathcal{C}$). Using the
notation of Remark \ref{remark-variant-cotangent-complex} we see that
$$
S_\bullet = \Mor_\mathcal{S}((E \to B), i(P_\bullet \to B))
$$
By Simplicial, Lemma \ref{simplicial-lemma-standard-simplicial-homotopy}
the map $i(P_\bullet \to B) \to i(B \to B)$ is a homotopy equivalence
in $\mathcal{S}$. Hence $S_\bullet$ is homotopy equivalent to
$$
\Mor_\mathcal{S}((E \to B), (B \to B)) = \{*\}
$$
as desired.
\end{proof}
\begin{lemma}
\label{lemma-compute-cotangent-complex}
Let $A \to B$ be a ring map. Let $\pi$ and $i$ be as in (\ref{equation-pi}).
There is a canonical isomorphism
$$
L_{B/A} = L\pi_!(Li^*\Omega_{\mathcal{O}/A}) =
L\pi_!(i^*\Omega_{\mathcal{O}/A}) =
L\pi_!(\Omega_{\mathcal{O}/A} \otimes_\mathcal{O} \underline{B})
$$
in $D(B)$.
\end{lemma}
\begin{proof}
For an object $\alpha : P \to B$ of the category $\mathcal{C}$
the module $\Omega_{P/A}$ is a free $P$-module. Thus
$\Omega_{\mathcal{O}/A}$ is a flat $\mathcal{O}$-module. Hence
$Li^*\Omega_{\mathcal{O}/A} = i^*\Omega_{\mathcal{O}/A}$ is the sheaf
of $\underline{B}$-modules which associates to $\alpha : P \to A$ the
$B$-module $\Omega_{P/A} \otimes_{P, \alpha} B$.
By Lemma \ref{lemma-pi-shriek-standard}
we see that the right hand side is computed by
the value of this sheaf on the standard resolution which is our
definition of the left hand side
(Definition \ref{definition-cotangent-complex-ring-map}).
\end{proof}
\begin{lemma}
\label{lemma-pi-lower-shriek-constant-sheaf}
If $A \to B$ is a ring map, then $L\pi_!(\pi^{-1}M) = M$
with $\pi$ as in (\ref{equation-pi}).
\end{lemma}
\begin{proof}
This follows from Lemma \ref{lemma-identify-pi-shriek} which tells us
$L\pi_!(\pi^{-1}M)$ is computed by $(\pi^{-1}M)(P_\bullet, \epsilon)$
which is the constant simplicial object on $M$.
\end{proof}
\begin{lemma}
\label{lemma-identify-H0}
If $A \to B$ is a ring map, then $H^0(L_{B/A}) = \Omega_{B/A}$.
\end{lemma}
\begin{proof}
We will prove this by a direct calculation.
We will use the identification of Lemma \ref{lemma-compute-cotangent-complex}.
There is clearly a map from $\Omega_{\mathcal{O}/A} \otimes \underline{B}$
to the constant sheaf with value $\Omega_{B/A}$. Thus this map induces
a map
$$
H^0(L_{B/A}) = H^0(L\pi_!(\Omega_{\mathcal{O}/A} \otimes \underline{B}))
= \pi_!(\Omega_{\mathcal{O}/A} \otimes \underline{B}) \to \Omega_{B/A}
$$
By choosing an object $P \to B$ of $\mathcal{C}_{B/A}$ with $P \to B$
surjective we see that this map is surjective (by
Algebra, Lemma \ref{algebra-lemma-differential-surjective}).
To show that it is injective, suppose that $P \to B$ is an object
of $\mathcal{C}_{B/A}$ and that $\xi \in \Omega_{P/A} \otimes_P B$
is an element which maps to zero in $\Omega_{B/A}$.
We first choose factorization $P \to P' \to B$ such that $P' \to B$
is surjective and $P'$ is a polynomial algebra over $A$.
We may replace $P$ by $P'$. If $B = P/I$, then the kernel
$\Omega_{P/A} \otimes_P B \to \Omega_{B/A}$ is the image of
$I/I^2$ (Algebra, Lemma \ref{algebra-lemma-differential-seq}).
Say $\xi$ is the image of $f \in I$.
Then we consider the two maps $a, b : P' = P[x] \to P$, the first of which
maps $x$ to $0$ and the second of which maps $x$ to $f$ (in both
cases $P[x] \to B$ maps $x$ to zero). We see that $\xi$ and $0$
are the image of $\text{d}x \otimes 1$ in $\Omega_{P'/A} \otimes_{P'} B$.
Thus $\xi$ and $0$ have the same image in the colimit (see
Cohomology on Sites, Example \ref{sites-cohomology-example-category-to-point})
$\pi_!(\Omega_{\mathcal{O}/A} \otimes \underline{B})$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-pi-lower-shriek-polynomial-algebra}
If $B$ is a polynomial algebra over the ring $A$, then
with $\pi$ as in (\ref{equation-pi}) we have that
$\pi_!$ is exact and $\pi_!\mathcal{F} = \mathcal{F}(B \to B)$.
\end{lemma}
\begin{proof}
This follows from Lemma \ref{lemma-identify-pi-shriek} which tells us
the constant simplicial algebra on $B$ can be used to compute $L\pi_!$.
\end{proof}
\begin{lemma}
\label{lemma-cotangent-complex-polynomial-algebra}
If $B$ is a polynomial algebra over the ring $A$, then
$L_{B/A}$ is quasi-isomorphic to $\Omega_{B/A}[0]$.
\end{lemma}
\begin{proof}
Immediate from
Lemmas \ref{lemma-compute-cotangent-complex} and
\ref{lemma-pi-lower-shriek-polynomial-algebra}.
\end{proof}
\section{Constructing a resolution}
\label{section-polynomial}
\noindent
In the Noetherian finite type case we can construct a ``small'' simplicial
resolution for finite type ring maps.
\begin{lemma}
\label{lemma-polynomial}
Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map.
Let $\mathcal{A}$ be the category of $A$-algebra maps $C \to B$. Let
$n \geq 0$ and let $P_\bullet$ be a simplicial object of $\mathcal{A}$
such that
\begin{enumerate}
\item $P_\bullet \to B$ is a trivial Kan fibration of simplicial sets,
\item $P_k$ is finite type over $A$ for $k \leq n$,
\item $P_\bullet = \text{cosk}_n \text{sk}_n P_\bullet$ as simplicial
objects of $\mathcal{A}$.
\end{enumerate}
Then $P_{n + 1}$ is a finite type $A$-algebra.
\end{lemma}
\begin{proof}
Although the proof we give of this lemma is straightforward, it is a bit
messy. To clarify the idea we explain what happens for low $n$ before giving
the proof in general. For example, if $n = 0$, then (3) means that
$P_1 = P_0 \times_B P_0$. Since the ring map $P_0 \to B$ is surjective, this
is of finite type over $A$ by
More on Algebra, Lemma \ref{more-algebra-lemma-fibre-product-finite-type}.
\medskip\noindent
If $n = 1$, then (3) means that
$$
P_2 = \{(f_0, f_1, f_2) \in P_1^3 \mid
d_0f_0 = d_0f_1,\ d_1f_0 = d_0f_2,\ d_1f_1 = d_1f_2 \}
$$
where the equalities take place in $P_0$. Observe that the triple
$$
(d_0f_0, d_1f_0, d_1f_1) = (d_0f_1, d_0f_2, d_1f_2)
$$
is an element of the fibre product $P_0 \times_B P_0 \times_B P_0$ over $B$
because the maps $d_i : P_1 \to P_0$ are morphisms over $B$. Thus we get
a map
$$
\psi : P_2 \longrightarrow P_0 \times_B P_0 \times_B P_0
$$
The fibre of $\psi$ over an element
$(g_0, g_1, g_2) \in P_0 \times_B P_0 \times_B P_0$
is the set of triples $(f_0, f_1, f_2)$ of $1$-simplices
with $(d_0, d_1)(f_0) = (g_0, g_1)$, $(d_0, d_1)(f_1) = (g_0, g_2)$,
and $(d_0, d_1)(f_2) = (g_1, g_2)$. As $P_\bullet \to B$ is a trivial
Kan fibration the map $(d_0, d_1) : P_1 \to P_0 \times_B P_0$ is
surjective. Thus we see that $P_2$ fits into the cartesian diagram
$$
\xymatrix{
P_2 \ar[d] \ar[r] & P_1^3 \ar[d] \\
P_0 \times_B P_0 \times_B P_0 \ar[r] & (P_0 \times_B P_0)^3
}
$$
By More on Algebra, Lemma \ref{more-algebra-lemma-formal-consequence}
we conclude. The general case is similar, but requires a bit more notation.
\medskip\noindent
The case $n > 1$. By Simplicial, Lemma \ref{simplicial-lemma-cosk-above-object}
the condition $P_\bullet = \text{cosk}_n \text{sk}_n P_\bullet$
implies the same thing is true in the category of simplicial
$A$-algebras and hence in the category of sets (as the forgetful
functor from $A$-algebras to sets commutes with limits). Thus
$$
P_{n + 1} =
\Mor(\Delta[n + 1], P_\bullet) =
\Mor(\text{sk}_n \Delta[n + 1], \text{sk}_n P_\bullet)
$$
by Simplicial, Lemma \ref{simplicial-lemma-simplex-map} and
Equation (\ref{simplicial-equation-cosk}). We will prove by induction
on $1 \leq k < m \leq n + 1$ that the ring
$$
Q_{k, m} = \Mor(\text{sk}_k \Delta[m], \text{sk}_k P_\bullet)
$$
is of finite type over $A$. The case $k = 1$, $1 < m \leq n + 1$
is entirely similar to the discussion above in the case $n = 1$.
Namely, there is a cartesian diagram
$$
\xymatrix{
Q_{1, m} \ar[d] \ar[r] & P_1^N \ar[d] \\
P_0 \times_B \ldots \times_B P_0 \ar[r] & (P_0 \times_B P_0)^N
}
$$
where $N = {m + 1 \choose 2}$. We conclude as before.
\medskip\noindent
Let $1 \leq k_0 \leq n$ and assume $Q_{k, m}$ is of finite type
over $A$ for all $1 \leq k \leq k_0$ and $k < m \leq n + 1$.
For $k_0 + 1 < m \leq n + 1$ we claim there is a cartesian square
$$
\xymatrix{
Q_{k_0 + 1, m} \ar[d] \ar[r] & P_{k_0 + 1}^N \ar[d] \\
Q_{k_0, m} \ar[r] & Q_{k_0, k_0 + 1}^N
}
$$
where $N$ is the number of nondegenerate $(k_0 + 1)$-simplices
of $\Delta[m]$. Namely, to see this is true, think of an element of
$Q_{k_0 + 1, m}$ as a function $f$ from the $(k_0 + 1)$-skeleton
of $\Delta[m]$ to $P_\bullet$. We can restrict $f$ to the $k_0$-skeleton
which gives the left vertical map of the diagram. We can also restrict
to each nondegenerate $(k_0 + 1)$-simplex which gives the top horizontal
arrow. Moreover, to give such an $f$ is the same thing as giving its
restriction to $k_0$-skeleton and to each nondegenerate
$(k_0 + 1)$-face, provided these agree on the overlap, and this
is exactly the content of the diagram. Moreover, the fact that
$P_\bullet \to B$ is a trivial Kan fibration implies that
the map
$$
P_{k_0} \to Q_{k_0, k_0 + 1} = \Mor(\partial \Delta[k_0 + 1], P_\bullet)
$$
is surjective as every map $\partial \Delta[k_0 + 1] \to B$ can be extended
to $\Delta[k_0 + 1] \to B$ for $k_0 \geq 1$ (small argument about constant
simplicial sets omitted). Since by induction hypothesis the rings
$Q_{k_0, m}$, $Q_{k_0, k_0 + 1}$ are finite type $A$-algebras, so is
$Q_{k_0 + 1, m}$ by
More on Algebra, Lemma \ref{more-algebra-lemma-formal-consequence}
once more.
\end{proof}
\begin{proposition}
\label{proposition-polynomial}
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.
\end{proposition}
\begin{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.
\medskip\noindent
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 \ref{simplicial-lemma-cosk-minus-one-equivalence}
the map $P_\bullet \to B$ is a trivial Kan fibration of simplicial sets.
Also, observe that $P_\bullet = \text{cosk}_0 \text{sk}_0 P_\bullet$.
\medskip\noindent
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
\begin{enumerate}
\item[(a)] $P_\bullet \to B$ is a trivial Kan fibration of simplicial sets,
\item[(b)] $P_k$ is a finitely generated polynomial algebra for
$0 \leq k \leq n$, and
\item[(c)] $P_\bullet = \text{cosk}_n \text{sk}_n P_\bullet$
\end{enumerate}
By Lemma \ref{lemma-polynomial}
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.)
\medskip\noindent
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 \ref{simplicial-lemma-section}
with $n + 1$ instead of $n$. Conditions (1) and (2) hold by construction.
By Simplicial, Lemma \ref{simplicial-lemma-cosk-above-object}
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 \ref{simplicial-lemma-trivial-kan-composition}
we conclude that $P'_\bullet \to B$ is a trivial Kan fibration and (a)
holds as well.
\medskip\noindent
To finish the proof we take the inverse limit $P_\bullet = \lim P^n_\bullet$
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 \ref{simplicial-lemma-limit-trivial-kan}.
However, the construction above stabilizes in each degree
to a fixed finitely generated polynomial algebra as desired.
\end{proof}
\begin{lemma}
\label{lemma-pi-shriek-finite}
Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map.
Let $\pi$, $\underline{B}$ be as in (\ref{equation-pi}).
If $\mathcal{F}$ is an $\underline{B}$-module such that
$\mathcal{F}(P, \alpha)$ is a finite $B$-module for all
$\alpha : P = A[x_1, \ldots, x_n] \to B$, then the cohomology modules
of $L\pi_!(\mathcal{F})$ are finite $B$-modules.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-identify-pi-shriek} and
Proposition \ref{proposition-polynomial}
we can compute $L\pi_!(\mathcal{F})$ by a complex
constructed out of the values of $\mathcal{F}$ on finite type
polynomial algebras.
\end{proof}
\begin{lemma}
\label{lemma-cotangent-finite}
Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map.
Then $H^n(L_{B/A})$ is a finite $B$-module for all $n \in \mathbf{Z}$.
\end{lemma}
\begin{proof}
Apply Lemmas \ref{lemma-compute-cotangent-complex} and
\ref{lemma-pi-shriek-finite}.
\end{proof}
\begin{remark}[Resolutions]
\label{remark-resolution}
Let $A \to B$ be any ring map. Let us call an augmented simplicial $A$-algebra
$\epsilon : P_\bullet \to B$ a {\it resolution of $B$ over $A$} if
each $P_n$ is a polynomial algebra and $\epsilon$ is a trivial Kan fibration
of simplicial sets. If $P_\bullet \to B$ is an augmentation of a simplicial
$A$-algebra with each $P_n$ a polynomial algebra surjecting onto $B$, then
the following are equivalent
\begin{enumerate}
\item $\epsilon : P_\bullet \to B$ is a resolution of $B$ over $A$,
\item $\epsilon : P_\bullet \to B$ is a quasi-isomorphism on
associated complexes,
\item $\epsilon : P_\bullet \to B$ induces a homotopy equivalence
of simplicial sets.
\end{enumerate}
To see this use Simplicial, Lemmas
\ref{simplicial-lemma-trivial-kan-homotopy},
\ref{simplicial-lemma-homotopy-equivalence}, and
\ref{simplicial-lemma-qis-simplicial-abelian-groups}.
A resolution $P_\bullet$ of $B$ over $A$ gives a cosimplicial object
$U_\bullet$ of $\mathcal{C}_{B/A}$ as in Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}
and it follows that
$$
L\pi_!\mathcal{F} = \mathcal{F}(P_\bullet)
$$
functorially in $\mathcal{F}$, see Lemma \ref{lemma-identify-pi-shriek}.
The (formal part of the) proof of Proposition \ref{proposition-polynomial}
shows that resolutions exist. We also have seen in the first proof of
Lemma \ref{lemma-pi-shriek-standard} that the standard resolution of $B$
over $A$ is a resolution (so that this terminology doesn't lead to a conflict).
However, the argument in the proof of Proposition \ref{proposition-polynomial}
shows the existence of resolutions without appealing to the simplicial
computations in Simplicial, Section \ref{simplicial-section-standard}.
Moreover, for {\it any} choice of resolution we have a canonical isomorphism
$$
L_{B/A} = \Omega_{P_\bullet/A} \otimes_{P_\bullet, \epsilon} B
$$
in $D(B)$ by Lemma \ref{lemma-compute-cotangent-complex}. The freedom to
choose an arbitrary resolution can be quite useful.
\end{remark}
\begin{lemma}
\label{lemma-O-homology-B-homology}
Let $A \to B$ be a ring map. Let $\pi$, $\mathcal{O}$, $\underline{B}$
be as in (\ref{equation-pi}). For any $\mathcal{O}$-module $\mathcal{F}$
we have
$$
L\pi_!(\mathcal{F}) = L\pi_!(Li^*\mathcal{F}) =
L\pi_!(\mathcal{F} \otimes_\mathcal{O}^\mathbf{L} \underline{B})
$$
in $D(\textit{Ab})$.
\end{lemma}
\begin{proof}
It suffices to verify the assumptions of Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-O-homology-qis}
hold for $\mathcal{O} \to \underline{B}$ on $\mathcal{C}_{B/A}$.
We will use the results of Remark \ref{remark-resolution} without
further mention. Choose a resolution $P_\bullet$ of $B$ over $A$ to get a
suitable cosimplicial object $U_\bullet$ of $\mathcal{C}_{B/A}$.
Since $P_\bullet \to B$ induces a quasi-isomorphism on associated
complexes of abelian groups we see that $L\pi_!\mathcal{O} = B$.
On the other hand $L\pi_!\underline{B}$ is computed by
$\underline{B}(U_\bullet) = B$. This verifies the second assumption of
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-O-homology-qis}
and we are done with the proof.
\end{proof}
\begin{lemma}
\label{lemma-apply-O-B-comparison}
Let $A \to B$ be a ring map. Let $\pi$, $\mathcal{O}$, $\underline{B}$
be as in (\ref{equation-pi}). We have
$$
L\pi_!(\mathcal{O}) = L\pi_!(\underline{B}) = B
\quad\text{and}\quad
L_{B/A} = L\pi_!(\Omega_{\mathcal{O}/A} \otimes_\mathcal{O} \underline{B}) =
L\pi_!(\Omega_{\mathcal{O}/A})
$$
in $D(\textit{Ab})$.
\end{lemma}
\begin{proof}
This is just an application of Lemma \ref{lemma-O-homology-B-homology}
(and the first equality on the right is
Lemma \ref{lemma-compute-cotangent-complex}).
\end{proof}
\noindent
Here is a special case of the fundamental triangle that is easy to prove.
\begin{lemma}
\label{lemma-special-case-triangle}
Let $A \to B \to C$ be ring maps. If $B$ is a polynomial algebra over
$A$, then there is a distinguished triangle
$L_{B/A} \otimes_B^\mathbf{L} C \to L_{C/A} \to L_{C/B} \to
L_{B/A} \otimes_B^\mathbf{L} C[1]$ in $D(C)$.
\end{lemma}
\begin{proof}
We will use the observations of Remark \ref{remark-resolution}
without further mention. Choose a resolution $\epsilon : P_\bullet \to C$
of $C$ over $B$ (for example the standard resolution). Since $B$ is a
polynomial algebra over $A$ we see that $P_\bullet$ is also a resolution of
$C$ over $A$. Hence $L_{C/A}$ is computed by
$\Omega_{P_\bullet/A} \otimes_{P_\bullet, \epsilon} C$
and $L_{C/B}$ is computed by
$\Omega_{P_\bullet/B} \otimes_{P_\bullet, \epsilon} C$.
Since for each $n$ we have the short exact sequence
$0 \to \Omega_{B/A} \otimes_B P_n \to \Omega_{P_n/A} \to \Omega_{P_n/B}$
(Algebra, Lemma \ref{algebra-lemma-ses-formally-smooth})
and since $L_{B/A} = \Omega_{B/A}[0]$
(Lemma \ref{lemma-cotangent-complex-polynomial-algebra})
we obtain the result.
\end{proof}
\begin{example}
\label{example-resolution-length-two}
Let $A \to B$ be a ring map. In this example we
will construct an ``explicit'' resolution $P_\bullet$ of $B$ over $A$ of
length $2$. To do this we follow the procedure of the proof of
Proposition \ref{proposition-polynomial}, see also the discussion in
Remark \ref{remark-resolution}.
\medskip\noindent
We choose a surjection $P_0 = A[u_i] \to B$ where $u_i$ is a set of
variables. Choose generators $f_t \in P_0$, $t \in T$ of the ideal
$\Ker(P_0 \to B)$. We choose $P_1 = A[u_i, x_t]$ with face maps
$d_0$ and $d_1$ the unique $A$-algebra maps with $d_j(u_i) = u_i$
and $d_0(x_t) = 0$ and $d_1(x_t) = f_t$. The map $s_0 : P_0 \to P_1$
is the unique $A$-algebra map with $s_0(u_i) = u_i$. It is clear that
$$
P_1 \xrightarrow{d_0 - d_1} P_0 \to B \to 0
$$
is exact, in particular the map $(d_0, d_1) : P_1 \to P_0 \times_B P_0$
is surjective. Thus, if $P_\bullet$ denotes the $1$-truncated
simplicial $A$-algebra given by $P_0$, $P_1$, $d_0$, $d_1$, and $s_0$, then
the augmentation $\text{cosk}_1(P_\bullet) \to B$ is a trivial Kan fibration.
The next step of the procedure in the proof of
Proposition \ref{proposition-polynomial}
is to choose a polynomial algebra $P_2$ and a surjection
$$
P_2 \longrightarrow \text{cosk}_1(P_\bullet)_2
$$
Recall that
$$
\text{cosk}_1(P_\bullet)_2 =
\{(g_0, g_1, g_2) \in P_1^3 \mid d_0(g_0) = d_0(g_1),
d_1(g_0) = d_0(g_2), d_1(g_1) = d_1(g_2)\}
$$
Thinking of $g_i \in P_1$ as a polynomial in $x_t$ the conditions
are
$$
g_0(0) = g_1(0),\quad
g_0(f_t) = g_2(0),\quad
g_1(f_t) = g_2(f_t)
$$
Thus $\text{cosk}_1(P_\bullet)_2$ contains the elements
$y_t = (x_t, x_t, f_t)$ and $z_t = (0, x_t, x_t)$.
Every element $G$ in $\text{cosk}_1(P_\bullet)_2$ is
of the form $G = H + (0, 0, g)$ where $H$ is in the image
of $A[u_i, y_t, z_t] \to \text{cosk}_1(P_\bullet)_2$. Here
$g \in P_1$ is a polynomial with vanishing
constant term such that $g(f_t) = 0$ in $P_0$. Observe that
\begin{enumerate}
\item $g = x_t x_{t'} - f_t x_{t'}$ and
\item $g = \sum r_t x_t$ with $r_t \in P_0$ if $\sum r_t f_t = 0$ in $P_0$
\end{enumerate}
are elements of $P_1$ of the desired form. Let
$$
Rel = \Ker(\bigoplus\nolimits_{t \in T} P_0 \longrightarrow P_0),\quad
(r_t) \longmapsto \sum r_tf_t
$$
We set $P_2 = A[u_i, y_t, z_t, v_r, w_{t, t'}]$ where
$r = (r_t) \in Rel$, with map
$$
P_2 \longrightarrow \text{cosk}_1(P_\bullet)_2
$$
given by $y_t \mapsto (x_t, x_t, f_t)$,
$z_t \mapsto (0, x_t, x_t)$,
$v_r \mapsto (0, 0, \sum r_t x_t)$, and
$w_{t, t'} \mapsto (0, 0, x_t x_{t'} - f_t x_{t'})$. A calculation
(omitted) shows that this map is surjective. Our choice of the
map displayed above determines the maps $d_0, d_1, d_2 : P_2 \to P_1$.
Finally, the procedure in the proof of
Proposition \ref{proposition-polynomial}
tells us to choose the maps $s_0, s_1 : P_1 \to P_2$ lifting the two
maps $P_1 \to \text{cosk}_1(P_\bullet)_2$. It is clear that we can take
$s_i$ to be the unique $A$-algebra maps determined by
$s_0(x_t) = y_t$ and $s_1(x_t) = z_t$.
\end{example}
\section{Functoriality}
\label{section-functoriality}
\noindent
In this section we consider a commutative square
\begin{equation}
\label{equation-commutative-square}
\vcenter{
\xymatrix{
B \ar[r] & B' \\
A \ar[u] \ar[r] & A' \ar[u]
}
}
\end{equation}
of ring maps. We claim there is a canonical $B$-linear map of complexes
$$
L_{B/A} \longrightarrow L_{B'/A'}
$$
associated to this diagram. Namely, if $P_\bullet \to B$ is the
standard resolution of $B$ over $A$ and $P'_\bullet \to B'$ is the
standard resolution of $B'$ over $A'$, then there is a canonical map
$P_\bullet \to P'_\bullet$
of simplicial $A$-algebras compatible with the augmentations
$P_\bullet \to B$ and $P'_\bullet \to B'$. This can be seen in terms
of the construction of standard resolutions in
Simplicial, Section \ref{simplicial-section-standard}
but in the special case at hand it probably suffices to say simply
that the maps
$$
P_0 = A[B] \longrightarrow A'[B'] = P'_0,\quad
P_1 = A[A[B]] \longrightarrow A'[A'[B']] = P'_1,
$$
and so on are given by the given maps $A \to A'$ and $B \to B'$.
The desired map $L_{B/A} \to L_{B'/A'}$ then comes from the associated
maps $\Omega_{P_n/A} \to \Omega_{P'_n/A'}$.
\medskip\noindent
Another description of the functoriality map can be given as follows.
Let $\mathcal{C} = \mathcal{C}_{B/A}$ and $\mathcal{C}' = \mathcal{C}_{B'/A}'$
be the categories considered in Section \ref{section-compute-L-pi-shriek}.
There is a functor
$$
u : \mathcal{C} \longrightarrow \mathcal{C}',\quad
(P, \alpha) \longmapsto (P \otimes_A A', c \circ (\alpha \otimes 1))
$$
where $c : B \otimes_A A' \to B'$ is the obvious map. As discussed in
Cohomology on Sites, Example
\ref{sites-cohomology-example-morphism-categories}
we obtain a morphism of topoi $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$
and a commutative diagram of maps of ringed topoi
\begin{equation}
\label{equation-double-square}
\vcenter{
\xymatrix{
(\Sh(\mathcal{C}), \underline{B}) \ar[d]_\pi &
(\Sh(\mathcal{C}'), \underline{B'}) \ar[d]_\pi \ar[l]^h \ar[r]_g &
(\Sh(\mathcal{C}), \underline{B'}) \ar[d]_{\pi'} \\
(\Sh(*), B) &
(\Sh(*), B') \ar[l]_f \ar[r] &
(\Sh(*), B')
}
}
\end{equation}
Here $h$ is the identity on underlying topoi and given by the ring map
$B \to B'$ on sheaves of rings.
By Cohomology on Sites, Remark
\ref{sites-cohomology-remark-morphism-fibred-categories}
given $\mathcal{F}$ on $\mathcal{C}$ and $\mathcal{F}'$ on $\mathcal{C}'$
and a transformation $t : \mathcal{F} \to g^{-1}\mathcal{F}'$
we obtain a canonical map $L\pi_!(\mathcal{F}) \to L\pi'_!(\mathcal{F}')$.
If we apply this to the sheaves
$$
\mathcal{F} : (P, \alpha) \mapsto \Omega_{P/A} \otimes_P B,\quad
\mathcal{F}' : (P', \alpha') \mapsto \Omega_{P'/A'} \otimes_{P'} B',
$$
and the transformation $t$ given by the canonical maps
$$
\Omega_{P/A} \otimes_P B \longrightarrow
\Omega_{P \otimes_A A'/A'} \otimes_{P \otimes_A A'} B'
$$
to get a canonical map
$$
L\pi_!(\Omega_{\mathcal{O}/A} \otimes_\mathcal{O} \underline{B})
\longrightarrow
L\pi'_!(\Omega_{\mathcal{O}'/A'} \otimes_{\mathcal{O}'} \underline{B'})
$$
By Lemma \ref{lemma-compute-cotangent-complex} this gives
$L_{B/A} \to L_{B'/A'}$. We omit the verification that this map
agrees with the map defined above in terms of simplicial
resolutions.
\begin{lemma}
\label{lemma-flat-base-change}
Assume (\ref{equation-commutative-square}) induces a quasi-isomorphism
$B \otimes_A^\mathbf{L} A' = B'$. Then, with notation as in
(\ref{equation-double-square}) and
$\mathcal{F}' \in \textit{Ab}(\mathcal{C}')$,
we have $L\pi_!(g^{-1}\mathcal{F}') = L\pi'_!(\mathcal{F}')$.
\end{lemma}
\begin{proof}
We use the results of Remark \ref{remark-resolution} without
further mention. We will apply Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-get-it-now}. Let $P_\bullet \to B$ be a resolution.
If we can show that $u(P_\bullet) = P_\bullet \otimes_A A' \to B'$
is a quasi-isomorphism, then we are done. The complex of $A$-modules
$s(P_\bullet)$ associated to $P_\bullet$ (viewed as a simplicial $A$-module)
is a free $A$-module resolution of $B$. Namely, $P_n$ is a free $A$-module and
$s(P_\bullet) \to B$ is a quasi-isomorphism. Thus $B \otimes_A^\mathbf{L} A'$
is computed by $s(P_\bullet) \otimes_A A' = s(P_\bullet \otimes_A A')$.
Therefore the assumption of the lemma signifies that
$\epsilon' : P_\bullet \otimes_A A' \to B'$ is a quasi-isomorphism.
\end{proof}
\noindent
The following lemma in particular applies when $A \to A'$ is flat
and $B' = B \otimes_A A'$ (flat base change).
\begin{lemma}
\label{lemma-flat-base-change-cotangent-complex}
If (\ref{equation-commutative-square}) induces a quasi-isomorphism
$B \otimes_A^\mathbf{L} A' = B'$, then the functoriality map
induces an isomorphism
$$
L_{B/A} \otimes_B^\mathbf{L} B' \longrightarrow L_{B'/A'}
$$
\end{lemma}
\begin{proof}
We will use the notation introduced in Equation (\ref{equation-double-square}).
We have
$$
L_{B/A} \otimes_B^\mathbf{L} B' =
L\pi_!(\Omega_{\mathcal{O}/A} \otimes_\mathcal{O} \underline{B})
\otimes_B^\mathbf{L} B' =
L\pi_!(Lh^*(\Omega_{\mathcal{O}/A} \otimes_\mathcal{O} \underline{B}))
$$
the first equality by Lemma \ref{lemma-compute-cotangent-complex}