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\title{Obsolete}

\maketitle

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\label{section-phantom}

\tableofcontents




\section{Introduction}
\label{section-introduction}

\noindent
In this chapter we put some lemmas that have become ``obsolete''
(see \cite{Miller}).



\section{Obsolete algebra lemmas}
\label{section-algebra}

\begin{lemma}
\label{lemma-finite-presentation-module-independent}
Let $M$ be an $R$-module of finite presentation.
For any surjection $\alpha : R^{\oplus n} \to M$ the
kernel of $\alpha$ is a finite $R$-module.
\end{lemma}

\begin{proof}
This is a special case of Algebra, Lemma \ref{algebra-lemma-extension}.
\end{proof}

\noindent
The following technical lemma says that you can lift any sequence
of relations from a fibre to the whole space of a ring
map which is essentially of finite type, in a suitable sense.

\begin{lemma}
\label{lemma-lift-elements-ideal}
Let $R \to S$ be a ring map.
Let $\mathfrak p \subset R$ be a prime.
Let $\mathfrak q \subset S$ be a prime lying over $\mathfrak p$.
Assume $S_{\mathfrak q}$ is essentially of finite type over $R_\mathfrak p$.
Assume given
\begin{enumerate}
\item an integer $n \geq 0$,
\item a prime $\mathfrak a \subset \kappa(\mathfrak p)[x_1, \ldots, x_n]$,
\item a surjective $\kappa(\mathfrak p)$-homomorphism
$$
\psi : (\kappa(\mathfrak p)[x_1, \ldots, x_n])_{\mathfrak a}
\longrightarrow
S_{\mathfrak q}/\mathfrak p S_{\mathfrak q},
$$
and
\item elements $\overline{f}_1, \ldots, \overline{f}_e$ in $\text{Ker}(\psi)$.
\end{enumerate}
Then there exist
\begin{enumerate}
\item an integer $m \geq 0$,
\item and element $g \in S$, $g \not\in \mathfrak q$,
\item a map
$$
\Psi :
R[x_1, \ldots, x_n, x_{n + 1}, \ldots, x_{n + m}]
\longrightarrow
S_g,
$$
and
\item elements $f_1, \ldots, f_e, f_{e + 1}, \ldots, f_{e + m}$
of $\text{Ker}(\Psi)$
\end{enumerate}
such that
\begin{enumerate}
\item the following diagram commutes
$$
\xymatrix{
R[x_1, \ldots, x_{n + m}] \ar[d]_\Psi
\ar[rr]_-{x_{n + j} \mapsto 0} & &
(\kappa(\mathfrak p)[x_1, \ldots, x_n])_{\mathfrak a} \ar[d]^\psi \\
S_g \ar[rr] & &
S_{\mathfrak q}/\mathfrak p S_{\mathfrak q}
},
$$
\item the element $f_i$, $i \leq n$ maps to a unit times
$\overline{f}_i$ in the local ring
$$
(\kappa(\mathfrak p)[x_1, \ldots, x_{n + m}])_{
(\mathfrak a, x_{n + 1}, \ldots, x_{n + m})},
$$
\item the element $f_{e + j}$ maps to
a unit times $x_{n + j}$ in the same local ring, and
\item the induced map $R[x_1, \ldots, x_{n + m}]_{\mathfrak b}
\to S_{\mathfrak q}$ is surjective, where
$\mathfrak b = \Psi^{-1}(\mathfrak qS_g)$.
\end{enumerate}
\end{lemma}

\begin{proof}
We claim that it suffices to prove the lemma in case $R$
and $S$ are local with maximal ideals $\mathfrak p$ and $\mathfrak q$.
Namely, suppose we have constructed
$$
\Psi' : R_{\mathfrak p}[x_1, \ldots, x_{n + m}]
\longrightarrow
S_{\mathfrak q}
$$
and $f_1', \ldots, f_{e + m}' \in R_{\mathfrak p}[x_1, \ldots, x_{n + m}]$
with all the required properties. Then there exists an element
$f \in R$, $f \not \in \mathfrak p$ such that each
$ff_k'$ comes from an element $f_k \in R[x_1, \ldots, x_{n + m}]$.
Moreover, for a suitable $g \in S$, $g \not \in \mathfrak q$
the elements $\Psi'(x_i)$ are the image of elements
$y_i \in S_g$. Let $\Psi$ be the $R$-algebra map defined
by the rule $\Psi(x_i) = y_i$. Since $\Psi(f_i)$ is zero
in the localization $S_{\mathfrak q}$ we may after possibly
replacing $g$ assume that $\Psi(f_i) = 0$. This proves the claim.

\medskip\noindent
Thus we may assume $R$ and $S$ are local
with maximal ideals $\mathfrak p$ and $\mathfrak q$.
Pick $y_1, \ldots, y_n \in S$ such that
$y_i \bmod \mathfrak pS = \psi(x_i)$.
Let $y_{n + 1}, \ldots, y_{n + m} \in S$ be elements which generate
an $R$-subalgebra of which $S$ is the localization.
These exist by the assumption that $S$ is essentially of
finite type over $R$. Since $\psi$ is surjective we
may write $y_{n + j} \bmod \mathfrak pS = \psi(h_j)$ for
some $h_j \in \kappa(\mathfrak p)[x_1, \ldots, x_n]_{\mathfrak a}$.
Write $h_j = g_j/d$, $g_j \in \kappa(\mathfrak p)[x_1, \ldots, x_n]$
for some common denominator $d \in \kappa(\mathfrak p)[x_1, \ldots, x_n]$,
$d \not \in \mathfrak a$. Choose lifts $G_j, D \in R[x_1, \ldots, x_n]$
of $g_j$ and $d$. Set
$y_{n + j}' = D(y_1, \ldots, y_n) y_{n + j} - G_j(y_1, \ldots, y_n)$.
By construction $y_{n + j}' \in \mathfrak p S$.
It is clear that $y_1, \ldots, y_n, y_n', \ldots, y_{n + m}'$
generate an $R$-subalgebra of $S$ whose localization is $S$.
We define
$$
\Psi : R[x_1, \ldots, x_{n + m}] \to S
$$
to be the map that sends $x_i$ to $y_i$ for $i = 1, \ldots, n$
and $x_{n + j}$ to $y'_{n + j}$ for $j = 1, \ldots, m$. Properties
(1) and (4) are clear by construction. Moreover the ideal
$\mathfrak b$ maps onto the ideal
$(\mathfrak a, x_{n + 1}, \ldots, x_{n + m})$
in the polynomial ring $\kappa(\mathfrak p)[x_1, \ldots, x_{n + m}]$.

\medskip\noindent
Denote $J = \text{Ker}(\Psi)$. We have a short exact sequence
$$
0 \to J_{\mathfrak b}
\to R[x_1, \ldots, x_{n + m}]_{\mathfrak b}
\to S_{\mathfrak q}
\to 0.
$$
The surjectivity comes from our choice of
$y_1, \ldots, y_n, y_n', \ldots, y_{n + m}'$ above.
This implies that
$$
J_{\mathfrak b}/ \mathfrak pJ_{\mathfrak b}
\to \kappa(\mathfrak p)[x_1, \ldots, x_{n + m}]_{
(\mathfrak a, x_{n + 1}, \ldots, x_{n + m})}
\to S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}
\to 0
$$
is exact. By construction $x_i$ maps to $\psi(x_i)$ and
$x_{n + j}$ maps to zero under the last map.
Thus it is easy to choose $f_i$ as in
(2) and (3) of the lemma.
\end{proof}


\begin{remark}[Projective resolutions]
\label{remark-projective-resolution}
Let $R$ be a ring.
For any set $S$ we let $F(S)$ denote the free $R$-module on $S$.
Then any left $R$-module has the following two step resolution
$$
F(M \times M) \oplus F(R \times M) \to F(M) \to M \to 0.
$$
The first map is given by the rule
$$
[m_1, m_2] \oplus [r, m] \mapsto [m_1 + m_2] - [m_1] - [m_2] + [rm] - r[m].
$$
\end{remark}


\section{Lemmas related to ZMT}
\label{section-ZMT}

\noindent
The lemmas in this section were originally used in the proof of the
(algebraic version of) Zariski's Main Theorem,
Algebra, Theorem \ref{algebra-theorem-main-theorem}.

\begin{lemma}
\label{lemma-make-integral-less-trivial}
Let $\varphi : R \to S$ be a ring map.
Suppose $t \in S$ satisfies the
relation $\varphi(a_0) + \varphi(a_1)t + \ldots + \varphi(a_n) t^n = 0$.
Set $u_n = \varphi(a_n)$, $u_{n-1} = u_n t + \varphi(a_{n-1})$,
and so on till $u_1 = u_2 t + \varphi(a_1)$.
Then all of $u_n, u_{n-1}, \ldots, u_1$ and
$u_nt, u_{n-1}t, \ldots, u_1t$ are integral over $R$,
and the ideals $(\varphi(a_0), \ldots, \varphi(a_n))$ and
$(u_n, \ldots, u_1)$ of $S$ are equal.
\end{lemma}

\begin{proof}
We prove this by induction on $n$. As $u_n = \varphi(a_n)$ we
conclude from
Algebra, Lemma \ref{algebra-lemma-make-integral-trivial}
that $u_nt$ is integral over $R$. Of course
$u_n = \varphi(a_n)$ is integral over $R$. Then
$u_{n - 1} = u_n t  + \varphi(a_{n - 1})$ is integral over $R$ (see
Algebra, Lemma \ref{algebra-lemma-integral-closure-is-ring})
and we have
$$
\varphi(a_0) + \varphi(a_1)t + \ldots + \varphi(a_{n - 1})t^{n - 1} +
u_{n - 1}t^{n - 1} = 0.
$$
Hence by the induction hypothesis applied to the map
$S' \to S$ where $S'$ is the integral closure of $R$ in $S$
and the displayed equation we see that
$u_{n-1}, \ldots, u_1$ and $u_{n-1}t, \ldots, u_1t$
are all in $S'$ too. The statement on the ideals is immediate from the
shape of the elements and the fact that $u_1t + \varphi(a_0) = 0$.
\end{proof}

\begin{lemma}
\label{lemma-make-integral-not-in-ideal}
Let $\varphi : R \to S$ be a ring map.
Suppose $t \in S$ satisfies the
relation $\varphi(a_0) + \varphi(a_1)t + \ldots + \varphi(a_n) t^n = 0$.
Let $J \subset S$ be an ideal such that for at
least one $i$ we have $\varphi(a_i) \not \in J$.
Then there exists a $u \in S$, $u \not\in J$ such
that both $u$ and $ut$ are integral over $R$.
\end{lemma}

\begin{proof}
This is immediate from Lemma \ref{lemma-make-integral-less-trivial}
since one of the elements $u_i$ will not be in $J$.
\end{proof}

\noindent
The following two lemmas are a way of describing closed
subschemes of $\mathbf{P}^1_R$ cut out by one (nondegenerate)
equation.

\begin{lemma}
\label{lemma-P1}
Let $R$ be a ring.
Let $F(X, Y) \in R[X, Y]$ be homogeneous of degree
$d$. Assume that for every prime $\mathfrak p$ of $R$
at least one coefficient of $F$ is not in $\mathfrak p$.
Let $S = R[X, Y]/(F)$ as a graded ring.
Then for all $n \geq d$ the $R$-module $S_n$
is finite locally free of rank $d$.
\end{lemma}

\begin{proof}
The $R$-module $S_n$ has a presentation
$$
R[X, Y]_{n-d} \to R[X, Y]_n \to S_n \to 0.
$$
Thus by Algebra, Lemma \ref{algebra-lemma-cokernel-flat}
it is enough to show that multiplication
by $F$ induces an injective map
$\kappa(\mathfrak p)[X, Y]
\to \kappa(\mathfrak p)[X, Y]$
for all primes $\mathfrak p$.
This is clear from the assumption that
$F$ does not map to the zero polynomial mod $\mathfrak p$.
The assertion on ranks is clear from this as well.
\end{proof}

\begin{lemma}
\label{lemma-rel-prime-pols}
Let $k$ be a field. Let $F, G \in k[X, Y]$ be homogeneous
of degrees $d, e$. Assume $F, G$ relatively prime.
Then multiplication by $G$ is injective on $S = k[X, Y]/(F)$.
\end{lemma}

\begin{proof}
This is one way to define ``relatively prime''. If you have another
definition, then you can show it is equivalent to this one.
\end{proof}

\begin{lemma}
\label{lemma-P1-localize}
Let $R$ be a ring. Let $F(X, Y) \in R[X, Y]$ be homogeneous of degree
$d$. Let $S = R[X, Y]/(F)$ as a graded ring.
Let $\mathfrak p \subset R$ be a prime such that
some coefficient of $F$ is not in $\mathfrak p$.
There exists an $f \in R$ $f \not\in \mathfrak p$,
an integer $e$, and a $G \in R[X, Y]_e$
such that multiplication by $G$ induces isomorphisms
$(S_n)_f \to (S_{n + e})_f$ for all $n \geq d$.
\end{lemma}

\begin{proof}
During the course of the proof we may replace $R$ by $R_f$
for $f\in R$, $f\not\in \mathfrak p$ (finitely often).
As a first step we do such a replacement such that
some coefficient of $F$ is invertible in $R$.
In particular the modules $S_n$ are now locally
free of rank $d$ for $n \geq d$ by Lemma \ref{lemma-P1}.
Pick any $G \in R[X, Y]_e$ such that the image of
$G$ in $\kappa(\mathfrak p)[X, Y]$ is relatively
prime to the image of $F(X, Y)$ (this is possible for some $e$).
Apply Algebra, Lemma \ref{algebra-lemma-cokernel-flat} to the map
induced by multiplication by $G$ from $S_d \to S_{d + e}$.
By our choice of $G$ and Lemma \ref{lemma-rel-prime-pols}
we see
$S_d \otimes \kappa(\mathfrak p) \to S_{d + e} \otimes \kappa(\mathfrak p)$
is bijective. Thus, after replacing $R$ by $R_f$ for a suitable
$f$ we may assume that $G : S_d \to S_{d + e}$
is bijective. This in turn implies that the image
of $G$ in $\kappa(\mathfrak p')[X, Y]$ is relatively
prime to the image of $F$ for all primes $\mathfrak p'$
of $R$. And then by Algebra, Lemma \ref{algebra-lemma-cokernel-flat}
again we see that all the maps
$G : S_d \to S_{d + e}$, $n \geq d$ are isomorphisms.
\end{proof}

\begin{remark}
\label{remark-algebra}
Let $R$ be a ring. Suppose that we have $F \in R[X, Y]_d$
and $G \in R[X, Y]_e$ such that, setting $S = R[X, Y]/(F)$
we have (1) $S_n$ is finite locally free of rank $d$ for
all $n \geq d$, and (2) multiplication by $G$ defines
isomorphisms $S_n \to S_{n + e}$ for all $n \geq d$. In this
case we may define a finite, locally free $R$-algebra
$A$ as follows:
\begin{enumerate}
\item as an $R$-module $A = S_{ed}$, and
\item multiplication $A \times A \to A$ is given by
the rule that $H_1 H_2 = H_3$ if and only if $G^d H_3 = H_1 H_2$
in $S_{2ed}$.
\end{enumerate}
This makes sense because multiplication by $G^d$
induces a bijective map $S_{de} \to S_{2de}$.
It is easy to see that this defines a ring structure.
Note the confusing fact that the element $G^d$
defines the unit element of the ring $A$.
\end{remark}

\begin{lemma}
\label{lemma-finite-after-localization}
Let $R$ be a ring, let $f \in R$.
Suppose we have $S$, $S'$ and the solid arrows
forming the following commutative diagram of rings
$$
\xymatrix{
& S'' \ar@{-->}[rd] \ar@{-->}[dd] &
\\
R \ar[rr] \ar@{-->}[ru] \ar[d] &  & S \ar[d]
\\
R_f \ar[r] & S' \ar[r] & S_f
}
$$
Assume that $R_f \to S'$ is finite. Then we can find
a finite ring map $R \to S''$ and dotted arrows as
in the diagram such that $S' = (S'')_f$.
\end{lemma}

\begin{proof}
Namely, suppose that $S'$ is generated by
$x_i$ over $R_f$, $i = 1, \ldots, w$. Let $P_i(t) \in R_f[t]$
be a monic polynomial such that $P_i(x_i) = 0$.
Say $P_i$ has degree $d_i > 0$. Write
$P_i(t) = t^{d_i} + \sum_{j < d_i} (a_{ij}/f^n) t^j$
for some uniform $n$. Also write
the image of $x_i$ in $S_f$ as $g_i / f^n$
for suitable $g_i \in S$. Then we know
that the element
$\xi_i = f^{nd_i} g_i^{d_i} + \sum_{j < d_i} f^{n(d_i - j)} a_{ij} g_i^j$
of $S$ is killed by a power of $f$.
Hence upon increasing $n$ to $n'$, which replaces
$g_i$ by $f^{n' - n}g_i$ we may assume $\xi_i = 0$.
Then $S'$ is generated by the elements
$f^n x_i$, each of which is a zero of the
monic polynomial $Q_i(t) = t^{d_i} +
\sum_{j < d_i} f^{n(d_i - j)} a_{ij} t^j$
with coefficients in $R$. Also, by construction
$Q_i(f^ng_i) = 0$ in $S$. Thus we get a finite $R$-algebra
$S'' = R[z_1, \ldots, z_w]/(Q_1(z_1), \ldots, Q_w(z_w))$
which fits into a commutative diagram as above.
The map $\alpha : S'' \to S$ maps $z_i$ to $f^ng_i$ and
the map $\beta : S'' \to S'$ maps $z_i$ to $f^nx_i$.
It may not yet be the case that $\beta$ induces an
isomorphism $(S'')_f \cong S'$.
For the moment we only know that this map
is surjective. The problem is that there could be
elements $h/f^n \in (S'')_f$ which map to zero
in $S'$ but are not zero. In this case $\beta(h)$
is an element of $S$ such that $f^N \beta(h) = 0$
for some $N$. Thus $f^N h$ is an element ot the ideal
$J = \{h \in S'' \mid \alpha(h) = 0 \text{ and }
\beta(h) = 0\}$ of $S''$. OK, and it is easy to see that
$S''/J$ does the job.
\end{proof}


\section{Formally smooth ring maps}
\label{section-formally-smooth}

\begin{lemma}
\label{lemma-formally-smooth-smooth}
Let $R$ be a ring. Let $S$ be a $R$-algebra.
If $S$ is of finite presentation and formally smooth over $R$
then $S$ is smooth over $R$.
\end{lemma}

\begin{proof}
See Algebra, Proposition \ref{algebra-proposition-smooth-formally-smooth}.
\end{proof}


\section{Simplicial methods}
\label{section-simplicial}


\begin{lemma}
\label{lemma-equiv}
Assumptions and notation as in
Simplicial, Lemma \ref{simplicial-lemma-section}.
There exists a section $g : U \to V$ to the morphism $f$ and
the composition $g \circ f$ is homotopy equivalent to the identity
on $V$. In particular, the morphism $f$ is a homotopy equivalence.
\end{lemma}

\begin{proof}
Immediate from Simplicial, Lemmas \ref{simplicial-lemma-section} and
\ref{simplicial-lemma-trivial-kan-homotopy}.
\end{proof}



\section{Devissage of coherent sheaves}
\label{section-devissage}

\noindent
Lemmas that seem superfluous.

\begin{lemma}
\label{lemma-property-irreducible-higher-rank}
Let $X$ be a Noetherian scheme.
Let $Z_0 \subset X$ be an irreducible closed subset with generic point $\xi$.
Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that
\begin{enumerate}
\item For any short exact sequence of coherent sheaves if two
out of three of them have property $\mathcal{P}$ then so does the
third.
\item If $\mathcal{P}$ holds for a direct sum of coherent sheaves
then it holds for both.
\item For every integral closed subscheme $Z \subset Z_0 \subset X$,
$Z \not = Z_0$ and every quasi-coherent sheaf of ideals
$\mathcal{I} \subset \mathcal{O}_Z$ we have
$\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$.
\item There exists some coherent sheaf $\mathcal{G}$ on $X$ such that
\begin{enumerate}
\item $\text{Supp}(\mathcal{G}) = Z_0$,
\item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, and
\item property $\mathcal{P}$ holds for $\mathcal{G}$.
\end{enumerate}
\end{enumerate}
Then property $\mathcal{P}$ holds for every coherent sheaf
$\mathcal{F}$ on $X$ whose support is contained in $Z_0$.
\end{lemma}

\begin{proof}
The proof is a variant on the proof of
Cohomology of Schemes, Lemma \ref{coherent-lemma-property-irreducible}.
In exactly the same manner as in that proof we see that
any coherent sheaf whose support is strictly contained in $Z_0$
has property $\mathcal{P}$.

\medskip\noindent
Consider a coherent sheaf $\mathcal{G}$ as in (3).
By Cohomology of Schemes, Lemma \ref{coherent-lemma-prepare-filter-irreducible}
there exists a sheaf of ideals $\mathcal{I}$ on $Z_0$ and
a short exact sequence
$$
0 \to
\left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r} \to
\mathcal{G} \to
\mathcal{Q} \to 0
$$
where the support of $\mathcal{Q}$ is strictly contained in $Z_0$.
In particular $r > 0$ and $\mathcal{I}$ is nonzero
because the support of $\mathcal{G}$ is equal to $Z$.
Since $\mathcal{Q}$ has property $\mathcal{P}$ we conclude that
also $\left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r}$
has property $\mathcal{P}$.
By (2) we deduce property $\mathcal{P}$ for
$(Z_0 \to X)_*\mathcal{I}$. Slotting this into the proof of
Cohomology of Schemes, Lemma \ref{coherent-lemma-property-irreducible}
at the appropriate point gives the lemma.
Some details omitted.
\end{proof}

\begin{lemma}
\label{lemma-property-higher-rank}
Let $X$ be a Noetherian scheme.
Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that
\begin{enumerate}
\item For any short exact sequence of coherent sheaves if two
out of three of them have property $\mathcal{P}$ then so does the
third.
\item If $\mathcal{P}$ holds for a direct sum of coherent sheaves
then it holds for both.
\item For every integral closed subscheme $Z \subset X$
with generic point $\xi$ there exists
some coherent sheaf $\mathcal{G}$ such that
\begin{enumerate}
\item $\text{Supp}(\mathcal{G}) = Z$,
\item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, and
\item property $\mathcal{P}$ holds for $\mathcal{G}$.
\end{enumerate}
\end{enumerate}
Then property $\mathcal{P}$ holds for every coherent sheaf
on $X$.
\end{lemma}

\begin{proof}
This follows from Lemma \ref{lemma-property-irreducible-higher-rank}
in exactly the same way that
Cohomology of Schemes, Lemma \ref{coherent-lemma-property} follows from
Cohomology of Schemes, Lemma \ref{coherent-lemma-property-irreducible}.
\end{proof}






\section{Functor of quotients}
\label{section-quotients}

\begin{lemma}
\label{lemma-factors-through-quotient}
Let $S = \Spec(R)$ be an affine scheme. Let $X$ be an algebraic space over
$S$. Let $q_i : \mathcal{F} \to \mathcal{Q}_i$, $i = 1, 2$
be surjective maps of quasi-coherent $\mathcal{O}_X$-modules.
Assume $\mathcal{Q}_1$ flat over $S$. Let $T \to S$ be a quasi-compact
morphism of schemes such that there exists a factorization
$$
\xymatrix{
& \mathcal{F}_T \ar[rd]^{q_{2, T}} \ar[ld]_{q_{1, T}} \\
\mathcal{Q}_{1, T} & & \mathcal{Q}_{2, T} \ar@{..>}[ll]
}
$$
Then exists a closed subscheme $Z \subset S$ such that
(a) $T \to S$ factors through $Z$ and (b)
$q_{1, Z}$ factors through $q_{2, Z}$.
If $\text{ker}(q_2)$ is a finite type $\mathcal{O}_X$-module and $X$
quasi-compact, then we can take $Z \to S$ of finite presentation.
\end{lemma}

\begin{proof}
Apply Quot, Lemma \ref{quot-lemma-F-zero-somewhat-closed}
to the map $\text{Ker}(q_2) \to \mathcal{Q}_1$.
\end{proof}








\section{Very reasonable algebraic spaces}
\label{section-very-reasonable}

\noindent
Material that is somewhat obsolete.

\begin{lemma}
\label{lemma-reasonable-kolmogorov}
Let $S$ be a scheme.
Let $X$ be a reasonable algebraic space over $S$.
Then $|X|$ is Kolmogorov (see
Topology, Definition \ref{topology-definition-generic-point}).
\end{lemma}

\begin{proof}
Follows from the definitions and
Decent Spaces, Lemma \ref{decent-spaces-lemma-kolmogorov}.
\end{proof}

\noindent
In the rest of this section we make some remarks about very reasonable
algebraic spaces. If there exists a scheme $U$ and a
surjective, \'etale, quasi-compact
morphism $U \to X$, then $X$ is very reasonable, see
Decent Spaces, Lemma \ref{decent-spaces-lemma-characterize-very-reasonable}.

\begin{lemma}
\label{lemma-scheme-very-reasonable}
A scheme is very reasonable.
\end{lemma}

\begin{proof}
This is true because the identity map is a quasi-compact, surjective
\'etale morphism.
\end{proof}

\begin{lemma}
\label{lemma-very-reasonable-Zariski-local}
Let $S$ be a scheme.
Let $X$ be an algebraic space over $S$.
If there exists a Zariski open covering $X = \bigcup X_i$ such that
each $X_i$ is very reasonable, then $X$ is very reasonable.
\end{lemma}

\begin{proof}
This is case $(\epsilon)$ of
Decent Spaces, Lemma \ref{decent-spaces-lemma-properties-local}.
\end{proof}

\begin{lemma}
\label{lemma-quasi-separated-very-reasonable}
An algebraic space which is Zariski locally quasi-separated is very reasonable.
In particular any quasi-separated algebraic space is very reasonable.
\end{lemma}

\begin{proof}
This is one of the implications of
Decent Spaces, Lemma \ref{decent-spaces-lemma-bounded-fibres}.
\end{proof}

\begin{lemma}
\label{lemma-representable-very-reasonable}
Let $S$ be a scheme.
Let $X$, $Y$ be algebraic spaces over $S$.
Let $Y \to X$ be a representable morphism.
If $X$ is very reasonable, so is $Y$.
\end{lemma}

\begin{proof}
This is case $(\epsilon)$ of
Decent Spaces, Lemma \ref{decent-spaces-lemma-representable-properties}.
\end{proof}

\begin{remark}
\label{remark-very-reasonable-Zariski-locally-quasi-separated}
Very reasonable algebraic spaces form a strictly larger collection than
Zariski locally quasi-separated algebraic spaces. Consider
an algebraic space of the form $X = [U/G]$ (see
Spaces, Definition \ref{spaces-definition-quotient})
where $G$ is a finite group acting without fixed points on a
non-quasi-separated scheme $U$. Namely, in this case
$U \times_X U = U \times G$ and clearly both projections to $U$ are
quasi-compact, hence $X$ is very reasonable. On the other hand, the diagonal
$U \times_X U \to U \times U$ is not quasi-compact, hence this
algebraic space is not quasi-separated. Now, take $U$ the infinite
affine space over a field $k$ of characteristic $\not = 2$ with
zero doubled, see
Schemes, Example \ref{schemes-example-not-quasi-separated}.
Let $0_1, 0_2$ be the two zeros of $U$. Let $G = \{+1, -1\}$, and
let $-1$ act by $-1$ on all coordinates, and by switching
$0_1$ and $0_2$. Then $[U/G]$ is very reasonable but not Zariski locally
quasi-separated (details omitted).
\end{remark}

\noindent
Warning: The following lemma should be used with caution, as the schemes
$U_i$ in it are not necessarily separated or even quasi-separated.

\begin{lemma}
\label{lemma-very-reasonable-quasi-compact-pieces}
Let $S$ be a scheme.
Let $X$ be a very reasonable algebraic space over $S$.
There exists a set of schemes
$U_i$ and morphisms $U_i \to X$ such that
\begin{enumerate}
\item each $U_i$ is a quasi-compact scheme,
\item each $U_i \to X$ is \'etale,
\item both projections $U_i \times_X U_i \to U_i$ are quasi-compact, and
\item the morphism $\coprod U_i \to X$ is surjective (and \'etale).
\end{enumerate}
\end{lemma}

\begin{proof}
Decent Spaces, Definition \ref{decent-spaces-definition-very-reasonable}
says that there exist $U_i \to X$ such that (2), (3) and (4) hold.
Fix $i$, and set $R_i = U_i \times_X U_i$, and denote $s, t : R_i \to U_i$
the projections.
For any affine open $W \subset U_i$ the open $W' = t(s^{-1}(W)) \subset U_i$
is a quasi-compact $R_i$-invariant open (see
Groupoids, Lemma \ref{groupoids-lemma-constructing-invariant-opens}).
Hence $W'$ is a quasi-compact scheme, $W' \to X$ is \'etale, and
$W' \times_X W' = s^{-1}(W') = t^{-1}(W')$ so both projections
$W' \times_X W' \to W'$ are quasi-compact. This means the family of
$W' \to X$, where $W \subset U_i$ runs through the members of affine
open coverings of the $U_i$ gives what we want.
\end{proof}



\section{Variants of cotangent complexes for schemes}
\label{section-cotangent-schemes-variant}

\noindent
This section gives an alternative construction of the cotangent complex
of a morphism of schemes. This section is currently in the obsolete
chapter as we can get by with the easier version discussed in
Cotangent, Section \ref{cotangent-section-cotangent-schemes-variant}
for applications.

\medskip\noindent
Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{C}_{X/Y}$ be the
category whose objects are commutative diagrams
\begin{equation}
\label{equation-object}
\vcenter{
\xymatrix{
X \ar[d] & U \ar[l] \ar[d] \ar[r]_i & A \ar[ld] \\
Y & V \ar[l]
}
}
\end{equation}
of schemes where
\begin{enumerate}
\item $U$ is an open subscheme of $X$,
\item $V$ is an open subscheme of $Y$, and
\item there exists an isomorphism $A = V \times \Spec(P)$ over $V$
where $P$ is a polynomial algebra over $\mathbf{Z}$ (on some set
of variables).
\end{enumerate}
In other words, $A$ is an (infinite dimensional) affine space over $V$.
Morphisms are given by commutative diagrams.

\medskip\noindent
{\bf Notation.} An object of $\mathcal{C}_{X/Y}$, i.e., a diagram
(\ref{equation-object}), is often denoted $U \to A$ where it is
understood that (a) $U$ is an open subscheme of $X$, (b)
$U \to A$ is a morphism over $Y$, (c) the image of the
structure morphism $A \to Y$ is an open $V \subset Y$, and (d)
$A \to V$ is an affine space. We'll write $U \to A/V$ to indicate
$V \subset Y$ is the image of $A \to Y$.
Recall that $X_{Zar}$ denotes the small Zariski site $X$.
There are forgetful functors
$$
\mathcal{C}_{X/Y} \to X_{Zar},\ (U \to A) \mapsto U
\quad\text{and}\quad
\mathcal{C}_{X/Y} \mapsto Y_{Zar},\ (U \to A/V) \mapsto V.
$$

\begin{lemma}
\label{lemma-category-fibred}
Let $X \to Y$ be a morphism of schemes.
\begin{enumerate}
\item The category $\mathcal{C}_{X/Y}$ is fibred over $X_{Zar}$.
\item The category $\mathcal{C}_{X/Y}$ is fibred over $Y_{Zar}$.
\item The category $\mathcal{C}_{X/Y}$ is fibred over the
category of pairs $(U, V)$ where $U \subset X$, $V \subset Y$ are
open and $f(U) \subset V$.
\end{enumerate}
\end{lemma}

\begin{proof}
Ad (1). Given an object $U \to A$ of $\mathcal{C}_{X/Y}$ and a morphism
$U' \to U$ of $X_{Zar}$ consider the object $i' : U' \to A$ of
$\mathcal{C}_{X/Y}$ where $i'$ is the composition of $i$ and $U' \to U$.
The morphism $(U' \to A) \to (U \to A)$ of $\mathcal{C}_{X/Y}$
is strongly cartesian over $X_{Zar}$.

\medskip\noindent
Ad (2). Given an object $U \to A/V$ and $V' \to V$ we can set
$U' = U \cap f^{-1}(V')$ and $A' = V' \times_V A$ to obtain a strongly
cartesian morphism $(U' \to A') \to (U \to A)$ over $V' \to V$.

\medskip\noindent
Ad (3). Denote $(X/Y)_{Zar}$ the category in (3). Given $U \to A/V$
and a morphism $(U', V') \to (U, V)$ in $(X/Y)_{Zar}$ we can consider
$A' = V' \times_V A$. Then the morphism $(U' \to A'/V') \to (U \to A/V)$
is strongly cartesian in $\mathcal{C}_{X/Y}$ over $(X/Y)_{Zar}$.
\end{proof}

\noindent
We obtain a topology $\tau_X$ on $\mathcal{C}_{X/Y}$ by
using the topology inherited from $X_{Zar}$ (see
Stacks, Section \ref{stacks-section-topology}). If not otherwise
stated this is the topology on $\mathcal{C}_{X/Y}$ we will consider.
To be precise, a family of morphisms $\{(U_i \to A_i) \to (U \to A)\}$
is a covering of $\mathcal{C}_{X/Y}$ if and only if
\begin{enumerate}
\item $U = \bigcup U_i$, and
\item $A_i = A$ for all $i$.
\end{enumerate}
We obtain the same collection of sheaves if we allow $A_i \cong A$ in (2).
The functor $u$ defines a morphism of topoi
$\pi : \Sh(\mathcal{C}_{X/Y}) \to \Sh(X_{Zar})$.

\medskip\noindent
The site $\mathcal{C}_{X/Y}$ comes with several sheaves of rings.
\begin{enumerate}
\item The sheaf $\mathcal{O}$ given by the rule
$(U \to A) \mapsto \mathcal{O}(A)$.
\item The sheaf $\underline{\mathcal{O}}_X = \pi^{-1}\mathcal{O}_X$ given by
the rule $(U \to A) \mapsto \mathcal{O}(U)$.
\item The sheaf $\underline{\mathcal{O}}_Y$ given by the rule
$(U \to A/V) \mapsto \mathcal{O}(V)$.
\end{enumerate}
We obtain morphisms of ringed topoi
\begin{equation}
\label{equation-pi-schemes}
\vcenter{
\xymatrix{
(\Sh(\mathcal{C}_{X/Y}), \underline{\mathcal{O}}_X) \ar[r]_i \ar[d]_\pi &
(\Sh(\mathcal{C}_{X/Y}), \mathcal{O}) \\
(\Sh(X_{Zar}), \mathcal{O}_X)
}
}
\end{equation}
The morphism $i$ is the identity on underlying topoi and
$i^\sharp : \mathcal{O} \to \underline{\mathcal{O}}_X$
is the obvious map.
The map $\pi$ is a special case of Cohomology on Sites, Situation
\ref{sites-cohomology-situation-fibred-category}.
An important role will be played in the following
by the derived functors
$
Li^* : D(\mathcal{O}) \longrightarrow D(\underline{\mathcal{O}}_X)
$
left adjoint to $Ri_* = i_* : D(\underline{\mathcal{O}}_X) \to D(\mathcal{O})$
and
$
L\pi_! : D(\underline{\mathcal{O}}_X) \longrightarrow D(\mathcal{O}_X)
$
left adjoint to
$\pi^* = \pi^{-1} : D(\mathcal{O}_X) \to D(\underline{\mathcal{O}}_X)$.

\begin{remark}
\label{remark-different-topologies}
We obtain a second topology $\tau_Y$ on $\mathcal{C}_{X/Y}$
by taking the topology inherited from $Y_{Zar}$.
There is a third topology $\tau_{X \to Y}$ where a family of morphisms
$\{(U_i \to A_i) \to (U \to A)\}$ is a covering if and only
if $U = \bigcup U_i$, $V = \bigcup V_i$ and $A_i \cong V_i \times_V A$.
This is the topology inherited from the topology on the site
$(X/Y)_{Zar}$ whose underlying category is the category of pairs
$(U, V)$ as in Lemma \ref{lemma-category-fibred} part (3). The coverings
of $(X/Y)_{Zar}$ are families $\{(U_i, V_i) \to (U, V)\}$ such that
$U = \bigcup U_i$ and $V = \bigcup V_i$. There are morphisms of topoi
$$
\xymatrix{
\Sh(\mathcal{C}_{X/Y})
= \Sh(\mathcal{C}_{X/Y}, \tau_X) &
\Sh(\mathcal{C}_{X/Y}, \tau_{X \to Y}) \ar[l] \ar[r] &
\Sh(\mathcal{C}_{X/Y}, \tau_Y)
}
$$
(recall that $\tau_X$ is our ``default'' topology). The pullback functors
for these arrows are sheafification and pushforward is the identity on
underlying presheaves. The diagram of topoi
$$
\xymatrix{
\Sh(X_{Zar}) \ar[d]^f & \Sh(\mathcal{C}_{X/Y}) \ar[l]^\pi &
\Sh(\mathcal{C}_{X/Y}, \tau_{X \to Y}) \ar[l] \ar[d] \\
\Sh(Y_{Zar}) & & \Sh(\mathcal{C}_{X/Y}, \tau_Y) \ar[ll]
}
$$
is {\bf not} commutative. Namely, the pullback of a nonzero abelian sheaf on
$Y$ is a nonzero abelian sheaf on $(\mathcal{C}_{X/Y}, \tau_{X \to Y})$,
but we can certainly find examples where such a sheaf pulls back to zero
on $X$. Note that any presheaf $\mathcal{F}$ on
$Y_{Zar}$ gives a sheaf $\underline{\mathcal{F}}$ on $\mathcal{C}_{Y/X}$
by the rule which assigns to $(U \to A/V)$ the set $\mathcal{F}(V)$.
Even if $\mathcal{F}$ happens to be a sheaf it isn't true in general that
$\underline{\mathcal{F}} = \pi^{-1}f^{-1}\mathcal{F}$. This is related
to the noncommutativity of the diagram above, as we can describe
$\underline{\mathcal{F}}$ as the pushforward of the pullback
of $\mathcal{F}$ to $\Sh(\mathcal{C}_{X/Y}, \tau_{X \to Y})$ via
the lower horizontal and right vertical arrows. An
example is the sheaf $\underline{\mathcal{O}}_Y$.
But what is true is that there is a map
$\underline{\mathcal{F}} \to \pi^{-1}f^{-1}\mathcal{F}$
which is transformed (as we shall see later)
into an isomorphism after applying $\pi_!$.
\end{remark}










\section{Deformations and obstructions of flat modules}
\label{section-modules}

\noindent
In this section we sketch a construction of a deformation theory for the
stack of coherent sheaves for any algebraic space $X$ over a ring $\Lambda$.
This material is obsolete due to the improved discussion in
Quot, Section \ref{quot-section-not-flat}.

\medskip\noindent
Our setup will be the following. We assume given
\begin{enumerate}
\item a ring $\Lambda$,
\item an algebraic space $X$ over $\Lambda$,
\item a $\Lambda$-algebra $A$, set
$X_A = X \times_{\Spec(\Lambda)} \Spec(A)$, and
\item a finitely presented $\mathcal{O}_{X_A}$-module $\mathcal{F}$
flat over $A$.
\end{enumerate}
In this situation we will consider all possible surjections
$$
0 \to I \to A' \to A \to 0
$$
where $A'$ is a $\Lambda$-algebra whose kernel $I$ is an ideal of square zero
in $A'$. Given $A'$ we obtain a first order thickening $X_A \to X_{A'}$
of algebraic spaces over $\Spec(\Lambda)$. For each of these we consider
the problem of lifting $\mathcal{F}$ to a finitely presented module
$\mathcal{F}'$ on $X_{A'}$ flat over $A'$. We would like to replicate the
results of Deformation Theory, Lemma \ref{defos-lemma-flat-ringed-topoi}
in this setting.

\medskip\noindent
To be more precise let $\textit{Lift}(\mathcal{F}, A')$ denote the category
of pairs $(\mathcal{F}', \alpha)$ where $\mathcal{F}'$ is a
finitely presented module on $X_{A'}$ flat over $A'$ and
$\alpha : \mathcal{F}'|_{X_A} \to \mathcal{F}$ is an isomorphism.
Morphisms $(\mathcal{F}'_1, \alpha_1) \to (\mathcal{F}'_2, \alpha_2)$
are isomorphisms $\mathcal{F}'_1 \to \mathcal{F}'_2$ which are compatible
with $\alpha_1$ and $\alpha_2$.
The set of isomorphism classes of $\textit{Lift}(\mathcal{F}, A')$
is denoted $\text{Lift}(\mathcal{F}, A')$.

\medskip\noindent
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_X \otimes_\Lambda A$-modules
on $X_\etale$ flat over $A$. We introduce the category
$\textit{Lift}(\mathcal{G}, A')$ of pairs
$(\mathcal{G}', \beta)$ where $\mathcal{G}'$ is a sheaf of
$\mathcal{O}_X \otimes_\Lambda A'$-modules flat over $A'$ and $\beta$
is an isomorphism $\mathcal{G}' \otimes_{A'} A \to \mathcal{G}$.

\begin{lemma}
\label{lemma-equivalence}
Notation and assumptions as above. Let $p : X_A \to X$ denote the projection.
Given $A'$ denote $p' : X_{A'} \to X$ the projection. The functor $p'_*$
induces an equivalence of categories between
\begin{enumerate}
\item the category $\textit{Lift}(\mathcal{F}, A')$, and
\item the category $\textit{Lift}(p_*\mathcal{F}, A')$.
\end{enumerate}
\end{lemma}

\begin{proof}
FIXME.
\end{proof}

\noindent
Let $\mathcal{H}$ be a sheaf of $\mathcal{O} \otimes_\Lambda A$-modules
on $\mathcal{C}_{X/\Lambda}$ flat over $A$. We introduce the category
$\textit{Lift}_\mathcal{O}(\mathcal{H}, A')$
whose objects are pairs $(\mathcal{H}', \gamma)$ where $\mathcal{H}'$
is a sheaf of $\mathcal{O} \otimes_\Lambda A'$-modules flat over $A'$
and $\gamma : \mathcal{H}' \otimes_A A' \to \mathcal{H}$
is an isomorphism of $\mathcal{O} \otimes_\Lambda A$-modules.

\medskip\noindent
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_X \otimes_\Lambda A$-modules
on $X_\etale$ flat over $A$.
Consider the morphisms $i$ and $\pi$ of
Cotangent, Equation (\ref{cotangent-equation-pi-spaces}).
Denote $\underline{\mathcal{G}} = \pi^{-1}(\mathcal{G})$. It is
simply given by the rule $(U \to \mathbf{A}) \mapsto \mathcal{G}(U)$
hence it is a sheaf of $\underline{\mathcal{O}}_X \otimes_\Lambda A$-modules.
Denote $i_*\underline{\mathcal{G}}$ the same sheaf but viewed as a
sheaf of $\mathcal{O} \otimes_\Lambda A$-modules.

\begin{lemma}
\label{lemma-second-equivalence}
Notation and assumptions as above.
The functor $\pi_!$ induces an equivalence of categories between
\begin{enumerate}
\item the category
$\textit{Lift}_\mathcal{O}(i_*\underline{\mathcal{G}}, A')$, and
\item the category $\textit{Lift}(\mathcal{G}, A')$.
\end{enumerate}
\end{lemma}

\begin{proof}
FIXME.
\end{proof}

\begin{lemma}
\label{lemma-second-equivalence-obs}
Notation and assumptions as in Lemma \ref{lemma-second-equivalence}.
Consider the object
$$
L = L(\Lambda, X, A, \mathcal{G}) = L\pi_!(Li^*(i_*(\underline{\mathcal{G}})))
$$
of $D(\mathcal{O}_X \otimes_\Lambda A)$. Given a surjection $A' \to A$ of
$\Lambda$-algebras with square zero kernel $I$ we have
\begin{enumerate}
\item The category $\textit{Lift}(\mathcal{G}, A')$ is nonempty
if and only if a certain class
$\xi \in \text{Ext}^2_{\mathcal{O}_X \otimes A}(L, \mathcal{G} \otimes_A I)$
is zero.
\item If $\textit{Lift}(\mathcal{G}, A')$ is nonempty, then
$\text{Lift}(\mathcal{G}, A')$ is principal homogeneous under
$\text{Ext}^1_{\mathcal{O}_X \otimes A}(L, \mathcal{G} \otimes_A I)$.
\item Given a lift $\mathcal{G}'$, the set of automorphisms of
$\mathcal{G}'$ which pull back to $\text{id}_\mathcal{G}$ is canonically
isomorphic to
$\text{Ext}^0_{\mathcal{O}_X \otimes A}(L, \mathcal{G} \otimes_A I)$.
\end{enumerate}
\end{lemma}

\begin{proof}
FIXME.
\end{proof}

\noindent
Finally, we put everything together as follows.

\begin{proposition}
\label{proposition-conclusion}
With $\Lambda$, $X$, $A$, $\mathcal{F}$ as above. There exists a canonical
object $L = L(\Lambda, X, A, \mathcal{F})$ of $D(X_A)$ such that given a
surjection $A' \to A$ of $\Lambda$-algebras with square zero kernel $I$ we
have
\begin{enumerate}
\item The category $\textit{Lift}(\mathcal{F}, A')$ is nonempty
if and only if a certain class $\xi \in \text{Ext}^2_{X_A}(L,
\mathcal{F} \otimes_A I)$ is zero.
\item If $\textit{Lift}(\mathcal{F}, A')$ is nonempty, then
$\text{Lift}(\mathcal{F}, A')$ is principal homogeneous under
$\text{Ext}^1_{X_A}(L, \mathcal{F} \otimes_A I)$.
\item Given a lift $\mathcal{F}'$, the set of automorphisms of
$\mathcal{F}'$ which pull back to $\text{id}_\mathcal{F}$ is canonically
isomorphic to
$\text{Ext}^0_{X_A}(L, \mathcal{F} \otimes_A I)$.
\end{enumerate}
\end{proposition}

\begin{proof}
FIXME.
\end{proof}

\begin{lemma}
\label{lemma-pseudo-coherent}
In the situation of Proposition \ref{proposition-conclusion}, if
$X \to \Spec(\Lambda)$ is locally of finite type and $\Lambda$ is Noetherian,
then $L$ is pseudo-coherent.
\end{lemma}

\begin{proof}
FIXME.
\end{proof}












\section{Duplicate references}
\label{section-duplicates}

\noindent
This section is a place where we collect duplicates.

\begin{lemma}
\label{lemma-points-monomorphism}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
The map $\{\Spec(k) \to X \text{ monomorphism}\} \to |X|$ is injective.
\end{lemma}

\begin{proof}
This is a duplicate of
Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-monomorphism}.
\end{proof}

\begin{theorem}
\label{theorem-equivalence-sheaves-point}
Let $S = \Spec(K)$ with $K$ a field.
Let $\overline{s}$ be a geometric point of $S$.
Let $G = \text{Gal}_{\kappa(s)}$ denote the absolute Galois group.
Then there is an equivalence of categories
$\Sh(S_{\acute{e}tale}) \to G\textit{-Sets}$,
$\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$.
\end{theorem}

\begin{proof}
This is a duplicate of \'Etale Cohomology, Theorem
\ref{etale-cohomology-theorem-equivalence-sheaves-point}.
\end{proof}

\begin{remark}
\label{remark-tangent-spaces}
You got here because of a duplicate tag. Please see
Formal Deformation Theory, Section \ref{formal-defos-section-tangent-spaces}
for the actual content.
\end{remark}








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