more-groupoids-lemma-two-fibres

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $r, r' \in R$ with $t(r) = t(r')$ in $U$. Set $u = s(r)$, $u' = s(r')$. Denote $F_u = s^{-1}(u)$ and $F_{u'} = s^{-1}(u')$ the scheme theoretic fibres. \begin{enumerate} \item There exists a common field extension $\kappa(u) \subset k$, $\kappa(u') \subset k$ and an isomorphism $(F_u)_k \cong (F_{u'})_k$. \item We may choose the isomorphism of (1) such that a point lying over $r$ maps to a point lying over $r'$. \item If the morphisms $s$, $t$ are flat then the morphisms of germs $s : (R, r) \to (U, u)$ and $s : (R, r') \to (U, u')$ are flat locally on the base isomorphic. \item If the morphisms $s$, $t$ are \'etale (resp.\ smooth, syntomic, or flat and locally of finite presentation) then the morphisms of germs $s : (R, r) \to (U, u)$ and $s : (R, r') \to (U, u')$ are locally on the base isomorphic in the \'etale (resp.\ smooth, syntomic, or fppf) topology. \end{enumerate}


more-groupoids-lemma-make-CM

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Assume $s$ and $t$ are flat and locally of finite presentation. Then there exists an open $U' \subset U$ such that \begin{enumerate} \item $t^{-1}(U') \subset R$ is the largest open subscheme of $R$ on which the morphism $s$ is Cohen-Macaulay, \item $s^{-1}(U') \subset R$ is the largest open subscheme of $R$ on which the morphism $t$ is Cohen-Macaulay, \item the morphism $t|_{s^{-1}(U')} : s^{-1}(U') \to U$ is surjective, \item the morphism $s|_{t^{-1}(U')} : t^{-1}(U') \to U$ is surjective, and \item the restriction $R' = s^{-1}(U') \cap t^{-1}(U')$ of $R$ to $U'$ defines a groupoid $(U', R', s', t', c')$ which has the property that the morphisms $s'$ and $t'$ are Cohen-Macaulay and locally of finite presentation. \end{enumerate}


more-groupoids-lemma-restrict-preserves-type

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$. \begin{enumerate} \item If $s, t$ are locally of finite type and $g$ is locally of finite type, then $s', t'$ are locally of finite type. \item If $s, t$ are locally of finite presentation and $g$ is locally of finite presentation, then $s', t'$ are locally of finite presentation. \item If $s, t$ are flat and $g$ is flat, then $s', t'$ are flat. \item Add more here. \end{enumerate}


more-groupoids-lemma-restrict-property

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$, and let $h = s \circ \text{pr}_1 : U' \times_{g, U, t} R \to U$. If $\mathcal{P}$ is a property of morphisms of schemes such that \begin{enumerate} \item $h$ has property $\mathcal{P}$, and \item $\mathcal{P}$ is preserved under base change, \end{enumerate} then $s', t'$ have property $\mathcal{P}$.


more-groupoids-lemma-double-restrict

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ and $g' : U'' \to U'$ be morphisms of schemes. Set $g'' = g \circ g'$. Let $(U', R', s', t', c')$ be the restriction of $R$ to $U'$. Let $h = s \circ \text{pr}_1 : U' \times_{g, U, t} R \to U$, let $h' = s' \circ \text{pr}_1 : U'' \times_{g', U', t} R \to U'$, and let $h'' = s \circ \text{pr}_1 : U'' \times_{g'', U, t} R \to U$. The following diagram is commutative $$ \xymatrix{ U'' \times_{g', U', t} R' \ar[d]^{h'} & (U' \times_{g, U, t} R) \times_U (U'' \times_{g'', U, t} R) \ar[l] \ar[r] \ar[d] & U'' \times_{g'', U, t} R \ar[d]_{h''} \\ U' & U' \times_{g, U, t} R \ar[l]_{\text{pr}_0} \ar[r]^h & U } $$ with both squares cartesian where the left upper horizontal arrow is given by the rule $$ \begin{matrix} (U' \times_{g, U, t} R) \times_U (U'' \times_{g'', U, t} R) & \longrightarrow & U'' \times_{g', U', t} R' \\ ((u', r_0), (u'', r_1)) & \longmapsto & (u'', (c(r_1, i(r_0)), (g'(u''), u'))) \end{matrix} $$ with notation as explained in the proof.


more-groupoids-lemma-double-restrict-property

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ and $g' : U'' \to U'$ be morphisms of schemes. Set $g'' = g \circ g'$. Let $(U', R', s', t', c')$ be the restriction of $R$ to $U'$. Let $h = s \circ \text{pr}_1 : U' \times_{g, U, t} R \to U$, let $h' = s' \circ \text{pr}_1 : U'' \times_{g', U', t} R \to U'$, and let $h'' = s \circ \text{pr}_1 : U'' \times_{g'', U, t} R \to U$. Let $\tau \in \{Zariski, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf, \linebreak[0] fpqc\}$. Let $\mathcal{P}$ be a property of morphisms of schemes which is preserved under base change, and which is local on the target for the $\tau$-topology. If \begin{enumerate} \item $h(U' \times_U R)$ is open in $U$, \item $\{h : U' \times_U R \to h(U' \times_U R)\}$ is a $\tau$-covering, \item $h'$ has property $\mathcal{P}$, \end{enumerate} then $h''$ has property $\mathcal{P}$. Conversely, if \begin{enumerate} \item[(a)] $\{t : R \to U\}$ is a $\tau$-covering, \item[(d)] $h''$ has property $\mathcal{P}$, \end{enumerate} then $h'$ has property $\mathcal{P}$.


more-groupoids-lemma-groupoid-on-field-homogeneous

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \Spec(k)$ with $k$ a field. For any points $r, r' \in R$ there exists a field extension $k \subset k'$ and points $r_1, r_2 \in R \times_{s, \Spec(k)} \Spec(k')$ and a diagram $$ \xymatrix{ R & R \times_{s, \Spec(k)} \Spec(k') \ar[l]_-{\text{pr}_0} \ar[r]^\varphi & R \times_{s, \Spec(k)} \Spec(k') \ar[r]^-{\text{pr}_0} & R } $$ such that $\varphi$ is an isomorphism of schemes over $\Spec(k')$, we have $\varphi(r_1) = r_2$, $\text{pr}_0(r_1) = r$, and $\text{pr}_0(r_2) = r'$.


more-groupoids-lemma-restrict-groupoid-on-field

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \Spec(k)$ with $k$ a field. Let $k \subset k'$ be a field extension, $U' = \Spec(k')$ and let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $U' \to U$. In the defining diagram $$ \xymatrix{ R' \ar[d] \ar[r] \ar@/_3pc/[dd]_{t'} \ar@/^1pc/[rr]^{s'} \ar@{..>}[rd] & R \times_{s, U} U' \ar[r] \ar[d] & U' \ar[d] \\ U' \times_{U, t} R \ar[d] \ar[r] & R \ar[r]^s \ar[d]_t & U \\ U' \ar[r] & U } $$ all the morphisms are surjective, flat, and universally open. The dotted arrow $R' \to R$ is in addition affine.


more-groupoids-lemma-groupoid-on-field-explain-points

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \Spec(k)$ with $k$ a field. For any point $r \in R$ there exist \begin{enumerate} \item a field extension $k \subset k'$ with $k'$ algebraically closed, \item a point $r' \in R'$ where $(U', R', s', t', c')$ is the restriction of $(U, R, s, t, c)$ via $\Spec(k') \to \Spec(k)$ \end{enumerate} such that \begin{enumerate} \item the point $r'$ maps to $r$ under the morphism $R' \to R$, and \item the maps $s', t' : R' \to \Spec(k')$ induce isomorphisms $k' \to \kappa(r')$. \end{enumerate}


more-groupoids-lemma-groupoid-on-field-move-point

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \Spec(k)$ with $k$ a field. If $r \in R$ is a point such that $s, t$ induce isomorphisms $k \to \kappa(r)$, then the map $$ R \longrightarrow R, \quad x \longmapsto c(r, x) $$ (see proof for precise notation) is an automorphism $R \to R$ which maps $e$ to $r$.


more-groupoids-lemma-groupoid-on-field-geometrically-irreducible

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \Spec(k)$ with $k$ a field. By abuse of notation denote $e \in R$ the image of the identity morphism $e : U \to R$. Then \begin{enumerate} \item every local ring $\mathcal{O}_{R, r}$ of $R$ has a unique minimal prime ideal, \item there is exactly one irreducible component $Z$ of $R$ passing through $e$, and \item $Z$ is geometrically irreducible over $k$ via either $s$ or $t$. \end{enumerate}


more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \Spec(k)$ with $k$ a field. Assume $s, t$ are locally of finite type. Then \begin{enumerate} \item $R$ is equidimensional, \item $\dim(R) = \dim_r(R)$ for all $r \in R$, \item for any $r \in R$ we have $\text{trdeg}_{s(k)}(\kappa(r)) = \text{trdeg}_{t(k)}(\kappa(r))$, and \item for any closed point $r \in R$ we have $\dim(R) = \dim(\mathcal{O}_{R, r})$. \end{enumerate}


more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \Spec(k)$ with $k$ a field. Assume $s, t$ are locally of finite type. Then $\dim(R) = \dim(G)$ where $G$ is the stabilizer group scheme of $R$.


more-groupoids-lemma-open-image-is-closed

Notation and assumptions as in Situation \ref{situation-morphism-groupoids-on-field}. If $a(R_1)$ is open in $R_2$, then $a(R_1)$ is closed in $R_2$.


more-groupoids-lemma-map-groupoids-on-field-image

Notation and assumptions as in Situation \ref{situation-morphism-groupoids-on-field}. Let $Z \subset R_2$ be the reduced closed subscheme (see Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}) whose underlying topological space is the closure of the image of $a : R_1 \to R_2$. Then $c_2(Z \times_{s_2, U, t_2} Z) \subset Z$ set theoretically.


more-groupoids-lemma-map-groupoids-on-perfect-field-image

Notation and assumptions as in Situation \ref{situation-morphism-groupoids-on-field}. Assume that $k$ is perfect. Let $Z \subset R_2$ be the reduced closed subscheme (see Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}) whose underlying topological space is the closure of the image of $a : R_1 \to R_2$. Then $$ (U, Z, s_2|_Z, t_2|_Z, c_2|_Z) $$ is a groupoid scheme over $S$.


more-groupoids-lemma-locally-closed-image-is-closed

Notation and assumptions as in Situation \ref{situation-morphism-groupoids-on-field}. If the image $a(R_1)$ is a locally closed subset of $R_2$ then it is a closed subset.


more-groupoids-lemma-quasi-compact-map-groupoids-on-field-image

Notation and assumptions as in Situation \ref{situation-morphism-groupoids-on-field}. Assume that $a : R_1 \to R_2$ is a quasi-compact morphism. Let $Z \subset R_2$ be the scheme theoretic image (see Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-image}) of $a : R_1 \to R_2$. Then $$ (U, Z, s_2|_Z, t_2|_Z, c_2|_Z) $$ is a groupoid scheme over $S$.


more-groupoids-lemma-groupoid-on-field-image

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U$ is the spectrum of a field. Let $Z \subset U \times_S U$ be the reduced closed subscheme (see Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}) whose underlying topological space is the closure of the image of $j = (t, s) : R \to U \times_S U$. Then $\text{pr}_{02}(Z \times_{\text{pr}_1, U, \text{pr}_0} Z) \subset Z$ set theoretically.


more-groupoids-lemma-groupoid-on-perfect-field-image

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U$ is the spectrum of a perfect field. Let $Z \subset U \times_S U$ be the reduced closed subscheme (see Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}) whose underlying topological space is the closure of the image of $j = (t, s) : R \to U \times_S U$. Then $$ (U, Z, \text{pr}_0|_Z, \text{pr}_1|_Z, \text{pr}_{02}|_{Z \times_{\text{pr}_1, U, \text{pr}_0} Z}) $$ is a groupoid scheme over $S$.


more-groupoids-lemma-slice

Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$. Let $G \to U$ be the stabilizer group scheme. Assume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation. Let $u \in U$ be a finite type point of the scheme $U$, see Morphisms, Definition \ref{morphisms-definition-finite-type-point}. With notation as in Situation \ref{situation-slice}, set $$ d_1 = \dim(G_u), \quad d_2 = \dim_{e(u)}(F_u). $$ If $d_2 > d_1$, then there exist an affine scheme $U'$ and a morphism $g : U' \to U$ such that (with notation as in Situation \ref{situation-slice}) \begin{enumerate} \item $g$ is an immersion \item $u \in U'$, \item $g$ is locally of finite presentation, \item the morphism $h : U' \times_{g, U, t} R \longrightarrow U$ is Cohen-Macaulay at $(u, e(u))$, and \item we have $\dim_{e'(u)}(F'_u) = d_2 - 1$. \end{enumerate}


more-groupoids-lemma-max-slice

Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$. Let $G \to U$ be the stabilizer group scheme. Assume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation. Let $u \in U$ be a finite type point of the scheme $U$, see Morphisms, Definition \ref{morphisms-definition-finite-type-point}. With notation as in Situation \ref{situation-slice} there exist an affine scheme $U'$ and a morphism $g : U' \to U$ such that \begin{enumerate} \item $g$ is an immersion, \item $u \in U'$, \item $g$ is locally of finite presentation, \item the morphism $h : U' \times_{g, U, t} R \longrightarrow U$ is Cohen-Macaulay and locally of finite presentation, \item the morphisms $s', t' : R' \to U'$ are Cohen-Macaulay and locally of finite presentation, and \item $\dim_{e(u)}(F'_u) = \dim(G'_u)$. \end{enumerate}


more-groupoids-lemma-max-slice-quasi-finite

Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$. Let $G \to U$ be the stabilizer group scheme. Assume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation. Let $u \in U$ be a finite type point of the scheme $U$, see Morphisms, Definition \ref{morphisms-definition-finite-type-point}. Assume that $G \to U$ is locally quasi-finite. With notation as in Situation \ref{situation-slice} there exist an affine scheme $U'$ and a morphism $g : U' \to U$ such that \begin{enumerate} \item $g$ is an immersion, \item $u \in U'$, \item $g$ is locally of finite presentation, \item the morphism $h : U' \times_{g, U, t} R \longrightarrow U$ is flat, locally of finite presentation, and locally quasi-finite, and \item the morphisms $s', t' : R' \to U'$ are flat, locally of finite presentation, and locally quasi-finite. \end{enumerate}


more-groupoids-lemma-quasi-finite-over-base

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $p \in S$ be a point, and let $u \in U$ be a point lying over $p$. Assume that \begin{enumerate} \item $U \to S$ is locally of finite type, \item $U \to S$ is quasi-finite at $u$, \item $U \to S$ is separated, \item $R \to S$ is separated, \item $s$, $t$ are flat and locally of finite presentation, and \item $s^{-1}(\{u\})$ is finite. \end{enumerate} Then there exists an \'etale neighbourhood $(S', p') \to (S, p)$ with $\kappa(p) = \kappa(p')$ and a base change diagram $$ \xymatrix{ R' \amalg W' \ar@{=}[r] & S' \times_S R \ar[r] \ar@<2ex>[d]^{s'} \ar@<-2ex>[d]_{t'} & R \ar@<1ex>[d]^s \ar@<-1ex>[d]_t \\ U' \amalg W \ar@{=}[r] & S' \times_S U \ar[r] \ar[d] & U \ar[d] \\  & S' \ar[r] & S } $$ where the equal signs are decompositions into open and closed subschemes such that \begin{enumerate} \item[(a)] there exists a point $u'$ of $U'$ mapping to $u$ in $U$, \item[(b)] the fibre $(U')_{p'}$ equals $t'\big((s')^{-1}(\{u'\})\big)$ set theoretically, \item[(c)] the fibre $(R')_{p'}$ equals $(s')^{-1}\big((U')_{p'}\big)$ set theoretically, \item[(d)] the schemes $U'$ and $R'$ are finite over $S'$, \item[(e)] we have $s'(R') \subset U'$ and $t'(R') \subset U'$, \item[(f)] we have $c'(R' \times_{s', U', t'} R') \subset R'$ where $c'$ is the base change of $c$, and \item[(g)] the morphisms $s', t', c'$ determine a groupoid structure by taking the system $(U', R', s'|_{R'}, t'|_{R'}, c'|_{R' \times_{s', U', t'} R'})$. \end{enumerate}


more-groupoids-lemma-quasi-finite-over-base-j-proper

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $p \in S$ be a point, and let $u \in U$ be a point lying over $p$. Assume assumptions (1) -- (6) of Lemma \ref{lemma-quasi-finite-over-base} hold as well as \begin{enumerate} \item[(7)] $j : R \to U \times_S U$ is universally closed\footnote{In view of the other conditions this is equivalent to requiring $j$ to be proper.}. \end{enumerate} Then we can choose $(S', p') \to (S, p)$ and decompositions $S' \times_S U = U' \amalg W$ and $S' \times_S R = R' \amalg W'$ and $u' \in U'$ such that (a) -- (g) of Lemma \ref{lemma-quasi-finite-over-base} hold as well as \begin{enumerate} \item[(h)] $R'$ is the restriction of $S' \times_S R$ to $U'$. \end{enumerate}


more-groupoids-lemma-finite-stratify

Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ are finite. There exists a sequence of $R$-invariant closed subschemes $$ U = Z_0 \supset Z_1 \supset Z_2 \supset \ldots $$ such that $\bigcap Z_r = \emptyset$ and such that $s^{-1}(Z_{r - 1}) \setminus s^{-1}(Z_r) \to Z_{r - 1} \setminus Z_r$ is finite locally free of rank $r$.


more-groupoids-lemma-finite-flat-over-almost-dense-subscheme

Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ are finite. There exists an open subscheme $W \subset U$ and a closed subscheme $W' \subset W$ such that \begin{enumerate} \item $W$ and $W'$ are $R$-invariant, \item $U = t(s^{-1}(\overline{W}))$ set theoretically, \item $W$ is a thickening of $W'$, and \item the maps $s'$, $t'$ of the restriction $(W', R', s', t', c')$ are finite locally free. \end{enumerate}


more-groupoids-lemma-finite-flat-over-almost-dense-subscheme-addendum

In Lemma \ref{lemma-finite-flat-over-almost-dense-subscheme} assume in addition that $s$ and $t$ are of finite presentation. Then \begin{enumerate} \item the morphism $W' \to W$ is of finite presentation, and \item if $u \in U$ is a point whose $R$-orbit consists of generic points of irreducible components of $U$, then $u \in W$. \end{enumerate}


more-groupoids-lemma-invariant-affine-open-around-generic-point

Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ are finite and of finite presentation and $U$ quasi-separated. Let $u_1, \ldots, u_m \in U$ be points whose orbits consist of generic points of irreducible components of $U$. Then there exist $R$-invariant subschemes $V' \subset V \subset U$ such that \begin{enumerate} \item $u_1, \ldots, u_m \in V'$, \item $V$ is open in $U$, \item $V'$ and $V$ are affine, \item $V' \subset V$ is a thickening of finite presentation, \item the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$ are finite locally free. \end{enumerate}


more-groupoids-lemma-find-affine-integral

Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$ with $s, t$ integral. Let $g : U' \to U$ be an integral morphism such that every $R$-orbit in $U$ meets $g(U')$. Let $(U', R', s', t', c')$ be the restriction of $R$ to $U'$. If $u' \in U'$ is contained in an $R'$-invariant affine open, then the image $u \in U$ is contained in an $R$-invariant affine open of $U$.


more-groupoids-lemma-find-almost-invariant-function

Let $(U, R, s, t, c)$ be a groupoid scheme with $s, t$ finite and of finite presentation. Let $u_1, \ldots, u_m \in U$ be points whose $R$-orbits consist of generic points of irreducible components of $U$. Let $j : U \to \Spec(A)$ be an immersion. Let $I \subset A$ be an ideal such that $j(U) \cap V(I) = \emptyset$ and $V(I) \cup j(U)$ is closed in $\Spec(A)$. Then there exists an $h \in I$ such that $j^{-1}D(h)$ is an $R$-invariant affine open subscheme of $U$ containing $u_1, \ldots, u_m$.


more-groupoids-lemma-no-specializations-map-to-same-point

Let $(U, R, s, t, c)$ be a groupoid scheme. If $s, t$ are finite, and $u, u' \in R$ are distinct points in the same orbit, then $u'$ is not a specialization of $u$.


more-groupoids-lemma-get-affine

Let $j : V \to \Spec(A)$ be a quasi-compact immersion of schemes. Let $f \in A$ be such that $j^{-1}D(f)$ is affine and $j(V) \cap V(f)$ is closed. Then $V$ is affine.


more-groupoids-lemma-find-affine-codimension-1

Let $(U, R, s, t, c)$ be a groupoid scheme. Let $u \in U$. Assume \begin{enumerate} \item $s, t$ are finite morphisms, \item $U$ is separated and locally Noetherian, \item $\dim(\mathcal{O}_{U, u'}) \leq 1$ for every point $u'$ in the orbit of $u$. \end{enumerate} Then $u$ is contained in an $R$-invariant affine open of $U$.


more-groupoids-lemma-sits-in-functions

Let $X$ be an ind-quasi-affine scheme. Let $E \subset X$ be an intersection of a nonempty family of quasi-compact opens of $X$. Set $A = \Gamma(E, \mathcal{O}_X|_E)$ and $Y = \Spec(A)$. Then the canonical morphism $$ j : (E, \mathcal{O}_X|_E) \longrightarrow (Y, \mathcal{O}_Y) $$ of Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} determines an isomorphism $(E, \mathcal{O}_X|_E) \to (E', \mathcal{O}_Y|_{E'})$ where $E' \subset Y$ is an intersection of quasi-compact opens. If $W \subset E$ is open in $X$, then $j(W)$ is open in $Y$.


more-groupoids-lemma-affine-base-change

Suppose given a cartesian diagram $$ \xymatrix{ X \ar[d]_f \ar[r] & \Spec(B) \ar[d] \\ Y \ar[r] & \Spec(A) } $$ of schemes. Let $E \subset Y$ be an intersection of a nonempty family of quasi-compact opens of $Y$. Then $$ \Gamma(f^{-1}(E), \mathcal{O}_X|_{f^{-1}(E)}) = \Gamma(E, \mathcal{O}_Y|_E) \otimes_A B $$ provided $Y$ is quasi-separated and $A \to B$ is flat.


more-groupoids-lemma-ind-quasi-affine

Let $S$ be a scheme. Let $\{X_i \to S\}_{i\in I}$ be an fpqc covering. Let $(V_i/X_i, \varphi_{ij})$ be a descent datum relative to $\{X_i \to S\}$, see Descent, Definition \ref{descent-definition-descent-datum-for-family-of-morphisms}.  If each morphism $V_i \to X_i$ is ind-quasi-affine, then the descent datum is effective.


examples-lemma-lim-not-quasi-compact

There exists an inverse system of quasi-compact topological spaces over $\mathbf{N}$ whose limit is not quasi-compact.


examples-lemma-noncomplete-completion

There exists a local ring $R$ and a maximal ideal $\mathfrak m$ such that the completion $R^\wedge$ of $R$ with respect to $\mathfrak m$ has the following properties \begin{enumerate} \item $R^\wedge$ is local, but its maximal ideal is not equal to $\mathfrak m R^\wedge$, \item $R^\wedge$ is not a complete local ring, and \item $R^\wedge$ is not $\mathfrak m$-adically complete as an $R$-module. \end{enumerate}


examples-lemma-noncomplete-quotient

There exists a ring $R$ complete with respect to a principal ideal $I$ and a principal ideal $J$ such that $R/J$ is not $I$-adically complete.


examples-lemma-completion-not-exact

\begin{slogan} Completion is neither left nor right exact in general. \end{slogan} Completion is not an exact functor in general; it is not even right exact in general. This holds even when $I$ is finitely generated on the category of finitely presented modules.


examples-lemma-complete-modules-not-abelian

Let $R$ be a ring and let $I \subset R$ be a finitely generated ideal. The category of $I$-adically complete $R$-modules has kernels and cokernels but is not abelian in general.


examples-lemma-derived-complete-modules

Let $A$ be a ring and let $I \subset A$ be an ideal. The category $\mathcal{C}$ of derived complete modules is abelian and the inclusion functor $F : \mathcal{C} \to \text{Mod}_A$ is exact and commutes with arbitrary limits. If $I$ is finitely generated, then $\mathcal{C}$ has arbitrary direct sums and colimits, but $F$ does not commute with these in general. Finally, filtered colimits are not exact in $\mathcal{C}$ in general, hence $\mathcal{C}$ is not a Grothendieck abelian category.


examples-lemma-countable-fg-tensor

Let $R$ be a ring. Let $M$ be an $R$-module which is countable. Then $M$ is a finite $R$-module if and only if $M \otimes_R R^\mathbf{N} \to M^\mathbf{N}$ is surjective.


examples-lemma-countable-fp-tensor

Let $R$ be a countable ring. Let $M$ be a countable $R$-module. Then $M$ is finitely presented if and only if the canonical map $M \otimes_R R^\mathbf{N} \to M^\mathbf{N}$ is an isomorphism.


examples-lemma-completion-polynomial-ring-not-flat

There exists a ring such that the completion $R[[x]]$ of $R[x]$ at $(x)$ is not flat over $R$ and a fortiori not flat over $R[x]$.


examples-lemma-nonflat-completion-localization

There exists a ring $A$ complete with respect to a principal ideal $I$ and an element $f \in A$ such that the $I$-adic completion $A_f^\wedge$ of $A_f$ is not flat over $A$.


examples-lemma-quasi-coherent-not-abelian

The category of quasi-coherent\footnote{With quasi-coherent modules as defined above. Due to how things are setup in the Stacks project, this is really the correct definition; as seen above our definition agrees with what one would naively have defined to be quasi-coherent modules on $\text{Spf}(A)$, namely complete $A$-modules.} modules on a formal algebraic space $X$ is not abelian in general, even if $X$ is a Noetherian affine formal algebraic space.


examples-lemma-strange-regular-sequence

There exists a local ring $R$ and a regular sequence $x, y, z$ (in the maximal ideal) such that there exists a nonzero element $\delta \in R/zR$ with $x\delta = y\delta = 0$.


examples-lemma-nonreduced-recompletion

There exists a local Noetherian $2$-dimensional domain $(B, \mathfrak m)$ complete with respect to a principal ideal $I = (b)$ and an element $f \in \mathfrak m$, $f \not \in I$ such that the $I$-adic completion $C = (B_f)^\wedge$ of the principal localization $B_f$ is nonreduced and even such that $C_b = C[1/b] = (B_f)^\wedge[1/b]$ is nonreduced.


examples-lemma-quasi-affine-normalization-not-quasi-affine

Let $k$ be a field. There exists a variety $X$ whose normalization is quasi-affine but which is itself not quasi-affine.


examples-lemma-complement-of-affine-does-not-contain-qc-dense-open

Nonexistence quasi-compact opens of affines: \begin{enumerate} \item There exist an affine scheme $S$ and affine open $U \subset S$ such that there is no quasi-compact open $V \subset S$ with $U \cap V = \emptyset$ and $U \cup V$ dense in $S$. \item There exists an affine scheme $S$ and a closed point $s \in S$ such that $S \setminus \{s\}$ does not contain a quasi-compact dense open. \end{enumerate}


examples-lemma-no-dense-separated-quasi-compact-open-in-qcqs

There exists a quasi-compact and quasi-separated scheme $X$ which does not contain a separated quasi-compact dense open.


examples-lemma-nonexistence-qc-dense-open-subscheme

There exists a quasi-compact and quasi-separated algebraic space which does not contain a quasi-compact dense open subscheme.


examples-lemma-cannot-embed-into-affine

There exists a finite type morphism of algebraic spaces $Y \to X$ with $Y$ affine and $X$ quasi-separated, such that there does not exist an immersion $Y \to \mathbf{A}^n_X$ over $X$.


examples-lemma-pushforward-quasi-coherent

Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} is sharp in the sense that one can neither drop the assumption of quasi-compactness nor the assumption of quasi-separatedness.


examples-lemma-locally-principal-not-invertible

There exists a domain $A$ and a nonzero ideal $I \subset A$ such that $I_\mathfrak q \subset A_\mathfrak q$ is a principal ideal for all primes $\mathfrak q \subset A$ but $I$ is not an invertible $A$-module.


examples-lemma-finite-flat-non-projective

Strange flat modules. \begin{enumerate} \item There exists a ring $R$ and a finite flat $R$-module $M$ which is not projective. \item There exists a closed immersion which is flat but not open. \end{enumerate}


examples-lemma-ideal-projective-not-locally-free

There exists a ring $R$ and an ideal $I$ such that $I$ is projective as an $R$-module but not locally free as an $R$-module.


examples-lemma-chow-group-product

Let $K$ be a field. Let $C_i$, $i = 1, \ldots, n$ be smooth, projective, geometrically irreducible curves over $K$. Let $P_i \in C_i(K)$ be a rational point and let $Q_i \in C_i$ be a point such that $[\kappa(Q_i) : K] = 2$. Then $[P_1 \times \ldots \times P_n]$ is nonzero in $\CH_0(U_1 \times_K \ldots \times_K U_n)$ where $U_i = C_i \setminus \{Q_i\}$.


examples-lemma-projective-not-locally-free

There exists a countable ring $R$ and a projective module $M$ which is a direct sum of countably many locally free rank $1$ modules such that $M$ is not locally free.


examples-lemma-zero-dimensional-flat-ideal

\begin{slogan} Zero dimensional ring with flat ideal. \end{slogan} There exists a local ring $R$ with a unique prime ideal and a nonzero ideal $I \subset R$ which is a flat $R$-module


examples-lemma-epi-not-surjective

There exists an epimorphism of local rings of dimension $0$ which is not a surjection.


examples-lemma-example-raynaud-gruson

There exists a local ring $A$, a finite type ring map $A \to B$ and a prime $\mathfrak q$ lying over $\mathfrak m_A$ such that $B_{\mathfrak q}$ is flat over $A$, and for any element $g \in B$, $g \not \in \mathfrak q$ the ring $B_g$ is neither finitely presented over $A$ nor flat over $A$.


examples-lemma-finite-type-flat-not-finitely-presented

There exist examples of \begin{enumerate} \item a flat finite type ring map with geometrically irreducible complete intersection fibre rings which is not of finite presentation, \item a flat finite type ring map with geometrically connected, geometrically reduced, dimension 1, complete intersection fibre rings which is not of finite presentation, \item a proper flat morphism of schemes $X \to S$ each of whose fibres is isomorphic to either $\mathbf{P}^1_s$ or to the vanishing locus of $X_1X_2$ in $\mathbf{P}^2_s$ which is not of finite presentation, and \item a proper flat morphism of schemes $X \to S$ each of whose fibres is isomorphic to either $\mathbf{P}^1_s$ or $\mathbf{P}^2_s$ which is not of finite presentation. \end{enumerate}


examples-lemma-topology-finite-type

There exists a local homomorphism $A \to B$ of local domains which is essentially of finite type and such that $A/\mathfrak m_A \to B/\mathfrak m_B$ is finite such that for every prime $\mathfrak q \not = \mathfrak m_B$ of $B$ the ring map $A \to B/\mathfrak q$ is not the localization of a quasi-finite ring map.


examples-lemma-pure-not-universally-pure

There exists a morphism of affine schemes of finite presentation $X \to S$ and an $\mathcal{O}_X$-module $\mathcal{F}$ of finite presentation such that $\mathcal{F}$ is pure relative to $S$, but not universally pure relative to $S$.


examples-lemma-formally-smooth-nonflat

There exists a formally smooth ring map which is not flat.


examples-lemma-formally-etale-not-presented

There exist formally \'etale nonflat ring maps.


examples-lemma-formally-etale-nontrivial-cotangent-complex

There exists a formally \'etale surjective ring map $A \to B$ with $L_{B/A}$ not equal to zero.


examples-lemma-completion-etale

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring of prime characteristic $p > 0$ such that $[\kappa : \kappa^p] < \infty$. Then the  canonical map  $A \to A^\wedge$ to the completion of $A$ is flat and formally unramified. However, if $A$ is regular but not excellent, then this map is not formally \'etale.


examples-lemma-not-generated-idempotents

There exists an affine scheme $X = \Spec(A)$ and a closed subscheme $T \subset X$ such that $T$ is Zariski locally on $X$ cut out by ideals generated by idempotents, but $T$ is not cut out by an ideal generated by idempotents.


examples-lemma-not-ind-etale

There is a ring map $A \to B$ which identifies local rings but which is not ind-\'etale. A fortiori it is not ind-Zariski.


examples-lemma-nonflasque

There exists an affine scheme $X = \Spec(A)$ and an injective $A$-module $J$ such that $\widetilde{J}$ is not a flasque sheaf on $X$. Even the restriction $\Gamma(X, \widetilde{J}) \to \Gamma(U, \widetilde{J})$ with $U$ a standard open need not be surjective.


examples-lemma-nonvanishing

There exists an affine scheme $X = \Spec(A)$ whose underlying topological space is Noetherian and an injective $A$-module $I$ such that $\widetilde{I}$ has nonvanishing $H^1$ on some quasi-compact open $U$ of $X$.


examples-lemma-non-separated-group-scheme

There exists a flat group scheme of finite type over the affine line which is not separated.


examples-lemma-non-quasi-separated-group-scheme

There exists a flat group scheme of finite type over the infinite dimensional affine space which is not quasi-separated.


examples-lemma-non-flat-group-scheme

There exists a group scheme $G$ over a base $S$ whose identity component is flat over $S$ but which is not flat over $S$.


examples-lemma-non-separated-group-space

There exists a group algebraic space of finite type over a field which is not separated (and not even quasi-separated or locally separated).


examples-lemma-specializations-fibre-etale

There exists an \'etale morphism of algebraic spaces $f : X \to Y$ and a nontrivial specialization of points $x \leadsto x'$ in $|X|$ with $f(x) = f(x')$ in $|Y|$.


examples-lemma-torsors-principal-spaces-not-equal

Let $S$ be a scheme. Let $G$ be a group scheme over $S$. The stack $G\textit{-Principal}$ classifying principal homogeneous $G$-spaces (see Examples of Stacks, Subsection \ref{examples-stacks-subsection-principal-homogeneous-spaces}) and the stack $G\textit{-Torsors}$ classifying fppf $G$-torsors (see Examples of Stacks, Subsection \ref{examples-stacks-subsection-fppf-torsors}) are not equivalent in general.


examples-lemma-BG-algebraic

Let $k$ be a field. Let $G$ be an affine group scheme over $k$. If the stack $[\Spec(k)/G]$ has a smooth covering by a scheme, then $G$ is of finite type over $k$.


examples-lemma-limit-preserving-on-objects-not-limit-preserving

Let $S$ be a nonempty scheme. There exists a stack in groupoids $p : \mathcal{X} \to (\Sch/S)_{fppf}$ such that $p$ is limit preserving on objects, but $\mathcal{X}$ is not limit preserving.


examples-lemma-not-algebraic

There exists a functor $F : \Sch^{opp} \to \textit{Sets}$ which satisfies the sheaf condition for the fpqc topology, has representable diagonal $\Delta : F \to F \times F$, and such that there exists a surjective, flat, universally open, quasi-compact morphism $U \to F$ where $U$ is a scheme, but such that $F$ is not an algebraic space.


examples-lemma-sheaf-zero-on-low-dimension

There exists a sheaf of abelian groups $G$ on $\Sch_\etale$ with the following properties \begin{enumerate} \item $G(X) = 0$ whenever $\dim(X) < n$, \item $G(X)$ is not zero if $\dim(X) \geq n$, and \item if $X \subset X'$ is a thickening, then $G(X) = G(X')$. \end{enumerate}


examples-lemma-weird-sheaf

There exists a sheaf of abelian groups $G$ on $\Sch_\etale$ with the following properties \begin{enumerate} \item $G(\Spec(k)) = 0$ whenever $k$ is a field, \item $G$ is limit preserving, \item if $X \subset X'$ is a thickening, then $G(X) = G(X')$, and \item $G$ is not zero. \end{enumerate}


examples-lemma-lisse-etale-not-functorial

The lisse-\'etale site is not functorial, even for morphisms of schemes.


examples-lemma-not-a-morphism-of-sites-noetherian-to-all

With $S = \Spec(\mathbf{F}_p)$ the inclusion functor $(\textit{Noetherian}/S)_{fppf} \to (\Sch/S)_{fppf}$ does not define a morphism of sites.


examples-lemma-is-limit

Let $\mathcal{X}$ be an algebraic stack. Let $K$ be an object of $D(\mathcal{O}_\mathcal{X})$ whose cohomology sheaves are locally quasi-coherent (Sheaves on Stacks, Definition \ref{stacks-sheaves-definition-locally-quasi-coherent}) and satisfy the flat base change property (Cohomology of Stacks, Definition \ref{stacks-cohomology-definition-flat-base-change}). Then there exists a distinguished triangle $$ K \to \prod\nolimits_{n \geq 0} \tau_{\geq -n} K \to \prod\nolimits_{n \geq 0} \tau_{\geq -n} K \to K[1] $$ in $D(\mathcal{O}_\mathcal{X})$. In other words, $K$ is the derived limit of its canonical truncations.


examples-lemma-push-not-OK

A quasi-compact and quasi-separated morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks need not induce a functor $Rf_* : D_\QCoh(\mathcal{O}_\mathcal{X}) \to D_\QCoh(\mathcal{O}_\mathcal{Y})$.


examples-lemma-big-abelian-category

There exists a ``big'' abelian category $\mathcal{A}$ whose $\Ext$-groups are proper classes.


examples-lemma-example-schematically-dense-missing-weakly-associated-point

There exists a reduced scheme $X$ and a schematically dense open $U \subset X$ such that some weakly associated point $x \in X$ is not in $U$.


examples-lemma-non-descending-property-projective

The properties \begin{enumerate} \item[] $\mathcal{P}(f) =$``$f$ is projective'', and \item[] $\mathcal{P}(f) =$``$f$ is quasi-projective'' \end{enumerate} are not Zariski local on the base. A fortiori, they are not fpqc local on the base.


examples-lemma-family-of-curves-not-scheme

There exists a field $k$ and a family of curves $X \to \mathbf{A}^1_k$ such that $X$ is not a scheme.


examples-lemma-no-derived-base-change

Let $R \to R'$ and $R \to A$ be ring maps. In general there does not exist a functor $T : D(A) \to D(A \otimes_R R')$ of triangulated categories such that an $A$-module $M$ gives an object $T(M)$ of $D(A \otimes_R R')$ which maps to $M \otimes_R^\mathbf{L} R'$ under the map $D(A \otimes_R R') \to D(R')$.


examples-lemma-no-good-representatif-compact-object

There exists a differential graded algebra $(A, \text{d})$ and a compact object $E$ of $D(A, \text{d})$ such that $E$ cannot be represented by a finite and graded projective differential graded $A$-module.


examples-lemma-torsors-over-two-dimensional-regular

Let $W$ be a two dimensional regular integral Noetherian scheme with function field $K$. Let $G \to W$ be an abelian scheme. Then the map $H^1_{fppf}(W, G) \to H^1_{fppf}(\Spec(K), G)$ is injective.


examples-lemma-torsors-over-field-torsion

Let $G$ be a smooth commutative group algebraic space over a field $K$. Then $H^1_{fppf}(\Spec(K), G)$ is torsion.


examples-lemma-non-formal-effectiveness

Let $k$ be an algebraically closed field which is not the closure of a finite field. Let $A$ be an abelian variety over $k$. Let $\mathcal{X} = [\Spec(k)/A]$. There exists an inverse system of $k$-algebras $R_n$ with surjective transition maps whose kernels are locally nilpotent and a system $(\xi_n)$ of $\mathcal{X}$ lying over the system $(\Spec(R_n))$ such that this system is not effective in the sense of Artin's Axioms, Remark \ref{artin-remark-strong-effectiveness}.


examples-lemma-affine-not-mcquillan

There exists an affine formal algebraic space which is not McQuillan.


examples-lemma-affine-formal-functions-do-not-separate-points

There exists an affine formal algebraic space $X$ whose regular functions do not separate points, in the following sense: If we write $X = \colim X_\lambda$ as in Formal Spaces, Definition \ref{formal-spaces-definition-affine-formal-algebraic-space} then $\lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})$ is a field, but $X_{red}$ has infinitely many points.


examples-lemma-representable-morphism-affine-formal-not-mcquillan-top

There exists a representable morphism $f : X \to Y$ of affine formal algebraic spaces with $Y$ McQuillan, but $X$ not McQuillan.


examples-lemma-weird-flat-map

There exists a commutative ring $A$ and a flat $A$-algebra $B$ which cannot be written as a filtered colimit of finitely presented flat $A$-algebras. In fact, we may either choose $A$ to be a finite type $\mathbf{F}_p$-algebra or a $1$-dimensional Noetherian local ring with residue field of characteristic $0$.


examples-lemma-colimit-topology

There exists a system $G_1 \to G_2 \to G_3 \to \ldots$ of (abelian) topological groups such that $\colim G_n$ taken in the category of topological spaces is different from $\colim G_n$ taken in the category of topological groups.


examples-lemma-universally-submersive-not-V

There exists a morphism $X \to Y$ of affine schemes which is universally submersive such that $\{X \to Y\}$ is not a V covering.


examples-lemma-non-fpqc-descent

There exists a ring $A$ and an infinite family of flat ring maps $\{A \to A_i\}_{i \in I}$ such that for every $A$-module $M$  $$ M = \text{Equalizer}\left( \xymatrix{ \prod\nolimits_{i \in I} M \otimes_A A_i \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{i, j \in I} M \otimes_A A_i \otimes_A A_j } \right) $$ but there is no finite subfamily where the same thing is true.


examples-proposition-nonalghomstack

The stack $\mathcal{X} = \underline{\Mor}_S(X, [S/A])$ is not algebraic.


examples-proposition-localization-and-serre-quotients

Let $A$ be a ring. Let $S$ be a multiplicative subset of $A$. Let $\text{Mod}_A$ denote the category of $A$-modules and $\mathcal{T}$ its Serre subcategory of modules for which any element is annihilated by some element of $S$. Then there is a canonical equivalence $\text{Mod}_A/\mathcal{T} \rightarrow \text{Mod}_{S^{-1}A}$.


examples-proposition-quotient-by-torsion-modules

Let $A$ be a ring. Let $Q(A)$ denote its total quotient ring (as in Algebra, Example \ref{algebra-example-localize-at-prime}). Let $\text{Mod}_A$ denote the category of $A$-modules and $\mathcal{T}$ its Serre subcategory of torsion modules. Let $\text{Mod}_{Q(A)}$ denote the category of $Q(A)$-modules. Then there is a canonical equivalence $\text{Mod}_A/\mathcal{T} \rightarrow \text{Mod}_{Q(A)}$.


bootstrap-theorem-bootstrap

Let $S$ be a scheme. Let $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Assume that \begin{enumerate} \item the presheaf $F$ is a sheaf, \item the diagonal morphism $F  \to F \times F$ is representable by algebraic spaces, and \item there exists an algebraic space $X$ and a map $X \to F$ which is surjective, and \'etale. \end{enumerate} Then $F$ is an algebraic space.


bootstrap-theorem-final-bootstrap

Let $S$ be a scheme. Let $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Any one of the following conditions implies that $F$ is an algebraic space: \begin{enumerate} \item $F = U/R$ where $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \to U \times_S U$ is an equivalence relation, \item $F = U/R$ where $(U, R, s, t, c)$ is a groupoid scheme over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \to U \times_S U$ is an equivalence relation, \item $F$ is a sheaf and there exists an algebraic space $U$ and a morphism $U \to F$ which is representable by algebraic spaces, surjective, flat and locally of finite presentation, \item $F$ is a sheaf and there exists a scheme $U$ and a morphism $U \to F$ which is representable (by algebraic spaces or schemes), surjective, flat and locally of finite presentation, \item $F$ is a sheaf, $\Delta_F$ is representable by algebraic spaces, and there exists an algebraic space $U$ and a morphism $U \to F$ which is surjective, flat, and locally of finite presentation, or \item $F$ is a sheaf, $\Delta_F$ is representable, and there exists a scheme $U$ and a morphism $U \to F$ which is surjective, flat, and locally of finite presentation. \end{enumerate}


bootstrap-lemma-base-change-transformation

\begin{slogan} A base change of a representable by algebraic spaces morphism of presheaves is representable by algebraic spaces. \end{slogan} Let $S$ be a scheme. Let $$ \xymatrix{ G' \times_G F \ar[r] \ar[d]^{a'} & F \ar[d]^a \\ G' \ar[r] & G } $$ be a fibre square of presheaves on $(\Sch/S)_{fppf}$. If $a$ is representable by algebraic spaces so is $a'$.


bootstrap-lemma-representable-by-spaces-transformation-to-sheaf

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be representable by algebraic spaces. If $G$ is a sheaf, then so is $F$.


bootstrap-lemma-representable-by-spaces-over-space

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be representable by algebraic spaces. If $G$ is an algebraic space, then so is $F$.


bootstrap-lemma-representable-by-spaces

Let $S$ be a scheme. Let $a : F \to G$ be a map of presheaves on $(\Sch/S)_{fppf}$. Suppose $a : F \to G$ is representable by algebraic spaces. If $X$ is an algebraic space over $S$, and $X \to G$ is a map of presheaves then $X \times_G F$ is an algebraic space.


bootstrap-lemma-composition-transformation

Let $S$ be a scheme. Let $$ \xymatrix{ F \ar[r]^a & G \ar[r]^b & H } $$ be maps of presheaves on $(\Sch/S)_{fppf}$. If $a$ and $b$ are representable by algebraic spaces, so is $b \circ a$.


bootstrap-lemma-product-transformations

Let $S$ be a scheme. Let $F_i, G_i : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$, $i = 1, 2$. Let $a_i : F_i \to G_i$, $i = 1, 2$ be representable by algebraic spaces. Then $$ a_1 \times a_2 : F_1 \times F_2 \longrightarrow G_1 \times G_2 $$ is a representable by algebraic spaces.


bootstrap-lemma-representable-by-spaces-permanence

Let $S$ be a scheme. Let $a : F \to G$ and $b : G \to H$ be transformations of functors $(\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Assume \begin{enumerate} \item $\Delta : G \to G \times_H G$ is representable by algebraic spaces, and \item $b \circ a : F \to H$ is representable by algebraic spaces. \end{enumerate} Then $a$ is representable by algebraic spaces.


bootstrap-lemma-glueing-sheaves

Let $S \in \Ob(\Sch_{fppf})$. Let $F$ be a presheaf of sets on $(\Sch/S)_{fppf}$. Assume \begin{enumerate} \item $F$ is a sheaf for the Zariski topology on $(\Sch/S)_{fppf}$, \item there exists an index set $I$ and subfunctors $F_i \subset F$ such that \begin{enumerate} \item each $F_i$ is an fppf sheaf, \item each $F_i \to F$ is representable by algebraic spaces, \item $\coprod F_i \to F$ becomes surjective after fppf sheafification. \end{enumerate} \end{enumerate} Then $F$ is an fppf sheaf.


bootstrap-lemma-base-change-transformation-property

Let $S$ be a scheme. Let $\mathcal{P}$ be a property as in Definition \ref{definition-property-transformation}. Let $$ \xymatrix{ G' \times_G F \ar[r] \ar[d]^{a'} & F \ar[d]^a \\ G' \ar[r] & G } $$ be a fibre square of presheaves on $(\Sch/S)_{fppf}$. If $a$ is representable by algebraic spaces and has $\mathcal{P}$ so does $a'$.


bootstrap-lemma-composition-transformation-property

Let $S$ be a scheme. Let $\mathcal{P}$ be a property as in Definition \ref{definition-property-transformation}, and assume $\mathcal{P}$ is stable under composition. Let $$ \xymatrix{ F \ar[r]^a & G \ar[r]^b & H } $$ be maps of presheaves on $(\Sch/S)_{fppf}$. If $a$, $b$ are representable by algebraic spaces and has $\mathcal{P}$ so does $b \circ a$.


bootstrap-lemma-product-transformations-property

Let $S$ be a scheme. Let $F_i, G_i : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$, $i = 1, 2$. Let $a_i : F_i \to G_i$, $i = 1, 2$ be representable by algebraic spaces. Let $\mathcal{P}$ be a property as in Definition \ref{definition-property-transformation} which is stable under composition. If $a_1$ and $a_2$ have property $\mathcal{P}$ so does $a_1 \times a_2 : F_1 \times F_2 \longrightarrow G_1 \times G_2$.


bootstrap-lemma-transformations-property-implication

Let $S$ be a scheme. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be a transformation of functors representable by algebraic spaces. Let $\mathcal{P}$, $\mathcal{P}'$ be properties as in Definition \ref{definition-property-transformation}. Suppose that for any morphism $f : X \to Y$ of algebraic spaces over $S$ we have $\mathcal{P}(f) \Rightarrow \mathcal{P}'(f)$. If $a$ has property $\mathcal{P}$, then $a$ has property $\mathcal{P}'$.


bootstrap-lemma-surjective-flat-locally-finite-presentation

Let $S$ be a scheme. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be sheaves. Let $a : F \to G$ be representable by algebraic spaces, flat, locally of finite presentation, and surjective. Then $a : F \to G$ is surjective as a map of sheaves.


bootstrap-lemma-representable-diagonal

\begin{slogan} The diagonal of a presheaf is representable by algebraic spaces if and only if every map from a scheme to the presheaf is representable by algebraic spaces. \end{slogan} Let $S$ be a scheme. If $F$ is a presheaf on $(\Sch/S)_{fppf}$. The following are equivalent: \begin{enumerate} \item $\Delta_F : F \to F \times F$ is representable by algebraic spaces, \item for every scheme $T$ any map $T \to F$ is representable by algebraic spaces, and \item for every algebraic space $X$ any map $X \to F$ is representable by algebraic spaces. \end{enumerate}


bootstrap-lemma-after-fppf-sep-lqf

Let $S$ be a scheme. Let $$ \xymatrix{ E \ar[r]_a \ar[d]_f & F \ar[d]^g \\ H \ar[r]^b & G } $$ be a cartesian diagram of sheaves on $(\Sch/S)_{fppf}$, so $E = H \times_G F$. If \begin{enumerate} \item $g$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation, and \item $a$ is representable by algebraic spaces, separated, and locally quasi-finite \end{enumerate} then $b$ is representable (by schemes) as well as separated and locally quasi-finite.


bootstrap-lemma-bootstrap-diagonal

Let $S$ be a scheme. Let $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Assume that \begin{enumerate} \item the presheaf $F$ is a sheaf, \item there exists an algebraic space $X$ and a map $X \to F$ which is representable by algebraic spaces, surjective, flat and locally of finite presentation. \end{enumerate} Then $\Delta_F$ is representable (by schemes).


bootstrap-lemma-better-finding-opens

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism. Assume \begin{enumerate} \item the composition $$ \xymatrix{ U' \times_{g, U, t} R \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^h & R \ar[r]_s & U } $$ has an open image $W \subset U$, and \item the resulting map $h : U' \times_{g, U, t} R \to W$ defines a surjection of sheaves in the fppf topology. \end{enumerate} Let $R' = R|_{U'}$ be the restriction of $R$ to $U$. Then the map of quotient sheaves $$ U'/R' \to U/R $$ in the fppf topology is representable, and is an open immersion.


bootstrap-lemma-slice-equivalence-relation

Let $S$ be a scheme. Let $j : R \to U \times_S U$ be an equivalence relation on schemes over $S$. Assume $s, t : R \to U$ are flat and locally of finite presentation. Then there exists an equivalence relation $j' : R' \to U'\times_S U'$ on schemes over $S$, and an isomorphism $$ U'/R' \longrightarrow U/R $$ induced by a morphism $U' \to U$ which maps $R'$ into $R$ such that $s', t' : R \to U$ are flat, locally of finite presentation and locally quasi-finite.


bootstrap-lemma-divide-subgroupoid

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $P \to R$ be monomorphism of schemes. Assume that \begin{enumerate} \item $(U, P, s|_P, t|_P, c|_{P \times_{s, U, t}P})$ is a groupoid scheme, \item $s|_P, t|_P : P \to U$ are finite locally free, \item $j|_P : P \to U \times_S U$ is a monomorphism. \item $U$ is affine, and \item $j : R \to U \times_S U$ is separated and locally quasi-finite, \end{enumerate} Then $U/P$ is representable by an affine scheme $\overline{U}$, the quotient morphism $U \to \overline{U}$ is finite locally free, and $P = U \times_{\overline{U}} U$. Moreover, $R$ is the restriction of a groupoid scheme $(\overline{U}, \overline{R}, \overline{s}, \overline{t}, \overline{c})$ on $\overline{U}$ via the quotient morphism $U \to \overline{U}$.


bootstrap-lemma-locally-algebraic-space

\begin{slogan} The definition of an algebraic space is fppf local. \end{slogan} Let $S$ be a scheme. Let $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Let $\{S_i \to S\}_{i \in I}$ be a covering of $(\Sch/S)_{fppf}$. Assume that \begin{enumerate} \item $F$ is a sheaf, \item each $F_i = h_{S_i} \times F$ is an algebraic space, and \item $\coprod_{i \in I} F_i$ is an algebraic space (see Spaces, Lemma \ref{spaces-lemma-coproduct-algebraic-spaces}). \end{enumerate} Then $F$ is an algebraic space.


bootstrap-lemma-locally-algebraic-space-finite-type

Let $S$ be a scheme. Let $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Let $\{S_i \to S\}_{i \in I}$ be a covering of $(\Sch/S)_{fppf}$. Assume that \begin{enumerate} \item $F$ is a sheaf, \item each $F_i = h_{S_i} \times F$ is an algebraic space, and \item the morphisms $F_i \to S_i$ are of finite type. \end{enumerate} Then $F$ is an algebraic space.


bootstrap-lemma-descend-algebraic-space

\begin{slogan} Fppf descent data for algebraic spaces are effective. \end{slogan} Let $S$ be a scheme. Let $\{X_i \to X\}_{i \in I}$ be an fppf covering of algebraic spaces over $S$. \begin{enumerate} \item If $I$ is countable\footnote{The restriction on countablility can be ignored by those who do not care about set theoretical issues. We can allow larger index sets here if we can bound the size of the algebraic spaces which we are descending. See for example Lemma \ref{lemma-locally-algebraic-space-finite-type}.}, then any descent datum for algebraic spaces relative to $\{X_i \to X\}$ is effective. \item Any descent datum $(Y_i, \varphi_{ij})$ relative to $\{X_i \to X\}_{i \in I}$ (Descent on Spaces, Definition \ref{spaces-descent-definition-descent-datum-for-family-of-morphisms}) with $Y_i \to X_i$ of finite type is effective. \end{enumerate}


bootstrap-lemma-representable-by-spaces-cover

Let $S$ be a scheme. Let $a : F \to G$ and $b : G \to H$ be transformations of functors $(\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Assume \begin{enumerate} \item $F, G, H$ are sheaves, \item $a : F \to G$ is representable by algebraic spaces, flat, locally of finite presentation, and surjective, and \item $b \circ a : F \to H$ is representable by algebraic spaces. \end{enumerate} Then $b$ is representable by algebraic spaces.


bootstrap-lemma-quotient-stack-isom

Assume $B \to S$ and $(U, R, s, t, c)$ are as in Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-quotient-stack} (1). For any scheme $T$ over $S$ and objects $x, y$ of $[U/R]$ over $T$ the sheaf $\mathit{Isom}(x, y)$ on $(\Sch/T)_{fppf}$ is an algebraic space.


bootstrap-lemma-covering-quotient

Let $S$ be a scheme. Consider an algebraic space $F$ of the form $F = U/R$ where $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \to U \times_S U$ is an equivalence relation. Then $U \to F$ is surjective, flat, and locally of finite presentation.


bootstrap-lemma-quotient-free-action

Let $S$ be a scheme. Let $X \to B$ be a morphism of algebraic spaces over $S$. Let $G$ be a group algebraic space over $B$ and let $a : G \times_B X \to X$ be an action of $G$ on $X$ over $B$. If \begin{enumerate} \item $a$ is a free action, and \item $G \to B$ is flat and locally of finite presentation, \end{enumerate} then $X/G$ (see Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-quotient-sheaf}) is an algebraic space and $X \to X/G$ is surjective, flat, and locally of finite presentation.


bootstrap-lemma-descent-torsor

Let $\{S_i \to S\}_{i \in I}$ be a covering of $(\Sch/S)_{fppf}$. Let $G$ be a group algebraic space over $S$, and denote $G_i = G_{S_i}$ the base changes. Suppose given \begin{enumerate} \item for each $i \in I$ an fppf $G_i$-torsor $X_i$ over $S_i$, and \item for each $i, j \in I$ a $G_{S_i \times_S S_j}$-equivariant isomorphism $\varphi_{ij} : X_i \times_S S_j \to S_i \times_S X_j$ satisfying the cocycle condition over every $S_i \times_S S_j \times_S S_j$. \end{enumerate} Then there exists an fppf $G$-torsor $X$ over $S$ whose base change to $S_i$ is isomorphic to $X_i$ such that we recover the descent datum $\varphi_{ij}$.


bootstrap-lemma-spaces-etale

Denote the common underlying category of $\Sch_{fppf}$ and $\Sch_\etale$ by $\Sch_\alpha$ (see Topologies, Remark \ref{topologies-remark-choice-sites}). Let $S$ be an object of $\Sch_\alpha$. Let $$ F : (\Sch_\alpha/S)^{opp} \longrightarrow \textit{Sets} $$ be a presheaf with the following properties: \begin{enumerate} \item $F$ is a sheaf for the \'etale topology, \item the diagonal $\Delta : F \to F \times F$ is representable, and \item there exists $U \in \Ob(\Sch_\alpha/S)$ and $U \to F$ which is surjective and \'etale. \end{enumerate} Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition \ref{spaces-definition-algebraic-space}.


bootstrap-lemma-spaces-etale-locally-representable

Denote the common underlying category of $\Sch_{fppf}$ and $\Sch_\etale$ by $\Sch_\alpha$ (see Topologies, Remark \ref{topologies-remark-choice-sites}). Let $S$ be an object of $\Sch_\alpha$. Let $$ F : (\Sch_\alpha/S)^{opp} \longrightarrow \textit{Sets} $$ be a presheaf with the following properties: \begin{enumerate} \item $F$ is a sheaf for the \'etale topology, \item there exists an algebraic space $U$ over $S$ and a map $U \to F$ which is representable by algebraic spaces, surjective, and \'etale. \end{enumerate} Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition \ref{spaces-definition-algebraic-space}.


bootstrap-lemma-spaces-etale-smooth-cover

Denote the common underlying category of $\Sch_{fppf}$ and $\Sch_\etale$ by $\Sch_\alpha$ (see Topologies, Remark \ref{topologies-remark-choice-sites}). Let $S$ be an object of $\Sch_\alpha$.  $$ F : (\Sch_\alpha/S)^{opp} \longrightarrow \textit{Sets} $$ be a presheaf with the following properties: \begin{enumerate} \item $F$ is a sheaf for the \'etale topology, \item the diagonal $\Delta : F \to F \times F$ is representable by algebraic spaces, and \item there exists $U \in \Ob(\Sch_\alpha/S)$ and $U \to F$ which is surjective and smooth. \end{enumerate} Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition \ref{spaces-definition-algebraic-space}.


spaces-perfect-theorem-approximation

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Then approximation by perfect complexes holds on $X$.


spaces-perfect-theorem-bondal-van-den-Bergh

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The category $D_\QCoh(\mathcal{O}_X)$ can be generated by a single perfect object. More precisely, there exists a perfect object $P$ of $D(\mathcal{O}_X)$ such that for  $E \in D_\QCoh(\mathcal{O}_X)$ the following are equivalent \begin{enumerate} \item $E = 0$, and \item $\Hom_{D(\mathcal{O}_X)}(P[n], E) = 0$ for all $n \in \mathbf{Z}$. \end{enumerate}


spaces-perfect-theorem-DQCoh-is-Ddga

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Then there exist a differential graded algebra $(E, \text{d})$ with only a finite number of nonzero cohomology groups $H^i(E)$ such that $D_\QCoh(\mathcal{O}_X)$ is equivalent to $D(E, \text{d})$.


spaces-perfect-lemma-restrict-direct-image-open

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Given an \'etale morphism $V \to Y$, set $U = V \times_Y X$ and denote $g : U \to V$ the projection morphism. Then $(Rf_*E)|_V = Rg_*(E|_U)$ for $E$ in $D(\mathcal{O}_X)$.


spaces-perfect-lemma-epsilon-flat

The morphism $\epsilon$ of (\ref{equation-epsilon}) is a flat morphism of ringed sites. In particular the functor $\epsilon^* : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_\etale)$ is exact. Moreover, if $\epsilon^*\mathcal{F} = 0$, then $\mathcal{F} = 0$.


spaces-perfect-lemma-derived-quasi-coherent-small-etale-site

Let $X$ be a scheme. The functor $\epsilon^* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_\etale)$ defined above is an equivalence.


spaces-perfect-lemma-check-quasi-coherence-on-covering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $E$ is in $D_\QCoh(\mathcal{O}_X)$, \item for every \'etale morphism $\varphi : U \to X$ where $U$ is an affine scheme $\varphi^*E$ is an object of $D_\QCoh(\mathcal{O}_U)$, \item for every \'etale morphism $\varphi : U \to X$ where $U$ is a scheme $\varphi^*E$ is an object of $D_\QCoh(\mathcal{O}_U)$, \item there exists a surjective \'etale morphism $\varphi : U \to X$ where $U$ is a scheme such that $\varphi^*E$ is an object of $D_\QCoh(\mathcal{O}_U)$, and \item there exists a surjective \'etale morphism of algebraic spaces $f : Y \to X$ such that $Lf^*E$ is an object of $D_\QCoh(\mathcal{O}_Y)$. \end{enumerate}


spaces-perfect-lemma-quasi-coherence-direct-sums

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $D_\QCoh(\mathcal{O}_X)$ has direct sums.


spaces-perfect-lemma-Rlim-quasi-coherent

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $(K_n)$ be an inverse system of $D_\QCoh(\mathcal{O}_X)$ with derived limit $K = R\lim K_n$ in $D(\mathcal{O}_X)$. Assume $H^q(K_{n + 1}) \to H^q(K_n)$ is surjective for all $q \in \mathbf{Z}$ and $n \geq 1$. Then \begin{enumerate} \item $H^q(K) = \lim H^q(K_n)$, \item $R\lim H^q(K_n) = \lim H^q(K_n)$, and \item for every affine open $U \subset X$ we have $H^p(U, \lim H^q(K_n)) = 0$ for $p > 0$. \end{enumerate}


spaces-perfect-lemma-quasi-coherence-pullback

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. The functor $Lf^*$ sends $D_\QCoh(\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_Y)$.


spaces-perfect-lemma-quasi-coherence-tensor-product

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For objects $K, L$ of $D_\QCoh(\mathcal{O}_X)$ the derived tensor product $K \otimes^\mathbf{L} L$ is in $D_\QCoh(\mathcal{O}_X)$.


spaces-perfect-lemma-nice-K-injective

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. Then the canonical map $E \to R\lim \tau_{\geq -n}E$ is an isomorphism\footnote{In particular, $E$ has a K-injective representative as in Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-K-injective}.}.


spaces-perfect-lemma-application-nice-K-injective

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $F : \textit{Mod}(\mathcal{O}_X) \to \textit{Ab}$ be a functor and $N \geq 0$ an integer. Assume that \begin{enumerate} \item $F$ is left exact, \item $F$ commutes with countable direct products, \item $R^pF(\mathcal{F}) = 0$ for all $p \geq N$ and $\mathcal{F}$ quasi-coherent. \end{enumerate} Then for $E \in D_\QCoh(\mathcal{O}_X)$ \begin{enumerate} \item $H^i(RF(\tau_{\leq a}E) \to H^i(RF(E))$ is an isomorphism for $i \leq a$, \item $H^i(RF(E)) \to H^i(RF(\tau_{\geq b - N + 1}E))$ is an isomorphism for $i \geq b$, \item if $H^i(E) = 0$ for $i \not \in [a, b]$ for some $-\infty \leq a \leq b \leq \infty$, then $H^i(RF(E)) = 0$ for $i \not \in [a, b + N - 1]$. \end{enumerate}


spaces-perfect-lemma-quasi-coherence-direct-image

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$. \begin{enumerate} \item The functor $Rf_*$ sends $D_\QCoh(\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_Y)$. \item If $Y$ is quasi-compact, there exists an integer $N = N(X, Y, f)$ such that for an object $E$ of $D_\QCoh(\mathcal{O}_X)$ with $H^m(E) = 0$ for $m > 0$ we have $H^m(Rf_*E) = 0$ for $m \geq N$. \item In fact, if $Y$ is quasi-compact we can find $N = N(X, Y, f)$ such that for every morphism of algebraic spaces $Y' \to Y$ the same conclusion holds for the functor $R(f')_*$ where $f' : X' \to Y'$ is the base change of $f$. \end{enumerate}


spaces-perfect-lemma-quasi-coherence-pushforward-direct-sums

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$. Then $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ commutes with direct sums.


spaces-perfect-lemma-affine-morphism

Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Then $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ reflects isomorphisms.


spaces-perfect-lemma-affine-morphism-pull-push

Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. For $E$ in $D_\QCoh(\mathcal{O}_Y)$ we have $Rf_* Lf^* E = E \otimes^\mathbf{L}_{\mathcal{O}_Y} f_*\mathcal{O}_X$.


spaces-perfect-lemma-closed-proper-over-base

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $T \subset |X|$ be a closed subset. The following are equivalent \begin{enumerate} \item the morphism $Z \to Y$ is proper if $Z$ is the reduced induced algebraic space structure on $T$ (Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}), \item for some closed subspace $Z \subset X$ with $|Z| = T$ the morphism $Z \to Y$ is proper, and \item for any closed subspace $Z \subset X$ with $|Z| = T$ the morphism $Z \to Y$ is proper. \end{enumerate}


spaces-perfect-lemma-closed-closed-proper-over-base

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $T' \subset T \subset |X|$ be closed subsets. If $T$ is proper over $Y$, then the same is true for $T'$.


spaces-perfect-lemma-base-change-closed-proper-over-base

Let $S$ be a scheme. Consider a cartesian diagram of algebraic spaces over $S$ $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ with $f$ locally of finite type. If $T$ is a closed subset of $|X|$ proper over $Y$, then $|g'|^{-1}(T)$ is a closed subset of $|X'|$ proper over $Y'$.


spaces-perfect-lemma-functoriality-closed-proper-over-base

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces which are locally of finite type over $B$. \begin{enumerate} \item If $Y$ is separated over $B$ and $T \subset |X|$ is a closed subset proper over $B$, then $|f|(T)$ is a closed subset of $|Y|$ proper over $B$. \item If $f$ is universally closed and $T \subset |X|$ is a closed subset proper over $B$, then $|f|(T)$ is a closed subset of $Y$ proper over $B$. \item If $f$ is proper and $T \subset |Y|$ is a closed subset proper over $B$, then $|f|^{-1}(T)$ is a closed subset of $|X|$ proper over $B$. \end{enumerate}


spaces-perfect-lemma-union-closed-proper-over-base

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $T_i \subset |X|$, $i = 1, \ldots, n$ be closed subsets. If $T_i$, $i = 1, \ldots, n$ are proper over $Y$, then the same is true for $T_1 \cup \ldots \cup T_n$.


spaces-perfect-lemma-module-support-proper-over-base

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item the support of $\mathcal{F}$ is proper over $Y$, \item the scheme theoretic support of $\mathcal{F}$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-scheme-theoretic-support}) is proper over $Y$, and \item there exists a closed subspace $Z \subset X$ and a finite type, quasi-coherent $\mathcal{O}_Z$-module $\mathcal{G}$ such that (a) $Z \to Y$ is proper, and (b) $(Z \to X)_*\mathcal{G} = \mathcal{F}$. \end{enumerate}


spaces-perfect-lemma-support-proper-over-base-pushforward

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ locally Noetherian. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module with support proper over $Y$. Then $R^pf_*\mathcal{F}$ is a coherent $\mathcal{O}_Y$-module for all $p \geq 0$.


spaces-perfect-lemma-direct-image-coherent

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian. Let $E$ be an object of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ such that the support of $H^i(E)$ is proper over $Y$ for all $i$. Then $Rf_*E$ is an object of $D^b_{\textit{Coh}}(\mathcal{O}_Y)$.


spaces-perfect-lemma-direct-image-coherent-bdd-below

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian. Let $E$ be an object of $D^+_{\textit{Coh}}(\mathcal{O}_X)$ such that the support of $H^i(E)$ is proper over $S$ for all $i$. Then $Rf_*E$ is an object of $D^+_{\textit{Coh}}(\mathcal{O}_Y)$.


spaces-perfect-lemma-coherent-internal-hom

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. If $L$ is in $D^+_{\textit{Coh}}(\mathcal{O}_X)$ and $K$ in $D^-_{\textit{Coh}}(\mathcal{O}_X)$, then $R\SheafHom(K, L)$ is in $D^+_{\textit{Coh}}(\mathcal{O}_X)$.


spaces-perfect-lemma-ext-finite

Let $A$ be a Noetherian ring. Let $X$ be a proper algebraic space over $A$. For $L$ in $D^+_{\textit{Coh}}(\mathcal{O}_X)$ and $K$ in $D^-_{\textit{Coh}}(\mathcal{O}_X)$, the $A$-modules $\Ext_{\mathcal{O}_X}^n(K, L)$ are finite.


spaces-perfect-lemma-make-more-elementary-distinguished-squares

Let $S$ be a scheme. Let $(U \subset W, f : V \to W)$ be an elementary distinguished square of algebraic spaces over $S$. \begin{enumerate} \item If $V' \subset V$ and $U \subset U' \subset W$ are open subspaces and $W' = U' \cup f(V')$ then $(U' \subset W', f|_{V'} : V' \to W')$ is an elementary distinguished square. \item If $p : W' \to W$ is a morphism of algebraic spaces, then $(p^{-1}(U) \subset W', V \times_W W' \to W')$ is an elementary distinguished square. \item If $S' \to S$ is a morphism of schemes, then $(S' \times_S U \subset S' \times_S W, S' \times_S V \to S' \times_S W)$ is an elementary distinguished square. \end{enumerate}


spaces-perfect-lemma-induction-principle

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $P$ be a property of the quasi-compact and quasi-separated objects of $X_{spaces, \etale}$. Assume that \begin{enumerate} \item $P$ holds for every affine object of $X_{spaces, \etale}$, \item for every elementary distinguished square $(U \subset W, f : V \to W)$ such that \begin{enumerate} \item $W$ is a quasi-compact and quasi-separated object of $X_{spaces, \etale}$, \item $U$ is quasi-compact, \item $V$ is affine, and \item $P$ holds for $U$, $V$, and $U \times_W V$, \end{enumerate} then $P$ holds for $W$. \end{enumerate} Then $P$ holds for every quasi-compact and quasi-separated object of $X_{spaces, \etale}$ and in particular for $X$.


spaces-perfect-lemma-induction-principle-separated

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{B} \subset \Ob(X_{spaces, \etale})$. Let $P$ be a property of the elements of $\mathcal{B}$. Assume that \begin{enumerate} \item every $W \in \mathcal{B}$ is quasi-compact and quasi-separated, \item if $W \in \mathcal{B}$ and $U \subset W$ is quasi-compact open, then $U \in \mathcal{B}$, \item if $V \in \Ob(X_{spaces, \etale})$ is affine, then (a) $V \in \mathcal{B}$ and (b) $P$ holds for $V$, \item for every elementary distinguished square $(U \subset W, f : V \to W)$ such that \begin{enumerate} \item $W \in \mathcal{B}$, \item $U$ is quasi-compact, \item $V$ is affine, and \item $P$ holds for $U$, $V$, and $U \times_W V$, \end{enumerate} then $P$ holds for $W$. \end{enumerate} Then $P$ holds for every $W \in \mathcal{B}$.


spaces-perfect-lemma-induction-principle-enlarge

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $W \subset X$ be a quasi-compact open subspace. Let $P$ be a property of quasi-compact open subspaces of $X$. Assume that \begin{enumerate} \item $P$ holds for $W$, and \item for every elementary distinguished square $(W_1 \subset W_2, f : V \to W_2)$ where  such that \begin{enumerate} \item $W_1$, $W_2$ are quasi-compact open subspaces of $X$, \item $W \subset W_1$, \item $V$ is affine, and \item $P$ holds for $W_1$, \end{enumerate} then $P$ holds for $W_2$. \end{enumerate} Then $P$ holds for $X$.


spaces-perfect-lemma-exact-sequence-lower-shriek

Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. \begin{enumerate} \item For a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ we have a short exact sequence $$ 0 \to j_{U \times_X V!}\mathcal{F}|_{U \times_X V} \to j_{U!}\mathcal{F}|_U \oplus j_{V!}\mathcal{F}|_V \to \mathcal{F} \to 0 $$ \item For an object $E$ of $D(\mathcal{O}_X)$ we have a distinguished triangle $$ j_{U \times_X V!}E|_{U \times_X V} \to j_{U!}E|_U \oplus j_{V!}E|_V \to E \to  j_{U \times_X V!}E|_{U \times_X V}[1] $$ in $D(\mathcal{O}_X)$. \end{enumerate}


spaces-perfect-lemma-exact-sequence-j-star

Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. \begin{enumerate} \item For every sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ we have a short exact sequence $$ 0 \to \mathcal{F} \to j_{U, *}\mathcal{F}|_U \oplus j_{V, *}\mathcal{F}|_V \to j_{U \times_X V, *}\mathcal{F}|_{U \times_X V} \to 0 $$ \item For any object $E$ of $D(\mathcal{O}_X)$ we have a distinguished triangle $$ E \to  Rj_{U, *}E|_U \oplus Rj_{V, *}E|_V \to Rj_{U \times_X V, *}E|_{U \times_X V} \to E[1] $$ in $D(\mathcal{O}_X)$. \end{enumerate}


spaces-perfect-lemma-unbounded-relative-mayer-vietoris

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $(U \subset X, V \to X)$ be an elementary distinguished square. Denote $a = f|_U : U \to Y$, $b = f|_V : V \to Y$, and $c = f|_{U \times_X V} : U \times_X V \to Y$ the restrictions. For every object $E$ of $D(\mathcal{O}_X)$ there exists a distinguished triangle $$ Rf_*E \to Ra_*(E|_U) \oplus Rb_*(E|_V) \to Rc_*(E|_{U \times_X V}) \to Rf_*E[1] $$ in $D(\mathcal{O}_Y)$. This triangle is functorial in $E$.


spaces-perfect-lemma-mayer-vietoris-hom

Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. For objects $E$, $F$ of $D(\mathcal{O}_X)$ we have a Mayer-Vietoris sequence $$ \xymatrix{ & \ldots \ar[r] & \Ext^{-1}(E_{U \times_X V}, F_{U \times_X V}) \ar[lld] \\ \Hom(E, F) \ar[r] & \Hom(E_U, F_U) \oplus \Hom(E_V, F_V) \ar[r] & \Hom(E_{U \times_X V}, F_{U \times_X V}) } $$ where the subscripts denote restrictions to the relevant opens and the $\Hom$'s are taken in the relevant derived categories.


spaces-perfect-lemma-unbounded-mayer-vietoris

Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. For an object $E$ of $D(\mathcal{O}_X)$ we have a distinguished triangle $$ R\Gamma(X, E) \to R\Gamma(U, E) \oplus R\Gamma(V, E) \to R\Gamma(U \times_X V, E) \to R\Gamma(X, E)[1] $$ and in particular a long exact cohomology sequence $$ \ldots \to H^n(X, E) \to H^n(U, E) \oplus H^n(V, E) \to H^n(U \times_X V, E) \to H^{n + 1}(X, E) \to \ldots $$ The construction of the distinguished triangle and the long exact sequence is functorial in $E$.


spaces-perfect-lemma-restrict-lower-shriek

Let $S$ be a scheme. Let $j : U \to X$ be a \'etale morphism of algebraic spaces over $S$. Given an \'etale morphism $V \to Y$, set $W = V \times_X U$ and denote $j_W : W \to V$ the projection morphism. Then $(j_!E)|_V = j_{W!}(E|_W)$ for $E$ in $D(\mathcal{O}_U)$.


spaces-perfect-lemma-pushforward-with-support-in-open

Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Set $T = |X| \setminus |U|$. \begin{enumerate} \item If $E$ is an object of $D(\mathcal{O}_X)$ supported on $T$, then (a) $E \to Rj_*(E|_V)$ and (b) $j_!(E|_V) \to E$ are isomorphisms. \item If $F$ is an object of $D(\mathcal{O}_V)$ supported on $j^{-1}T$, then (a) $F \to (j_!F)|_V$, (b) $(Rj_*F)|_V \to F$, and (c) $j_!F \to Rj_*F$ are isomorphisms. \end{enumerate}


spaces-perfect-lemma-glue

Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Suppose given \begin{enumerate} \item an object $A$ of $D(\mathcal{O}_U)$, \item an object $B$ of $D(\mathcal{O}_V)$, and \item an isomorphism $c : A|_{U \times_X V} \to B|_{U \times_X V}$. \end{enumerate} Then there exists an object $F$ of $D(\mathcal{O}_X)$ and isomorphisms $f : F|_U \to A$, $g : F|_V \to B$ such that $c = g|_{U \times_X V} \circ f^{-1}|_{U \times_X V}$. Moreover, given \begin{enumerate} \item an object $E$ of $D(\mathcal{O}_X)$, \item a morphism $a : A \to E|_U$ of $D(\mathcal{O}_U)$, \item a morphism $b : B \to E|_V$ of $D(\mathcal{O}_V)$, \end{enumerate} such that $$ a|_{U \times_X V}  = b|_{U \times_X V} \circ c. $$ Then there exists a morphism $F \to E$ in $D(\mathcal{O}_X)$ whose restriction to $U$ is $a \circ f$ and whose restriction to $V$ is $b \circ g$.


spaces-perfect-lemma-affine-pushforward

Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Then $f_*$ defines a derived functor $f_* : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))$. This functor has the property that $$ \xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[d]_{f_*} \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(\QCoh(\mathcal{O}_Y)) \ar[r] & D_\QCoh(\mathcal{O}_Y) } $$ commutes.


spaces-perfect-lemma-flat-pushforward-coherator

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact, quasi-separated, and flat. Then, denoting $$ \Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y)) $$ the right derived functor of $f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)$ we have $RQ_Y \circ Rf_* = \Phi \circ RQ_X$.


spaces-perfect-lemma-affine-coherator

Let $S$ be a scheme. Let $X$ be an affine algebraic space over $S$. Set $A = \Gamma(X, \mathcal{O}_X)$. Then \begin{enumerate} \item $Q_X : \textit{Mod}(\mathcal{O}_X) \to \QCoh(\mathcal{O}_X)$ is the functor which sends $\mathcal{F}$ to the quasi-coherent $\mathcal{O}_X$-module associated to the $A$-module $\Gamma(X, \mathcal{F})$, \item $RQ_X : D(\mathcal{O}_X) \to D(\QCoh(\mathcal{O}_X))$ is the functor which sends $E$ to the complex of quasi-coherent $\mathcal{O}_X$-modules associated to the object $R\Gamma(X, E)$ of $D(A)$, \item restricted to $D_\QCoh(\mathcal{O}_X)$ the functor $RQ_X$ defines a quasi-inverse to (\ref{equation-compare}). \end{enumerate}


spaces-perfect-lemma-argument-proves

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Suppose that for every \'etale morphism $j : V \to W$ with $W \subset X$ quasi-compact open and $V$ affine the right derived functor $$ \Phi : D(\QCoh(\mathcal{O}_U)) \to D(\QCoh(\mathcal{O}_W)) $$ of the left exact functor $j_* : \QCoh(\mathcal{O}_V) \to \QCoh(\mathcal{O}_W)$ fits into a commutative diagram $$ \xymatrix{ D(\QCoh(\mathcal{O}_V)) \ar[d]_\Phi \ar[r]_{i_V} & D_\QCoh(\mathcal{O}_V) \ar[d]^{Rj_*} \\ D(\QCoh(\mathcal{O}_W)) \ar[r]^{i_W} & D_\QCoh(\mathcal{O}_W) } $$ Then the functor (\ref{equation-compare}) $$ D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X) $$ is an equivalence with quasi-inverse given by $RQ_X$.


spaces-perfect-lemma-affine-injective-colimit-direct-sum-pushforwards-artin

Let $S$ be a Noetherian affine scheme. Every injective object of $\QCoh(\mathcal{O}_S)$ is a filtered colimit $\colim_i \mathcal{F}_i$ of quasi-coherent sheaves of the form $$ \mathcal{F}_i = (Z_i \to S)_*\mathcal{G}_i $$ where $Z_i$ is the spectrum of an Artinian ring and $\mathcal{G}_i$ is a coherent module on $Z_i$.


spaces-perfect-lemma-injective-colimit-direct-sum-pushforwards-artin

Let $S$ be an affine scheme. Let $X$ be a Noetherian algebraic space over $S$. Every injective object of $\QCoh(\mathcal{O}_X)$ is a direct summand of a filtered colimit $\colim_i \mathcal{F}_i$ of quasi-coherent sheaves of the form $$ \mathcal{F}_i = (Z_i \to X)_*\mathcal{G}_i $$ where $Z_i$ is the spectrum of an Artinian ring and $\mathcal{G}_i$ is a coherent module on $Z_i$.


spaces-perfect-lemma-injective-pushforward

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. If $\mathcal{J}$ is an injective object of $\QCoh(\mathcal{O}_X)$, then \begin{enumerate} \item $H^p(U, \mathcal{J}|_U) = 0$ for $p > 0$ and for every quasi-compact and quasi-separated algebraic space $U$ \'etale over $X$, \item for any morphism $f : X \to Y$ of algebraic spaces over $S$ with $Y$ quasi-separated we have $R^pf_*\mathcal{J} = 0$ for $p > 0$. \end{enumerate}


spaces-perfect-lemma-Noetherian-pushforward

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of Noetherian algebraic spaces over $S$. Then $f_*$ on quasi-coherent sheaves has a right derived extension $\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))$ such that the diagram $$ \xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[d]_{\Phi} \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(\QCoh(\mathcal{O}_Y)) \ar[r] & D_\QCoh(\mathcal{O}_Y) } $$ commutes.


spaces-perfect-lemma-descend-finite-type

Let $X$ be a scheme. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is of finite type as an $\mathcal{O}_X$-module, and \item $\epsilon^*\mathcal{F}$ is of finite type as an $\mathcal{O}_\etale$-module on the small \'etale site of $X$. \end{enumerate} Here $\epsilon$ is as in (\ref{equation-epsilon}).


spaces-perfect-lemma-descend-pseudo-coherent

Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $E$ is $m$-pseudo-coherent, and \item $\epsilon^*E$ is $m$-pseudo-coherent on the small \'etale site of $X$. \end{enumerate} Here $\epsilon$ is as in (\ref{equation-epsilon}).


spaces-perfect-lemma-descend-tor-amplitude

Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_X)$. Then \begin{enumerate} \item $E$ has tor amplitude in $[a, b]$ if and only if $\epsilon^*E$ has tor amplitude in $[a, b]$. \item $E$ has finite tor dimension if and only if $\epsilon^*E$ has finite tor dimension. \end{enumerate} Here $\epsilon$ is as in (\ref{equation-epsilon}).


spaces-perfect-lemma-tor-dimension-rel

Let $f : X \to Y$ be a morphism of schemes. Let $E$ be an object of $D(\mathcal{O}_X)$. Then \begin{enumerate} \item $E$ as an object of $D(f^{-1}\mathcal{O}_Y)$ has tor amplitude in $[a, b]$ if and only if $\epsilon^*E$ has tor amplitude in $[a, b]$ as an object of $D(f_{small}^{-1}\mathcal{O}_{Y_\etale})$. \item $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_Y)$ if and only if $\epsilon^*E$ locally has finite tor dimension as an object of $D(f_{small}^{-1}\mathcal{O}_{Y_\etale})$. \end{enumerate} Here $\epsilon$ is as in (\ref{equation-epsilon}).


spaces-perfect-lemma-descend-perfect

Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_X)$. Then $E$ is a perfect object of $D(\mathcal{O}_X)$ if and only if $\epsilon^*E$ is a perfect object of $D(\mathcal{O}_\etale)$. Here $\epsilon$ is as in (\ref{equation-epsilon}).


spaces-perfect-lemma-pseudo-coherent

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $E$ is an $m$-pseudo-coherent object of $D(\mathcal{O}_X)$, then $H^i(E)$ is a quasi-coherent $\mathcal{O}_X$-module for $i > m$. If $E$ is pseudo-coherent, then $E$ is an object of $D_\QCoh(\mathcal{O}_X)$.


spaces-perfect-lemma-identify-pseudo-coherent-noetherian

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. For $m \in \mathbf{Z}$ the following are equivalent \begin{enumerate} \item $H^i(E)$ is coherent for $i \geq m$ and zero for $i \gg 0$, and \item $E$ is $m$-pseudo-coherent. \end{enumerate} In particular, $E$ is pseudo-coherent if and only if $E$ is an object of $D^-_{\textit{Coh}}(\mathcal{O}_X)$.


spaces-perfect-lemma-tor-qc-qs

Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. Let $a \leq b$. The following are equivalent \begin{enumerate} \item $E$ has tor amplitude in $[a, b]$, and \item for all $\mathcal{F}$ in $\QCoh(\mathcal{O}_X)$ we have $H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}) = 0$ for $i \not \in [a, b]$. \end{enumerate}


spaces-perfect-lemma-descend-RHom

Let $X$ be a scheme. Let $E, F$ be objects of $D(\mathcal{O}_X)$. Assume either \begin{enumerate} \item $E$ is pseudo-coherent and $F$ lies in $D^+(\mathcal{O}_X)$, or \item $E$ is perfect and $F$ arbitrary, \end{enumerate} then there is a canonical isomorphism $$ \epsilon^*R\SheafHom(E, F) \longrightarrow R\SheafHom(\epsilon^*E, \epsilon^*F) $$ Here $\epsilon$ is as in (\ref{equation-epsilon}).


spaces-perfect-lemma-quasi-coherence-internal-hom

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $L, K$ be objects of $D(\mathcal{O}_X)$. If either \begin{enumerate} \item $L$ in $D^+_\QCoh(\mathcal{O}_X)$ and $K$ is pseudo-coherent, \item $L$ in $D_\QCoh(\mathcal{O}_X)$ and $K$ is perfect, \end{enumerate} then $R\SheafHom(K, L)$ is in $D_\QCoh(\mathcal{O}_X)$.


spaces-perfect-lemma-internal-hom-evaluate-tensor-isomorphism

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $K, L, M$ be objects of $D_\QCoh(\mathcal{O}_X)$. The map $$ K \otimes_{\mathcal{O}_X}^\mathbf{L} R\SheafHom(M, L) \longrightarrow R\SheafHom(M, K \otimes_{\mathcal{O}_X}^\mathbf{L} L) $$ of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-internal-hom-diagonal-better} is an isomorphism in the following cases \begin{enumerate} \item $M$ perfect, or \item $K$ is perfect, or \item $M$ is pseudo-coherent, $L \in D^+(\mathcal{O}_X)$, and $K$ has finite tor dimension. \end{enumerate}


spaces-perfect-lemma-pushforward-perfect

Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic space over $S$. Let $E$ be a perfect object of $D(\mathcal{O}_V)$ supported on $j^{-1}(T)$ where $T = |X| \setminus |U|$. Then $Rj_*E$ is a perfect object of $D(\mathcal{O}_X)$.


spaces-perfect-lemma-open

Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Let $T$ be a closed subset of $|X| \setminus |U|$ and let $(T, E, m)$ be a triple as in Definition \ref{definition-approximation-holds}. If \begin{enumerate} \item approximation holds for $(j^{-1}T, E|_V, m)$, and \item the sheaves $H^i(E)$ for $i \geq m$ are supported on $T$, \end{enumerate} then approximation holds for $(T, E, m)$.


spaces-perfect-lemma-approximation-affine

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ which is representable by an affine scheme. Then approximation holds for every triple $(T, E, m)$ as in Definition \ref{definition-approximation-holds} such that there exists an integer $r \geq 0$ with \begin{enumerate} \item $E$ is $m$-pseudo-coherent, \item $H^i(E)$ is supported on $T$ for $i \geq m - r + 1$, \item $X \setminus T$ is the union of $r$ affine opens. \end{enumerate} In particular, approximation by perfect complexes holds for affine schemes.


spaces-perfect-lemma-induction-step

Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Assume $U$ quasi-compact, $V$ affine, and $U \times_X V$ quasi-compact. If approximation by perfect complexes holds on $U$, then approximation by perfect complexes holds on $X$.


spaces-perfect-lemma-lift-map-from-perfect-complex-with-support

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $W \subset X$ be a quasi-compact open. Let $T \subset |X|$ be a closed subset such that $X \setminus T \to X$ is a quasi-compact morphism. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. Let $\alpha : P \to E|_W$ be a map where $P$ is a perfect object of $D(\mathcal{O}_W)$ supported on $T \cap W$. Then there exists a map $\beta : R \to E$ where $R$ is a perfect object of $D(\mathcal{O}_X)$ supported on $T$ such that $P$ is a direct summand of $R|_W$ in $D(\mathcal{O}_W)$ compatible $\alpha$ and $\beta|_W$.


spaces-perfect-lemma-direct-summand-of-a-restriction

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $W$ be a quasi-compact open subspace of $X$. Let $P$ be a perfect object of $D(\mathcal{O}_W)$. Then $P$ is a direct summand of the restriction of a perfect object of $D(\mathcal{O}_X)$.


spaces-perfect-lemma-generator-with-support

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \subset |X|$ be a closed subset such that $|X| \setminus T$ is quasi-compact. With notation as above, the category $D_{\QCoh, T}(\mathcal{O}_X)$ is generated by a single perfect object.


spaces-perfect-lemma-compact-is-perfect-with-support

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \subset |X|$ be a closed subset such that $|X| \setminus T$ is quasi-compact. An object of $D_{\QCoh, T}(\mathcal{O}_X)$ is compact if and only if it is perfect as an object of $D(\mathcal{O}_X)$.


spaces-perfect-lemma-tensor-with-QCoh-complex

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $K^\bullet$ be a complex of $\mathcal{O}_X$-modules whose cohomology sheaves are quasi-coherent. Let $(E, d) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet)$ be the endomorphism differential graded algebra. Then the functor $$ - \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}_X) $$ of Differential Graded Algebra, Lemma \ref{dga-lemma-tensor-with-complex-derived} has image contained in $D_\QCoh(\mathcal{O}_X)$.


spaces-perfect-lemma-ext-from-perfect-into-bounded-QCoh

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $K$ be a perfect object of $D(\mathcal{O}_X)$. Then \begin{enumerate} \item there exist integers $a \leq b$ such that $\Hom_{D(\mathcal{O}_X)}(K, L) = 0$ for $L \in D_\QCoh(\mathcal{O}_X)$ with $H^i(L) = 0$ for $i \in [a, b]$, and \item if $L$ is bounded, then $\Ext^n_{D(\mathcal{O}_X)}(K, L)$ is zero for all but finitely many $n$. \end{enumerate}


spaces-perfect-lemma-pseudo-coherent-hocolim

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $K \in D(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $K$ is pseudo-coherent, and \item $K = \text{hocolim} K_n$ where $K_n$ is perfect and $\tau_{\geq -n}K_n \to \tau_{\geq -n}K$ is an isomorphism for all $n$. \end{enumerate}


spaces-perfect-lemma-better-coherator

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The inclusion functor $D_\QCoh(\mathcal{O}_X) \to D(\mathcal{O}_X)$ has a right adjoint.


spaces-perfect-lemma-pushforward-better-coherator

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. If the right adjoints $DQ_X$ and $DQ_Y$ of the inclusion functors $D_\QCoh \to D$ exist for $X$ and $Y$, then $$ Rf_* \circ DQ_X = DQ_Y \circ Rf_* $$


spaces-perfect-lemma-boundedness-better-coherator

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The functor $DQ_X$ of Lemma \ref{lemma-better-coherator} has the following boundedness property: there exists an integer $N = N(X)$ such that, if $K$ in $D(\mathcal{O}_X)$ with $H^i(U, K) = 0$ for $U$ affine \'etale over $X$ and $i \not \in [a, b]$, then the cohomology sheaves $H^i(DQ_X(K))$ are zero for $i \not \in [a, b + N]$.


spaces-perfect-lemma-cohomology-base-change

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. For $E$ in $D_\QCoh(\mathcal{O}_X)$ and $K$ in $D_\QCoh(\mathcal{O}_Y)$ we have $$ Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K) $$


spaces-perfect-lemma-tor-independent

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X$, $Y$ be algebraic spaces over $B$. The following are equivalent \begin{enumerate} \item $X$ and $Y$ are Tor independent over $B$, \item for every commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & W \ar[d] & V \ar[d] \ar[l] \\ X \ar[r] & B & Y \ar[l] } $$ with \'etale vertical arrows $U$ and $V$ are Tor independent over $W$, \item for some commutative diagram as in (2) with (a) $W \to B$ \'etale surjective, (b) $U \to X \times_B W$ \'etale surjective, (c) $V \to Y \times_B W$ \'etale surjective, the spaces $U$ and $V$ are Tor independent over $W$, and \item for some commutative diagram as in (3) with $U$, $V$, $W$ schemes, the schemes $U$ and $V$ are Tor independent over $W$ in the sense of Derived Categories of Schemes, Definition \ref{perfect-definition-tor-independent}. \end{enumerate}


spaces-perfect-lemma-compare-base-change

Let $S$ be a scheme. Let $g : Y' \to Y$ be a morphism of algebraic spaces over $S$. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Consider the base change diagram $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ If $X$ and $Y'$ are Tor independent over $Y$, then for all $E \in D_\QCoh(\mathcal{O}_X)$ we have $Rf'_*L(g')^*E = Lg^*Rf_*E$.


spaces-perfect-lemma-affine-morphism-and-hom-out-of-perfect

Let $g : S' \to S$ be a morphism of affine schemes. Consider a cartesian square $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ of quasi-compact and quasi-separated algebraic spaces. Assume $g$ and $f$ Tor independent. Write $S = \Spec(R)$ and $S' = \Spec(R')$. For $M, K \in D(\mathcal{O}_X)$ the canonical map $$ R\Hom_X(M, K) \otimes^\mathbf{L}_R R' \longrightarrow R\Hom_{X'}(L(g')^*M, L(g')^*K) $$ in $D(R')$ is an isomorphism in the following two cases \begin{enumerate} \item $M \in D(\mathcal{O}_X)$ is perfect and $K \in D_\QCoh(X)$, or \item $M \in D(\mathcal{O}_X)$ is pseudo-coherent, $K \in D_\QCoh^+(X)$, and $R'$ has finite tor dimension over $R$. \end{enumerate}


spaces-perfect-lemma-tor-independence-and-tor-amplitude

Let $S$ be a scheme. Consider a cartesian square of algebraic spaces $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ over $S$. Assume $g$ and $f$ Tor independent. \begin{enumerate} \item If $E \in D(\mathcal{O}_X)$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\mathcal{O}_Y$-modules, then $L(g')^*E$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\mathcal{O}_{Y'}$-modules. \item If $\mathcal{G}$ is an $\mathcal{O}_X$-module flat over $Y$, then $L(g')^*\mathcal{G} = (g')^*\mathcal{G}$. \end{enumerate}


spaces-perfect-lemma-single-complex-base-change-condition

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Let $K \in D_\QCoh(\mathcal{O}_X)$ and let $L(g')^*K \to K'$ be a map in $D_\QCoh(\mathcal{O}_{X'})$. The following are equivalent \begin{enumerate} \item for any $x' \in X'$ and $i \in \mathbf{Z}$ the map (\ref{equation-bc}) is an isomorphism, \item for any commutative diagram $$ \xymatrix{ & U \ar[d] \ar[rd]^a \\ V' \ar[r] \ar[rd]^c & V \ar[rd]^b & X \ar[d]^f \\ & Y' \ar[r]^g & Y } $$ with $a, b, c$ \'etale, $U, V, V'$ schemes, and with $U' = V' \times_V U$ the equivalent conditions of Derived Categories of Schemes, Lemma \ref{lemma-single-complex-base-change-condition} hold for $(U \to X)^*K$ and $(U' \to X')^*K'$, and \item there is some diagram as in (2) with $U' \to X'$ surjective. \end{enumerate}


spaces-perfect-lemma-single-complex-base-change

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Let $K \in D_\QCoh(\mathcal{O}_X)$ and let $L(g')^*K \to K'$ be a map in $D_\QCoh(\mathcal{O}_{X'})$. If \begin{enumerate} \item the equivalent conditions of Lemma \ref{lemma-single-complex-base-change-condition} hold, and \item $f$ is quasi-compact and quasi-separated, \end{enumerate} then the composition $Lg^*Rf_*K \to Rf'_*L(g')^*K \to Rf'_*K'$ is an isomorphism.


spaces-perfect-lemma-single-complex-base-change-condition-inherited

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ be a cartesian diagram of algebraic spaces over $S$. Let $K \in D_\QCoh(\mathcal{O}_X)$ and let $L(g')^*K \to K'$ be a map in $D_\QCoh(\mathcal{O}_{X'})$. If the equivalent conditions of Lemma \ref{lemma-single-complex-base-change-condition} hold, then \begin{enumerate} \item for $E \in D_\QCoh(\mathcal{O}_X)$ the equivalent conditions of Lemma \ref{lemma-single-complex-base-change-condition} hold for $L(g')^*(E \otimes^\mathbf{L} K) \to L(g')^*E \otimes^\mathbf{L} K'$, \item if $E$ in $D(\mathcal{O}_X)$ is perfect the equivalent conditions of Lemma \ref{lemma-single-complex-base-change-condition} hold for $L(g')^*R\SheafHom(E, K) \to R\SheafHom(L(g')^*E, K')$, and \item if $K$ is bounded below and $E$ in $D(\mathcal{O}_X)$ pseudo-coherent the equivalent conditions of Lemma \ref{lemma-single-complex-base-change-condition} hold for $L(g')^*R\SheafHom(E, K) \to R\SheafHom(L(g')^*E, K')$. \end{enumerate}


spaces-perfect-lemma-base-change-tensor

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $E \in D_\QCoh(\mathcal{O}_X)$. Let $\mathcal{G}^\bullet$ be a bounded above complex of quasi-coherent $\mathcal{O}_X$-modules flat over $Y$. Then formation of $$ Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet) $$ commutes with arbitrary base change (see proof for precise statement).


spaces-perfect-lemma-base-change-RHom

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $\mathcal{G}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_X$-modules. If \begin{enumerate} \item $E$ is perfect, $\mathcal{G}^\bullet$ is a bounded above, and $\mathcal{G}^n$ is flat over $Y$, or \item $E$ is pseudo-coherent, $\mathcal{G}^\bullet$ is bounded, and $\mathcal{G}^n$ is flat over $Y$, \end{enumerate} then formation of $$ Rf_*R\SheafHom(E, \mathcal{G}^\bullet) $$ commutes with arbitrary base change (see proof for precise statement).


spaces-perfect-lemma-perfect-direct-image

Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let $E \in D(\mathcal{O}_X)$ such that \begin{enumerate} \item $E \in D^b_{\textit{Coh}}(\mathcal{O}_X)$, \item the support of $H^i(E)$ is proper over $Y$ for all $i$, \item $E$ has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_Y)$. \end{enumerate} Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_Y)$.


spaces-perfect-lemma-tensor-perfect

Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$. Let $f : X \to B$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let $E \in D(\mathcal{O}_X)$ be perfect. Let $\mathcal{G}^\bullet$ be a bounded complex of coherent $\mathcal{O}_X$-modules flat over $B$ with support proper over $B$. Then $K = Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet)$ is a perfect object of $D(\mathcal{O}_B)$.