obsolete-lemma-lift-elements-ideal

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


obsolete-lemma-relative-effective-cartier-algebra

Let $A \to B$ be a ring map. Let $f \in B$. Assume that \begin{enumerate} \item $A \to B$ is flat, \item $f$ is a nonzerodivisor, and \item $A \to B/fB$ is flat. \end{enumerate} Then for every ideal $I \subset A$ the map $f : B/IB \to B/IB$ is injective.


obsolete-lemma-make-integral-less-trivial

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


obsolete-lemma-P1

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


obsolete-lemma-rel-prime-pols

Let $k$ be a field. Let $F, G \in k[X, Y]$ be homogeneous of degrees $d, e$. Assume $F, G$ relatively prime. Then multiplication by $G$ is injective on $S = k[X, Y]/(F)$.


obsolete-lemma-finite-after-localization

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


obsolete-lemma-property-irreducible-higher-rank

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


obsolete-lemma-section-maps-back-into

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$ be a section. Let $\mathcal{F}' \subset \mathcal{F}$ be quasi-coherent $\mathcal{O}_X$-modules. Assume that \begin{enumerate} \item $X$ is quasi-compact, \item $\mathcal{F}$ is of finite type, and \item $\mathcal{F}'|_{X_s} = \mathcal{F}|_{X_s}$. \end{enumerate} Then there exists an $n \geq 0$ such that multiplication by $s^n$ on $\mathcal{F}$ factors through $\mathcal{F}'$.


obsolete-lemma-sheaf-fpqc-quasi-separated

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Zariski locally quasi-separated over $S$, then $X$ satisfies the sheaf condition for the fpqc topology.


obsolete-lemma-scheme-very-reasonable

A scheme is very reasonable.


obsolete-lemma-category-fibred

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


obsolete-lemma-equivalence

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


obsolete-lemma-second-equivalence

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


obsolete-lemma-second-equivalence-obs

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


obsolete-lemma-pseudo-coherent

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


obsolete-lemma-ob-is-obstruction

In the situation of Remark \ref{remark-construction-ob} assume that $\mathcal{F}$ is flat over $U$. Then the vanishing of the class $\xi_{U'}$ is a necessary and sufficient condition for the existence of a $\mathcal{O}_{X \times_B U'}$-module $\mathcal{F}'$ flat over $U'$ with $i^*\mathcal{F}' \cong \mathcal{F}$.


obsolete-lemma-coherent-defo-thy-general

In Quot, Situation \ref{quot-situation-coherent} assume that $S$ is a locally Noetherian scheme and $S = B$. Let $\mathcal{X} = \textit{Coh}_{X/B}$. Then we have openness of versality for $\mathcal{X}$ (see Artin's Axioms, Definition \ref{artin-definition-openness-versality}).


obsolete-lemma-push-pull-effective-Cartier

Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a proper morphism. Let $D \subset Y$ be an effective Cartier divisor. Assume $X$, $Y$ integral, $n = \dim_\delta(X) = \dim_\delta(Y)$ and $f$ dominant. Then $$ f_*[f^{-1}(D)]_{n - 1} = [R(X) : R(Y)] [D]_{n - 1}. $$ In particular if $f$ is birational then $f_*[f^{-1}(D)]_{n - 1} = [D]_{n - 1}$.


obsolete-lemma-two-divisors

Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim_\delta(X) = n$. Let $D_1, D_2$ be two effective Cartier divisors in $X$. Let $Z$ be an open and closed subscheme of the scheme $D_1 \cap D_2$. Assume $\dim_\delta(D_1 \cap D_2 \setminus Z) \leq n - 2$. Then there exists a morphism $b : X' \to X$, and Cartier divisors $D_1', D_2', E$ on $X'$ with the following properties \begin{enumerate} \item $X'$ is integral, \item $b$ is projective, \item $b$ is the blowup of $X$ in the closed subscheme $Z$, \item $E = b^{-1}(Z)$, \item $b^{-1}(D_1) = D'_1 + E$, and $b^{-1}D_2 = D_2' + E$, \item $\dim_\delta(D'_1 \cap D'_2) \leq n - 2$, and if $Z = D_1 \cap D_2$ then $D'_1 \cap D'_2 = \emptyset$, \item for every integral closed subscheme $W'$ with $\dim_\delta(W') = n - 1$ we have \begin{enumerate} \item if $\epsilon_{W'}(D'_1, E) > 0$, then setting $W = b(W')$ we have $\dim_\delta(W) = n - 1$ and $$ \epsilon_{W'}(D'_1, E) < \epsilon_W(D_1, D_2), $$ \item if $\epsilon_{W'}(D'_2, E) > 0$, then setting $W = b(W')$ we have $\dim_\delta(W) = n - 1$ and $$ \epsilon_{W'}(D'_2, E) < \epsilon_W(D_1, D_2), $$ \end{enumerate} \end{enumerate}


obsolete-lemma-sum-divisors-associated-Weil

Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim_\delta(X) = n$. Let $\{D_i\}_{i \in I}$ be a locally finite collection of effective Cartier divisors on $X$. Suppose given $n_i \geq 0$ for $i \in I$. Then $$ [D]_{n - 1} = \sum\nolimits_i n_i[D_i]_{n - 1} $$ in $Z_{n - 1}(X)$.


obsolete-lemma-blowing-up-intersections

Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim_\delta(X) = d$. Let $\{D_i\}_{i \in I}$ be a locally finite collection of effective Cartier divisors on $X$. Assume that for all $\{i, j, k\} \subset I$, $\#\{i, j, k\} = 3$ we have $D_i \cap D_j \cap D_k = \emptyset$. Then there exist \begin{enumerate} \item an open subscheme $U \subset X$ with $\dim_\delta(X \setminus U) \leq d - 3$, \item a morphism $b : U' \to U$, and \item effective Cartier divisors $\{D'_j\}_{j \in J}$ on $U'$ \end{enumerate} with the following properties: \begin{enumerate} \item $b$ is proper morphism $b : U' \to U$, \item $U'$ is integral, \item $b$ is an isomorphism over the complement of the union of the pairwise intersections of the $D_i|_U$, \item $\{D'_j\}_{j \in J}$ is a locally finite collection of effective Cartier divisors on $U'$, \item $\dim_\delta(D'_j \cap D'_{j'}) \leq d - 2$ if $j \not = j'$, and \item $b^{-1}(D_i|_U) = \sum n_{ij} D'_j$ for certain $n_{ij} \geq 0$. \end{enumerate} Moreover, if $X$ is quasi-compact, then we may assume $U = X$ in the above.


obsolete-lemma-improved-additivity

Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Let $\{i_j : D_j \to X \}_{j \in J}$ be a locally finite collection of effective Cartier divisors on $X$. Let $n_j > 0$, $j\in J$. Set $D = \sum_{j \in J} n_j D_j$, and denote $i : D \to X$ the inclusion morphism. Let $\alpha \in Z_{k + 1}(X)$. Then $$ p : \coprod\nolimits_{j \in J} D_j \longrightarrow D $$ is proper and $$ i^*\alpha = p_*\left(\sum n_j i_j^*\alpha\right) $$ in $\CH_k(D)$.


obsolete-lemma-commutativity-effective-Cartier-proper-intersection

Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim_\delta(X) = n$. Let $D$, $D'$ be effective Cartier divisors on $X$. Assume $\dim_\delta(D \cap D') = n - 2$. Let $i : D \to X$, resp.\ $i' : D' \to X$ be the corresponding closed immersions. Then \begin{enumerate} \item there exists a cycle $\alpha \in Z_{n - 2}(D \cap D')$ whose pushforward to $D$ represents $i^*[D']_{n - 1} \in \CH_{n - 2}(D)$ and whose pushforward to $D'$ represents $(i')^*[D]_{n - 1} \in \CH_{n - 2}(D')$, and \item we have $$ D \cdot [D']_{n - 1} = D' \cdot [D]_{n - 1} $$ in $\CH_{n - 2}(X)$. \end{enumerate}


obsolete-lemma-commutativity-effective-Cartier-proper-intersection-infinite

Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim_\delta(X) = n$. Let $\{D_j\}_{j \in J}$ be a locally finite collection of effective Cartier divisors on $X$. Let $n_j, m_j \geq 0$ be collections of nonnegative integers. Set $D = \sum n_j D_j$ and $D' = \sum m_j D_j$. Assume that $\dim_\delta(D_j \cap D_{j'}) = n - 2$ for every $j \not = j'$. Then $D \cdot [D']_{n - 1} = D' \cdot [D]_{n - 1}$ in $\CH_{n - 2}(X)$.


obsolete-proposition-lqf-shriek

Let $f : X \to Y$ be a locally quasi-finite morphism. There exist adjoint functors $f_! : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ and $f^! : \textit{Ab}(Y_\etale) \to \textit{Ab}(X_\etale)$ with the following properties \begin{enumerate} \item the functor $f^!$ is the one constructed in More \'Etale Cohomology, Lemma \ref{more-etale-lemma-lqf-f-upper-shriek}, \item for any open $j : U \to X$ with $f \circ j$ separated there is a canonical isomorphism $f_! \circ j_! = (f \circ j)_!$, and \item these isomorphisms for $U \subset U' \subset X$ are compatible with the isomorphisms in More \'Etale Cohomology, Lemma \ref{more-etale-lemma-f-shriek-composition}. \end{enumerate}


obsolete-proposition-conclusion

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


stacks-sheaves-lemma-1-morphisms-presheaves

Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Then $(g \circ f)^p = f^p \circ g^p$ and there is a canonical isomorphism ${}_p(g \circ f) \to {}_pg \circ {}_pf$ compatible with adjointness of $(f^p, {}_pf)$, $(g^p, {}_pg)$, and $((g \circ f)^p, {}_p(g \circ f))$.


stacks-sheaves-lemma-functoriality-sheaves

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. The functors ${}_pf$ and $f^p$ of (\ref{equation-pushforward-pullback}) transform $\tau$ sheaves into $\tau$ sheaves and define a morphism of topoi $f : \Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)$.


stacks-sheaves-lemma-base-change

Let $S$ be a scheme. Let $$ \xymatrix{ \mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \ar[r]_{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Y}' \ar[r]^g & \mathcal{Y} } $$ be a $2$-cartesian diagram of categories fibred in groupoids over $S$. Then we have a canonical isomorphism $$ g^{-1}f_*\mathcal{F} \longrightarrow f'_*(g')^{-1}\mathcal{F} $$ functorial in the presheaf $\mathcal{F}$ on $\mathcal{X}$.


stacks-sheaves-lemma-representable

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. The following are equivalent \begin{enumerate} \item $f$ is representable, and \item for every $y \in \Ob(\mathcal{Y})$ the functor $\mathcal{X}^{opp} \to \textit{Sets}$, $x \mapsto \Mor_\mathcal{Y}(f(x), y)$ is representable. \end{enumerate}


stacks-sheaves-lemma-representable-pushforward

Let $f : \mathcal{X} \to \mathcal{Y}$ be a representable $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Then the functor $u : \mathcal{Y}_\tau \to \mathcal{X}_\tau$ is continuous and defines a morphism of sites $\mathcal{X}_\tau \to \mathcal{Y}_\tau$ which induces the same morphism of topoi $\Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)$ as the morphism $f$ constructed in Lemma \ref{lemma-functoriality-sheaves}. Moreover, $f_*\mathcal{F}(y) = \mathcal{F}(u(y))$ for any presheaf $\mathcal{F}$ on $\mathcal{X}$.


stacks-sheaves-lemma-functoriality-structure-sheaf

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. There is a canonical identification $f^{-1}\mathcal{O}_\mathcal{X} = \mathcal{O}_\mathcal{Y}$ which turns $f : \Sh(\mathcal{X}_\tau) \to \Sh(\mathcal{Y}_\tau)$ into a morphism of ringed topoi.


stacks-sheaves-lemma-compare-with-scheme

Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)$. Assume $\mathcal{X}$ is representable by a scheme $X$. For $\tau \in \{Zar,\linebreak[0] \etale,\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$ there is a canonical equivalence $$ (\mathcal{X}_\tau, \mathcal{O}_\mathcal{X}) = ((\Sch/X)_\tau, \mathcal{O}_X) $$ of ringed sites.


stacks-sheaves-lemma-compare-with-morphism-of-schemes

Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories fibred in groupoids over $S$. Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by schemes $X$, $Y$. Let $f : X \to Y$ be the morphism of schemes corresponding to $f$. For $\tau \in \{Zar,\linebreak[0] \etale,\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$ the morphism of ringed topoi $f : (\Sh(\mathcal{X}_\tau), \mathcal{O}_\mathcal{X}) \to (\Sh(\mathcal{Y}_\tau), \mathcal{O}_\mathcal{Y})$ agrees with the morphism of ringed topoi $f : (\Sh((\Sch/X)_\tau), \mathcal{O}_X) \to  (\Sh((\Sch/Y)_\tau), \mathcal{O}_Y)$ via the identifications of Lemma \ref{lemma-compare-with-scheme}.


stacks-sheaves-lemma-localizing

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$. The functor $p$ induces an equivalence of sites $\mathcal{X}_\tau/x \to (\Sch/U)_\tau$.


stacks-sheaves-lemma-comparison

Let $\mathcal{F}$ be an \'etale sheaf on $\mathcal{X} \to (\Sch/S)_{fppf}$. \begin{enumerate} \item If $\varphi : x \to y$ and $\psi : y \to z$ are morphisms of $\mathcal{X}$ lying over $a : U \to V$ and $b : V \to W$, then the composition $$ a_{small}^{-1}(b_{small}^{-1} (\mathcal{F}|_{W_\etale})) \xrightarrow{a_{small}^{-1}c_\psi} a_{small}^{-1}(\mathcal{F}|_{V_\etale}) \xrightarrow{c_\varphi} \mathcal{F}|_{U_\etale} $$ is equal to $c_{\psi \circ \varphi}$ via the identification $$ (b \circ a)_{small}^{-1}(\mathcal{F}|_{W_\etale}) = a_{small}^{-1}(b_{small}^{-1} (\mathcal{F}|_{W_\etale})). $$ \item If $\varphi : x \to y$ lies over an \'etale morphism of schemes $a : U \to V$, then (\ref{equation-comparison}) is an isomorphism. \item Suppose $f : \mathcal{Y} \to \mathcal{X}$ is a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$ and $y$ is an object of $\mathcal{Y}$ lying over the scheme $U$ with image $x = f(y)$. Then there is a canonical identification $f^{-1}\mathcal{F}|_{U_\etale} = \mathcal{F}|_{U_\etale}$. \item Moreover, given $\psi : y' \to y$ in $\mathcal{Y}$ lying over $a : U' \to U$ the comparison map $c_\psi : a_{small}^{-1}(F^{-1}\mathcal{F}|_{U_\etale}) \to F^{-1}\mathcal{F}|_{U'_\etale}$ is equal to the comparison map $c_{f(\psi)} : a_{small}^{-1}\mathcal{F}|_{U_\etale} \to \mathcal{F}|_{U'_\etale}$ via the identifications in (3). \end{enumerate}


stacks-sheaves-lemma-localizing-structure-sheaf

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Let $x \in \Ob(\mathcal{X})$ lying over $U = p(x)$. The equivalence of Lemma \ref{lemma-localizing} extends to an equivalence of ringed sites $(\mathcal{X}_\tau/x, \mathcal{O}_\mathcal{X}|_x) \to ((\Sch/U)_\tau, \mathcal{O})$.


stacks-sheaves-lemma-compare

Let $S$ be a scheme. Let $\mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Assume $\mathcal{X}$ is representable by an algebraic space $F$. Then there exists a continuous and cocontinuous functor $ F_\etale \to \mathcal{X}_\etale $ which induces a morphism of ringed sites $$ \pi_F : (\mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \longrightarrow (F_\etale, \mathcal{O}_F) $$ and a morphism of ringed topoi $$ i_F : (\Sh(F_\etale), \mathcal{O}_F) \longrightarrow (\Sh(\mathcal{X}_\etale), \mathcal{O}_\mathcal{X}) $$ such that $\pi_F \circ i_F = \text{id}$. Moreover $\pi_{F, *} = i_F^{-1}$.


stacks-sheaves-lemma-compare-morphism

Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by algebraic spaces $F$, $G$. Denote $f : F \to G$ the induced morphism of algebraic spaces, and $f_{small} : F_\etale \to G_\etale$ the corresponding morphism of ringed topoi. Then $$ \xymatrix{ (\Sh(\mathcal{X}_\etale), \mathcal{O}_\mathcal{X}) \ar[d]_{\pi_F} \ar[rr]_f & & (\Sh(\mathcal{Y}_\etale), \mathcal{O}_\mathcal{Y}) \ar[d]^{\pi_G} \\ (\Sh(F_\etale), \mathcal{O}_F) \ar[rr]^{f_{small}} & & (\Sh(G_\etale), \mathcal{O}_G) } $$ is a commutative diagram of ringed topoi.


stacks-sheaves-lemma-pullback-quasi-coherent

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \textit{Mod}(\mathcal{O}_\mathcal{Y}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})$ preserves quasi-coherent sheaves.


stacks-sheaves-lemma-characterize-quasi-coherent

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules. Then $\mathcal{F}$ is quasi-coherent if and only if $x^*\mathcal{F}$ is a quasi-coherent sheaf on $(\Sch/U)_{fppf}$ for every object $x$ of $\mathcal{X}$ with $U = p(x)$.


stacks-sheaves-lemma-characterize-quasi-coherent-bis

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of modules on $\mathcal{X}$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal{X})$ and $\mathcal{F}$ is a quasi-coherent module on $(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal{X})$ in the sense of Modules on Sites, Definition \ref{sites-modules-definition-site-local}, \item $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ and $\mathcal{F}$ is a quasi-coherent module on $(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ in the sense of Modules on Sites, Definition \ref{sites-modules-definition-site-local}, and \item $\mathcal{F}$ is a quasi-coherent module on $\mathcal{X}$ in the sense of Definition \ref{definition-quasi-coherent}. \end{enumerate}


stacks-sheaves-lemma-quasi-coherent

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal{X}$-modules. Then $\mathcal{F}$ is quasi-coherent if and only if the following two conditions hold \begin{enumerate} \item $\mathcal{F}$ is locally quasi-coherent, and \item for any morphism $\varphi : x \to y$ of $\mathcal{X}$ lying over $f : U \to V$ the comparison map $c_\varphi : f_{small}^*\mathcal{F}|_{V_\etale} \to \mathcal{F}|_{U_\etale}$ of (\ref{equation-comparison-modules}) is an isomorphism. \end{enumerate}


stacks-sheaves-lemma-pullback-lqc

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \textit{Mod}(\mathcal{Y}_\etale, \mathcal{O}_\mathcal{Y}) \to \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ preserves locally quasi-coherent sheaves.


stacks-sheaves-lemma-lqc-colimits

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. \begin{enumerate} \item The category $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. \item The category $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ is abelian with kernels and cokernels computed in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$, in other words the inclusion functor is exact. \item Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ if two out of three are locally quasi-coherent so is the third. \item Given $\mathcal{F}, \mathcal{G}$ in $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ the tensor product $\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ is an object of $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$. \item Given $\mathcal{F}, \mathcal{G}$ in $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_\etale$ the sheaf $\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ is an object of $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$. \end{enumerate}


stacks-sheaves-lemma-qc-colimits

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. \begin{enumerate} \item The category $\QCoh(\mathcal{O}_\mathcal{X})$ has colimits and they agree with colimits in the categories $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal{X})$, $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$, $\textit{Mod}(\mathcal{O}_\mathcal{X})$, and $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$. \item Given $\mathcal{F}, \mathcal{G}$ in $\QCoh(\mathcal{O}_\mathcal{X})$ the tensor product $\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ is an object of $\QCoh(\mathcal{O}_\mathcal{X})$. \item Given $\mathcal{F}, \mathcal{G}$ in $\QCoh(\mathcal{O}_\mathcal{X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_{fppf}$ the sheaf $\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ is an object of $\QCoh(\mathcal{O}_\mathcal{X})$. \end{enumerate}


stacks-sheaves-lemma-stackification

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $f$ induces an equivalence of stackifications, then the morphism of topoi $f : \Sh(\mathcal{X}_{fppf}) \to \Sh(\mathcal{Y}_{fppf})$ is an equivalence.


stacks-sheaves-lemma-stackification-quasi-coherent

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $f$ induces an equivalence of stackifications, then $f^*$ induces equivalences $\textit{Mod}(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{Y})$ and $\QCoh(\mathcal{O}_\mathcal{X}) \to \QCoh(\mathcal{O}_\mathcal{Y})$.


stacks-sheaves-lemma-map-from-quasi-coherent

Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Let $\mathcal{X} = [U/R]$ be the quotient stack. Denote $\pi : \mathcal{S}_U \to \mathcal{X}$ the quotient map. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_\mathcal{X}$-module, and let $\mathcal{H}$ be any object of $\textit{Mod}(\mathcal{O}_\mathcal{X})$. The map $$ \Hom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{H}) \longrightarrow \Hom_{\mathcal{O}_U}(x^*\mathcal{F}|_{U_\etale}, x^*\mathcal{H}|_{U_\etale}), \quad \phi \longmapsto x^*\phi|_{U_\etale} $$ is injective and its image consists of exactly those $\varphi : x^*\mathcal{F}|_{U_\etale} \to x^*\mathcal{H}|_{U_\etale}$ which give rise to a commutative diagram $$ \xymatrix{ s_{small}^*(x^*\mathcal{F}|_{U_\etale}) \ar[r] \ar[d]^{s_{small}^*\varphi} & (x \circ s)^*\mathcal{F}|_{R_\etale} = (x \circ t)^*\mathcal{F}|_{R_\etale} & t_{small}^*(x^*\mathcal{F}|_{U_\etale}) \ar[l] \ar[d]_{t_{small}^*\varphi} \\ s_{small}^*(x^*\mathcal{H}|_{U_\etale}) \ar[r] & (x \circ s)^*\mathcal{H}|_{R_\etale} = (x \circ t)^*\mathcal{H}|_{R_\etale} & t_{small}^*(x^*\mathcal{H}|_{U_\etale}) \ar[l] } $$ of modules on $R_\etale$ where the horizontal arrows are the comparison maps (\ref{equation-comparison-algebraic-spaces-modules}).


stacks-sheaves-lemma-quasi-coherent-algebraic-stack

Let $\mathcal{X}$ be an algebraic stack over $S$. \begin{enumerate} \item If $[U/R] \to \mathcal{X}$ is a presentation of $\mathcal{X}$ then there is a canonical equivalence $\QCoh(\mathcal{O}_\mathcal{X}) \cong \QCoh(U, R, s, t, c)$. \item The category $\QCoh(\mathcal{O}_\mathcal{X})$ is abelian. \item The category $\QCoh(\mathcal{O}_\mathcal{X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{O}_\mathcal{X})$. \item Given $\mathcal{F}, \mathcal{G}$ in $\QCoh(\mathcal{O}_\mathcal{X})$ the tensor product $\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ is an object of $\QCoh(\mathcal{O}_\mathcal{X})$. \item Given $\mathcal{F}, \mathcal{G}$ in $\QCoh(\mathcal{O}_\mathcal{X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_{fppf}$ the sheaf $\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ is an object of $\QCoh(\mathcal{O}_\mathcal{X})$. \end{enumerate}


stacks-sheaves-lemma-cohomology-restriction

Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. Let $x \in \Ob(\mathcal{X})$ be an object lying over the scheme $U$. Let $\mathcal{F}$ be an object of $\textit{Ab}(\mathcal{X}_\tau)$ or $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$. Then $$ H^p_\tau(x, \mathcal{F}) = H^p((\Sch/U)_\tau, x^{-1}\mathcal{F}) $$ and if $\tau = \etale$, then we also have $$ H^p_\etale(x, \mathcal{F}) = H^p(U_\etale, \mathcal{F}|_{U_\etale}). $$


stacks-sheaves-lemma-pushforward-injective

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. \begin{enumerate} \item $f_*\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{Y}_\tau)$ for $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{X}_\tau)$, and \item $f_*\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})$ for $\mathcal{I}$ injective in $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$. \end{enumerate}


stacks-sheaves-lemma-fibre-products

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. \begin{enumerate} \item The category $\mathcal{X}$ has fibre products. \item If the $\mathit{Isom}$-presheaves of $\mathcal{X}$ are representable by algebraic spaces, then $\mathcal{X}$ has equalizers. \item If $\mathcal{X}$ is an algebraic stack (or more generally a quotient stack), then $\mathcal{X}$ has equalizers. \end{enumerate}


stacks-sheaves-lemma-fibre-products-morphism

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item The functor $f$ transforms fibre products into fibre products. \item If $f$ is faithful, then $f$ transforms equalizers into equalizers. \end{enumerate}


stacks-sheaves-lemma-fibre-products-preserve-properties

Let $f : \mathcal{X} \to \mathcal{Y}$, $g : \mathcal{Z} \to \mathcal{Y}$ be faithful $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item the functor $\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Y}$ is faithful, and \item if $\mathcal{X}, \mathcal{Z}$ have equalizers, so does $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$. \end{enumerate}


stacks-sheaves-lemma-pullback-injective

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. The functor $f^{-1} : \textit{Ab}(\mathcal{Y}_\tau) \to \textit{Ab}(\mathcal{X}_\tau)$ has a left adjoint $f_! : \textit{Ab}(\mathcal{X}_\tau) \to \textit{Ab}(\mathcal{Y}_\tau)$. If $f$ is faithful and $\mathcal{X}$ has equalizers, then \begin{enumerate} \item $f_!$ is exact, and \item $f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{X}_\tau)$ for $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{Y}_\tau)$. \end{enumerate}


stacks-sheaves-lemma-pullback-injective-modules

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. The functor $f^* : \textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y}) \to \textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$ has a left adjoint $f_! : \textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})$ which agrees with the functor $f_!$ of Lemma \ref{lemma-pullback-injective} on underlying abelian sheaves. If $f$ is faithful and $\mathcal{X}$ has equalizers, then \begin{enumerate} \item $f_!$ is exact, and \item $f^{-1}\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$ for $\mathcal{I}$ injective in $\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{X})$. \end{enumerate}


stacks-sheaves-lemma-generalities

Generalities on {\v C}ech complexes. \begin{enumerate} \item If $$ \xymatrix{ \mathcal{V} \ar[d]_g \ar[r]_h & \mathcal{U} \ar[d]^f \\ \mathcal{Y} \ar[r]^e & \mathcal{X} } $$ is $2$-commutative diagram of categories fibred in groupoids over $(\Sch/S)_{fppf}$, then there is a morphism of {\v C}ech complexes $$ \check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{V} \to \mathcal{Y}, e^{-1}\mathcal{F}) $$ \item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism, \item if $f, f' : \mathcal{U} \to \mathcal{X}$ are $2$-isomorphic, then the associated {\v C}ech complexes are isomorphic, \end{enumerate}


stacks-sheaves-lemma-homotopy

If there exists a $1$-morphism $s : \mathcal{X} \to \mathcal{U}$ such that $f \circ s$ is $2$-isomorphic to $\text{id}_\mathcal{X}$ then the extended {\v C}ech complex is homotopic to zero.


stacks-sheaves-lemma-generalities-sheafified

Generalities on relative {\v C}ech complexes. \begin{enumerate} \item If $$ \xymatrix{ \mathcal{V} \ar[d]_g \ar[r]_h & \mathcal{U} \ar[d]^f \\ \mathcal{Y} \ar[r]^e & \mathcal{X} } $$ is $2$-commutative diagram of categories fibred in groupoids over $(\Sch/S)_{fppf}$, then there is a morphism $e^{-1}\mathcal{K}^\bullet(f, \mathcal{F}) \to \mathcal{K}^\bullet(g, e^{-1}\mathcal{F})$. \item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism, \item if $f, f' : \mathcal{U} \to \mathcal{X}$ are $2$-isomorphic, then the associated relative {\v C}ech complexes are isomorphic, \end{enumerate}


stacks-sheaves-lemma-homotopy-sheafified

If there exists a $1$-morphism $s : \mathcal{X} \to \mathcal{U}$ such that $f \circ s$ is $2$-isomorphic to $\text{id}_\mathcal{X}$ then the extended relative {\v C}ech complex is homotopic to zero.


stacks-sheaves-lemma-base-change-cech-complex

Let $$ \xymatrix{ \mathcal{V} \ar[d]_g \ar[r]_h & \mathcal{U} \ar[d]^f \\ \mathcal{Y} \ar[r]^e & \mathcal{X} } $$ be a $2$-fibre product of categories fibred in groupoids over $(\Sch/S)_{fppf}$ and let $\mathcal{F}$ be an abelian presheaf on $\mathcal{X}$. Then the map $e^{-1}\mathcal{K}^\bullet(f, \mathcal{F}) \to \mathcal{K}^\bullet(g, e^{-1}\mathcal{F})$ of Lemma \ref{lemma-generalities-sheafified} is an isomorphism of complexes of abelian presheaves.


stacks-sheaves-lemma-check-exactness-covering

Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Let $$ \mathcal{F} \to \mathcal{G} \to \mathcal{H} $$ be a complex in $\textit{Ab}(\mathcal{X}_\tau)$. Assume that \begin{enumerate} \item for every object $x$ of $\mathcal{X}$ there exists a covering $\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, and \item $f^{-1}\mathcal{F} \to f^{-1}\mathcal{G} \to f^{-1}\mathcal{H}$ is exact. \end{enumerate} Then the sequence $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ is exact.


stacks-sheaves-lemma-cech-to-cohomology

Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Assume \begin{enumerate} \item $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$, \item for every object $x$ of $\mathcal{X}$ there exists a covering $\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, \item the category $\mathcal{U}$ has equalizers, and \item the functor $f$ is faithful. \end{enumerate} Then there is a first quadrant spectral sequence of abelian groups $$ E_1^{p, q} = H^q((\mathcal{U}_p)_\tau, f_p^{-1}\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}_\tau, \mathcal{F}) $$ converging to the cohomology of $\mathcal{F}$ in the $\tau$-topology.


stacks-sheaves-lemma-surjective-flat-locally-finite-presentation

Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then for any object $y$ of $\mathcal{Y}$ there exists an fppf covering $\{y_i \to y\}$ and objects $x_i$ of $\mathcal{X}$ such that $f(x_i) \cong y_i$ in $\mathcal{Y}$. \item Assume that $f$ is representable by algebraic spaces, surjective, and smooth. Then for any object $y$ of $\mathcal{Y}$ there exists an \'etale covering $\{y_i \to y\}$ and objects $x_i$ of $\mathcal{X}$ such that $f(x_i) \cong y_i$ in $\mathcal{Y}$. \end{enumerate}


stacks-sheaves-lemma-cech-to-cohomology-relative

Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, \linebreak[0] fppf\}$. Assume \begin{enumerate} \item $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$, \item for every object $x$ of $\mathcal{X}$ there exists a covering $\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, \item the category $\mathcal{U}$ has equalizers, and \item the functor $f$ is faithful. \end{enumerate} Then there is a first quadrant spectral sequence of abelian sheaves on $\mathcal{Y}_\tau$ $$ E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} $$ where all higher direct images are computed in the $\tau$-topology.


stacks-sheaves-lemma-cech-to-cohomology-relative-modules

Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, \linebreak[0] fppf\}$. Assume \begin{enumerate} \item $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$, \item for every object $x$ of $\mathcal{X}$ there exists a covering $\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, \item the category $\mathcal{U}$ has equalizers, and \item the functor $f$ is faithful. \end{enumerate} Then there is a first quadrant spectral sequence in $\textit{Mod}(\mathcal{Y}_\tau, \mathcal{O}_\mathcal{Y})$ $$ E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} $$ where all higher direct images are computed in the $\tau$-topology.


stacks-sheaves-lemma-pushforward-restriction

Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks\footnote{This result should hold for any $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$.} over $S$. Let $\tau \in \{Zariski,\linebreak[0] \etale,\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$. Let $\mathcal{F}$ be an object of $\textit{Ab}(\mathcal{X}_\tau)$ or $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$. Then the sheaf $R^if_*\mathcal{F}$ is the sheaf associated to the presheaf $$ y \longmapsto H^i_\tau\Big((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big) $$ Here $y$ is a typical object of $\mathcal{Y}$ lying over the scheme $V$.


stacks-sheaves-lemma-base-change-higher-direct-images

Let $S$ be a scheme. Let $\tau \in \{Zariski,\linebreak[0] \etale,\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\}$. Let $$ \xymatrix{ \mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \ar[r]_{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Y}' \ar[r]^g & \mathcal{Y} } $$ be a $2$-cartesian diagram of algebraic stacks over $S$. Then the base change map is an isomorphism $$ g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F} $$ functorial for $\mathcal{F}$ in $\textit{Ab}(\mathcal{X}_\tau)$ or $\mathcal{F}$ in $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$.


stacks-sheaves-lemma-compare-injectives

Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$ representable by the algebraic space $F$. \begin{enumerate} \item If $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{X}_\etale)$, then $\mathcal{I}|_{F_\etale}$ is injective in $\textit{Ab}(F_\etale)$, \item If $\mathcal{I}^\bullet$ is a K-injective complex in $\textit{Ab}(\mathcal{X}_\etale)$, then $\mathcal{I}^\bullet|_{F_\etale}$ is a K-injective complex in $\textit{Ab}(F_\etale)$. \end{enumerate} The same does not hold for modules.


stacks-sheaves-lemma-compare-morphism-cohomology

Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over $S$. Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by algebraic spaces $F$, $G$. Denote $f : F \to G$ the induced morphism of algebraic spaces. \begin{enumerate} \item For any $\mathcal{F} \in \textit{Ab}(\mathcal{X}_\etale)$ we have $$ (Rf_*\mathcal{F})|_{G_\etale} = Rf_{small, *}(\mathcal{F}|_{F_\etale}) $$ in $D(G_\etale)$. \item For any object $\mathcal{F}$ of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ we have $$ (Rf_*\mathcal{F})|_{G_\etale} = Rf_{small, *}(\mathcal{F}|_{F_\etale}) $$ in $D(\mathcal{O}_G)$. \end{enumerate}


stacks-sheaves-lemma-compare-representable-morphism-cohomology

Let $S$ be a scheme. Consider a $2$-fibre product square $$ \xymatrix{ \mathcal{X}' \ar[r]_{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Y}' \ar[r]^g & \mathcal{Y} } $$ of algebraic stacks over $S$. Assume that $f$ is representable by algebraic spaces and that $\mathcal{Y}'$ is representable by an algebraic space $G'$. Then $\mathcal{X}'$ is representable by an algebraic space $F'$ and denoting $f' : F' \to G'$ the induced morphism of algebraic spaces we have $$ g^{-1}(Rf_*\mathcal{F})|_{G'_\etale} = Rf'_{small, *}((g')^{-1}\mathcal{F}|_{F'_\etale}) $$ for any $\mathcal{F}$ in $\textit{Ab}(\mathcal{X}_\etale)$ or in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$


stacks-sheaves-lemma-lqc-flat-base-change-fppf-sheaf

Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal{X}$-modules. Assume \begin{enumerate} \item[(a)] $\mathcal{F}$ is locally quasi-coherent, and \item[(b)] for any morphism $\varphi : x \to y$ of $\mathcal{X}$ which lies over a morphism of schemes $f : U \to V$ which is flat and locally of finite presentation the comparison map $c_\varphi : f_{small}^*\mathcal{F}|_{V_\etale} \to \mathcal{F}|_{U_\etale}$ of (\ref{equation-comparison-modules}) is an isomorphism. \end{enumerate} Then $\mathcal{F}$ is a sheaf for the fppf topology.


stacks-sheaves-lemma-compare-fppf-etale

Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Let $\mathcal{F}$ be a presheaf $\mathcal{O}_\mathcal{X}$-module such that \begin{enumerate} \item[(a)] $\mathcal{F}$ is locally quasi-coherent, and \item[(b)] for any morphism $\varphi : x \to y$ of $\mathcal{X}$ which lies over a morphism of schemes $f : U \to V$ which is flat and locally of finite presentation, the comparison map $c_\varphi : f_{small}^*\mathcal{F}|_{V_\etale} \to \mathcal{F}|_{U_\etale}$ of (\ref{equation-comparison-modules}) is an isomorphism. \end{enumerate} Then $\mathcal{F}$ is an $\mathcal{O}_\mathcal{X}$-module and we have the following \begin{enumerate} \item If $\epsilon : \mathcal{X}_{fppf} \to \mathcal{X}_\etale$ is the comparison morphism, then $R\epsilon_*\mathcal{F} = \epsilon_*\mathcal{F}$. \item The cohomology groups $H^p_{fppf}(\mathcal{X}, \mathcal{F})$ are equal to the cohomology groups computed in the \'etale topology on $\mathcal{X}$. Similarly for the cohomology groups $H^p_{fppf}(x, \mathcal{F})$ and the derived versions $R\Gamma(\mathcal{X}, \mathcal{F})$ and $R\Gamma(x, \mathcal{F})$. \item If $f : \mathcal{X} \to \mathcal{Y}$ is a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$ then $R^if_*\mathcal{F}$ is equal to the fppf-sheafification of the higher direct image computed in the \'etale cohomology. Similarly for derived pullback. \end{enumerate}


stacks-sheaves-lemma-cohomology-on-subcategory

Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Let $\tau = \etale$ (resp.\ $\tau = fppf$). Let $\mathcal{X}' \subset \mathcal{X}$ be a full subcategory with the following properties \begin{enumerate} \item if $x \to x'$ is a morphism of $\mathcal{X}$ which lies over a smooth (resp.\ flat and locally finitely presented) morphism of schemes and $x' \in \Ob(\mathcal{X}')$, then $x \in \Ob(\mathcal{X}')$, and \item there exists an object $x \in \Ob(\mathcal{X}')$ lying over a scheme $U$ such that the associated $1$-morphism $x : (\Sch/U)_{fppf} \to \mathcal{X}$ is smooth and surjective. \end{enumerate} We get a site $\mathcal{X}'_\tau$ by declaring a covering of $\mathcal{X}'$ to be any family of morphisms $\{x_i \to x\}$ in $\mathcal{X}'$ which is a covering in $\mathcal{X}_\tau$. Then the inclusion functor $\mathcal{X}' \to \mathcal{X}_\tau$ is fully faithful, cocontinuous, and continuous, whence defines a morphism of topoi $$ g : \Sh(\mathcal{X}'_\tau) \longrightarrow \Sh(\mathcal{X}_\tau) $$ and $H^p(\mathcal{X}'_\tau, g^{-1}\mathcal{F}) = H^p(\mathcal{X}_\tau, \mathcal{F})$ for all $p \geq 0$ and all $\mathcal{F} \in \textit{Ab}(\mathcal{X}_\tau)$.


stacks-sheaves-proposition-quasi-coherent

Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Let $\mathcal{X} = [U/R]$ be the quotient stack. The category of quasi-coherent modules on $\mathcal{X}$ is equivalent to the category of quasi-coherent modules on $(U, R, s, t, c)$.


stacks-sheaves-proposition-exactness-cech-complex

Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. If \begin{enumerate} \item $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$, and \item for every object $x$ of $\mathcal{X}$ there exists a covering $\{x_i \to x\}$ in $\mathcal{X}_\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\mathcal{U}$, \end{enumerate} then the extended relative {\v C}ech complex $$ \ldots \to 0 \to \mathcal{F} \to f_{0, *}f_0^{-1}\mathcal{F} \to f_{1, *}f_1^{-1}\mathcal{F} \to f_{2, *}f_2^{-1}\mathcal{F} \to \ldots $$ is exact in $\textit{Ab}(\mathcal{X}_\tau)$.


stacks-sheaves-proposition-smooth-covering-compute-cohomology

Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of algebraic stacks. \begin{enumerate} \item Let $\mathcal{F}$ be an abelian \'etale sheaf on $\mathcal{X}$. Assume that $f$ is representable by algebraic spaces, surjective, and smooth. Then there is a spectral sequence $$ E_1^{p, q} = H^q_\etale(\mathcal{U}_p, f_p^{-1}\mathcal{F}) \Rightarrow H^{p + q}_\etale(\mathcal{X}, \mathcal{F}) $$ \item Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{X}$. Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then there is a spectral sequence $$ E_1^{p, q} = H^q_{fppf}(\mathcal{U}_p, f_p^{-1}\mathcal{F}) \Rightarrow H^{p + q}_{fppf}(\mathcal{X}, \mathcal{F}) $$ \end{enumerate}


stacks-sheaves-proposition-smooth-covering-compute-direct-image

Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable $1$-morphisms of algebraic stacks. \begin{enumerate} \item Assume that $f$ is representable by algebraic spaces, surjective and smooth. \begin{enumerate} \item If $\mathcal{F}$ is in $\textit{Ab}(\mathcal{X}_\etale)$ then there is a spectral sequence $$ E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} $$ in $\textit{Ab}(\mathcal{Y}_\etale)$ with higher direct images computed in the \'etale topology. \item If $\mathcal{F}$ is in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ then there is a spectral sequence $$ E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} $$ in $\textit{Mod}(\mathcal{Y}_\etale, \mathcal{O}_\mathcal{Y})$. \end{enumerate} \item Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. \begin{enumerate} \item If $\mathcal{F}$ is in $\textit{Ab}(\mathcal{X})$ then there is a spectral sequence $$ E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} $$ in $\textit{Ab}(\mathcal{Y})$ with higher direct images computed in the fppf topology. \item If $\mathcal{F}$ is in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ then there is a spectral sequence $$ E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} $$ in $\textit{Mod}(\mathcal{O}_\mathcal{Y})$. \end{enumerate} \end{enumerate}


resolve-theorem-resolve

\begin{reference} \cite[Theorem on page 151]{Lipman} \end{reference} Let $Y$ be a two dimensional integral Noetherian scheme. The following are equivalent \begin{enumerate} \item there exists an alteration $X \to Y$ with $X$ regular, \item there exists a resolution of singularities of $Y$, \item $Y$ has a resolution of singularities by normalized blowups, \item the normalization $Y^\nu \to Y$ is finite, $Y^\nu$ has finitely many singular points $y_1, \ldots, y_m$, and for each $y_i$ the completion of $\mathcal{O}_{Y^\nu, y_i}$ is normal. \end{enumerate}


resolve-lemma-trace-well-defined

Let $\varphi : R[x]/(x^p - a) \to R[y]/(y^p - b)$ be an $R$-algebra homomorphism. Then $\text{Tr}_x = \text{Tr}_y \circ \varphi$.


resolve-lemma-trace-higher

Let $\mathbf{F}_p \subset \Lambda \subset R \subset S$ be ring extensions and assume that $S$ is isomorphic to $R[x]/(x^p - a)$ for some $a \in R$. Then there are canonical $R$-linear maps $$ \text{Tr} : \Omega^{t + 1}_{S/\Lambda} \longrightarrow \Omega_{R/\Lambda}^{t + 1} $$ for $t \geq 0$ such that $$ \eta_1 \wedge \ldots \wedge \eta_t \wedge x^i\text{d}x \longmapsto \left\{ \begin{matrix} 0 & \text{if} & 0 \leq i \leq p - 2, \\ \eta_1 \wedge \ldots \wedge \eta_t \wedge \text{d}a & \text{if} & i = p - 1 \end{matrix} \right. $$ for $\eta_i \in \Omega_{R/\Lambda}$ and such that $\text{Tr}$ annihilates the image of $S \otimes_R \Omega_{R/\Lambda}^{t + 1} \to \Omega_{S/\Lambda}^{t + 1}$.


resolve-lemma-trace-extends

Let $S$ be a scheme over $\mathbf{F}_p$. Let $f : Y \to X$ be a finite morphism of Noetherian normal integral schemes over $S$. Assume \begin{enumerate} \item the extension of function fields is purely inseparable of degree $p$, and \item $\Omega_{X/S}$ is a coherent $\mathcal{O}_X$-module (for example if $X$ is of finite type over $S$). \end{enumerate} For $i \geq 1$ there is a canonical map $$ \text{Tr} : f_*\Omega^i_{Y/S} \longrightarrow (\Omega_{X/S}^i)^{**} $$ whose stalk in the generic point of $X$ recovers the trace map of Lemma \ref{lemma-trace-higher}.


resolve-lemma-blowup

Let $(A, \mathfrak m, \kappa)$ be a regular local ring of dimension $2$. Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$ wotj exceptional divisor $E$. There is a closed immersion $$ r : X \longrightarrow \mathbf{P}^1_S $$ over $S$ such that \begin{enumerate} \item $r|_E : E \to \mathbf{P}^1_\kappa$ is an isomorphism, \item $\mathcal{O}_X(E) = \mathcal{O}_X(-1) = r^*\mathcal{O}_{\mathbf{P}^1}(-1)$, and \item $\mathcal{C}_{E/X} = (r|_E)^*\mathcal{O}_{\mathbf{P}^1}(1)$ and $\mathcal{N}_{E/X} = (r|_E)^*\mathcal{O}_{\mathbf{P}^1}(-1)$. \end{enumerate}


resolve-lemma-blowup-regular

Let $(A, \mathfrak m, \kappa)$ be a regular local ring of dimension $2$. Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$. Then $X$ is an irreducible regular scheme.


resolve-lemma-blowup-pic

Let $(A, \mathfrak m, \kappa)$ be a regular local ring of dimension $2$. Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$. Then $\Pic(X) = \mathbf{Z}$ generated by $\mathcal{O}_X(E)$.


resolve-lemma-cohomology-of-blowup

Let $(A, \mathfrak m, \kappa)$ be a regular local ring of dimension $2$. Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item $H^p(X, \mathcal{F}) = 0$ for $p \not \in \{0, 1\}$, \item $H^1(X, \mathcal{O}_X(n)) = 0$ for $n \geq -1$, \item $H^1(X, \mathcal{F}) = 0$ if $\mathcal{F}$ or $\mathcal{F}(1)$ is globally generated, \item $H^0(X, \mathcal{O}_X(n)) = \mathfrak m^{\max(0, n)}$, \item $\text{length}_A H^1(X, \mathcal{O}_X(n)) = -n(-n - 1)/2$ if $n < 0$. \end{enumerate}


resolve-lemma-blowup-improve

Let $(A, \mathfrak m)$ be a regular local ring of dimension $2$. Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$. Let $\mathfrak m^n \subset I \subset \mathfrak m$ be an ideal. Let $d \geq 0$ be the largest integer such that $$ I \mathcal{O}_X \subset \mathcal{O}_X(-dE) $$ where $E$ is the exceptional divisor. Set $\mathcal{I}' = I\mathcal{O}_X(dE) \subset \mathcal{O}_X$. Then $d > 0$, the sheaf $\mathcal{O}_X/\mathcal{I}'$ is supported in finitely many closed points $x_1, \ldots, x_r$ of $X$, and \begin{align*} \text{length}_A(A/I) & > \text{length}_A \Gamma(X, \mathcal{O}_X/\mathcal{I}') \\ & \geq \sum\nolimits_{i = 1, \ldots, r} \text{length}_{\mathcal{O}_{X, x_i}} (\mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i}) \end{align*}


resolve-lemma-differentials-of-blowup

Let $(A, \mathfrak m, \kappa)$ be a regular local ring of dimension $2$. Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$. Then $\Omega_{X/S} = i_*\Omega_{E/\kappa}$, where $i : E \to X$ is the immersion of the exceptional divisor.


resolve-lemma-make-ideal-principal

Let $X$ be a Noetherian scheme. Let $T \subset X$ be a finite set of closed points $x$ such that $\mathcal{O}_{X, x}$ is regular of dimension $2$ for $x \in T$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals such that $\mathcal{O}_X/\mathcal{I}$ is supported on $T$. Then there exists a sequence $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X $$ where $X_{i + 1} \to X_i$ is the blowing up of $X_i$ at a closed point $x_i$ lying above a point of $T$ such that $\mathcal{I}\mathcal{O}_{X_n}$ is an invertible ideal sheaf.


resolve-lemma-dominate-by-blowing-up-in-points

Let $X$ be a Noetherian scheme. Let $T \subset X$ be a finite set of closed points $x$ such that $\mathcal{O}_{X, x}$ is a regular local ring of dimension $2$. Let $f : Y \to X$ be a proper morphism of schemes which is an isomorphism over $U = X \setminus T$. Then there exists a sequence $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X $$ where $X_{i + 1} \to X_i$ is the blowing up of $X_i$ at a closed point $x_i$ lying above a point of $T$ and a factorization $X_n \to Y \to X$ of the composition.


resolve-lemma-extend-rational-map-blowing-up

Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is regular and has dimension $2$. Let $Y$ be a proper scheme over $S$. Given an $S$-rational map $f : U \to Y$ from $X$ to $Y$ there exists a sequence $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X $$ and an $S$-morphism $f_n : X_n \to Y$ such that $X_{i + 1} \to X_i$ blowing up of $X_i$ at a closed point not lying over $U$ and $f_n$ and $f$ agree.


resolve-lemma-dominate-by-normalized-blowing-up

Let $X$ be a scheme which is Noetherian, Nagata, and has dimension $2$. Let $f : Y \to X$ be a proper birational morphism. Then there exists a commutative diagram $$ \xymatrix{ X_n \ar[r] \ar[d] & X_{n - 1} \ar[r] & \ldots \ar[r] & X_1 \ar[r] & X_0 \ar[d] \\ Y \ar[rrrr]  & & & & X } $$ where $X_0 \to X$ is the normalization and where $X_{i + 1} \to X_i$ is the normalized blowing up of $X_i$ at a closed point.


resolve-lemma-equivalence-fibre

Let $S, s_i, S_i$ be as in (\ref{equation-equivalence}). If $f : X \to S$ corresponds to $g_i : Y_i \to S_i$ under $F$, then $X_{s_i} \cong (Y_i)_{s_i}$ as schemes over $\kappa(s_i)$.


resolve-lemma-equivalence-sequence-blowups

Let $S, s_i, S_i$ be as in (\ref{equation-equivalence}) and assume $f : X \to S$ corresponds to $g_i : Y_i \to S_i$ under $F$. Then there exists a factorization $$ X = Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S $$ of $f$ where $Z_{j + 1} \to Z_j$ is the blowing up of $Z_j$ at a closed point $z_j$ lying over $\{s_1, \ldots, s_n\}$ if and only if for each $i$ there exists a factorization $$ Y_i = Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_i $$ of $g_i$ where $Z_{i, j + 1} \to Z_{i, j}$ is the blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $s_i$.


resolve-lemma-equivalence-sequence-normalized-blowups

Let $S, s_i, S_i$ be as in (\ref{equation-equivalence}) and assume $f : X \to S$ corresponds to $g_i : Y_i \to S_i$ under $F$. Assume every quasi-compact open of $S$ has finitely many irreducible components.  Then there exists a factorization $$ X = Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S $$ of $f$ where $Z_{j + 1} \to Z_j$ is the normalized blowing up of $Z_j$ at a closed point $z_j$ lying over $\{x_1, \ldots, x_n\}$ if and only if for each $i$ there exists a factorization $$ Y_i = Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_i $$ of $g_i$ where $Z_{i, j + 1} \to Z_{i, j}$ is the normalized blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $s_i$.


resolve-lemma-dominate-by-scheme-modification

In Situation \ref{situation-vanishing} there exists a $U$-admissible blowup $X' \to S$ which dominates $X$.


resolve-lemma-nice-meromorphic-function

In Situation \ref{situation-vanishing} there exists a nonzero $f \in \mathfrak m$ such that for every $i = 1, \ldots, r$ there exist \begin{enumerate} \item a closed point $x_i \in C_i$ with $x_i \not \in C_j$ for $j \not = i$, \item a factorization $f = g_i f_i$ of $f$ in $\mathcal{O}_{X, x_i}$ such that $g_i \in \mathfrak m_{x_i}$ maps to a nonzero element of $\mathcal{O}_{C_i, x_i}$. \end{enumerate}


resolve-lemma-nontrivial-normal-bundle

In Situation \ref{situation-vanishing} assume $X$ is normal. Let $Z \subset X$ be a nonempty effective Cartier divisor such that $Z \subset X_s$ set theoretically. Then the conormal sheaf of $Z$ is not trivial. More precisely, there exists an $i$ such that $C_i \subset Z$ and $\deg(\mathcal{C}_{Z/X}|_{C_i}) > 0$.


resolve-lemma-H1-injective

In Situation \ref{situation-vanishing} assume $X$ is normal and $A$ Nagata. The map $$ H^1(X, \mathcal{O}_X) \longrightarrow H^1(f^{-1}(U), \mathcal{O}_X) $$ is injective.


resolve-lemma-R1-injective

In Situation \ref{situation-vanishing} assume $X$ is normal and $A$ Nagata. Then $$ \Hom_{D(A)}(\kappa[-1], Rf_*\mathcal{O}_X) $$ is zero. This uses $D(A) = D_\QCoh(\mathcal{O}_S)$ to think of $Rf_*\mathcal{O}_X$ as an object of $D(A)$.


resolve-lemma-exact-sequence

Let $(A, \mathfrak m, \kappa)$ be a Noetherian normal local domain of dimension $2$. Consider a commutative diagram $$ \xymatrix{ X' \ar[rd]_{f'} \ar[rr]_g & & X \ar[ld]^f \\ & \Spec(A) } $$ where $f$ and $f'$ are modifications as in Situation \ref{situation-vanishing} and $X$ normal. Then we have a short exact sequence $$ 0 \to H^1(X, \mathcal{O}_X) \to H^1(X', \mathcal{O}_{X'}) \to H^0(X, R^1g_*\mathcal{O}_{X'}) \to 0 $$ Also $\dim(\text{Supp}(R^1g_*\mathcal{O}_{X'})) = 0$ and $R^1g_*\mathcal{O}_{X'}$ is generated by global sections.


resolve-lemma-bound-a-torsion

Let $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain of dimension $2$. Let $a \in A$ be nonzero. There exists an integer $N$ such that for every modification $f : X \to \Spec(A)$ with $X$ normal the $A$-module $$ M_{X, a} = \Coker(A \longrightarrow H^0(Z, \mathcal{O}_Z)) $$ where $Z \subset X$ is cut out by $a$ has length bounded by $N$.


resolve-lemma-rational-propagates

Let $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain of dimension $2$ which defines a rational singularity. Let $A \subset B$ be a local extension of domains with the same fraction field which is essentially of finite type such that $\dim(B) = 2$ and $B$ normal. Then $B$ defines a rational singularity.


resolve-lemma-reduce-to-rational

Let $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain of dimension $2$. If reduction to rational singularities is possible for $A$, then there exists a finite sequence of normalized blowups $$ X = X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \Spec(A) $$ in closed points such that for any closed point $x \in X$ the local ring $\mathcal{O}_{X, x}$ defines a rational singularity. In particular $X \to \Spec(A)$ is a modification and $X$ is a normal scheme projective over $A$.


resolve-lemma-go-up-separable

Let $A \to B$ be a finite injective local ring map of local normal Nagata domains of dimension $2$. Assume that the induced extension of fraction fields is separable. If reduction to rational singularities is possible for $A$ then it is possible for $B$.


resolve-lemma-regular-rational

Let $A$ be a Nagata regular local ring of dimension $2$. Then $A$ defines a rational singularity.


resolve-lemma-bound-dualizing-implies-bound

Let $A$ be a local normal Nagata domain of dimension $2$ which has a dualizing complex $\omega_A^\bullet$. If there exists a nonzero $d \in A$ such that for all normal modifications $X \to \Spec(A)$ the cokernel of the trace map $$ \Gamma(X, \omega_X) \to \omega_A $$ is annihilated by $d$, then reduction to rational singularities is possible for $A$.


resolve-lemma-compare-differentials-dualizing

Let $p$ be a prime number. Let $A$ be a regular local ring of dimension $2$ and characteristic $p$. Let $A_0 \subset A$ be a subring such that $\Omega_{A/A_0}$ is free of rank $r < \infty$. Set $\omega_A = \Omega^r_{A/A_0}$. If $X \to \Spec(A)$ is the result of a sequence of blowups in closed points, then there exists a map $$ \varphi_X : (\Omega^r_{X/\Spec(A_0)})^{**} \longrightarrow \omega_X $$ extending the given identification in the generic point.


resolve-lemma-go-up-degree-p

Let $p$ be a prime number. Let $A$ be a complete regular local ring of dimension $2$ and characteristic $p$. Let $L/K$ be a degree $p$ inseparable extension of the fraction field $K$ of $A$. Let $B \subset L$ be the integral closure of $A$. Then reduction to rational singularities is possible for $B$.


resolve-lemma-globally-generated

In Situation \ref{situation-rational}. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then \begin{enumerate} \item $H^p(X, \mathcal{F}) = 0$ for $p \not \in \{0, 1\}$, and \item $H^1(X, \mathcal{F}) = 0$ if $\mathcal{F}$ is globally generated. \end{enumerate}


resolve-lemma-sections-powers-I-rational

In Situation \ref{situation-rational} assume $E = X_s$ is an effective Cartier divisor. Let $\mathcal{I}$ be the ideal sheaf of $E$. Then $H^0(X, \mathcal{I}^n) = \mathfrak m^n$ and $H^1(X, \mathcal{I}^n) = 0$.


resolve-lemma-blow-up-normal-rational

In Situation \ref{situation-rational} the blowup of $\Spec(A)$ in $\mathfrak m$ is normal.


resolve-lemma-cohomology-blow-up-rational

In Situation \ref{situation-rational}. Let $X$ be the blowup of $\Spec(A)$ in $\mathfrak m$. Let $E \subset X$ be the exceptional divisor. With $\mathcal{O}_X(1) = \mathcal{I}$ as usual and $\mathcal{O}_E(1) = \mathcal{O}_X(1)|_E$ we have \begin{enumerate} \item $E$ is a proper Cohen-Macaulay curve over $\kappa$. \item $\mathcal{O}_E(1)$ is very ample \item $\deg(\mathcal{O}_E(1)) \geq 1$ and equality holds only if $A$ is a regular local ring, \item $H^1(E, \mathcal{O}_E(n)) = 0$ for $n \geq 0$, and \item $H^0(E, \mathcal{O}_E(n)) = \mathfrak m^n/\mathfrak m^{n + 1}$ for $n \geq 0$. \end{enumerate}


resolve-lemma-dualizing-rational

In Situation \ref{situation-rational} assume $A$ has a dualizing complex $\omega_A^\bullet$. With $\omega_X$ the dualizing module of $X$, the trace map $H^0(X, \omega_X) \to \omega_A$ is an isomorphism and consequently there is a canonical map $f^*\omega_A \to \omega_X$.


resolve-lemma-dualizing-blow-up-rational

In Situation \ref{situation-rational} assume $A$ has a dualizing complex $\omega_A^\bullet$ and is not regular. Let $X$ be the blowup of $\Spec(A)$ in $\mathfrak m$ with exceptional divisor $E \subset X$. Let $\omega_X$ be the dualizing module of $X$. Then \begin{enumerate} \item $\omega_E = \omega_X|_E \otimes \mathcal{O}_E(-1)$, \item $H^1(X, \omega_X(n)) = 0$ for $n \geq 0$, \item the map $f^*\omega_A \to \omega_X$ of Lemma \ref{lemma-dualizing-rational} is surjective. \end{enumerate}


resolve-lemma-rational-to-gorenstein

Let $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain of dimension $2$ which defines a rational singularity. Assume $A$ has a dualizing complex. Then there exists a finite sequence of blowups in singular closed points $$ X = X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \Spec(A) $$ such that $X_i$ is normal for each $i$ and such that the dualizing sheaf $\omega_X$ of $X$ is an invertible $\mathcal{O}_X$-module.


resolve-lemma-sequence-blowups-along-arc-becomes-nonsingular

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local domain of dimension $2$. Let $A \to R$ be a surjection onto a complete discrete valuation ring. This defines a nonsingular arc $a : T = \Spec(R) \to \Spec(A)$. Let $$ \Spec(A) = X_0 \leftarrow X_1 \leftarrow X_2 \leftarrow X_3 \leftarrow \ldots $$ be the sequence of blowing ups constructed from $a$. If $A_\mathfrak p$ is a regular local ring where $\mathfrak p = \Ker(A \to R)$, then for some $i$ the scheme $X_i$ is regular at $x_i$.


resolve-lemma-iso-completions

Let $(A, \mathfrak m, \kappa)$ be a local ring with finitely generated maximal ideal $\mathfrak m$. Let $X$ be a scheme over $A$. Let $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$ where $A^\wedge$ is the $\mathfrak m$-adic completion of $A$. For a point $q \in Y$ with image $p \in X$ lying over the closed point of $\Spec(A)$ the local ring map $\mathcal{O}_{X, p} \to \mathcal{O}_{Y, q}$ induces an isomorphism on completions.


resolve-lemma-port-regularity-to-completion

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $X \to \Spec(A)$ be a morphism which is locally of finite type. Set $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$. Let $y \in Y$ with image $x \in X$. Then \begin{enumerate} \item if $\mathcal{O}_{Y, y}$ is regular, then $\mathcal{O}_{X, x}$ is regular, \item if $y$ is in the closed fibre, then $\mathcal{O}_{Y, y}$ is regular $\Leftrightarrow \mathcal{O}_{X, x}$ is regular, and \item If $X$ is proper over $A$, then $X$ is regular if and only if $Y$ is regular. \end{enumerate}


resolve-lemma-descend-admissible-blowup

Let $(A, \mathfrak m)$ be a Noetherian local ring with completion $A^\wedge$. Let $U \subset \Spec(A)$ and $U^\wedge \subset \Spec(A^\wedge)$ be the punctured spectra. If $Y \to \Spec(A^\wedge)$ is a $U^\wedge$-admissible blowup, then there exists a $U$-admissible blowup $X \to \Spec(A)$ such that $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$.


resolve-lemma-formally-unramified

Let $(A, \mathfrak m)$ be a local Noetherian ring. Let $X$ be a scheme over $A$. Assume \begin{enumerate} \item $A$ is analytically unramified (Algebra, Definition \ref{algebra-definition-analytically-unramified}), \item $X$ is locally of finite type over $A$, and \item $X \to \Spec(A)$ is \'etale at the generic points of irreducible components of $X$. \end{enumerate} Then the normalization of $X$ is finite over $X$.


resolve-lemma-normalization-completion

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $X \to \Spec(A)$ be a morphism which is locally of finite type. Set $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$. If the complement of the special fibre in $Y$ is normal, then the normalization $X^\nu \to X$ is finite and the base change of $X^\nu$ to $\Spec(A^\wedge)$ recovers the normalization of $Y$.


resolve-lemma-normalized-blowup-completion

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local domain whose completion $A^\wedge$ is normal. Then given any sequence $$ Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to \Spec(A^\wedge) $$ of normalized blowups, there exists a sequence of (proper) normalized blowups $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to \Spec(A) $$ whose base change to $A^\wedge$ recovers the given sequence.


resolve-lemma-resolve-rational-double-points

Let $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain of dimension $2$ which defines a rational singularity, whose completion is normal, and which is Gorenstein. Then there exists a finite sequence of blowups in singular closed points $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \Spec(A) $$ such that $X_n$ is regular and such that each intervening schemes $X_i$ is normal with finitely many singular points of the same type.


resolve-lemma-regular-alteration-implies

Let $Y$ be a Noetherian integral scheme. Assume there exists an alteration $f : X \to Y$ with $X$ regular. Then the normalization $Y^\nu \to Y$ is finite and $Y$ has a dense open which is regular.


resolve-lemma-algebra-helper

Let $(A, \mathfrak m)$ be a local Noetherian ring. Let $B \subset C$ be finite $A$-algebras. Assume that (a) $B$ is a normal ring, and (b) the $\mathfrak m$-adic completion $C^\wedge$ is a normal ring. Then $B^\wedge$ is a normal ring.


resolve-lemma-regular-alteration-implies-local

Let $(A, \mathfrak m, \kappa)$ be a local Noetherian domain. Assume there exists an alteration $f : X \to \Spec(A)$ with $X$ regular. Then \begin{enumerate} \item there exists a nonzero $f \in A$ such that $A_f$ is regular, \item the integral closure $B$ of $A$ in its fraction field is finite over $A$, \item the $\mathfrak m$-adic completion of $B$ is a normal ring, i.e., the completions of $B$ at its maximal ideals are normal domains, and \item the generic formal fibre of $A$ is regular. \end{enumerate}


resolve-lemma-existence-implies-existence-by-normalized-blowing-ups

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Assume $A$ is normal and has dimension $2$. If $\Spec(A)$ has a resolution of singularities, then $\Spec(A)$ has a resolution by normalized blowups.


resolve-lemma-resolve-complete

Let $(A, \mathfrak m, \kappa)$ be a Noetherian complete local ring. Assume $A$ is a normal domain of dimension $2$. Then $\Spec(A)$ has a resolution of singularities.


resolve-lemma-resolve-curve

Let $Y$ be a one dimensional integral Noetherian scheme. The following are equivalent \begin{enumerate} \item there exists an alteration $X \to Y$ with $X$ regular, \item there exists a resolution of singularities of $Y$, \item there exists a finite sequence $Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to Y$ of blowups in closed points with $Y_n$ regular, and \item the normalization $Y^\nu \to Y$ is finite. \end{enumerate}


resolve-lemma-blowup-curve

Let $X$ be a Noetherian scheme. Let $Y \subset X$ be an integral closed subscheme of dimension $1$ satisfying the equivalent conditions of Lemma \ref{lemma-resolve-curve}. Then there exists a finite sequence $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to X $$ of blowups in closed points such that the strict transform of $Y$ in $X_n$ is a regular curve.


resolve-lemma-blowup-nonsingular-curves-meeting-at-point

In the situation above let $X' \to X$ be the blowing up of $X$ in $p$. Let $Y', Z' \subset X'$ be the strict transforms of $Y, Z$. If $\mathcal{O}_{Y, p}$ is regular, then \begin{enumerate} \item $Y' \to Y$ is an isomorphism, \item $Y'$ meets the exceptional fibre $E \subset X'$ in one point $q$ and $m_q(Y \cap E) = 1$, \item if $q \in Z'$ too, then $m_q(Y \cap Z') < m_p(Y \cap Z)$. \end{enumerate}


resolve-lemma-blowup-curves

Let $X$ be a Noetherian scheme. Let $Y_i \subset X$, $i = 1, \ldots, n$ be an integral closed subschemes of dimension $1$ each satisfying the equivalent conditions of Lemma \ref{lemma-resolve-curve}. Then there exists a finite sequence $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to X $$ of blowups in closed points such that the strict transform $Y'_i \subset X_n$ of $Y_i$ in $X_n$ are pairwise disjoint regular curves.


resolve-lemma-turn-into-effective-Cartier

Let $X$ be a regular scheme of dimension $2$. Let $Z \subset X$ be a proper closed subscheme. There exists a sequence $$ X_n \to \ldots \to X_1 \to X $$ of blowing ups in closed points such that the inverse image $Z_n$ of $Z$ in $X_n$ is an effective Cartier divisor.


resolve-lemma-factor-through-contraction

Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. If a contraction $X \to X'$ of $E$ exists, then it has the following universal property: for every morphism $\varphi : X \to Y$ such that $\varphi(E)$ is a point, there is a unique factorization $X \to X' \to Y$ of $\varphi$.


resolve-lemma-contraction-unique

Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. If there exists a contraction of $E$, then it is unique up to unique isomorphism.


resolve-lemma-exceptional-first-kind-local

Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. Let $E_n = nE$ and denote $\mathcal{O}_n$ its structure sheaf. Then $$ A = \lim H^0(E_n, \mathcal{O}_n) $$ is a complete local Noetherian regular local ring of dimension $2$ and $\Ker(A \to H^0(E_n, \mathcal{O}_n))$ is the $n$th power of its maximal ideal.


resolve-lemma-contraction

Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. If there exists a morphism $f : X \to Y$ such that \begin{enumerate} \item $Y$ is Noetherian, \item $f$ is proper, \item $f$ maps $E$ to a point $y$ of $Y$, \item $f$ is quasi-finite at every point not in $E$, \end{enumerate} Then there exists a contraction of $E$ and it is the Stein factorization of $f$.


resolve-lemma-pic-blowup

Let $b : X \to X'$ be the contraction of an exceptional curve of the first kind $E \subset X$. Then there is a short exact sequence $$ 0 \to \Pic(X') \to \Pic(X) \to \mathbf{Z} \to 0 $$ where the first map is pullback by $b$ and the second map sends $\mathcal{L}$ to the degree of $\mathcal{L}$ on the exceptional curve $E$. The sequence is split by the map $n \mapsto \mathcal{O}_X(-nE)$.


resolve-lemma-lift-sections-and-h1

Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $n$ be the integer such that $\mathcal{L}|_E$ has degree $n$ viewed as an invertible module on $\mathbf{P}^1$. Then \begin{enumerate} \item If $H^1(X, \mathcal{L}) = 0$ and $n \geq 0$, then $H^1(X, \mathcal{L}(iE)) = 0$ for $0 \leq i \leq n + 1$. \item If $n \leq 0$, then $H^1(X, \mathcal{L}) \subset H^1(X, \mathcal{L}(E))$. \end{enumerate}


resolve-lemma-contract-ample

Let $S = \Spec(R)$ be an affine Noetherian scheme. Let $X \to S$ be a proper morphism. Let $\mathcal{L}$ be an ample invertible sheaf on $X$. Let $E \subset X$ be an exceptional curve of the first kind. Then \begin{enumerate} \item there exists a contraction $b : X \to X'$ of $E$, \item $X'$ is proper over $S$, and \item the invertible $\mathcal{O}_{X'}$-module $\mathcal{L}'$ is ample with $\mathcal{L}'$ as in Remark \ref{remark-pic-blowup}. \end{enumerate}


resolve-lemma-contract-when-quasi-projective

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of finite type. Let $E \subset X$ be an exceptional curve of the first kind which is in a fibre of $f$. \begin{enumerate} \item If $X$ is projective over $S$, then there exists a contraction $X \to X'$ of $E$ and $X'$ is projective over $S$. \item If $X$ is quasi-projective over $S$, then there exists a contraction $X \to X'$ of $E$ and $X'$ is quasi-projective over $S$. \end{enumerate}


resolve-lemma-regular-dim-2-quasi-projective

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a separated morphism of finite type with $X$ regular of dimension $2$. Then $X$ is quasi-projective over $S$.


resolve-lemma-regular-dim-2-projective

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a proper morphism with $X$ regular of dimension $2$. Then $X$ is projective over $S$.


resolve-lemma-proper-birational-regular-surfaces

Let $f : X \to Y$ be a proper birational morphism between integral Noetherian schemes regular of dimension $2$. Then $f$ is a sequence of blowups in closed points.


resolve-lemma-birational-regular-surfaces

Let $S$ be a Noetherian scheme. Let $X$ and $Y$ be proper integral schemes over $S$ which are regular of dimension $2$. Then $X$ and $Y$ are $S$-birational if and only if there exists a diagram of $S$-morphisms $$ X = X_0 \leftarrow X_1 \leftarrow \ldots \leftarrow X_n = Y_m \to \ldots \to Y_1 \to Y_0 = Y $$ where each morphism is a blowup in a closed point.


resolve-proposition-Grauert-Riemenschneider

In Situation \ref{situation-vanishing} assume \begin{enumerate} \item $X$ is a normal scheme, \item $A$ is Nagata and has a dualizing complex $\omega_A^\bullet$. \end{enumerate} Let $\omega_X$ be the dualizing module of $X$ (Remark \ref{remark-dualizing-setup}). Then $R^1f_*\omega_X = 0$.


spaces-duality-lemma-equivalent-definitions

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $K$ be an object of $D_\QCoh(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item For every \'etale morphism $U \to X$ where $U$ is a scheme the restriction $K|_U$ is a dualizing complex for $U$ (as discussed above). \item There exists a surjective \'etale morphism $U \to X$ where $U$ is a scheme such that $K|_U$ is a dualizing complex for $U$. \end{enumerate}


spaces-duality-lemma-affine-duality

Let $A$ be a Noetherian ring and let $X = \Spec(A)$. Let $\mathcal{O}_\etale$ be the structure sheaf of $X$ on the small \'etale site of $X$. Let $K, L$ be objects of $D(A)$. If $K \in D_{\textit{Coh}}(A)$ and $L$ has finite injective dimension, then $$ \epsilon^*\widetilde{R\Hom_A(K, L)} = R\SheafHom_{\mathcal{O}_\etale}(\epsilon^*\widetilde{K}, \epsilon^*\widetilde{L}) $$ in $D(\mathcal{O}_\etale)$ where $\epsilon : (X_\etale, \mathcal{O}_\etale) \to (X, \mathcal{O}_X)$ is as in Derived Categories of Spaces, Section \ref{spaces-perfect-section-derived-quasi-coherent-etale}.


spaces-duality-lemma-twisted-inverse-image

\begin{reference} This is almost the same as \cite[Example 4.2]{Neeman-Grothendieck}. \end{reference} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism between quasi-separated and quasi-compact algebraic spaces over $S$. The functor $Rf_* : D_\QCoh(X) \to D_\QCoh(Y)$ has a right adjoint.


spaces-duality-lemma-twisted-inverse-image-bounded-below

Notation and assumptions as in Lemma \ref{lemma-twisted-inverse-image}. Let $a : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)$ be the right adjoint to $Rf_*$. Then $a$ maps $D^+_\QCoh(\mathcal{O}_Y)$ into $D^+_\QCoh(\mathcal{O}_X)$. In fact, there exists an integer $N$ such that $H^i(K) = 0$ for $i \leq c$ implies $H^i(a(K)) = 0$ for $i \leq c - N$.


spaces-duality-lemma-iso-on-RSheafHom

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Let $a$ be the right adjoint to $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$. Let $L \in D_\QCoh(\mathcal{O}_X)$ and $K \in D_\QCoh(\mathcal{O}_Y)$. Then the map (\ref{equation-sheafy-trace}) $$ Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K)) \longrightarrow R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K) $$ becomes an isomorphism after applying the functor $DQ_Y : D(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_Y)$ discussed in Derived Categories of Spaces, Section \ref{spaces-perfect-section-better-coherator}.


spaces-duality-lemma-iso-global-hom

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of quasi-separated and quasi-compact algebraic spaces over $S$. For all $L \in D_\QCoh(\mathcal{O}_X)$ and $K \in D_\QCoh(\mathcal{O}_Y)$ (\ref{equation-sheafy-trace}) induces an isomorphism $R\Hom_X(L, a(K)) \to R\Hom_Y(Rf_*L, K)$ of global derived homs.


spaces-duality-lemma-flat-precompose-pus

In diagram (\ref{equation-base-change}) the map $a \circ Rg_* \leftarrow Rg'_* \circ a'$ is an isomorphism.


spaces-duality-lemma-compose-base-change-maps

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ X' \ar[r]_k \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^l \ar[d]_{g'} & Y \ar[d]^g \\ Z' \ar[r]^m & Z } $$ of quasi-compact and quasi-separated algebraic spaces over $S$ where both diagrams are cartesian and where $f$ and $l$ as well as $g$ and $m$ are Tor independent. Then the maps (\ref{equation-base-change-map}) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement).


spaces-duality-lemma-compose-base-change-maps-horizontal

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ X'' \ar[r]_{g'} \ar[d]_{f''} & X' \ar[r]_g \ar[d]_{f'} & X \ar[d]^f \\ Y'' \ar[r]^{h'} & Y' \ar[r]^h & Y } $$ of quasi-compact and quasi-separated algebraic spaces over $S$ where both diagrams are cartesian and where $f$ and $h$ as well as $f'$ and $h'$ are Tor independent. Then the maps (\ref{equation-base-change-map}) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement).


spaces-duality-lemma-trace-map-and-base-change

Suppose we have a diagram (\ref{equation-base-change}). Then the maps $1 \star \text{Tr}_f : Lg^* \circ Rf_* \circ a \to Lg^*$ and $\text{Tr}_{f'} \star 1 : Rf'_* \circ a' \circ Lg^* \to Lg^*$ agree via the base change maps $\beta : Lg^* \circ Rf_* \to Rf'_* \circ L(g')^*$ (Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}) and $\alpha : L(g')^* \circ a \to a' \circ Lg^*$ (\ref{equation-base-change-map}). More precisely, the diagram $$ \xymatrix{ Lg^* \circ Rf_* \circ a \ar[d]_{\beta \star 1} \ar[r]_-{1 \star \text{Tr}_f} & Lg^* \\ Rf'_* \circ L(g')^* \circ a \ar[r]^{1 \star \alpha} & Rf'_* \circ a' \circ Lg^* \ar[u]_{\text{Tr}_{f'} \star 1} } $$ of transformations of functors commutes.


spaces-duality-lemma-unit-and-base-change

Suppose we have a diagram (\ref{equation-base-change}). Then the maps $1 \star \eta_f : L(g')^* \to L(g')^* \circ a \circ Rf_*$ and $\eta_{f'} \star 1 : L(g')^* \to a' \circ Rf'_* \circ L(g')^*$ agree via the base change maps $\beta : Lg^* \circ Rf_* \to Rf'_* \circ L(g')^*$ (Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}) and $\alpha : L(g')^* \circ a \to a' \circ Lg^*$ (\ref{equation-base-change-map}). More precisely, the diagram $$ \xymatrix{ L(g')^* \ar[r]_-{1 \star \eta_f} \ar[d]_{\eta_{f'} \star 1} & L(g')^* \circ a \circ Rf_* \ar[d]^\alpha \\ a' \circ Rf'_* \circ L(g')^* & a' \circ Lg^* \circ Rf_* \ar[l]_-\beta } $$ of transformations of functors commutes.


spaces-duality-lemma-compare-with-pullback-perfect

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. The map $Lf^*K \otimes^\mathbf{L}_{\mathcal{O}_X} a(L) \to a(K \otimes_{\mathcal{O}_Y}^\mathbf{L} L)$ defined above for $K, L \in D_\QCoh(\mathcal{O}_Y)$ is an isomorphism if $K$ is perfect. In particular, (\ref{equation-compare-with-pullback}) is an isomorphism if $K$ is perfect.


spaces-duality-lemma-proper-flat

Let $S$ be a scheme. Let $Y$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces which is proper, flat, and of finite presentation. Let $a$ be the right adjoint for $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ of Lemma \ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums.


spaces-duality-lemma-properties-relative-dualizing

Let $Y$ be an affine scheme. Let $f : X \to Y$ be a morphism of algebraic spaces which is proper, flat, and of finite presentation. Let $a$ be the right adjoint for $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ of Lemma \ref{lemma-twisted-inverse-image}. Then \begin{enumerate} \item $a(\mathcal{O}_Y)$ is a $Y$-perfect object of $D(\mathcal{O}_X)$, \item $Rf_*a(\mathcal{O}_Y)$ has vanishing cohomology sheaves in positive degrees, \item $\mathcal{O}_X \to R\SheafHom_{\mathcal{O}_X}(a(\mathcal{O}_Y), a(\mathcal{O}_Y))$ is an isomorphism. \end{enumerate}


spaces-duality-lemma-relative-dualizing-RHom

Let $S$ be a scheme. Let $X \to Y$ be a proper, flat morphism of algebraic spaces which is of finite presentation. If $(\omega_{X/Y}^\bullet, \tau)$ is a relative dualizing complex, then  $\mathcal{O}_X \to R\SheafHom_{\mathcal{O}_X}(\omega_{X/Y}^\bullet, \omega_{X/Y}^\bullet)$ is an isomorphism and $Rf_*\omega_{X/Y}^\bullet$ has vanishing cohomology sheaves in positive degrees.


spaces-duality-lemma-uniqueness-relative-dualizing

Let $S$ be a scheme. Let $X \to Y$ be a proper, flat morphism of algebraic spaces which is of finite presentation. If $(\omega_j^\bullet, \tau_j)$, $j = 1, 2$ are two relative dualizing complexes on $X/Y$, then there is a unique isomorphism $(\omega_1^\bullet, \tau_1) \to (\omega_2^\bullet, \tau_2)$.


spaces-duality-lemma-covering-enough

Let $S$ be a scheme. Let $X \to Y$ be a proper, flat morphism of algebraic spaces which is of finite presentation. Let $(\omega^\bullet, \tau)$ be a pair consisting of a $Y$-perfect object of $D(\mathcal{O}_X)$ and a map $\tau : Rf_*\omega^\bullet \to \mathcal{O}_Y$. Assume we have cartesian diagrams $$ \xymatrix{ X_i \ar[r]_{g_i'} \ar[d]_{f_i} & X \ar[d]^f \\ Y_i \ar[r]^{g_i} & Y } $$ with $Y_i$ affine such that $\{g_i : Y_i \to Y\}$ is an \'etale covering and isomorphisms of pairs $(\omega^\bullet|_{X_i}, \tau|_{Y_i}) \to (a_i(\mathcal{O}_{Y_i}), \text{Tr}_{f_i, \mathcal{O}_{Y_i}})$ as in Definition \ref{definition-relative-dualizing-proper-flat}. Then $(\omega^\bullet, \tau)$ is a relative dualizing complex for $X$ over $Y$.


spaces-properties-theorem-exactness-stalks

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A map $a : \mathcal{F} \to \mathcal{G}$ of sheaves of sets is injective (resp.\ surjective) if and only if the map on stalks $a_{\overline{x}} : \mathcal{F}_{\overline{x}} \to \mathcal{G}_{\overline{x}}$ is injective (resp.\ surjective) for all geometric points of $X$. A sequence of abelian sheaves on $X_\etale$ is exact if and only if it is exact on all stalks at geometric points of $S$.


spaces-properties-theorem-fully-faithful

Let $X$, $Y$ be algebraic spaces over $\Spec(\mathbf{Z})$. Let $$ (g, g^\sharp) : (\Sh(X_\etale), \mathcal{O}_X) \longrightarrow (\Sh(Y_\etale), \mathcal{O}_Y) $$ be a morphism of locally ringed topoi. Then there exists a unique morphism of algebraic spaces $f : X \to Y$ such that $(g, g^\sharp)$ is isomorphic to $(f_{small}, f^\sharp)$. In other words, the construction $$ \textit{Spaces}/\Spec(\mathbf{Z}) \longrightarrow \textit{Locally ringed topoi}, \quad X \longrightarrow (X_\etale, \mathcal{O}_X) $$ is fully faithful (morphisms up to $2$-isomorphisms on the right hand side).


spaces-properties-lemma-trivial-implications

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We have the following implications among the separation axioms of Definition \ref{definition-separated}: \begin{enumerate} \item separated implies all the others, \item quasi-separated implies Zariski locally quasi-separated. \end{enumerate}


spaces-properties-lemma-characterize-quasi-separated

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item $X$ is a quasi-separated algebraic space, \item for $U \to X$, $V \to X$ with $U$, $V$ quasi-compact schemes the fibre product $U \times_X V$ is quasi-compact, \item for $U \to X$, $V \to X$ with $U$, $V$ affine the fibre product $U \times_X V$ is quasi-compact. \end{enumerate}


spaces-properties-lemma-scheme-points

Let $S$ be a scheme. Let $X$ be a scheme over $S$. The points of $X$ as a scheme are in canonical 1-1 correspondence with the points of $X$ as an algebraic space.


spaces-properties-lemma-points-cartesian

Let $S$ be a scheme. Let $$ \xymatrix{ Z \times_Y X \ar[r] \ar[d] & X \ar[d] \\ Z \ar[r] & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Then the map of sets of points $$ |Z \times_Y X| \longrightarrow |Z| \times_{|Y|} |X| $$ is surjective.


spaces-properties-lemma-characterize-surjective

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f : T \to X$ be a morphism from a scheme to $X$. The following are equivalent \begin{enumerate} \item $f : T \to X$ is surjective (according to Spaces, Definition \ref{spaces-definition-relative-representable-property}), and \item $|f| : |T| \to |X|$ is surjective. \end{enumerate}


spaces-properties-lemma-points-presentation

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X = U/R$ be a presentation of $X$, see Spaces, Definition \ref{spaces-definition-presentation}. Then the image of $|R| \to |U| \times |U|$ is an equivalence relation and $|X|$ is the quotient of $|U|$ by this equivalence relation.


spaces-properties-lemma-topology-points

Let $S$ be a scheme. There exists a unique topology on the sets of points of algebraic spaces over $S$ with the following properties: \begin{enumerate} \item if $X$ is a scheme over $S$, then the topology on $|X|$ is the usual one (via the identification of Lemma \ref{lemma-scheme-points}), \item for every morphism of algebraic spaces $X \to Y$ over $S$ the map $|X| \to |Y|$ is continuous, and \item for every \'etale morphism $U \to X$ with $U$ a scheme the map of topological spaces $|U| \to |X|$ is continuous and open. \end{enumerate}


spaces-properties-lemma-open-subspaces

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item The rule $X' \mapsto |X'|$ defines an inclusion preserving bijection between open subspaces $X'$ (see Spaces, Definition \ref{spaces-definition-immersion}) of $X$, and opens of the topological space $|X|$. \item A family $\{X_i \subset X\}_{i \in I}$ of open subspaces of $X$ is a Zariski covering (see Spaces, Definition \ref{spaces-definition-Zariski-open-covering}) if and only if $|X| = \bigcup |X_i|$. \end{enumerate} In other words, the small Zariski site $X_{Zar}$ of $X$ is canonically identified with a site associated to the topological space $|X|$ (see Sites, Example \ref{sites-example-site-topological}).


spaces-properties-lemma-factor-through-open-subspace

Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $X' \subset X$ be an open subspace. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Then $f$ factors through $X'$ if and only if $|f| : |Y| \to |X|$ factors through $|X'| \subset |X|$.


spaces-properties-lemma-etale-image-open

Let $S$ be a scheme. Let $X$ be an algebraic spaces over $S$. Let $U$ be a scheme and let $f : U \to X$ be an \'etale morphism. Let $X' \subset X$ be the open subspace corresponding to the open $|f|(|U|) \subset |X|$ via Lemma \ref{lemma-open-subspaces}. Then $f$ factors through a surjective \'etale morphism $f' : U \to X'$. Moreover, if $R = U \times_X U$, then $R = U \times_{X'} U$ and $X'$ has the presentation $X' = U/R$.


spaces-properties-lemma-points-monomorphism

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the map $$ \{\Spec(k) \to X \text{ monomorphism where }k\text{ is a field}\} \longrightarrow |X| $$ This map is injective.


spaces-properties-lemma-quasi-compact-space

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is quasi-compact if and only if $|X|$ is quasi-compact.


spaces-properties-lemma-space-locally-quasi-compact

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Every point of $|X|$ has a fundamental system of open quasi-compact neighbourhoods. In particular $|X|$ is locally quasi-compact in the sense of Topology, Definition \ref{topology-definition-locally-quasi-compact}.


spaces-properties-lemma-cover-by-union-affines

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a surjective \'etale morphism $U \to X$ where $U$ is a disjoint union of affine schemes. We may in addition assume each of these affines maps into an affine open of $S$.


spaces-properties-lemma-union-of-quasi-compact

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a Zariski covering $X = \bigcup X_i$ such that each algebraic space $X_i$ has a surjective \'etale covering by an affine scheme. We may in addition assume each $X_i$ maps into an affine open of $S$.


spaces-properties-lemma-quasi-compact-affine-cover

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is quasi-compact if and only if there exists an \'etale surjective morphism $U \to X$ with $U$ an affine scheme.


spaces-properties-lemma-separated-cover

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a separated scheme and $U \to X$ \'etale. Then $U \to X$ is separated, and $R = U \times_X U$ is a separated scheme.


spaces-properties-lemma-quasi-separated

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a quasi-separated scheme $U$ and a surjective \'etale morphism $U \to X$ such that either of the projections $U \times_X U \to U$ is quasi-compact, then $X$ is quasi-separated.


spaces-properties-lemma-quasi-separated-quasi-compact-pieces

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item $X$ is Zariski locally quasi-separated over $S$, \item $X$ is Zariski locally quasi-separated, \item there exists a Zariski open covering $X = \bigcup X_i$ such that for each $i$ there exists an affine scheme $U_i$ and a quasi-compact surjective \'etale morphism $U_i \to X_i$, and \item there exists a Zariski open covering $X = \bigcup X_i$ such that for each $i$ there exists an affine scheme $U_i$ which maps into an affine open of $S$ and a quasi-compact surjective \'etale morphism $U_i \to X_i$. \end{enumerate}


spaces-properties-lemma-finite-fibres-presentation

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a scheme. Let $\varphi : U \to X$ be an \'etale morphism such that the projections $R = U \times_X U \to U$ are quasi-compact; for example if $\varphi$ is quasi-compact. Then the fibres of $$ |U| \to |X| \quad\text{and}\quad |R| \to |X| $$ are finite.


spaces-properties-lemma-type-property

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{P}$ be a property of schemes which is local in the \'etale topology, see Descent, Definition \ref{descent-definition-property-local}. The following are equivalent \begin{enumerate} \item for some scheme $U$ and surjective \'etale morphism $U \to X$ the scheme $U$ has property $\mathcal{P}$, and \item for every scheme $U$ and every \'etale morphism $U \to X$ the scheme $U$ has property $\mathcal{P}$. \end{enumerate} If $X$ is representable this is equivalent to $\mathcal{P}(X)$.


spaces-properties-lemma-local-source-target-at-point

Let $\mathcal{P}$ be a property of germs of schemes which is \'etale local, see Descent, Definition \ref{descent-definition-local-at-point}. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point of $X$. Consider \'etale morphisms $a : U \to X$ where $U$ is a scheme. The following are equivalent \begin{enumerate} \item for any $U \to X$ as above and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$, and \item for some $U \to X$ as above and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$. \end{enumerate} If $X$ is representable, then this is equivalent to $\mathcal{P}(X, x)$.


spaces-properties-lemma-locally-constructible

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \subset |X|$ be a subset. The following are equivalent \begin{enumerate} \item for every \'etale morphism $U \to X$ where $U$ is a scheme the inverse image of $E$ in $U$ is a locally constructible subset of $U$, \item for every \'etale morphism $U \to X$ where $U$ is an affine scheme the inverse image of $E$ in $U$ is a constructible subset of $U$, \item for some surjective \'etale morphism $U \to X$ where $U$ is a scheme the inverse image of $E$ in $U$ is a locally constructible subset of $U$. \end{enumerate}


spaces-properties-lemma-pre-dimension-local-ring

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. Let $d \in \{0, 1, 2, \ldots, \infty\}$. The following are equivalent \begin{enumerate} \item for some scheme $U$ and \'etale morphism $a : U \to X$ and point $u \in U$ with $a(u) = x$ we have $\dim(\mathcal{O}_{U, u}) = d$, \item for any scheme $U$, any \'etale morphism $a : U \to X$, and any point $u \in U$ with $a(u) = x$ we have $\dim(\mathcal{O}_{U, u}) = d$. \end{enumerate} If $X$ is a scheme, this is equivalent to $\dim(\mathcal{O}_{X, x}) = d$.


spaces-properties-lemma-dimension

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following quantities are equal: \begin{enumerate} \item The dimension of $X$. \item The supremum of the dimensions of the local rings of $X$. \item The supremum of $\dim_x(X)$ for $x \in |X|$. \end{enumerate}