obsolete-lemma-bound-degree-in-nbhd-generic-point

Let $f : X \to Y$ be a morphism schemes. Assume \begin{enumerate} \item $X$ and $Y$ are integral schemes, \item $f$ is locally of finite type and dominant, \item $f$ is either quasi-compact or separated, \item $f$ is generically finite, i.e., one of (1) -- (5) of Morphisms, Lemma \ref{morphisms-lemma-finite-degree} holds. \end{enumerate} Then there is a nonempty open $V \subset Y$ such that $f^{-1}(V) \to V$ is finite locally free of degree $\deg(X/Y)$. In particular, the degrees of the fibres of $f^{-1}(V) \to V$ are bounded by $\deg(X/Y)$.


obsolete-lemma-factors-through-quotient

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


obsolete-lemma-sheaf-fpqc-open-covering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X = \bigcup_{j \in J} X_j$ be a Zariski covering, see Spaces, Definition \ref{spaces-definition-Zariski-open-covering}. If each $X_j$ satisfies the sheaf property for the fpqc topology then $X$ satisfies the sheaf property for the fpqc topology.


obsolete-lemma-reasonable-kolmogorov

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


obsolete-lemma-very-reasonable-Zariski-local

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a Zariski open covering $X = \bigcup X_i$ such that each $X_i$ is very reasonable, then $X$ is very reasonable.


obsolete-lemma-representable-very-reasonable

Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $Y \to X$ be a representable morphism. If $X$ is very reasonable, so is $Y$.


obsolete-lemma-very-reasonable-quasi-compact-pieces

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


obsolete-lemma-vanishing-surjective

In Cohomology of Spaces, Situation \ref{spaces-cohomology-situation-vanishing} the morphism $p : X \to \Spec(A)$ is surjective.


obsolete-lemma-vanishing-universally-closed

In Cohomology of Spaces, Situation \ref{spaces-cohomology-situation-vanishing} the morphism $p : X \to \Spec(A)$ is universally closed.


obsolete-lemma-blowing-up-denominators

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 with $\dim_\delta(X) = n$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. Let $U \subset X$ be the maximal open subscheme such that $s$ corresponds to a section of $\mathcal{L}$ over $U$. There exists a projective morphism $$ \pi : X' \longrightarrow X $$ such that \begin{enumerate} \item $X'$ is integral, \item $\pi|_{\pi^{-1}(U)} : \pi^{-1}(U) \to U$ is an isomorphism, \item there exist effective Cartier divisors $D, E \subset X'$ such that $$ \pi^*\mathcal{L} = \mathcal{O}_{X'}(D - E), $$ \item the meromorphic section $s$ corresponds, via the isomorphism above, to the meromorphic section $1_D \otimes (1_E)^{-1}$ (see Divisors, Definition \ref{divisors-definition-invertible-sheaf-effective-Cartier-divisor}), \item we have $$ \pi_*([D]_{n - 1} - [E]_{n - 1}) = \text{div}_\mathcal{L}(s) $$ in $Z_{n - 1}(X)$. \end{enumerate}


obsolete-lemma-directed-colimit-finite-type

Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is the directed colimit of its finite type quasi-coherent submodules.


obsolete-lemma-points-monomorphism

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


obsolete-lemma-locally-ringed-space-direct-summand-free

Let $X$ be a locally ringed space. A direct summand of a finite free $\mathcal{O}_X$-module is finite locally free.


obsolete-lemma-characterize-injective

Let $R$ be a ring. Let $E$ be an $R$-module. The following are equivalent \begin{enumerate} \item $E$ is an injective $R$-module, and \item given an ideal $I \subset R$ and a module map $\varphi : I \to E$ there exists an extension of $\varphi$ to an $R$-module map $R \to E$. \end{enumerate}


obsolete-lemma-periodic-length

Let $R$ be a local ring. \begin{enumerate} \item If $(M, N, \varphi, \psi)$ is a $2$-periodic complex such that $M$, $N$ have finite length. Then $e_R(M, N, \varphi, \psi) = \text{length}_R(M) - \text{length}_R(N)$. \item If $(M, \varphi, \psi)$ is a $(2, 1)$-periodic complex such that $M$ has finite length. Then $e_R(M, \varphi, \psi) = 0$. \item Suppose that we have a short exact sequence of $2$-periodic complexes $$ 0 \to (M_1, N_1, \varphi_1, \psi_1) \to (M_2, N_2, \varphi_2, \psi_2) \to (M_3, N_3, \varphi_3, \psi_3) \to 0 $$ If two out of three have cohomology modules of finite length so does the third and we have $$ e_R(M_2, N_2, \varphi_2, \psi_2) = e_R(M_1, N_1, \varphi_1, \psi_1) + e_R(M_3, N_3, \varphi_3, \psi_3). $$ \end{enumerate}


obsolete-lemma-examples-have-RS

Deformation Problems, Examples \ref{examples-defos-example-finite-projective-modules}, \ref{examples-defos-example-representations}, \ref{examples-defos-example-continuous-representations}, and \ref{examples-defos-example-graded-algebras} satisfy the Rim-Schlessinger condition (RS).


stacks-sheaves-lemma-enough-points

Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. The site $\mathcal{X}_\tau$ has enough points.


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

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 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 of $\Gamma(\mathcal{O}_\mathcal{X})$-modules $$ E_1^{p, q} = H^q((\mathcal{U}_p)_\tau, f_p^*\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}_\tau, \mathcal{F}) $$ converging to the cohomology of $\mathcal{F}$ in the $\tau$-topology.


stacks-sheaves-proposition-coherator

Let $\mathcal{X}$ be an algebraic stack over $S$. \begin{enumerate} \item The category $\QCoh(\mathcal{O}_\mathcal{X})$ is a Grothendieck abelian category. Consequently, $\QCoh(\mathcal{O}_\mathcal{X})$ has enough injectives and all limits. \item The inclusion functor $\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})$ has a right adjoint\footnote{This functor is sometimes called the {\it coherator}.} $$ Q : \textit{Mod}(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(\mathcal{O}_\mathcal{X}) $$ such that for every quasi-coherent sheaf $\mathcal{F}$ the adjunction mapping $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism. \end{enumerate}


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

Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is Noetherian, Nagata, 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 $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 \to X $$ and an $S$-morphism $f_n : X_n \to Y$ such that $X_0 \to X$ is the normalization, $X_{i + 1} \to X_i$ is the normalized blowing up of $X_i$ at a closed point, and $f_n$ and $f$ agree.


resolve-lemma-equivalence

The functor $F$ (\ref{equation-equivalence}) is an equivalence.


resolve-lemma-sequence-blowups

Let $X$ be a locally Noetherian scheme. Let $$ (X, p) = (X_0, p_0) \leftarrow (X_1, p_1) \leftarrow (X_2, p_2) \leftarrow (X_3, p_3) \leftarrow \ldots $$ be a sequence of blowups such that \begin{enumerate} \item $p_i$ is closed, maps to $p_{i - 1}$, and $\kappa(p_i) = \kappa(p_{i - 1})$, \item there exists an $x_1 \in \mathfrak m_p$ whose image in $\mathfrak m_{p_i}$, $i > 0$ defines the exceptional divisor $E_i \subset X_i$. \end{enumerate} Then the sequence is obtained from a nonsingular arc $a : T \to X$ as above.


resolve-lemma-issquare

Let $\kappa$ be a field. Let $I \subset \kappa[x, y]$ be an ideal. Let $$ a + b x + c y + d x^2 + exy + f y^2 \in I^2 $$ for some $a, b, c, d, e, f \in k$ not all zero. If the colength of $I$ in $\kappa[x, y]$ is $> 1$, then $a + b x + c y + d x^2 + exy + f y^2 = j(g + hx + iy)^2$ for some $j, g, h, i \in \kappa$.


spaces-duality-lemma-dualizing-unique-spaces

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. If $K$ and $K'$ are dualizing complexes on $X$, then $K'$ is isomorphic to $K \otimes_{\mathcal{O}_X}^\mathbf{L} L$ for some invertible object $L$ of $D(\mathcal{O}_X)$.


spaces-duality-lemma-dimension-function-scheme

Let $S$ be a scheme. Let $X$ be a locally Noetherian quasi-separated algebraic space over $S$. Let $\omega_X^\bullet$ be a dualizing complex on $X$. Then $X$ the function $|X| \to \mathbf{Z}$ defined by $$ x \longmapsto \delta(x)\text{ such that } \omega_{X, \overline{x}}^\bullet[-\delta(x)] \text{ is a normalized dualizing complex over } \mathcal{O}_{X, \overline{x}} $$ is a dimension function on $|X|$.


spaces-duality-lemma-compare-with-pullback-flat-proper

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. The map (\ref{equation-compare-with-pullback}) is an isomorphism for every object $K$ of $D_\QCoh(\mathcal{O}_Y)$.


spaces-duality-lemma-existence-relative-dualizing

Let $S$ be a scheme. Let $X \to Y$ be a proper, flat morphism of algebraic spaces which is of finite presentation. There exists a relative dualizing complex $(\omega_{X/Y}^\bullet, \tau)$.


spaces-properties-lemma-characterize-separated

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item $X$ is a separated algebraic space, \item for $U \to X$, $V \to X$ with $U$, $V$ affine the fibre product $U \times_X V$ is affine and $$ \mathcal{O}(U) \otimes_\mathbf{Z} \mathcal{O}(V) \longrightarrow \mathcal{O}(U \times_X V) $$ is surjective. \end{enumerate}


spaces-properties-lemma-finite-disjoint-quasi-compact

A finite disjoint union of quasi-compact algebraic spaces is a quasi-compact algebraic space.


spaces-properties-lemma-reduced-local-ring

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. The following are equivalent \begin{enumerate} \item the local ring of $X$ at $x$ is reduced (Remark \ref{remark-list-properties-local-ring-local-etale-topology}), \item $\mathcal{O}_{X, \overline{x}}$ is reduced for some geometric point $\overline{x}$ lying over $x$, and \item $\mathcal{O}_{X, \overline{x}}$ is reduced for any geometric point $\overline{x}$ lying over $x$. \end{enumerate}


spaces-properties-lemma-regular-at-x

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. The following are equivalent \begin{enumerate} \item $X$ is regular at $x$, and \item the \'etale local ring $\mathcal{O}_{X, \overline{x}}$ is regular for any (equivalently some) geometric point $\overline{x}$ lying over $x$. \end{enumerate}


intersection-lemma-compute-tor-nonsingular

Let $X$ be a nonsingular variety. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. The $\mathcal{O}_X$-module $\text{Tor}_p^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is coherent, has stalk at $x$ equal to $\text{Tor}_p^{\mathcal{O}_{X, x}}(\mathcal{F}_x, \mathcal{G}_x)$, is supported on $\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G})$, and is nonzero only for $p \in \{0, \ldots, \dim(X)\}$.


intersection-lemma-exterior-product-rational-equivalence

Let $X$ and $Y$ be varieties. Let $\alpha \in Z_r(X)$ and $\beta \in Z_s(Y)$. If $\alpha \sim_{rat} 0$ or $\beta \sim_{rat} 0$, then $\alpha \times \beta \sim_{rat} 0$.


intersection-lemma-exterior-product

Let $X$ and $Y$ be nonsingular varieties. Let $\alpha \in Z_r(X)$ and $\beta \in Z_s(Y)$. Then \begin{enumerate} \item $\text{pr}_Y^*(\beta) = [X] \times \beta$ and $\text{pr}_X^*(\alpha) = \alpha \times [Y]$, \item $\alpha \times [Y]$ and $[X]\times \beta$ intersect properly on $X\times Y$, and \item we have $\alpha \times \beta = (\alpha \times [Y])\cdot ([X]\times\beta) = pr_Y^*(\alpha) \cdot pr_X^*(\beta)$ in $Z_{r + s}(X \times Y)$. \end{enumerate}


intersection-lemma-flat-pullback-and-intersections

Let $f : X \to Y$ be a flat morphism of nonsingular varieties. Let $\alpha$ be a $r$-cycle on $Y$ and $\beta$ an $s$-cycle on $Y$. Assume that $\alpha$ and $\beta$ intersect properly. Then $f^*\alpha$ and $f^*\beta$ intersect properly and $f^*( \alpha \cdot \beta ) = f^*\alpha \cdot f^*\beta$.


homology-lemma-composition-strict

Let $\mathcal{A}$ be an abelian category. Let $f : A \to B$, $g : B \to C$ be strict morphisms of filtered objects. \begin{enumerate} \item In general the composition $g \circ f$ is not strict. \item If $g$ is injective, then $g \circ f$ is strict. \item If $f$ is surjective, then $g \circ f$ is strict. \end{enumerate}


homology-lemma-fibre-product-filtered

Let $\mathcal{A}$ be an abelian category. Let $A, B, C \in \text{Fil}(\mathcal{A})$. Let $f : B \to A$ and $g : C \to A$ be morphisms. Then there exists a fibre product $$ \xymatrix{ B \times_A C \ar[r]_{g'} \ar[d]_{f'} & B \ar[d]^f \\ C \ar[r]^g & A } $$ in $\text{Fil}(\mathcal{A})$. If $f$ is strict, so is $f'$.


homology-lemma-spectral-sequence-filtered-differential-d1

Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. The spectral sequence $(E_r, d_r)_{r \geq 0}$ associated to $(K, F, d)$ has $$ d_1^p : E_1^p = H(\text{gr}^pK) \longrightarrow H(\text{gr}^{p + 1}K) = E_1^{p + 1} $$ equal to the boundary map in homology associated to the short exact sequence of differential objects $$ 0 \to \text{gr}^{p + 1}K \to F^pK/F^{p + 2}K \to \text{gr}^pK \to 0. $$


homology-lemma-right-resolution-gives-qis

Let $M^\bullet$ be a complex of abelian groups. Let $$ 0 \to M^\bullet \to A_0^\bullet \to A_1^\bullet \to A_2^\bullet \to \ldots $$ be an exact complex of complexes of abelian groups. Set $A^{p, q} = A_p^q$ to obtain a double complex. Then the map $M^\bullet \to \text{Tot}(A^{\bullet, \bullet})$ induced by $M^\bullet \to A_0^\bullet$ is a quasi-isomorphism.


homology-lemma-essentially-constant-into-karoubian

Let $\mathcal{I}$ be a category, let $\mathcal{A}$ be a pre-additive Karoubian category, and let $M : \mathcal{I} \to \mathcal{A}$ be a diagram. \begin{enumerate} \item Assume $\mathcal{I}$ is filtered. The following are equivalent \begin{enumerate} \item $M$ is essentially constant, \item $X = \colim M$ exists and there exists a cofinal filtered subcategory $\mathcal{I}' \subset \mathcal{I}$ and for $i' \in \Ob(\mathcal{I}')$ a direct sum decomposition $M_{i'} = X_{i'} \oplus Z_{i'}$ such that $X_{i'}$ maps isomorphically to $X$ and $Z_{i'}$ to zero in $M_{i''}$ for some $i' \to i''$ in $\mathcal{I}'$. \end{enumerate} \item Assume $\mathcal{I}$ is cofiltered. The following are equivalent \begin{enumerate} \item $M$ is essentially constant, \item $X = \lim M$ exists and there exists an initial cofiltered subcategory $\mathcal{I}' \subset \mathcal{I}$ and for $i' \in \Ob(\mathcal{I}')$ a direct sum decomposition $M_{i'} = X_{i'} \oplus Z_{i'}$ such that $X$ maps isomorphically to $X_{i'}$ and $M_{i''} \to Z_{i'}$ is zero for some $i'' \to i'$ in $\mathcal{I}'$. \end{enumerate} \end{enumerate}


categories-lemma-product-with-connected

Let $\mathcal{I}$ and $\mathcal{J}$ be a categories and denote $p : \mathcal{I} \times \mathcal{J} \to \mathcal{J}$ the projection. If $\mathcal{I}$ is connected, then for a diagram $M : \mathcal{J} \to \mathcal{C}$ the colimit $\colim_\mathcal{J} M$ exists if and only if $\colim_{\mathcal{I} \times \mathcal{J}} M \circ p$ exists and if so these colimits are equal.


categories-lemma-push-outs-coequalizers-exist

Let $\mathcal{C}$ be a category. The following are equivalent: \begin{enumerate} \item Connected finite colimits exist in $\mathcal{C}$. \item Coequalizers and pushouts exist in $\mathcal{C}$. \end{enumerate}


categories-lemma-right-localization-diagram

Let $\mathcal{C}$ be a category. Let $S$ be a right multiplicative system. If $f : X \to Y$, $f' : X' \to Y'$ are two morphisms of $\mathcal{C}$ and if $$ \xymatrix{ Q(X) \ar[d]_{Q(f)} \ar[r]_a & Q(X') \ar[d]^{Q(f')} \\ Q(Y) \ar[r]^b & Q(Y') } $$ is a commutative diagram in $S^{-1}\mathcal{C}$, then there exists a morphism $f'' : X'' \to Y''$ in $\mathcal{C}$ and a commutative diagram $$ \xymatrix{ X \ar[d]_f & X'' \ar[l]^s \ar[d]^{f''} \ar[r]_g & X' \ar[d]^{f'} \\ Y & Y'' \ar[l]_t \ar[r]^h & Y' } $$ in $\mathcal{C}$ with $s, t \in S$ and $a = gs^{-1}$, $b = ht^{-1}$.


categories-lemma-inertia-fibred-groupoids

Let $\mathcal{C}$ be a category. If $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids, then so is the inertia fibred category $\mathcal{I}_\mathcal{S} \to \mathcal{C}$.


categories-lemma-prepare-representable-map-stack-in-groupoids

Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism. Let $G : \mathcal{C}/U \to \mathcal{Y}$ be a $1$-morphism. Then $$ (\mathcal{C}/U) \times_\mathcal{Y} \mathcal{X} \longrightarrow \mathcal{C}/U $$ is a category fibred in groupoids.


categories-lemma-criterion-representable-map-stack-in-groupoids

Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism. Make a choice of pullbacks for $\mathcal{Y}$. Assume \begin{enumerate} \item each functor $F_U : \mathcal{X}_U \longrightarrow \mathcal{Y}_U$ between fibre categories is faithful, and \item for each $U$ and each $y \in \mathcal{Y}_U$ the presheaf $$ (f : V \to U) \longmapsto \{(x, \phi) \mid x \in \mathcal{X}_V, \phi : f^*y \to F(x)\}/\cong $$ is a representable presheaf on $\mathcal{C}/U$. \end{enumerate} Then $F$ is representable.


topologies-lemma-composition

Given schemes $X$, $Y$, $Y$ in $(\Sch/S)_{Zar}$ and morphisms $f : X \to Y$, $g : Y \to Z$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$ and $g_{small} \circ f_{small} = (g \circ f)_{small}$.


topologies-lemma-morphism-big-small-cartesian-diagram

Let $\Sch_{Zar}$ be a big Zariski site. Consider a cartesian diagram $$ \xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^f \\ S' \ar[r]^g & S } $$ in $\Sch_{Zar}$. Then $i_g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}$ and $g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}$.


topologies-lemma-affine-big-site-ph

Let $S$ be a scheme. Let $\Sch_{ph}$ be a big ph site containing $S$. The functor $(\textit{Aff}/S)_{ph} \to (\Sch/S)_{ph}$ is cocontinuous and induces an equivalence of topoi from $\Sh((\textit{Aff}/S)_{ph})$ to $\Sh((\Sch/S)_{ph})$.


topologies-lemma-composition-ph

Given schemes $X$, $Y$, $Y$ in $(\Sch/S)_{ph}$ and morphisms $f : X \to Y$, $g : Y \to Z$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$.


topologies-lemma-fpqc-covering-affines-mapping-in

Let $T$ be a scheme. Let $\{f_i : T_i \to T\}_{i \in I}$ be a family of morphisms of schemes with target $T$. Assume that \begin{enumerate} \item each $f_i$ is flat, and \item every affine scheme $Z$ and morphism $h : Z \to T$ there exists a standard fpqc covering $\{Z_j \to Z\}_{j = 1, \ldots, n}$ which refines the family $\{T_i \times_T Z \to Z\}_{i \in I}$. \end{enumerate} Then $\{f_i : T_i \to T\}_{i \in I}$ is an fpqc covering of $T$.


pic-lemma-hilb-d-limit-preserving

Let $X \to S$ be a morphism of schemes. If $X \to S$ is of finite presentation, then the functor $\Hilbfunctor^d_{X/S}$ is limit preserving (Limits, Remark \ref{limits-remark-limit-preserving}).


constructions-lemma-gkn-representable

Let $0 < k < n$. The functor $G(k, n)$ of (\ref{equation-gkn}) is representable by a scheme.


algebraization-lemma-restriction-derived-complete

Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}'$ be a homomorphism of sheaves of rings. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. The inverse image of $D_{comp}(\mathcal{O}, \mathcal{I})$ under the restriction functor $D(\mathcal{O}') \to D(\mathcal{O})$ is $D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}')$.


algebraization-lemma-restriction-derived-complete-equivalence

Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}'$ be a homomorphism of sheaves of rings. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. If $\mathcal{O} \to \mathcal{O}'$ is flat and $\mathcal{O}/\mathcal{I} \cong \mathcal{O}'/\mathcal{I}\mathcal{O}'$, then the restriction functor $D(\mathcal{O}') \to D(\mathcal{O})$ induces an equivalence $D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}') \to D_{comp}(\mathcal{O}, \mathcal{I})$.


algebraization-lemma-fully-faithful-very-general

Let $I \subset \mathfrak a \subset A$ be ideals of a Noetherian ring $A$. Let $U = \Spec(A) \setminus V(\mathfrak a)$. Let $\mathcal{V}$ be the set of open subschemes of $U$ containing $U \cap V(I)$ ordered by reverse inclusion. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent $\mathcal{O}_V$-modules for some $V \in \mathcal{V}$. The map $$ \colim_{V' \geq V} \Hom_V(\mathcal{G}|_{V'}, \mathcal{F}|_{V'}) \longrightarrow \Hom_{\textit{Coh}(U, I\mathcal{O}_U)}(\mathcal{G}^\wedge, \mathcal{F}^\wedge) $$ is bijective if the following assumptions hold: \begin{enumerate} \item $A$ is $I$-adically complete and has a dualizing complex, \item if $x \in \text{Ass}(\mathcal{F})$, $x \not \in V(I)$, $\overline{\{x\}} \cap V(I) \not \subset V(\mathfrak a)$ and $z \in \overline{\{x\}} \cap V(\mathfrak a)$, then $\dim(\mathcal{O}_{\overline{\{x\}}, z}) > \text{cd}(A, I) + 1$. \end{enumerate}


algebraization-lemma-when-ML

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. If the inverse system $H^0(U, \mathcal{F}_n)$ has Mittag-Leffler, then the canonical maps $$ \widetilde{M/I^nM}|_U \to \mathcal{F}_n $$ are surjective for all $n$ where $M$ is as in (\ref{equation-guess}).


algebraization-lemma-construct-unique

In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$ and $d \geq 1$. Assume \begin{enumerate} \item $A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$, \item $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module, \item $(\mathcal{F}_n)$ satisfies the strict $(1, 1 + d)$-inequalities. \end{enumerate} Then there exists a unique coherent $\mathcal{O}_U$-module $\mathcal{F}$ whose completion is $(\mathcal{F}_n)$ such that for $x \in U$ with $\overline{\{x\}} \cap Y \subset Z$ we have $\text{depth}(\mathcal{F}_x) \geq 2$.


algebraization-lemma-injective-torsion-in-pic

Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$ be a nonzerodivisor and assume that $\text{depth}(A/fA) \geq 2$, or equivalently $\text{depth}(A) \geq 3$. Let $U$, resp.\ $U_0$ be the punctured spectrum of $A$, resp.\ $A/fA$. The map $$ \Pic(U) \to \Pic(U_0) $$ is injective on torsion.


algebraization-proposition-application-higher

Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Let $\mathcal{F}$ be a coherent module on $U = \Spec(A) \setminus V(\mathfrak a)$. Let $s \geq 0$. Assume \begin{enumerate} \item $A$ is $I$-adically complete and has a dualizing complex, \item if $x \in U \setminus V(I)$ then $\text{depth}(\mathcal{F}_x) > s$ or $$ \text{depth}(\mathcal{F}_x) + \dim(\mathcal{O}_{\overline{\{x\}}, z}) > \text{cd}(A, I) + s + 1 $$ for all $z \in V(\mathfrak a) \cap \overline{\{x\}}$, \item one of the following conditions holds: \begin{enumerate} \item the restriction of $\mathcal{F}$ to $U \setminus V(I)$ is $(S_{s + 1})$, or \item the dimension of $V(\mathfrak a)$ is at most $2$\footnote{In the sense that the difference of the maximal and minimal values on $V(\mathfrak a)$ of a dimension function on $\Spec(A)$ is at most $2$.}. \end{enumerate} \end{enumerate} Then the maps $$ H^i(U, \mathcal{F}) \longrightarrow \lim H^i(U, \mathcal{F}/I^n\mathcal{F}) $$ are isomorphisms for $i < s$. Moreover we have an isomorphism $$ \colim H^s(V, \mathcal{F}) \longrightarrow \lim H^s(U, \mathcal{F}/I^n\mathcal{F}) $$ where the colimit is over opens $V \subset U$ containing $U \cap V(I)$.


algebraization-proposition-trivial-local-pic-complete-intersection

Let $(A, \mathfrak m)$ be a Noetherian local ring. If $A$ is a complete intersection of dimension $\geq 4$, then the Picard group of the punctured spectrum of $A$ is trivial.


spaces-over-fields-lemma-order-vanishing-agrees

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $f \in R(X)^*$. If the prime divisor $Z \subset X$ meets the schematic locus of $X$, then the order of vanishing $\text{ord}_Z(f)$ of Definition \ref{definition-order-vanishing} agrees with the order of vanishing of Divisors, Definition \ref{divisors-definition-order-vanishing}.


spaces-over-fields-lemma-normal-c1-injective

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. If $X$ is normal, then the map (\ref{equation-c1}) $\Pic(X) \to \text{Cl}(X)$ is injective.


spaces-over-fields-lemma-intersection-number-additive

In the situation of Definition \ref{definition-intersection-number} the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z)$ is additive: if $\mathcal{L}_i = \mathcal{L}_i' \otimes \mathcal{L}_i''$, then we have $$ (\mathcal{L}_1 \cdots \mathcal{L}_i \cdots \mathcal{L}_d \cdot Z) = (\mathcal{L}_1 \cdots \mathcal{L}_i' \cdots \mathcal{L}_d \cdot Z) + (\mathcal{L}_1 \cdots \mathcal{L}_i'' \cdots \mathcal{L}_d \cdot Z) $$


spaces-over-fields-lemma-intersection-number-in-terms-of-components

In the situation of Definition \ref{definition-intersection-number} let $Z_i \subset Z$ be the irreducible components of dimension $d$. Let $m_i = \text{length}_{\mathcal{O}_{X, \overline{x}_i}} (\mathcal{O}_{Z, \overline{x}_i})$ where $\overline{x}_i$ is a geometric generic point of $Z_i$. Then $$ (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) = \sum m_i(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z_i) $$


spaces-divisors-lemma-weakly-ass-support

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\text{WeakAss}(\mathcal{F}) \subset \text{Supp}(\mathcal{F})$.


spaces-divisors-lemma-restriction-injective-open-contains-weakly-ass

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $U \to X$ is an \'etale morphism such that $\text{WeakAss}(\mathcal{F}) \subset \Im(|U| \to |X|)$, then $\Gamma(X, \mathcal{F}) \to \Gamma(U, \mathcal{F})$ is injective.


spaces-divisors-lemma-finite-flat-weak-assassin-up-down

Let $S$ be a scheme. Let $f : X \to Y$ be a finite flat morphism of algebraic spaces. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. Let $x \in |X|$ be a point with image $y \in |Y|$. Then $$ x \in \text{WeakAss}(g^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}(\mathcal{G}) $$


spaces-divisors-lemma-locally-finite-type-locally-Noetherian-fibres

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, then  the fibres of $f$ are locally Noetherian.


spaces-divisors-lemma-bourbaki

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. Assume \begin{enumerate} \item $\mathcal{F}$ is flat over $Y$, \item $X$ and $Y$ are locally Noetherian, and \item the fibres of $f$ are locally Noetherian. \end{enumerate} Then $$ \text{Ass}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) = \{x \in \text{Ass}_{X/Y}(\mathcal{F})\text{ such that } f(x) \in \text{Ass}_Y(\mathcal{G}) \} $$


spaces-divisors-lemma-base-change-fitting-ideal

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_Y$-module. Then $f^{-1}\text{Fit}_i(\mathcal{F}) \cdot \mathcal{O}_X = \text{Fit}_i(f^*\mathcal{F})$.


spaces-divisors-lemma-fitting-ideal-finite-locally-free

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. Let $r \geq 0$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is finite locally free of rank $r$ \item $\text{Fit}_{r - 1}(\mathcal{F}) = 0$ and $\text{Fit}_r(\mathcal{F}) = \mathcal{O}_X$, and \item $\text{Fit}_k(\mathcal{F}) = 0$ for $k < r$ and $\text{Fit}_k(\mathcal{F}) = \mathcal{O}_X$ for $k \geq r$. \end{enumerate}


spaces-divisors-lemma-finite-presentation-module

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite presentation. Let $X = Z_{-1} \subset Z_0 \subset Z_1 \subset \ldots$ be as in Lemma \ref{lemma-locally-free-rank-r-pullback}. Set $X_r = Z_{r - 1} \setminus Z_r$. Then $X' = \coprod_{r \geq 0} X_r$ represents the functor $$ F_{flat} : \Sch/X \longrightarrow \textit{Sets},\quad\quad T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ flat over }T\\ \emptyset & \text{otherwise} \end{matrix} \right. $$ Moreover, $\mathcal{F}|_{X_r}$ is locally free of rank $r$ and the morphisms $X_r \to X$ and $X' \to X$ are of finite presentation.


spaces-divisors-lemma-effective-Cartier-makes-dimension-drop

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Let $x \in |D|$. If $\dim_x(X) < \infty$, then $\dim_x(D) < \dim_x(X)$.


spaces-divisors-lemma-pullback-effective-Cartier-defined

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $D \subset Y$ be an effective Cartier divisor. The pullback of $D$ by $f$ is defined in each of the following cases: \begin{enumerate} \item $f(x) \not \in |D|$ for any weakly associated point $x$ of $X$, \item $f$ is flat, and \item add more here as needed. \end{enumerate}


spaces-divisors-lemma-effective-Cartier-divisor-Sk

Let $S$ be a scheme and let $X$ be a locally Noetherian algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. If $X$ is $(S_k)$, then $D$ is $(S_{k - 1})$.


spaces-divisors-lemma-twists-of-structure-sheaf

In Situation \ref{situation-relative-proj}. The relative Proj comes equipped with a quasi-coherent sheaf of $\mathbf{Z}$-graded algebras $\bigoplus_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_X(\mathcal{A})}(n)$ and a canonical homomorphism of graded algebras $$ \psi : \pi^*\mathcal{A} \longrightarrow \bigoplus\nolimits_{n \geq 0} \mathcal{O}_{\underline{\text{Proj}}_X(\mathcal{A})}(n) $$ whose base change to any scheme over $X$ agrees with Constructions, Lemma \ref{constructions-lemma-glue-relative-proj-twists}.


spaces-divisors-lemma-relative-proj-quasi-compact

In Situation \ref{situation-relative-proj}. If one of the following holds \begin{enumerate} \item $\mathcal{A}$ is of finite type as a sheaf of $\mathcal{A}_0$-algebras, \item $\mathcal{A}$ is generated by $\mathcal{A}_1$ as an $\mathcal{A}_0$-algebra and $\mathcal{A}_1$ is a finite type $\mathcal{A}_0$-module, \item there exists a finite type quasi-coherent $\mathcal{A}_0$-submodule $\mathcal{F} \subset \mathcal{A}_{+}$ such that $\mathcal{A}_{+}/\mathcal{F}\mathcal{A}$ is a locally nilpotent sheaf of ideals of $\mathcal{A}/\mathcal{F}\mathcal{A}$, \end{enumerate} then $\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X$ is quasi-compact.


spaces-divisors-lemma-relative-proj-finite-type

In Situation \ref{situation-relative-proj}. If $\mathcal{A}$ is of finite type as a sheaf of $\mathcal{O}_X$-algebras, then $\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X$ is of finite type.


spaces-divisors-lemma-relative-proj-universally-closed

In Situation \ref{situation-relative-proj}. If $\mathcal{O}_X \to \mathcal{A}_0$ is an integral algebra map\footnote{In other words, the integral closure of $\mathcal{O}_X$ in $\mathcal{A}_0$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-integral-closure}, equals $\mathcal{A}_0$.} and $\mathcal{A}$ is of finite type as an $\mathcal{A}_0$-algebra, then $\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X$ is universally closed.


spaces-divisors-lemma-relative-proj-proper

In Situation \ref{situation-relative-proj}. The following conditions are equivalent \begin{enumerate} \item $\mathcal{A}_0$ is a finite type $\mathcal{O}_X$-module and $\mathcal{A}$ is of finite type as an $\mathcal{A}_0$-algebra, \item $\mathcal{A}_0$ is a finite type $\mathcal{O}_X$-module  and $\mathcal{A}$ is of finite type as an $\mathcal{O}_X$-algebra. \end{enumerate} If these conditions hold, then $\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X$ is proper.


spaces-divisors-lemma-blow-up-reduced-space

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. If $X$ is reduced, then the blowup $X'$ of $X$ in $\mathcal{I}$ is reduced.


spaces-divisors-lemma-blowup-finite-nr-irreducibles

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $b : X' \to X$ be the blowup of $X$ in a closed subspace. If $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-prepare-normalization} then so does $X'$.


spaces-divisors-lemma-strict-transform-different-centers

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $Z \subset B$ be a closed subspace. Let $D \subset B$ be an effective Cartier divisor. Let $Z' \subset B$ be the closed subspace cut out by the product of the ideal sheaves of $Z$ and $D$. Let $B' \to B$ be the blowup of $B$ in $Z$. \begin{enumerate} \item The blowup of $B$ in $Z'$ is isomorphic to $B' \to B$. \item Let $f : X \to B$ be a morphism of algebraic spaces and let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If the subsheaf of $\mathcal{F}$ of sections supported on $|f^{-1}D|$ is zero, then the strict transform of $\mathcal{F}$ relative to the blowing up in $Z$ agrees with the strict transform of $\mathcal{F}$ relative to the blowing up of $B$ in $Z'$. \end{enumerate}


spaces-divisors-lemma-strict-transform-blowup-fitting-ideal

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_B$-module. Let $Z_k \subset S$ be the closed subscheme cut out by $\text{Fit}_k(\mathcal{F})$, see Section \ref{section-fitting-ideals}. Let $B' \to B$ be the blowup of $B$ in $Z_k$ and let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$. Then $\mathcal{F}'$ can locally be generated by $\leq k$ sections.


dga-lemma-compose-homotopy

Let $(A, \text{d})$ be a differential graded algebra. Let $f, g : L \to M$ be homomorphisms of differential graded $A$-modules. Suppose given further homomorphisms $a : K \to L$, and $c : M \to N$. If $h : L \to M$ is an $A$-module map which defines a homotopy between $f$ and $g$, then $c \circ h \circ a$ defines a homotopy between $c \circ f \circ a$ and $c \circ g \circ a$.


dga-lemma-characterize-hom-other-side

Let $R$ be a ring. Let $(A, \text{d})$ be a differential graded $R$-algebra. Let $M$ be a right differential graded $A$-module and let $M'$ be a left differential graded $A$-module. Let $N^\bullet$ be a complex of $R$-modules. Then we have $$ \Hom_{\text{left diff graded }A\text{-modules}}(M', \Hom(M, N^\bullet)) = \Hom_{\text{Comp}(R)}(M \otimes_A M', N^\bullet) $$ where $M \otimes_A M'$ is viewed as a complex of $R$-modules as in Section \ref{section-tensor-product}.


dga-lemma-what-makes-a-bimodule-dg

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$. Let $M$ be a right differential graded $B$-module. There is a $1$-to-$1$ correspondence between $(A, B)$-bimodule structures on $M$ compatible with the given differential graded $B$-module structure and homomorphisms $$ A \longrightarrow \Hom_{\text{Mod}^{dg}_{(B, \text{d})}}(M, M) $$ of differential graded $R$-algebras.


dga-lemma-base-change-K-flat

Let $R \to R'$ be a ring map. Let $(A, \text{d})$ be a differential graded $R$-algebra. Let $(A', \text{d})$ be the base change, i.e., $A' = A \otimes_R R'$. If $A$ is K-flat as a complex of $R$-modules, then \begin{enumerate} \item $- \otimes_A^\mathbf{L} A' : D(A, \text{d}) \to D(A', \text{d})$ is equal to the right derived functor of $$ K(A, \text{d}) \longrightarrow K(A', \text{d}),\quad M \longmapsto M \otimes_R R' $$ \item the diagram $$ \xymatrix{ D(A, \text{d}) \ar[r]_{- \otimes_A^\mathbf{L} A'} \ar[d]_{restriction} & D(A', \text{d}) \ar[d]^{restriction} \\ D(R) \ar[r]^{- \otimes_R^\mathbf{L} R'} & D(R') } $$ commutes, and \item if $M$ is K-flat as a complex of $R$-modules, then the differential graded $A'$-module $M \otimes_R R'$ represents $M \otimes_A^\mathbf{L} A'$. \end{enumerate}


dga-lemma-compose-tensor-functors-tor

Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$, and $(C, \text{d})$ be differential graded $R$-algebras. Assume $A \otimes_R C$ represents $A \otimes^\mathbf{L}_R C$ in $D(R)$. Let $N$ be a differential graded $(A, B)$-bimodule. Let $N'$ be a differential graded $(B, C)$-bimodule. Then the composition $$ \xymatrix{ D(A, \text{d}) \ar[rr]^{- \otimes_A^\mathbf{L} N} & & D(B, \text{d}) \ar[rr]^{- \otimes_B^\mathbf{L} N'} & & D(C, \text{d}) } $$ is isomorphic to $- \otimes_A^\mathbf{L} N''$ for some differential graded $(A, C)$-bimodule $N''$.


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

In Situation \ref{situation-groupoid-on-field} the algebraic space $R$ is geometrically unibranch. In Situation \ref{situation-group-over-field} the algebraic space $G$ is geometrically unibranch.


modules-lemma-quasi-coherent-limit-finite-presentation

Let $(X, \mathcal{O}_X)$ be a ringed space. Set $R = \Gamma(X, \mathcal{O}_X)$. Let $M$ be an $R$-module. The $\mathcal{O}_X$-module $\mathcal{F}_M$ associated to $M$ is a directed colimit of finitely presented $\mathcal{O}_X$-modules.


modules-lemma-i-star-quasi-coherent

Let $i : (Z, \mathcal{O}_Z) \to (X, \mathcal{O}_X)$ be a closed immersion of ringed spaces. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_Z$-module. Then $i_*\mathcal{F}$ is locally on $X$ the cokernel of a map of quasi-coherent $\mathcal{O}_X$-modules.


modules-lemma-stalk-tensor-algebra

In the situation described above. Let $x \in X$. There are canonical isomorphisms of $\mathcal{O}_{X, x}$-modules $\text{T}(\mathcal{F})_x = \text{T}(\mathcal{F}_x)$, $\text{Sym}(\mathcal{F})_x = \text{Sym}(\mathcal{F}_x)$, and $\wedge(\mathcal{F})_x = \wedge(\mathcal{F}_x)$.


modules-lemma-pullback-invertible

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{L}$ of an invertible $\mathcal{O}_Y$-module is invertible.


modules-lemma-differential-seq

In Lemma \ref{lemma-functoriality-differentials} suppose that $\mathcal{O}_2 \to \mathcal{O}'_2$ is surjective with kernel $\mathcal{I} \subset \mathcal{O}_2$ and assume that $\mathcal{O}_1 = \mathcal{O}'_1$. Then there is a canonical exact sequence of $\mathcal{O}'_2$-modules $$ \mathcal{I}/\mathcal{I}^2 \longrightarrow \Omega_{\mathcal{O}_2/\mathcal{O}_1} \otimes_{\mathcal{O}_2} \mathcal{O}'_2 \longrightarrow \Omega_{\mathcal{O}'_2/\mathcal{O}_1} \longrightarrow 0 $$ The leftmost map is characterized by the rule that a local section $f$ of $\mathcal{I}$ maps to $\text{d}f \otimes 1$.


modules-lemma-differentials-relative-de-rham-complex-order-1

Let $f : X \to Y$ be a morphism of ringed spaces. The differentials $\text{d} : \Omega^i_{X/Y}  \to \Omega^{i + 1}_{X/Y}$ are differential operators of order $1$ on $X/Y$.


defos-lemma-inf-obs-map-rel

Let $(f, f')$ be a morphism of first order thickenings as in Situation \ref{situation-morphism-thickenings}. Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}_{X'}$-modules and set $\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}_X$-linear map. Assume that $\mathcal{F}'$ and $\mathcal{G}'$ are flat over $S'$ and that $(f, f')$ is a strict morphism of thickenings. There exists an element $$ o(\varphi) \in  \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{J}) $$ whose vanishing is a necessary and sufficient condition for the existence of a lift of $\varphi$ to an $\mathcal{O}_{X'}$-linear map $\varphi' : \mathcal{F}' \to \mathcal{G}'$.


groupoids-quotients-lemma-set-theoretic-equivalence-geometric

In the situation of Definition \ref{definition-set-theoretic-equivalence}. The following are equivalent: \begin{enumerate} \item The morphism $j$ is a set-theoretic equivalence relation. \item The morphism $j$ is universally injective and $j(|R|) \subset |U \times_B U|$ contains the image of $|j'|$ for any of the morphisms $j'$ as in Equation (\ref{equation-list}). \item For every algebraically closed field $k$ over $B$ of sufficiently large cardinality the map $j : R(k) \to U(k) \times U(k)$ is injective and its image is an equivalence relation. \end{enumerate} If $j$ is decent, or locally separated, or quasi-separated these are also equivalent to \begin{enumerate} \item[(4)] For every algebraically closed field $k$ over $B$ the map $j : R(k) \to U(k) \times U(k)$ is injective and its image is an equivalence relation. \end{enumerate}


groupoids-quotients-lemma-orbit-space

Let $B \to S$ as in Section \ref{section-conventions-notation}. Let $j : R \to U \times_B U$ be a set-theoretic pre-equivalence relation. A morphism $\phi : U \to X$ is an orbit space for $R$ if and only if \begin{enumerate} \item $\phi \circ s = \phi \circ t$, i.e., $\phi$ is invariant, \item the induced morphism $(t, s) : R \to U \times_X U$ is surjective, and \item the morphism $\phi : U \to X$ is surjective. \end{enumerate} This characterization applies for example if $j$ is a pre-equivalence relation, or comes from a groupoid in algebraic spaces over $B$, or comes from the action of a group algebraic space over $B$ on $U$.


spaces-resolve-lemma-modification

Let $(A, \mathfrak m, \kappa)$ be a $2$-dimensional Noetherian local domain such that $U = \Spec(A) \setminus \{\mathfrak m\}$ is a normal scheme. Then any modification $f : X \to \Spec(A)$ is a morphism as in (\ref{equation-modification}).


spaces-resolve-lemma-Nagata-normalized-blowup

In Definition \ref{definition-normalized-blowup} if $X$ is Nagata, then the normalized blowing up of $X$ at $x$ is a normal Nagata algebraic space proper over $X$.


duality-theorem-lichtenbaum

Let $X$ be a nonempty separated scheme of finite type over a field $k$. Let $d = \dim(X)$. The following are equivalent \begin{enumerate} \item $H^d(X, \mathcal{F}) = 0$ for all coherent $\mathcal{O}_X$-modules $\mathcal{F}$ on $X$, \item $H^d(X, \mathcal{F}) = 0$ for all quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F}$ on $X$, and \item no irreducible component $X' \subset X$ of dimension $d$ is proper over $k$. \end{enumerate}


duality-lemma-proper-perfect-base-change

Let $f : X \to Y$ be a perfect proper morphism of Noetherian schemes. Let $g : Y' \to Y$ be a morphism with $Y'$ Noetherian. If $X$ and $Y'$ are tor independent over $Y$, then the base change map (\ref{equation-base-change-map}) is an isomorphism for all $K \in D_\QCoh(\mathcal{O}_Y)$.


duality-lemma-restrict-before-or-after

In Situation \ref{situation-shriek} let $$ \xymatrix{ U \ar[r]_j \ar[d]_g & X \ar[d]^f \\ V \ar[r]^{j'} & Y } $$ be a commutative diagram of $\textit{FTS}_S$ where $j$ and $j'$ are open immersions. Then $j^* \circ f^! = g^! \circ (j')^*$ as functors $D^+_\QCoh(\mathcal{O}_Y) \to D^+(\mathcal{O}_U)$.


duality-lemma-good-dualizing-independence-covering

In Situation \ref{situation-dualizing} let $X$ be a scheme of finite type over $S$ and let $\mathcal{U}$, $\mathcal{V}$ be two finite open coverings of $X$ by schemes separated over $S$. If there exists a dualizing complex normalized relative to $\omega_S^\bullet$ and $\mathcal{U}$, then there exists a dualizing complex normalized relative to $\omega_S^\bullet$ and $\mathcal{V}$ and these complexes are canonically isomorphic.


duality-lemma-existence-good-dualizing

In Situation \ref{situation-dualizing} let $X$ be a scheme of finite type over $S$ and let $\mathcal{U}$ be a finite open covering of $X$ by schemes separated over $S$. Then there exists a dualizing complex normalized relative to $\omega_S^\bullet$ and $\mathcal{U}$.


duality-lemma-open-immersion-good-dualizing-complex

Let $(S, \omega_S^\bullet)$ be as in Situation \ref{situation-dualizing}. Let $j : X \to Y$ be an open immersion of schemes of finite type over $S$. Let $\omega_X^\bullet$ and $\omega_Y^\bullet$ be dualizing complexes normalized relative to $\omega_S^\bullet$. Then there is a canonical isomorphism $\omega_X^\bullet = \omega_Y^\bullet|_X$.


duality-lemma-gorenstein-local-source-and-target

The property $\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is Gorenstein'' is local in the fppf topology on the target and local in the syntomic topology on the source.


duality-lemma-lower-shriek-well-defined

The functor $Rf_!$ is, up to isomorphism, independent of the choice of the compactification.


more-morphisms-lemma-descending-property-thickening

The property of being a thickening is fpqc local. Similarly for first order thickenings.


more-morphisms-lemma-affine-formally-unramified

Let $f : X \to S$ be a morphism of schemes. Assume $X$ and $S$ are affine. Then $f$ is formally unramified if and only if $\mathcal{O}_S(S) \to \mathcal{O}_X(X)$ is a formally unramified ring map.


more-morphisms-lemma-formally-etale-not-affine

If $f : X \to S$ is a formally \'etale morphism, then given any solid commutative diagram $$ \xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] } $$ where $T \subset T'$ is a first order thickening of schemes over $S$ there exists exactly one dotted arrow making the diagram commute. In other words, in Definition \ref{definition-formally-etale} the condition that $T$ be affine may be dropped.


more-morphisms-lemma-differentials-formally-unramified-formally-smooth

Let $h : Z \to X$ be a formally unramified morphism of schemes over $S$. Assume that $Z$ is formally smooth over $S$. Then the canonical exact sequence $$ 0 \to \mathcal{C}_{Z/X} \to h^*\Omega_{X/S} \to \Omega_{Z/S} \to 0 $$ of Lemma \ref{lemma-universally-unramified-differentials-sequence} is short exact.


more-morphisms-lemma-base-change-NL

Consider a cartesian diagram of schemes $$ \xymatrix{ X' \ar[r]_{g'} \ar[d] & X \ar[d] \\ Y' \ar[r] & Y } $$ The canonical map $(g')^*\NL_{X/Y} \to \NL_{X'/Y'}$ induces an isomorphism on $H^0$ and a surjection on $H^{-1}$.


more-morphisms-lemma-flat-base-change-NL

Consider a cartesian diagram of schemes $$ \xymatrix{ X' \ar[d] \ar[r]_{g'} & X \ar[d] \\ Y' \ar[r] & Y } $$ If $Y' \to Y$ is flat, then the canonical map $(g')^*\NL_{X/Y} \to \NL_{X'/Y'}$ is a quasi-isomorphism.


more-morphisms-lemma-base-change-flatness-fibres

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Assume \begin{enumerate} \item $X$ is locally of finite presentation over $S$, \item $X$ is flat over $S$, and \item $Y$ is locally of finite type over $S$. \end{enumerate} Then the set $$ U = \{x \in X \mid X\text{ flat at }x \text{ over }Y\}. $$ is open in $X$ and its formation commutes with arbitrary base change.


more-morphisms-lemma-CM-dimension

Let $f : X \to Y$ be a morphism of locally Noetherian schemes which is locally of finite type and Cohen-Macaulay. For every point $x$ in $X$ with image $y$ in $Y$, $$ \dim_x(X) = \dim_y(Y) + \dim_x(X_y), $$ where $X_y$ denotes the fiber over $y$.


more-morphisms-lemma-connected-along-section-good

Let $f : X \to Y$, $s : Y \to X$ be as in Situation \ref{situation-connected-along-section}. Assume $f$ of finite type. Let $y \in Y$ be a point. Then there exists a nonempty open $V \subset \overline{\{y\}}$ such that the inverse image of $X^0$ in the base change $X_V$ is open and closed in $X_V$.


more-morphisms-lemma-connected-along-section-open-neighbourhood

Let $f : X \to Y$, $s : Y \to X$ be as in Situation \ref{situation-connected-along-section}. Let $y \in Y$ be a point. Assume \begin{enumerate} \item $f$ is of finite presentation and flat, and \item the fibre $X_y$ is geometrically reduced. \end{enumerate} Then $X^0$ is a neighbourhood of $X^0_y$ in $X$.


more-morphisms-lemma-trivial-on-fibres

Let $f : X \to S$ be a flat, proper morphism of finite presentation such that $f_*\mathcal{O}_X = \mathcal{O}_S$ and this remains true after arbitrary base change. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $\mathcal{E}|_{X_s}$ is isomorphic to $\mathcal{O}_{X_s}^{\oplus r_s}$ for all $s \in S$, and \item $S$ is reduced. \end{enumerate} Then $\mathcal{E} = f^*\mathcal{N}$ for some finite locally free $\mathcal{O}_S$-module $\mathcal{N}$.


more-morphisms-lemma-Noetherian-approximation-standard-syntomic

Let $f : X \to S$ be a morphism of affine schemes, which is standard syntomic (see Morphisms, Definition \ref{morphisms-definition-syntomic}). Then there exists a diagram as in Lemma \ref{lemma-Noetherian-approximation} such that in addition $f_0$ is standard syntomic.


more-morphisms-lemma-relative-isomorphism-approximation

With notation an assumptions as in Lemma \ref{lemma-relative-map-approximation} assume that $\varphi$ induces an isomorphism on completions. Then we can choose our diagram such that $f$ is \'etale.


more-morphisms-lemma-quasi-finite-separated-pass-through-finite-addendum

With notation and hypotheses as in Lemma \ref{lemma-quasi-finite-separated-pass-through-finite}. Assume moreover that $f$ is locally of finite presentation. Then we can choose the factorization such that $T$ is finite and of finite presentation over $S$.


more-morphisms-lemma-integral-over-quasi-affine

Let $X$ be a quasi-affine scheme. Let $f : U \to X$ be an integral morphism. Then $U$ is quasi-affine and the diagram $$ \xymatrix{ U \ar[r] \ar[d] & \Spec(\Gamma(U, \mathcal{O}_U)) \ar[d] \\ X \ar[r] & \Spec(\Gamma(X, \mathcal{O}_X)) } $$ is cartesian.


more-morphisms-lemma-equality-dimensions

Let $A \to B$ be a local homomorphism of local rings. Assume \begin{enumerate} \item $A$ and $B$ are domains and $A \subset B$, \item $B$ is essentially of finite type over $A$, and \item $B$ is flat over $A$. \end{enumerate} Then we have $$ \dim(B/\mathfrak m_AB) + \text{trdeg}_{\kappa(\mathfrak m_A)}(\kappa(\mathfrak m_B)) = \text{trdeg}_A(B). $$


more-morphisms-lemma-quasi-finite-quasi-section-meeting-nearby-open

Let $\varphi : A \to B$ be a local ring map of local rings. Let $V \subset \Spec(B)$ be an open subscheme which contains at least one prime not lying over $\mathfrak m_A$. Assume $A$ is Noetherian, $\varphi$ essentially of finite type, and $A/\mathfrak m_A \subset B/\mathfrak m_B$ is finite. Then there exists a $\mathfrak q \in V$, $\mathfrak m_A \not = \mathfrak q \cap A$ such that $A \to B/\mathfrak q$ is the localization of a quasi-finite ring map.


more-morphisms-lemma-proper-flat-geom-red

Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ is proper, flat, and of finite presentation, and \item the geometric fibres of $f$ are reduced. \end{enumerate} Then the function $n_{X/S} : S \to \mathbf{Z}$ counting the numbers of geometric connected components of fibres of $f$ is locally constant.


more-morphisms-lemma-finite-morphism-relative-finite-presentation

Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite type over a base scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is of finite presentation relative to $S$ if and only if $\pi_*\mathcal{F}$ is of finite presentation relative to $S$.


more-morphisms-lemma-pull-relative-finite-presentation

Let $X \to Y \to S$ be morphisms of schemes which are locally of finite type. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. If $f : X \to Y$ is locally of finite presentation and $\mathcal{G}$ of finite presentation relative to $S$, then $f^*\mathcal{G}$ is of finite presentation relative to $S$.


more-morphisms-lemma-composition-relative-finite-presentation

Let $X \to Y \to S$ be morphisms of schemes which are locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $Y \to S$ is locally of finite presentation and $\mathcal{F}$ is of finite presentation relative to $Y$, then $\mathcal{F}$ is of finite presentation relative to $S$.


more-morphisms-lemma-ses-relatively-finite-presentation

Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ be a short exact sequence of quasi-coherent $\mathcal{O}_X$-modules. \begin{enumerate} \item If $\mathcal{F}', \mathcal{F}''$ are finitely presented relative to $S$, then so is $\mathcal{F}$. \item If $\mathcal{F}'$ is a finite type $\mathcal{O}_X$-module and $\mathcal{F}$ is finitely presented relative to $S$, then $\mathcal{F}''$ is finitely presented relative to $S$. \end{enumerate}


more-morphisms-lemma-relative-pseudo-coherence

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. If $E$ in $D(\mathcal{O}_X)$ is $m$-pseudo-coherent relative to $S$, then $H^i(E)$ is a quasi-coherent $\mathcal{O}_X$-module for $i > m$. If $E$ is pseudo-coherent relative to $S$, then $E$ is an object of $D_\QCoh(\mathcal{O}_X)$.


more-morphisms-lemma-closed-morphism-relative-pseudo-coherence

Let $i : X \to Y$ morphism of schemes locally of finite type over a base scheme $S$. Assume that $i$ induces a homeomorphism of $X$ with a closed subset of $Y$. Let $E$ be an object of $D(\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $Ri_*E$ is $m$-pseudo-coherent relative to $S$.


more-morphisms-lemma-cohomology-relative-pseudo-coherent

Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_X)$. If $E$ is (locally) bounded above and $H^i(E)$ is $(m - i)$-pseudo-coherent relative to $S$ for all $i$, then $E$ is $m$-pseudo-coherent relative to $S$.


more-morphisms-lemma-descending-property-pseudo-coherent

The property $\mathcal{P}(f) =$``$f$ is pseudo-coherent'' is fpqc local on the base.


more-morphisms-lemma-pseudo-coherent-syntomic-local-source

The property $\mathcal{P}(f) =$``$f$ is pseudo-coherent'' is syntomic local on the source.


more-morphisms-lemma-perfect-closed-immersion-perfect-direct-image

Let $i : Z \to X$ be a perfect closed immersion of schemes. Then $i_*\mathcal{O}_X$ is a perfect $\mathcal{O}_X$-module, i.e., it is a perfect object of $D(\mathcal{O}_X)$.


more-morphisms-lemma-perfect-fppf-local-source

The property $\mathcal{P}(f) =$``$f$ is perfect'' is fppf local on the source.


more-morphisms-lemma-factor-regular-immersion

Let $i : Z \to Y$ and $j : Y \to X$ be immersions of schemes. Assume \begin{enumerate} \item $X$ is locally Noetherian, \item $j \circ i$ is a regular immersion, and \item $i$ is perfect. \end{enumerate} Then $i$ and $j$ are regular immersions.


more-morphisms-lemma-koszul-independence-factorization

Let $S$ be a scheme. Let $U$, $P$, $P'$ be schemes over $S$. Let $u \in U$. Let $i : U \to P$, $i' : U \to P'$ be immersions over $S$. Assume $P$ and $P'$ smooth over $S$. Then the following are equivalent \begin{enumerate} \item $i$ is a Koszul-regular immersion in a neighbourhood of $x$, and \item $i'$ is a Koszul-regular immersion in a neighbourhood of $x$. \end{enumerate}


more-morphisms-lemma-relative-frobenius-weakly-etale

Let $U \to X$ be a weakly \'etale morphism of schemes where $X$ is a scheme in characteristic $p$. Then the relative Frobenius $F_{U/X} : U \to U \times_{X, F_X} X$ is an isomorphism.


more-morphisms-lemma-pushout-separated

In Situation \ref{situation-pushout-along-closed-immersion-and-integral}. If $X$ and $Y$ are separated, then the pushout $Y \amalg_Z X$ (Proposition \ref{proposition-pushout-along-closed-immersion-and-integral}) is separated. Same with ``separated over $S$'', ``quasi-separated'', and ``quasi-separated over $S$''.


more-morphisms-lemma-improve-stratification

Let $X$ be a scheme. Let $X = \coprod_{i \in I} X_i$ be a finite affine stratification. There exists an affine stratification with index set $\{0, \ldots, n\}$ where $n$ is the length of $I$.


more-morphisms-lemma-qc-affine-stratification

Let $X$ be a scheme. The following are equivalent \begin{enumerate} \item $X$ has a finite affine stratification, and \item $X$ is quasi-compact and quasi-separated. \end{enumerate}


more-morphisms-lemma-affine-stratification-number-bound

Let $X$ be a separated scheme which has an open covering by $n + 1$ affines. Then the affine stratification number of $X$ is at most $n$.


more-morphisms-lemma-affine-stratification-number-bound-Noetherian

Let $X$ be a Noetherian scheme of dimension $\infty > d \geq 0$. Then the affine stratification number of $X$ is at most $d$.


more-morphisms-lemma-large-open

Let $f : X \to Y$ be a separated, locally quasi-finite, and universally open morphism of schemes. Let $n_{X/Y}$ be as in Lemma \ref{lemma-count-geometric-fibres}. If $n_{X/Y}$ attains a maximum $d < \infty$, then the set $$ Y_d = \{y \in Y \mid n_{X/Y}(y) = d\} $$ is open in $Y$ and the morphism $f^{-1}(Y_d) \to Y_d$ is finite.


more-morphisms-proposition-asn-weighting

Let $f : X \to Y$ be a surjective quasi-finite morphism of schemes. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Assume $X$ affine and $Y$ separated\footnote{It suffices if the diagonal of $Y$ is affine.}. Then the affine stratification number of $Y$ is at most the number of distinct values of $\int_f w$.


sites-modules-lemma-tensor-product-permanence

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}$-modules. \begin{enumerate} \item If $\mathcal{F}$, $\mathcal{G}$ are locally free, so is $\mathcal{F} \otimes_\mathcal{O} \mathcal{G}$. \item If $\mathcal{F}$, $\mathcal{G}$ are finite locally free, so is $\mathcal{F} \otimes_\mathcal{O} \mathcal{G}$. \item If $\mathcal{F}$, $\mathcal{G}$ are locally generated by sections, so is $\mathcal{F} \otimes_\mathcal{O} \mathcal{G}$. \item If $\mathcal{F}$, $\mathcal{G}$ are of finite type, so is $\mathcal{F} \otimes_\mathcal{O} \mathcal{G}$. \item If $\mathcal{F}$, $\mathcal{G}$ are quasi-coherent, so is $\mathcal{F} \otimes_\mathcal{O} \mathcal{G}$. \item If $\mathcal{F}$, $\mathcal{G}$ are of finite presentation, so is $\mathcal{F} \otimes_\mathcal{O} \mathcal{G}$. \item If $\mathcal{F}$ is of finite presentation and $\mathcal{G}$ is coherent, then $\mathcal{F} \otimes_\mathcal{O} \mathcal{G}$ is coherent. \item If $\mathcal{F}$, $\mathcal{G}$ are coherent, so is $\mathcal{F} \otimes_\mathcal{O} \mathcal{G}$. \end{enumerate}


sites-modules-lemma-pic-set

Let $(\mathcal{C}, \mathcal{O})$ be a ringed space. There exists a set of invertible modules $\{\mathcal{L}_i\}_{i \in I}$ such that each invertible module on $(\mathcal{C}, \mathcal{O})$ is isomorphic to exactly one of the $\mathcal{L}_i$.


derham-lemma-finite-de-Rham

Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $S = \Spec(A)$. Then $H^i_{dR}(X/S)$ is a finite $A$-module for all $i$.


derham-lemma-kunneth-de-rham-relative

Assume $X \to S$ and $Y \to S$ are smooth and quasi-compact and the morphisms $X \to X \times_S X$ and $Y \to Y \times_S Y$ are affine. Then the relative cup product $$ Ra_*\Omega^\bullet_{X/S} \otimes_{\mathcal{O}_S}^\mathbf{L} Rb_*\Omega^\bullet_{Y/S} \longrightarrow Rf_*\Omega^\bullet_{X \times_S Y/S} $$ is an isomorphism in $D(\mathcal{O}_S)$.


derham-lemma-splitting-on-omega-a

With notation as in Lemma \ref{lemma-blowup} for $a \geq 0$ there is a unique arrow $Rb_*\Omega^a_{X'/S} \to \Omega^a_{X/S}$ in $D(\mathcal{O}_X)$ whose composition with $\Omega^a_{X/S} \to Rb_*\Omega^a_{X'/S}$ is the identity on $\Omega^a_{X/S}$.


derham-lemma-gysin-differential-hodge

Let $X \to S$ and $i : Z \to X$ be as in Lemma \ref{lemma-gysin-global}. Assume $X \to S$ is smooth and $Z \to X$ Koszul regular. The gysin maps $\gamma^{p, q}$ are compatible with the de Rham differentials on $\Omega^\bullet_{X/S}$ and $\Omega^\bullet_{Z/S}$.


derham-lemma-gysin-projection-global

Let $X \to S$, $i : Z \to X$, and $c \geq 0$ be as in Lemma \ref{lemma-gysin-global}. Assume $X \to S$ smooth and $Z \to X$ Koszul regular. Given $\alpha \in H^q(X, \Omega^p_{X/S})$ we have $\gamma^{p, q}(\alpha|_Z) = \alpha \cup \gamma^{0, 0}(1)$ in $H^{q + c}(X, \Omega^{p + c}_{X/S})$ with $\gamma^{a, b}$ as in Remark \ref{remark-how-to-use}.


derham-lemma-class-of-a-point

Let $k$ be a field. Let $X$ be an irreducible smooth proper scheme over $k$ of dimension $d$. Let $Z \subset X$ be the reduced closed subscheme consisting of a single $k$-rational point $x$. Then the image of $1 \in k = H^0(Z, \mathcal{O}_Z) = H^0(Z, \Omega^0_{Z/k})$ by the map $H^0(Z, \Omega^0_{Z/k}) \to H^d(X, \Omega^d_{X/k})$ of Remark \ref{remark-how-to-use} is nonzero.


derham-proposition-relative-poincare-duality

Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a proper smooth morphism of schemes all of whose fibres are nonempty and equidimensional of dimension $d$. There exists an $\mathcal{O}_S$-module map $$ t : R^{2d}f_*\Omega^\bullet_{X/S} \longrightarrow \mathcal{O}_S $$ unique up to precomposing by multiplication by a unit of $H^0(X, \mathcal{O}_X)$ with the following property: the pairing $$ Rf_*\Omega^\bullet_{X/S} \otimes_{\mathcal{O}_S}^\mathbf{L} Rf_*\Omega^\bullet_{X/S}[2d] \longrightarrow \mathcal{O}_S, \quad (\xi, \xi') \longmapsto t(\xi \cup \xi') $$ is a perfect pairing of perfect complexes on $S$.


trace-lemma-sheaf-over-finite-field-has-frobenius-descent

Let $\mathcal{F}$ be a sheaf on $X_\etale$. Then there are canonical isomorphisms $\pi_X^{-1} \mathcal{F} \cong \mathcal{F}$ and $\mathcal{F} \cong {\pi_X}_*\mathcal{F}$.


trace-lemma-weil-mod

Consider the situation of Theorem \ref{theorem-weil-trace-formula} and let $\ell$ be a prime number invertible in $k$. Then $$ \sum\nolimits_{i = 0}^2 (-1)^i \text{Tr}(\varphi^* |_{H^i (C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}})}) = V(\varphi) \mod \ell^n. $$


trace-lemma-l-adic-abelian

The category of $\mathbf{Z}_\ell$-sheaves on $X$ is abelian.


trace-lemma-piece-together

Suppose we have $K_n\in D_{perf}(\mathbf{Z}/\ell^n\mathbf{Z})$, $\pi_n : K_n\to K_n$ and isomorphisms $\varphi_n : K_{n+1} \otimes^\mathbf{L}_{\mathbf{Z}/\ell^{n+1}\mathbf{Z}} \mathbf{Z}/\ell^n\mathbf{Z} \to K_n$ compatible with $\pi_{n+1}$ and $\pi_n$. Then \begin{enumerate} \item the elements $t_n = \text{Tr}(\pi_n |_{K_n})\in \mathbf{Z}/\ell^n\mathbf{Z}$ form an element $t_\infty = \{t_n\}$ of $\mathbf{Z}_\ell$, \item the $\mathbf{Z}_\ell$-module $H_\infty^i = \lim_n H^i(k_n)$ is finite and is nonzero for finitely many $i$ only, and \item the operators $H^i(\pi_n): H^i(K_n)\to H^i(K_n)$ are compatible and define $\pi_\infty^i : H_\infty^i\to H_\infty^i$ satisfying $$ \sum (-1)^i \text{Tr}( \pi_\infty^i |_{H_\infty^i \otimes_{\mathbf{Z}_\ell}\mathbf{Q}_\ell}) = t_\infty. $$ \end{enumerate}


trace-lemma-eigenvalues-algebraic

Algebraicity of eigenvalues. If $\Lambda$ is a field then the eigenvalues $t_v$ for $f\in C(\Lambda)$ are algebraic over the prime subfield $\mathbf{F} \subset \Lambda$.


sheaves-lemma-sheafify-presheaf-structures

Let $X$ be a topological space. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$ on $X$. Then there exists a sheaf $\mathcal{F}^\#$ with values in $\mathcal{C}$ and a morphism $\mathcal{F} \to \mathcal{F}^\#$ of presheaves with values in $\mathcal{C}$ with the following properties: \begin{enumerate} \item The map $\mathcal{F} \to \mathcal{F}^\#$ identifies the underlying sheaf of sets of $\mathcal{F}^\#$ with the sheafification of the underlying presheaf of sets of $\mathcal{F}$. \item For any morphism $\mathcal{F} \to \mathcal{G}$, where $\mathcal{G}$ is a sheaf with values in $\mathcal{C}$ there exists a unique factorization $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$. \end{enumerate}


sheaves-lemma-adjoint-pull-push-modules

Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a sheaf of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Let $\mathcal{G}$ be a sheaf of $f_*\mathcal{O}$-modules. Then $$ \Mor_{\textit{Mod}(\mathcal{O})}( \mathcal{O} \otimes_{f^{-1}f_*\mathcal{O}} f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{Mod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). $$ Here we use Lemmas \ref{lemma-pullback-module} and \ref{lemma-pushforward-module}, and we use the canonical map $f^{-1}f_*\mathcal{O} \to \mathcal{O}$ in the definition of the tensor product.


descent-lemma-equalizer-S

The diagram \begin{equation} \label{equation-equalizer-S} \xymatrix@C=8pc{ S_1 \ar[r]^{\delta^1_1} & S_2 \ar@<1ex>[r]^{\delta^2_2} \ar@<-1ex>[r]_{\delta^2_1} &  S_3 } \end{equation} is a split equalizer.


descent-lemma-equivalence-quasi-coherent-properties

Let $S$ be a scheme. Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. Let $\mathcal{P}$ be one of the properties of modules\footnote{The list is: free, finite free, generated by global sections, generated by $r$ global sections, generated by finitely many global sections, having a global presentation, having a global finite presentation, locally free, finite locally free, locally generated by sections, locally generated by $r$ sections, finite type, of finite presentation, coherent, or flat.} defined in Modules on Sites, Definitions \ref{sites-modules-definition-global}, \ref{sites-modules-definition-site-local}, and \ref{sites-modules-definition-flat}. The equivalences of categories $$ \QCoh(\mathcal{O}_S) \longrightarrow \QCoh((\Sch/S)_\tau, \mathcal{O}) \quad\text{and}\quad \QCoh(\mathcal{O}_S) \longrightarrow \QCoh(S_\tau, \mathcal{O}) $$ defined by the rule $\mathcal{F} \mapsto \mathcal{F}^a$ seen in Proposition \ref{proposition-equivalence-quasi-coherent} have the property $$ \mathcal{F}\text{ has }\mathcal{P} \Leftrightarrow \mathcal{F}^a\text{ has }\mathcal{P}\text{ as an }\mathcal{O}\text{-module} $$ except (possibly) when $\mathcal{P}$ is ``locally free'' or ``coherent''. If $\mathcal{P}=$``coherent'' the equivalence holds for $\QCoh(\mathcal{O}_S) \to \QCoh(S_\tau, \mathcal{O})$ when $S$ is locally Noetherian and $\tau$ is Zariski or \'etale.


descent-lemma-regular-local-ring-local

Let $f : U \to V$ be an \'etale morphism of schemes. Let $u \in U$ and $v = f(u)$. Then $\mathcal{O}_{U, u}$ is a regular local ring if and only if $\mathcal{O}_{V, v}$ is a regular local ring.


descent-lemma-morphism-with-section-equivalence

Let $f : X' \to X$ be a morphism of schemes over a base scheme $S$. Assume there exists a morphism $g : X \to X'$ over $S$, for example if $f$ has a section. Then the pullback functor of Lemma \ref{lemma-pullback} defines an equivalence of categories between the category of descent data relative to $X/S$ and $X'/S$.


simplicial-lemma-N-d-in-N

Let $\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object in $\mathcal{A}$. Let $N(U_m)$ as in Lemma \ref{lemma-splitting-abelian-category} above. Then $d^m_m(N(U_m)) \subset N(U_{m - 1})$.


simplicial-lemma-N-faithful

Let $\mathcal{A}$ be an abelian category. The functor $N$ is faithful, and reflects isomorphisms, injections and surjections.


simplicial-lemma-dual-dold-kan

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item The functor $s : \text{CoSimp}(\mathcal{A}) \to \text{CoCh}_{\geq 0}(\mathcal{A})$ is exact. \item The maps $s(U)^n \to Q(U)^n$ define a morphism of cochain complexes. \item There exists a functorial direct sum decomposition $s(U) = D(U) \oplus Q(U)$ in $\text{CoCh}_{\geq 0}(\mathcal{A})$. \item The functor $Q$ is exact. \item The morphism of complexes $s(U) \to Q(U)$ is a quasi-isomorphism. \item The functor $U \mapsto Q(U)^\bullet$ defines an equivalence of categories $\text{CoSimp}(\mathcal{A}) \to \text{CoCh}_{\geq 0}(\mathcal{A})$. \end{enumerate}


simplicial-lemma-contractible

Let $\mathcal{C}$ be a category with finite coproducts. Let $U$ be a simplicial object of $\mathcal{C}$. Consider the maps $e_1, e_0 : U \to U \times \Delta[1]$, and $\pi : U \times \Delta[1] \to U$, see Lemma \ref{lemma-back-to-U}. \begin{enumerate} \item We have $\pi \circ e_1 = \pi \circ e_0 = \text{id}_U$, and \item The morphisms $\text{id}_{U \times \Delta[1]}$, and $e_0 \circ \pi$ are homotopic. \item The morphisms $\text{id}_{U \times \Delta[1]}$, and $e_1 \circ \pi$ are homotopic. \end{enumerate}


simplicial-lemma-fibre-products-simplicial-object-w-section

Let $f : Y \to X$ be a morphism of a category $\mathcal{C}$ with fibre products. Assume $f$ has a section $s$. Consider the simplicial object $U$ constructed in Example \ref{example-fibre-products-simplicial-object} starting with $f$. The morphism $U \to U$ which in each degree is the self map $(s \circ f)^{n + 1}$ of $Y \times_X \ldots \times_X Y$ given by $s \circ f$ on each factor is homotopic to the identity on $U$. In particular, $U$ is homotopy equivalent to the constant simplicial object $X$.


simplicial-lemma-push-outs-simplicial-object-w-section

Let $f : X \to Y$ be a morphism of a category $\mathcal{C}$ with pushouts. Assume there is a morphism $s : Y \to X$ with $s \circ f = \text{id}_X$. Consider the cosimplicial object $U$ constructed in Example \ref{example-push-outs-simplicial-object} starting with $f$. The morphism $U \to U$ which in each degree is the self map of $Y \amalg_X \ldots \amalg_X Y$ given by $f \circ s$ on each factor is homotopic to the identity on $U$. In particular, $U$ is homotopy equivalent to the constant cosimplicial object $X$.


simplicial-lemma-backwards-homotopy

Let $\mathcal{A}$ be an abelian category. Let $U$, $V$ be simplicial objects of $\mathcal{A}$. Let $a, b : U \to V$ be a pair of morphisms. Assume the corresponding maps of chain complexes $N(a), N(b) : N(U) \to N(V)$ are homotopic by a homotopy $\{N_n : N(U)_n \to N(V)_{n + 1}\}$. Then there exists a homotopy from $a$ to $b$ as in Definition \ref{definition-homotopy}. Moreover, one can choose the homotopy $h : U \times \Delta[1] \to V$ such that $N_n = N(h)_n$ where $N(h)$ is the homotopy coming from $h$ as in Section \ref{section-homotopy-abelian}.


simplicial-lemma-kan-base-change

Let $f : X \to Y$ be a Kan fibration of simplicial sets. Let $Y' \to Y$ be a morphism of simplicial sets. Then $X \times_Y Y' \to Y'$ is a Kan fibration.


simplicial-lemma-surjection-simplicial-abelian-groups-kan

Let $f : X \to Y$ be a homomorphism of simplicial abelian groups which is termwise surjective. Then $f$ is a Kan fibration of simplicial sets.


simplicial-lemma-homotopy

Let $f^0, f^1 : V \to U$ be maps of simplicial sets. Let $n \geq 0$ be an integer. Assume \begin{enumerate} \item The maps $f^j_i : V_i \to U_i$, $j = 0, 1$ are equal for $i < n$. \item The canonical morphism $U \to \text{cosk}_n \text{sk}_n U$ is an isomorphism. \item The canonical morphism $V \to \text{cosk}_n \text{sk}_n V$ is an isomorphism. \end{enumerate} Then $f^0$ is homotopic to $f^1$.


discriminant-lemma-compare-dualizing-algebraic

Let $A \to B$ be a quasi-finite homomorphism of Noetherian rings. Let $\omega_{B/A}^\bullet \in D(B)$ be the algebraic relative dualizing complex discussed in Dualizing Complexes, Section \ref{dualizing-section-relative-dualizing-complexes-Noetherian}. Then there is a (nonunique) isomorphism $\omega_{B/A} = H^0(\omega_{B/A}^\bullet)$.


discriminant-lemma-noether-different

Let $A \to B$ be a finite type ring map. Let $\mathfrak{D} \subset B$ be the Noether different. Then $V(\mathfrak{D})$ is the set of primes $\mathfrak q \subset B$ such that $A \to B$ is not unramified at $\mathfrak q$.


discriminant-lemma-base-change-kahler-different

Consider a cartesian diagram of schemes $$ \xymatrix{ Y' \ar[d]_{f'} \ar[r] & Y \ar[d]^f \\ X' \ar[r]^g & X } $$ with $f$ locally of finite type. Let $R \subset Y$, resp.\ $R' \subset Y'$ be the closed subscheme cut out by the K\"ahler different of $f$, resp.\ $f'$. Then $Y' \to Y$ induces an isomorphism $R' \to R \times_Y Y'$.


discriminant-lemma-kahler-different

Let $f : Y \to X$ be a morphism of schemes which is locally of finite type. Let $R \subset Y$ be the closed subscheme defined by the K\"ahler different. Then $R \subset Y$ is exactly the set of points where $f$ is not unramified.


discriminant-lemma-flat-agree-dedekind

Let $f : Y \to X$ be a flat quasi-finite morphism of Noetherian schemes. Let $V = \Spec(B) \subset Y$, $U = \Spec(A) \subset X$ be affine open subschemes with $f(V) \subset U$. If the Dedekind different of $A \to B$ is defined, then $$ \mathfrak{D}_f|_V = \widetilde{\mathfrak{D}_{B/A}} $$ as coherent ideal sheaves on $V$.


limits-lemma-descend-section

In Situation \ref{situation-descent}. Suppose that $\mathcal{F}_0$ is a quasi-coherent sheaf on $S_0$. Set $\mathcal{F}_i = f_{i0}^*\mathcal{F}_0$ for $i \geq 0$ and set $\mathcal{F} = f_0^*\mathcal{F}_0$. Then $$ \Gamma(S, \mathcal{F}) = \colim_{i \geq 0} \Gamma(S_i, \mathcal{F}_i) $$


limits-lemma-chow-EGA

Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a separated morphism of finite type. Assume that $X$ has finitely many irreducible components. Then there exists an $n \geq 0$ and a diagram $$ \xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_S \ar[dl] \\ & S & } $$ where $X' \to \mathbf{P}^n_S$ is an immersion, and $\pi : X' \to X$ is proper and surjective. Moreover, there exists an open dense subscheme $U \subset X$ such that $\pi^{-1}(U) \to U$ is an isomorphism of schemes.


limits-lemma-refined-valuative-criterion-universally-closed

Let $f : X \to S$ and $h : U \to X$ be morphisms of schemes. Assume that $S$ is locally Noetherian, that $f$ and $h$ are of finite type, and that $h(U)$ is dense in $X$. If given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^f \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & S } $$ where $A$ is a discrete valuation ring with field of fractions $K$, there exists a unique dotted arrow making the diagram commute, then $f$ is proper.


limits-lemma-proper-top-cohomology-finite-presentation

Let $f : X \to Y$ be a morphism of schemes. Let $d \geq 0$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $f$ is a proper morphism of finite presentation all of whose fibres have dimension $\leq d$, \item $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation. \end{enumerate} Then $R^df_*\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation.


limits-lemma-morphism-good-diagram-proper

Notation and assumptions as in Lemma \ref{lemma-morphism-good-diagram}. If $f$ is proper, then there exists an $i_3 \geq i_0$ such that for $i \geq i_3$ we have $f_i$ is proper.