stacks-morphisms-remark-implication

Let $\mathcal{P}, \mathcal{P}'$ be properties of morphisms of algebraic spaces which are smooth local on the source-and-target. Suppose that we have $\mathcal{P} \Rightarrow \mathcal{P}'$ for morphisms of algebraic spaces. Then we also have $\mathcal{P} \Rightarrow \mathcal{P}'$ for the properties of morphisms of algebraic stacks defined in Definition \ref{definition-P} using $\mathcal{P}$ and $\mathcal{P}'$. This is clear from the definition.


stacks-morphisms-remark-property-automorphism-groups

Let $P$ be a property of algebraic spaces over fields which is invariant under ground field extensions. Given an algebraic stack $\mathcal{X}$ and $x \in |\mathcal{X}|$, we say the automorphism group of $\mathcal{X}$ at $x$ has $P$ if the equivalent conditions of Lemma \ref{lemma-property-automorphism-groups} are satisfied. For example, we say {\it the automorphism group of $\mathcal{X}$ at $x$ is finite}, if $G_x \to \Spec(k)$ is finite whenever $x : \Spec(k) \to \mathcal{X}$ is a representative of $x$. Similarly for smooth, proper, etc. (There is clearly an abuse of language going on here, but we believe it will not cause confusion or imprecision.)


stacks-morphisms-remark-identify-automorphism-groups

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $x \in |\mathcal{X}|$ be a point. To indicate the equivalent conditions of Lemma \ref{lemma-iso-automorphism-groups} are satisfied for $f$ and $x$ in the literature the terminology {\it $f$ is stabilizer preserving at $x$} or {\it $f$ is fixed-point reflecting at $x$} is used. We prefer to say {\it $f$ induces an isomorphism between automorphism groups at $x$ and $f(x)$}.


stacks-morphisms-remark-order-type

We can wonder about the order type of the canonical stratifications which occur as output of the stratifications of type (a) constructed in Lemma \ref{lemma-every-point-in-a-stratum}. A natural guess is that the well-ordered set $I$ has {\it cardinality} at most $\aleph_0$. We have no idea if this is true or false. If you do please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.


stacks-morphisms-remark-etale-smooth-composition

Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is \'etale-smooth local on the source-and-target and stable under composition. Then the property of DM morphisms of algebraic stacks defined in Definition \ref{definition-etale-smooth-P} is stable under composition. Namely, let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be DM morphisms of algebraic stacks having property $\mathcal{P}$. By Lemma \ref{lemma-composition-separated} the composition $g \circ f$ is DM. Choose an algebraic space $W$ and a surjective smooth morphism $W \to \mathcal{Z}$. Choose an algebraic space $V$ and a surjective \'etale morphism $V \to \mathcal{Y} \times_\mathcal{Z} W$ (Lemma \ref{lemma-DM}). Choose an algebraic space $U$ and a surjective \'etale morphism $U \to \mathcal{X} \times_\mathcal{Y} V$. Then the morphisms $V \to W$ and $U \to V$ have property $\mathcal{P}$ by definition. Whence $U \to W$ has property $\mathcal{P}$ as we assumed that $\mathcal{P}$ is stable under composition. Thus, by definition again, we see that $g \circ f : \mathcal{X} \to \mathcal{Z}$ has property $\mathcal{P}$.


stacks-morphisms-remark-etale-smooth-base-change

Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is \'etale-smooth local on the source-and-target and stable under base change. Then the property of DM morphisms of algebraic stacks defined in Definition \ref{definition-etale-smooth-P} is stable under arbitrary base change. Namely, let $f : \mathcal{X} \to \mathcal{Y}$ be a DM morphism of algebraic stacks and $g : \mathcal{Y}' \to \mathcal{Y}$ be a morphism of algebraic stacks and assume $f$ has property $\mathcal{P}$. Then the base change $\mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \to \mathcal{Y}'$ is a DM morphism by Lemma \ref{lemma-base-change-separated}. Choose an algebraic space $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Choose an algebraic space $U$ and a surjective \'etale morphism $U \to \mathcal{X} \times_\mathcal{Y} V$ (Lemma \ref{lemma-DM}). Finally, choose an algebraic space $V'$ and a surjective and smooth morphism $V' \to \mathcal{Y}' \times_\mathcal{Y} V$. Then the morphism $U \to V$ has property $\mathcal{P}$ by definition. Whence $V' \times_V U \to V'$ has property $\mathcal{P}$ as we assumed that $\mathcal{P}$ is stable under base change. Considering the diagram $$ \xymatrix{ V' \times_V U \ar[r] \ar[d] & \mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ V' \ar[r] & \mathcal{Y}' \ar[r] & \mathcal{Y} } $$ we see that the left top horizontal arrow is surjective and $$ V' \times_V U \to V' \times_\mathcal{Y} (\mathcal{Y}' \times_{\mathcal{Y}'} \mathcal{X}) = V' \times_V (\mathcal{X} \times_\mathcal{Y} V) $$ is \'etale as a base change of $U \to \mathcal{X} \times_\mathcal{Y} V$, whence by definition we see that the projection $\mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \to \mathcal{Y}'$ has property $\mathcal{P}$.


stacks-morphisms-remark-etale-smooth-implication

Let $\mathcal{P}, \mathcal{P}'$ be properties of morphisms of algebraic spaces which are \'etale-smooth local on the source-and-target. Suppose that we have $\mathcal{P} \Rightarrow \mathcal{P}'$ for morphisms of algebraic spaces. Then we also have $\mathcal{P} \Rightarrow \mathcal{P}'$ for the properties of morphisms of algebraic stacks defined in Definition \ref{definition-etale-smooth-P} using $\mathcal{P}$ and $\mathcal{P}'$. This is clear from the definition.


stacks-morphisms-remark-get-property-auts-from-diagonal

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $U \to \mathcal{X}$ be a morphism whose source is an algebraic space. Let $G \to H$ be the pullback of the morphism $\mathcal{I}_\mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{I}_\mathcal{Y}$ to $U$. If $\Delta_f$ is unramified, \'etale, etc, so is $G \to H$. This is true because $$ \xymatrix{ U \times_\mathcal{X} U \ar[r] \ar[d] & \mathcal{X} \ar[d]^{\Delta_f} \\ U \times_\mathcal{Y} U \ar[r] & \mathcal{X} \times_\mathcal{Y} \mathcal{X} } $$ is cartesian and the morphism $G \to H$ is the base change of the left vertical arrow by the diagonal $U \to U \times U$. Compare with the proof of Lemma \ref{lemma-separated-implies-isom}.


schemes-remark-not-reverse-open-closed

If $f : X \to Y$ is an immersion of schemes, then it is in general not possible to factor $f$ as an open immersion followed by a closed immersion. See Morphisms, Example \ref{morphisms-example-thibaut}.


schemes-remark-intersection-affine-opens

In general the intersection of two affine opens in $X$ is not affine open. See Example \ref{example-affine-space-zero-doubled}.


schemes-remark-reduced-induced-locally-closed

Let $X$ be a scheme. Let $T \subset X$ be a locally closed subset. In this situation we sometimes also use the phrase ``reduced induced scheme structure on $T$''. It refers to the reduced induced scheme structure from Definition \ref{definition-reduced-induced-scheme} when we view $T$ as a closed subset of the open subscheme  $X \setminus \partial T$ of $X$. Here $\partial T = \overline{T} \setminus T$ is the ``boundary'' of $T$ in the topological space of $X$.


schemes-remark-representable-locally-ringed

Suppose the functor $F$ is defined on all locally ringed spaces, and if conditions of Lemma \ref{lemma-glue-functors} are replaced by the following: \begin{enumerate} \item $F$ satisfies the sheaf property on the category of locally ringed spaces, \item there exists a set $I$ and a collection of subfunctors $F_i \subset F$ such that \begin{enumerate} \item each $F_i$ is representable by a scheme, \item each $F_i \subset F$ is representable by open immersions on the category of locally ringed spaces, and \item the collection $(F_i)_{i \in I}$ covers $F$ as a functor on the category of locally ringed spaces. \end{enumerate} \end{enumerate} We leave it to the reader to spell this out further. Then the end result is that the functor $F$ is representable in the category of locally ringed spaces and that the representing object is a scheme.


schemes-remark-fibre-product-schemes-locally-ringed

Using Remark \ref{remark-representable-locally-ringed} you can show that the fibre product of morphisms of schemes exists in the category of locally ringed spaces and is a scheme.


schemes-remark-quasi-compact-and-quasi-separated

The category of quasi-compact and quasi-separated schemes $\mathcal{C}$ has the following properties. If $X, Y \in \Ob(\mathcal{C})$, then any morphism of schemes $f : X \to Y$ is quasi-compact and quasi-separated by Lemmas \ref{lemma-quasi-compact-permanence} and \ref{lemma-compose-after-separated} with $Z = \Spec(\mathbf{Z})$. Moreover, if $X \to Y$ and $Z \to Y$ are morphisms $\mathcal{C}$, then $X \times_Y Z$ is an object of $\mathcal{C}$ too. Namely, the projection $X \times_Y Z \to Z$ is quasi-compact and quasi-separated as a base change of the morphism $Z \to Y$, see Lemmas \ref{lemma-separated-permanence} and \ref{lemma-quasi-compact-preserved-base-change}. Hence the composition $X \times_Y Z \to Z \to \Spec(\mathbf{Z})$ is quasi-compact and quasi-separated, see Lemmas \ref{lemma-separated-permanence} and \ref{lemma-composition-quasi-compact}.


injectives-remark-embedding

The Freyd-Mitchell embedding theorem says there exists a fully faithful exact functor from any abelian category $\mathcal{A}$ to the category of modules over a ring. Lemma \ref{lemma-embedding} is not quite as strong. But the result is suitable for the Stacks project as we have to understand sheaves of abelian groups on sites in detail anyway. Moreover, ``diagram chasing'' works in the category of abelian sheaves on $\mathcal{C}$, for example by working with sections over objects, or by working on the level of stalks using that $\mathcal{C}$ has enough points. To see how to deduce the Freyd-Mitchell embedding theorem from Lemma \ref{lemma-embedding} see Remark \ref{remark-embedding-freyd}.


injectives-remark-embedding-big

If $\mathcal{A}$ is a ``big'' abelian category, i.e., if $\mathcal{A}$ has a class of objects, then Lemma \ref{lemma-embedding} does not work. In this case, given any set of objects $E \subset \Ob(\mathcal{A})$ there exists an abelian full subcategory $\mathcal{A}' \subset \mathcal{A}$ such that $\Ob(\mathcal{A}')$ is a set and $E \subset \Ob(\mathcal{A}')$. Then one can apply Lemma \ref{lemma-embedding} to $\mathcal{A}'$. One can use this to prove that results depending on a diagram chase hold in $\mathcal{A}$.


injectives-remark-embedding-freyd

Let $\mathcal{C}$ be a site. Note that $\textit{Ab}(\mathcal{C})$ has enough injectives, see Theorem \ref{theorem-sheaves-injectives}. (In the case that $\mathcal{C}$ has enough points this is straightforward because  $p_*I$ is an injective sheaf if $I$ is an injective $\mathbf{Z}$-module and $p$ is a point.) Also, $\textit{Ab}(\mathcal{C})$ has a cogenerator (details omitted). Hence Lemma \ref{lemma-embedding} proves that we have a fully faithful, exact embedding $\mathcal{A} \to \mathcal{B}$ where $\mathcal{B}$ has a cogenerator and enough injectives. We can apply this to $\mathcal{A}^{opp}$ and we get a fully faithful exact functor $i : \mathcal{A} \to \mathcal{D} = \mathcal{B}^{opp}$ where $\mathcal{D}$ has enough projectives and a generator. Hence $\mathcal{D}$ has a projective generator $P$. Set $R = \Mor_\mathcal{D}(P, P)$. Then $$ \mathcal{A} \longrightarrow \text{Mod}_R, \quad X \longmapsto \Hom_\mathcal{D}(P, X). $$ One can check this is a fully faithful, exact functor. In other words, one retrieves the Freyd-Mitchell theorem mentioned in Remark \ref{remark-embedding} above.


injectives-remark-embed-exact-category

The arguments proving Lemmas \ref{lemma-site-abelian-category} and \ref{lemma-embedding} work also for {\it exact categories}, see \cite[Appendix A]{Buhler} and \cite[1.1.4]{BBD}. We quickly review this here and we add more details if we ever need it in the Stacks project. \medskip\noindent Let $\mathcal{A}$ be an additive category. A {\it kernel-cokernel} pair is a pair $(i, p)$ of morphisms of $\mathcal{A}$ with $i : A \to B$, $p : B \to C$ such that $i$ is the kernel of $p$ and $p$ is the cokernel of $i$. Given a set $\mathcal{E}$ of kernel-cokernel pairs we say $i : A \to B$ is an {\it admissible monomorphism} if $(i, p) \in \mathcal{E}$ for some morphism $p$. Similarly we say a morphism $p : B \to C$ is an {\it admissible epimorphism} if $(i, p) \in \mathcal{E}$ for some morphism $i$. The pair $(\mathcal{A}, \mathcal{E})$ is said to be an {\it exact category} if the following axioms hold \begin{enumerate} \item $\mathcal{E}$ is closed under isomorphisms of kernel-cokernel pairs, \item for any object $A$ the morphism $1_A$ is both an admissible epimorphism and an admissible monomorphism, \item admissible monomorphisms are stable under composition, \item admissible epimorphisms are stable under composition, \item the push-out of an admissible monomorphism $i : A \to B$ via any morphism $A \to A'$ exist and the induced morphism $i' : A' \to B'$ is an admissible monomorphism, and \item the base change of an admissible epimorphism $p : B \to C$ via any morphism $C' \to C$ exist and the induced morphism $p' : B' \to C'$ is an admissible epimorphism. \end{enumerate} Given such a structure let $\mathcal{C} = (\mathcal{A}, \text{Cov})$ where coverings (i.e., elements of $\text{Cov}$) are given by admissible epimorphisms. The axioms listed above immediately imply that this is a site. Consider the functor $$ F : \mathcal{A} \longrightarrow \textit{Ab}(\mathcal{C}), \quad X \longmapsto h_X $$ exactly as in Lemma \ref{lemma-embedding}. It turns out that this functor is fully faithful, exact, and reflects exactness. Moreover, any extension of objects in the essential image of $F$ is in the essential image of $F$.


injectives-remark-existence-D

In the chapter on derived categories we consistently work with ``small'' abelian categories (as is the convention in the Stacks project). For a ``big'' abelian category $\mathcal{A}$ it isn't clear that the derived category $D(\mathcal{A})$ exists because it isn't clear that morphisms in the derived category are sets. In general this isn't true, see Examples, Lemma \ref{examples-lemma-big-abelian-category}. However, if $\mathcal{A}$ is a Grothendieck abelian category, and given $K^\bullet, L^\bullet$ in $K(\mathcal{A})$, then by Theorem \ref{theorem-K-injective-embedding-grothendieck} there exists a quasi-isomorphism $L^\bullet \to I^\bullet$ to a K-injective complex $I^\bullet$ and Derived Categories, Lemma \ref{derived-lemma-K-injective} shows that $$ \Hom_{D(\mathcal{A})}(K^\bullet, L^\bullet) = \Hom_{K(\mathcal{A})}(K^\bullet, I^\bullet) $$ which is a set. Some examples of Grothendieck abelian categories are the category of modules over a ring, or more generally the category of sheaves of modules on a ringed site.


injectives-remark-direct-sum-product-derived

Let $R$ be a ring. Suppose that $M_n$, $n \in \mathbf{Z}$ are $R$-modules. Denote $E_n = M_n[-n] \in D(R)$. We claim that $E = \bigoplus M_n[-n]$ is {\it both} the direct sum and the product of the objects $E_n$ in $D(R)$. To see that it is the direct sum, take a look at the proof of Lemma \ref{lemma-derived-products}. To see that it is the direct product, take injective resolutions $M_n \to I_n^\bullet$. By the proof of Lemma \ref{lemma-derived-products} we have $$ \prod E_n = \prod I_n^\bullet[-n] $$ in $D(R)$. Since products in $\text{Mod}_R$ are exact, we see that $\prod I_n^\bullet$ is quasi-isomorphic to $E$. This works more generally in $D(\mathcal{A})$ where $\mathcal{A}$ is a Grothendieck abelian category with Ab4*.


injectives-remark-ext-into-filtered-complex

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $K^\bullet$ be a filtered complex of $\mathcal{A}$, see Homology, Definition \ref{homology-definition-filtered-complex}. For ease of notation denote $K$, $F^pK$, $\text{gr}^pK$ the object of $D(\mathcal{A})$ represented by $K^\bullet$, $F^pK^\bullet$, $\text{gr}^pK^\bullet$. Let $M \in D(\mathcal{A})$. Using Lemma \ref{lemma-K-injective-embedding-filtration} we can construct a spectral sequence $(E_r, d_r)_{r \geq 1}$ of bigraded objects of $\mathcal{A}$ with $d_r$ of bidgree $(r, -r + 1)$ and with $$ E_1^{p, q} = \Ext^{p + q}(M, \text{gr}^pK) $$ If for every $n$ we have $$ \Ext^n(M, F^pK) = 0 \text{ for } p \gg 0 \quad\text{and}\quad \Ext^n(M, F^pK) = \Ext^n(M, K) \text{ for } p \ll 0 $$ then the spectral sequence is bounded and converges to $\Ext^{p + q}(M, K)$. Namely, choose any complex $M^\bullet$ representing $M$, choose $j : K^\bullet \to J^\bullet$ as in the lemma, and consider the complex $$ \Hom^\bullet(M^\bullet, I^\bullet) $$ defined exactly as in More on Algebra, Section \ref{more-algebra-section-hom-complexes}. Setting $F^p\Hom^\bullet(M^\bullet, I^\bullet) = \Hom^\bullet(M^\bullet, F^pI^\bullet)$ we obtain a filtered complex. The spectral sequence of Homology, Section \ref{homology-section-filtered-complex} has differentials and terms as described above; details omitted. The boundedness and convergence follows from Homology, Lemma \ref{homology-lemma-ss-converges-trivial}.


injectives-remark-spectral-sequences-ext

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $M, K$ be objects of $D(\mathcal{A})$. For any choice of complex $K^\bullet$ representing $K$ we can use the filtration $F^pK^\bullet = \tau_{\leq -p}K^\bullet$ and the discussion in Remark \ref{remark-ext-into-filtered-complex} to get a spectral sequence with $$ E_1^{p, q} = \Ext^{2p + q}(M, H^{-p}(K)) $$ This spectral sequence is independent of the choice of complex $K^\bullet$ representing $K$. After renumbering $p = -j$ and $q = i + 2j$ we find a spectral sequence $(E'_r, d'_r)_{r \geq 2}$ with $d'_r$ of bidegree $(r, -r + 1)$, with $$ (E'_2)^{i, j} = \Ext^i(M, H^j(K)) $$ If $M \in D^-(\mathcal{A})$ and $K \in D^+(\mathcal{A})$ then both $E_r$ and $E'_r$ are bounded and converge to $\Ext^{p + q}(M, K)$. If we use the filtration $F^pK^\bullet = \sigma_{\geq p}K^\bullet$ then we get $$ E_1^{p, q} = \Ext^q(M, K^p) $$ If $M \in D^-(\mathcal{A})$ and $K^\bullet$ is bounded below, then this spectral sequence is bounded and converges to $\Ext^{p + q}(M, K)$.


injectives-remark-ext-from-filtered-complex

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $K \in D(\mathcal{A})$. Let $M^\bullet$ be a filtered complex of $\mathcal{A}$, see Homology, Definition \ref{homology-definition-filtered-complex}. For ease of notation denote $M$, $M/F^pM$, $\text{gr}^pM$ the object of $D(\mathcal{A})$ represented by $M^\bullet$, $M^\bullet/F^pM^\bullet$, $\text{gr}^pM^\bullet$. Dually to Remark \ref{remark-ext-into-filtered-complex} we can construct a spectral sequence $(E_r, d_r)_{r \geq 1}$ of bigraded objects of $\mathcal{A}$ with $d_r$ of bidgree $(r, -r + 1)$ and with $$ E_1^{p, q} = \Ext^{p + q}(\text{gr}^{-p}M, K) $$ If for every $n$ we have $$ \Ext^n(M/F^pM, K) = 0 \text{ for } p \ll 0 \quad\text{and}\quad \Ext^n(M/F^pM, K) = \Ext^n(M, K) \text{ for } p \gg 0 $$ then the spectral sequence is bounded and converges to $\Ext^{p + q}(M, K)$. Namely, choose a K-injective complex $I^\bullet$ with injective terms representing $K$, see Theorem \ref{theorem-K-injective-embedding-grothendieck}. Consider the complex $$ \Hom^\bullet(M^\bullet, I^\bullet) $$ defined exactly as in More on Algebra, Section \ref{more-algebra-section-hom-complexes}. Setting $$ F^p\Hom^\bullet(M^\bullet, I^\bullet) = \Hom^\bullet(M^\bullet/F^{-p + 1}M^\bullet, I^\bullet) $$ we obtain a filtered complex (note sign and shift in filtration). The spectral sequence of Homology, Section \ref{homology-section-filtered-complex} has differentials and terms as described above; details omitted. The boundedness and convergence follows from Homology, Lemma \ref{homology-lemma-ss-converges-trivial}.


injectives-remark-spectral-sequences-ext-variant

Let $\mathcal{A}$ be a Grothendieck abelian category. Let $M, K$ be objects of $D(\mathcal{A})$. For any choice of complex $M^\bullet$ representing $M$ we can use the filtration $F^pM^\bullet = \tau_{\leq -p}M^\bullet$ and the discussion in Remark \ref{remark-ext-into-filtered-complex} to get a spectral sequence with $$ E_1^{p, q} = \Ext^{2p + q}(H^p(M), K) $$ This spectral sequence is independent of the choice of complex $M^\bullet$ representing $M$. After renumbering $p = -j$ and $q = i + 2j$ we find a spectral sequence $(E'_r, d'_r)_{r \geq 2}$ with $d'_r$ of bidegree $(r, -r + 1)$, with $$ (E'_2)^{i, j} = \Ext^i(H^{-j}(M), K) $$ If $M \in D^-(\mathcal{A})$ and $K \in D^+(\mathcal{A})$ then $E_r$ and $E'_r$ are bounded and converge to $\Ext^{p + q}(M, K)$. If we use the filtration $F^pM^\bullet = \sigma_{\geq p}M^\bullet$ then we get $$ E_1^{p, q} = \Ext^q(M^{-p}, K) $$ If $K \in D^+(\mathcal{A})$ and $M^\bullet$ is bounded above, then this spectral sequence is bounded and converges to $\Ext^{p + q}(M, K)$.


divisors-remark-base-change-relative-assassin

With notation and assumptions as in Lemma \ref{lemma-base-change-relative-assassin} we see that it is always the case that $(g')^{-1}(\text{Ass}_{X/S}(\mathcal{F})) \supset \text{Ass}_{X'/S'}(\mathcal{F}')$. If the morphism $S' \to S$ is locally quasi-finite, then we actually have $$ (g')^{-1}(\text{Ass}_{X/S}(\mathcal{F})) = \text{Ass}_{X'/S'}(\mathcal{F}') $$ because in this case the field extensions $\kappa(s) \subset \kappa(s')$ are always finite. In fact, this holds more generally for any morphism $g : S' \to S$ such that all the field extensions $\kappa(s) \subset \kappa(s')$ are algebraic, because in this case all prime ideals of $\kappa(s') \otimes_{\kappa(s)} \kappa(x)$ are maximal (and minimal) primes, see Algebra, Lemma \ref{algebra-lemma-integral-over-field}.


divisors-remark-different-reflexive

If $X$ is a scheme of finite type over a field, then sometimes a different notion of reflexive modules is used (see for example \cite[bottom of page 5 and Definition 1.1.9]{HL}). This other notion uses $R\SheafHom$ into a dualizing complex $\omega_X^\bullet$ instead of into $\mathcal{O}_X$ and should probably have a different name because it can be different when $X$ is not Gorenstein. For example, if $X = \Spec(k[t^3, t^4, t^5])$, then a computation shows the dualizing sheaf $\omega_X$ is not reflexive in our sense, but it is reflexive in the other sense as $\omega_X \to \SheafHom(\SheafHom(\omega_X, \omega_X), \omega_X)$ is an isomorphism.


divisors-remark-tensor

Let $X$ be an integral locally Noetherian scheme. Thanks to Lemma \ref{lemma-dual-reflexive} we know that the reflexive hull $\mathcal{F}^{**}$ of a coherent $\mathcal{O}_X$-module is coherent reflexive. Consider the category $\mathcal{C}$ of coherent reflexive $\mathcal{O}_X$-modules. Taking reflexive hulls gives a left adjoint to the inclusion functor $\mathcal{C} \to \textit{Coh}(\mathcal{O}_X)$. Observe that $\mathcal{C}$ is an additive category with kernels and cokernels. Namely, given $\varphi : \mathcal{F} \to \mathcal{G}$ in $\mathcal{C}$, the usual kernel $\Ker(\varphi)$ is reflexive (Lemma \ref{lemma-sequence-reflexive}) and the reflexive hull $\Coker(\varphi)^{**}$ of the usual cokernel is the cokernel in $\mathcal{C}$. Moreover $\mathcal{C}$ inherits a tensor product $$ \mathcal{F} \otimes_\mathcal{C} \mathcal{G} = (\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G})^{**} $$ which is associative and symmetric. There is an internal Hom in the sense that for any three objects $\mathcal{F}, \mathcal{G}, \mathcal{H}$ of $\mathcal{C}$ we have the identity $$ \Hom_\mathcal{C}(\mathcal{F} \otimes_\mathcal{C} \mathcal{G}, \mathcal{H}) = \Hom_\mathcal{C}(\mathcal{F}, \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{H})) $$ see Modules, Lemma \ref{modules-lemma-internal-hom}. In $\mathcal{C}$ every object $\mathcal{F}$ has a {\it dual object} $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{O}_X)$. Without further conditions on $X$ it can happen that $$ \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) \not \cong \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{O}_X) \otimes_\mathcal{C} \mathcal{G} \quad\text{and}\quad \mathcal{F} \otimes_\mathcal{C} \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{O}_X) \not \cong \mathcal{O}_X $$ for $\mathcal{F}, \mathcal{G}$ of rank $1$ in $\mathcal{C}$. To make an example let $X = \Spec(R)$ where $R$ is as in More on Algebra, Example \ref{more-algebra-example-ring-not-S2} and let $\mathcal{F}, \mathcal{G}$ be the modules corresponding to $M$. Computation omitted.


divisors-remark-ses-regular-section

Let $X$ be a scheme, $\mathcal{L}$ an invertible $\mathcal{O}_X$-module, and $s$ a regular section of $\mathcal{L}$. Then the zero scheme $D = Z(s)$ is an effective Cartier divisor on $X$ and there are short exact sequences $$ 0 \to \mathcal{O}_X \to \mathcal{L} \to i_*(\mathcal{L}|_D) \to 0 \quad\text{and}\quad 0 \to \mathcal{L}^{\otimes -1} \to \mathcal{O}_X \to i_*\mathcal{O}_D \to 0. $$ Given an effective Cartier divisor $D \subset X$ using Lemmas \ref{lemma-characterize-OD} and \ref{lemma-conormal-effective-Cartier-divisor} we get $$ 0 \to \mathcal{O}_X \to \mathcal{O}_X(D) \to i_*(\mathcal{N}_{D/X}) \to 0 \quad\text{and}\quad 0 \to \mathcal{O}_X(-D) \to \mathcal{O}_X \to i_*(\mathcal{O}_D) \to 0 $$


divisors-remark-affine-punctured-spectrum-standard-proof

If $(A, \mathfrak m)$ is a Noetherian local normal domain of dimension $\geq 2$ and $U$ is the punctured spectrum of $A$, then $\Gamma(U, \mathcal{O}_U) = A$. This algebraic version of Hartogs's theorem follows from the fact that $A = \bigcap_{\text{height}(\mathfrak p) = 1} A_\mathfrak p$ we've seen in Algebra, Lemma \ref{algebra-lemma-normal-domain-intersection-localizations-height-1}. Thus in this case $U$ cannot be affine (since it would force $\mathfrak m$ to be a point of $U$). This is often used as the starting point of the proof of Lemma \ref{lemma-affine-punctured-spec}. To reduce the case of a general Noetherian local ring to this case, we first complete (to get a Nagata local ring), then replace $A$ by $A/\mathfrak q$ for a suitable minimal prime, and then normalize. Each of these steps does not change the dimension and we obtain a contradiction. You can skip the completion step, but then the normalization in general is not a Noetherian domain. However, it is still a Krull domain of the same dimension (this is proved using Krull-Akizuki) and one can apply the same argument.


divisors-remark-affine-puctured-spectrum-general

It is not clear how to characterize the non-Noetherian local rings $(A, \mathfrak m)$ whose punctured spectrum is affine. Such a ring has a finitely generated ideal $I$ with $\mathfrak m = \sqrt{I}$. Of course if we can take $I$ generated by $1$ element, then $A$ has an affine puncture spectrum; this gives lots of non-Noetherian examples. Conversely, it follows from the argument in the proof of Lemma \ref{lemma-affine-punctured-spec} that such a ring cannot possess a nonzerodivisor $f \in \mathfrak m$ with $H^0_I(A/fA) = 0$ (so $A$ cannot have a regular sequence of length $2$). Moreover, the same holds for any ring $A'$ which is the target of a local homomorphism of local rings $A \to A'$ such that $\mathfrak m_{A'} = \sqrt{\mathfrak mA'}$.


divisors-remark-not-always-extra-permanence

In the situation of Lemma \ref{lemma-extra-permanence-regular-immersion-noetherian} parts (1), (2), (3) are {\bf not} equivalent to ``$j \circ i$ and $j$ are regular immersions at $z$ and $y$''. An example is $X = \mathbf{A}^1_k = \Spec(k[x])$, $Y = \Spec(k[x]/(x^2))$ and $Z = \Spec(k[x]/(x))$.


divisors-remark-relative-regular-immersion-elements

The codimension of a relative quasi-regular immersion, if it is constant, does not change after a base change. In fact, if we have a ring map $A \to B$ and a quasi-regular sequence $f_1, \ldots, f_r \in B$ such that $B/(f_1, \ldots, f_r)$ is flat over $A$, then for any ring map $A \to A'$ we have a quasi-regular sequence  $f_1 \otimes 1, \ldots, f_r \otimes 1$ in $B' = B \otimes_A A'$ by More on Algebra, Lemma \ref{more-algebra-lemma-relative-regular-immersion-algebra} (which was used in the proof of Lemma \ref{lemma-relative-regular-immersion} above). Now the proof of Lemma \ref{lemma-relative-regular-immersion-flat-in-neighbourhood} shows that if $A \to B$ is flat and locally of finite presentation, then for every prime ideal $\mathfrak q' \subset B'$ the sequence $f_1 \otimes 1, \ldots, f_r \otimes 1$ is even a regular sequence in the local ring $B'_{\mathfrak q'}$.


divisors-remark-structure-sheaf-Xs

Let $A$ be a Noetherian normal domain. Let $M$ be a rank $1$ finite reflexive $A$-module. Let $s \in M$ be nonzero. Let $\mathfrak p_1, \ldots, \mathfrak p_r$ be the height $1$ primes of $A$ in the support of $M/As$. Then the open $U$ of Lemma \ref{lemma-structure-sheaf-Xs} is $$ U = \Spec(A) \setminus \left(V(\mathfrak p_1) \cup \ldots \cup \mathfrak p_r)\right) $$ by Lemma \ref{lemma-Xs-codim-complement}. Moreover, if $M^{[n]}$ denotes the reflexive hull of $M \otimes_A \ldots \otimes_A M$ ($n$-factors), then $$ \Gamma(U, \mathcal{O}_U) = \colim M^{[n]} $$ according to Lemma \ref{lemma-structure-sheaf-Xs}.


spaces-remark-list-properties-stable-base-change

Here is a list of properties/types of morphisms which are {\it stable under arbitrary base change}: \begin{enumerate} \item closed, open, and locally closed immersions, see Schemes, Lemma \ref{schemes-lemma-base-change-immersion}, \item quasi-compact, see Schemes, Lemma \ref{schemes-lemma-quasi-compact-preserved-base-change}, \item universally closed, see Schemes, Definition \ref{schemes-definition-universally-closed}, \item (quasi-)separated, see Schemes, Lemma \ref{schemes-lemma-separated-permanence}, \item monomorphism, see Schemes, Lemma \ref{schemes-lemma-base-change-monomorphism} \item surjective, see Morphisms, Lemma \ref{morphisms-lemma-base-change-surjective}, \item universally injective, see Morphisms, Lemma \ref{morphisms-lemma-universally-injective}, \item affine, see Morphisms, Lemma \ref{morphisms-lemma-base-change-affine}, \item quasi-affine, see Morphisms, Lemma \ref{morphisms-lemma-base-change-quasi-affine}, \item (locally) of finite type, see Morphisms, Lemma \ref{morphisms-lemma-base-change-finite-type}, \item (locally) quasi-finite, see Morphisms, Lemma \ref{morphisms-lemma-base-change-quasi-finite}, \item (locally) of finite presentation, see Morphisms, Lemma \ref{morphisms-lemma-base-change-finite-presentation}, \item locally of finite type of relative dimension $d$, see Morphisms, Lemma \ref{morphisms-lemma-base-change-relative-dimension-d}, \item universally open, see Morphisms, Definition \ref{morphisms-definition-open}, \item flat, see Morphisms, Lemma \ref{morphisms-lemma-base-change-flat}, \item syntomic, see Morphisms, Lemma \ref{morphisms-lemma-base-change-syntomic}, \item smooth, see Morphisms, Lemma \ref{morphisms-lemma-base-change-smooth}, \item unramified (resp.\ G-unramified), see Morphisms, Lemma \ref{morphisms-lemma-base-change-unramified}, \item \'etale, see Morphisms, Lemma \ref{morphisms-lemma-base-change-etale}, \item proper, see Morphisms, Lemma \ref{morphisms-lemma-base-change-proper}, \item H-projective, see Morphisms, Lemma \ref{morphisms-lemma-H-projective-base-change}, \item (locally) projective, see Morphisms, Lemma \ref{morphisms-lemma-base-change-projective}, \item finite or integral, see Morphisms, Lemma \ref{morphisms-lemma-base-change-finite}, \item finite locally free, see Morphisms, Lemma \ref{morphisms-lemma-base-change-finite-locally-free}, \item universally submersive, see Morphisms, Lemma \ref{morphisms-lemma-base-change-universally-submersive}, \item universal homeomorphism, see Morphisms, Lemma \ref{morphisms-lemma-base-change-universal-homeomorphism}. \end{enumerate} Add more as needed.


spaces-remark-list-properties-stable-composition

Of the properties of morphisms which are stable under base change (as listed in Remark \ref{remark-list-properties-stable-base-change}) the following are also {\it stable under compositions}: \begin{enumerate} \item closed, open and locally closed immersions, see Schemes, Lemma \ref{schemes-lemma-composition-immersion}, \item quasi-compact, see Schemes, Lemma \ref{schemes-lemma-composition-quasi-compact}, \item universally closed, see Morphisms, Lemma \ref{morphisms-lemma-composition-proper}, \item (quasi-)separated, see Schemes, Lemma \ref{schemes-lemma-separated-permanence}, \item monomorphism, see Schemes, Lemma \ref{schemes-lemma-composition-monomorphism}, \item surjective, see Morphisms, Lemma \ref{morphisms-lemma-composition-surjective}, \item universally injective, see Morphisms, Lemma \ref{morphisms-lemma-composition-universally-injective}, \item affine, see Morphisms, Lemma \ref{morphisms-lemma-composition-affine}, \item quasi-affine, see Morphisms, Lemma \ref{morphisms-lemma-composition-quasi-affine}, \item (locally) of finite type, see Morphisms, Lemma \ref{morphisms-lemma-composition-finite-type}, \item (locally) quasi-finite, see Morphisms, Lemma \ref{morphisms-lemma-composition-quasi-finite}, \item (locally) of finite presentation, see Morphisms, Lemma \ref{morphisms-lemma-composition-finite-presentation}, \item universally open, see Morphisms, Lemma \ref{morphisms-lemma-composition-open}, \item flat, see Morphisms, Lemma \ref{morphisms-lemma-composition-flat}, \item syntomic, see Morphisms, Lemma \ref{morphisms-lemma-composition-syntomic}, \item smooth, see Morphisms, Lemma \ref{morphisms-lemma-composition-smooth}, \item unramified (resp.\ G-unramified), see Morphisms, Lemma \ref{morphisms-lemma-composition-unramified}, \item \'etale, see Morphisms, Lemma \ref{morphisms-lemma-composition-etale}, \item proper, see Morphisms, Lemma \ref{morphisms-lemma-composition-proper}, \item H-projective, see Morphisms, Lemma \ref{morphisms-lemma-H-projective-composition}, \item finite or integral, see Morphisms, Lemma \ref{morphisms-lemma-composition-finite}, \item finite locally free, see Morphisms, Lemma \ref{morphisms-lemma-composition-finite-locally-free}, \item universally submersive, see Morphisms, Lemma \ref{morphisms-lemma-composition-universally-submersive}, \item universal homeomorphism, see Morphisms, Lemma \ref{morphisms-lemma-composition-universal-homeomorphism}. \end{enumerate} Add more as needed.


spaces-remark-list-properties-fpqc-local-base

Of the properties mentioned which are stable under base change (as listed in Remark \ref{remark-list-properties-stable-base-change}) the following are also {\it fpqc local on the base} (and a fortiori fppf local on the base): \begin{enumerate} \item for immersions we have this for \begin{enumerate} \item closed immersions, see Descent, Lemma \ref{descent-lemma-descending-property-closed-immersion}, \item open immersions, see Descent, Lemma \ref{descent-lemma-descending-property-open-immersion}, and \item quasi-compact immersions, see Descent, Lemma \ref{descent-lemma-descending-property-quasi-compact-immersion}, \end{enumerate} \item quasi-compact, see Descent, Lemma \ref{descent-lemma-descending-property-quasi-compact}, \item universally closed, see Descent, Lemma \ref{descent-lemma-descending-property-universally-closed}, \item (quasi-)separated, see Descent, Lemmas \ref{descent-lemma-descending-property-quasi-separated}, and \ref{descent-lemma-descending-property-separated}, \item monomorphism, see Descent, Lemma \ref{descent-lemma-descending-property-monomorphism}, \item surjective, see Descent, Lemma \ref{descent-lemma-descending-property-surjective}, \item universally injective, see Descent, Lemma \ref{descent-lemma-descending-property-universally-injective}, \item affine, see Descent, Lemma \ref{descent-lemma-descending-property-affine}, \item quasi-affine, see Descent, Lemma \ref{descent-lemma-descending-property-quasi-affine}, \item (locally) of finite type, see Descent, Lemmas \ref{descent-lemma-descending-property-locally-finite-type}, and \ref{descent-lemma-descending-property-finite-type}, \item (locally) quasi-finite, see Descent, Lemma \ref{descent-lemma-descending-property-quasi-finite}, \item (locally) of finite presentation, see Descent, Lemmas \ref{descent-lemma-descending-property-locally-finite-presentation}, and \ref{descent-lemma-descending-property-finite-presentation}, \item locally of finite type of relative dimension $d$, see Descent, Lemma \ref{descent-lemma-descending-property-relative-dimension-d}, \item universally open, see Descent, Lemma \ref{descent-lemma-descending-property-universally-open}, \item flat, see Descent, Lemma \ref{descent-lemma-descending-property-flat}, \item syntomic, see Descent, Lemma \ref{descent-lemma-descending-property-syntomic}, \item smooth, see Descent, Lemma \ref{descent-lemma-descending-property-smooth}, \item unramified (resp.\ G-unramified), see Descent, Lemma \ref{descent-lemma-descending-property-unramified}, \item \'etale, see Descent, Lemma \ref{descent-lemma-descending-property-etale}, \item proper, see Descent, Lemma \ref{descent-lemma-descending-property-proper}, \item finite or integral, see Descent, Lemma \ref{descent-lemma-descending-property-finite}, \item finite locally free, see Descent, Lemma \ref{descent-lemma-descending-property-finite-locally-free}, \item universally submersive, see Descent, Lemma \ref{descent-lemma-descending-property-universally-submersive}, \item universal homeomorphism, see Descent, Lemma \ref{descent-lemma-descending-property-universal-homeomorphism}. \end{enumerate} Note that the property of being an ``immersion'' may not be fpqc local on the base, but in Descent, Lemma \ref{descent-lemma-descending-fppf-property-immersion} we proved that it is fppf local on the base.


spaces-remark-warning

Consider the property $\mathcal{P}=$``surjective''. In this case there could be some ambiguity if we say ``let $F \to G$ be a surjective map''. Namely, we could mean the notion defined in Definition \ref{definition-relative-representable-property} above, or we could mean a surjective map of presheaves, see Sites, Definition \ref{sites-definition-presheaves-injective-surjective}, or, if both $F$ and $G$ are sheaves, we could mean a surjective map of sheaves, see Sites, Definition \ref{sites-definition-sheaves-injective-surjective}, If not mentioned otherwise when discussing morphisms of algebraic spaces we will always mean the first. See Lemma \ref{lemma-surjective-flat-locally-finite-presentation} for a case where surjectivity implies surjectivity as a map of sheaves.


topology-remark-quasi-components

\begin{reference} \cite[Example 6.1.24]{Engelking} \end{reference} Let $X$ be a topological space and $x \in X$. Let $Z \subset X$ be the connected component of $X$ passing through $x$. Consider the intersection $E$ of all open and closed subsets of $X$ containing $x$. It is clear that $Z \subset E$. In general $Z \not = E$. For example, let $X = \{x, y, z_1, z_2, \ldots\}$ with the topology with the following basis of opens, $\{z_n\}$, $\{x, z_n, z_{n + 1}, \ldots\}$, and $\{y, z_n, z_{n + 1}, \ldots\}$ for all $n$. Then $Z = \{x\}$ and $E = \{x, y\}$. We omit the details.


topology-remark-lemma-literature

Lemma \ref{lemma-characterize-quasi-compact} is a combination of \cite[I, p. 75, Lemme 1]{Bourbaki} and \cite[I, p. 76, Corollaire 1]{Bourbaki}.


topology-remark-proof-literature

Here are some references to the literature. In \cite[I, p. 75, Theorem 1]{Bourbaki} you can find: (2) $\Leftrightarrow$ (4). In \cite[I, p. 77, Proposition 6]{Bourbaki} you can find: (2) $\Rightarrow$ (1). Of course, trivially we have (1) $\Rightarrow$ (4). Thus (1), (2) and (4) are equivalent. Fan Zhou claimed and proved that (3) and (4) are equivalent; let me know if you find a reference in the literature.


topology-remark-obstruction-to-dimension-function

Combining Lemmas \ref{lemma-dimension-function-unique} and \ref{lemma-locally-dimension-function} we see that on a catenary, locally Noetherian, sober topological space the obstruction to having a dimension function is an element of $H^1(X, \mathbf{Z})$.


topology-remark-size-projective-cover

Let $X$ be a quasi-compact Hausdorff space. Let $\kappa$ be an infinite cardinal bigger or equal than the cardinality of $X$. Then the cardinality of the minimal quasi-compact, Hausdorff, extremally disconnected cover $X' \to X$ (Lemma \ref{lemma-existence-projective-cover}) is at most $2^{2^\kappa}$. Namely, choose a subset $S \subset X'$ mapping bijectively to $X$. By minimality of $X'$ the set $S$ is dense in $X'$. Thus $|X'| \leq 2^{2^\kappa}$ by Lemma \ref{lemma-dense-image}.


topology-remark-locally-finite-stratification

Given a locally finite stratification $X = \coprod X_i$ of a topological space $X$, we obtain a family of closed subsets $Z_i = \bigcup_{j \leq i} X_j$ of $X$ indexed by $I$ such that $$ Z_i \cap Z_j = \bigcup\nolimits_{k \leq i, j} Z_k $$ Conversely, given closed subsets $Z_i \subset X$ indexed by a partially ordered set $I$ such that $X = \bigcup Z_i$, such that every point has a neighbourhood meeting only finitely many $Z_i$, and such that the displayed formula holds, then we obtain a locally finite stratification of $X$ by setting $X_i = Z_i \setminus \bigcup_{j < i} Z_j$.


hypercovering-remark-hypercoverings-really-set

The lemma does not just say that there is a cofinal system of choices of hypercoverings that is a set, but that really the hypercoverings form a set.


hypercovering-remark-P-covering

A useful special case of Lemmas \ref{lemma-add-simplices} and \ref{lemma-degeneracy-maps-coverings} is the following. Suppose we have a category $\mathcal{C}$ having fibre products. Let $P \subset \text{Arrows}(\mathcal{C})$ be a subset stable under base change, stable under composition, and containing all isomorphisms. Then one says a {\it $P$-hypercovering} is an augmentation $a : U \to X$ from a simplicial object of $\mathcal{C}$ such that \begin{enumerate} \item $U_0 \to X$ is in $P$, \item $U_1 \to U_0 \times_X U_0$ is in $P$, \item $U_{n + 1} \to (\text{cosk}_n\text{sk}_n U)_{n + 1}$ is in $P$ for $n \geq 1$. \end{enumerate} The category $\mathcal{C}/X$ has all finite limits, hence the coskeleta used in the formulation above exist (see Categories, Lemma \ref{categories-lemma-finite-limits-exist}). Then we claim that the morphisms $U_n \to X$ and $d^n_i : U_n \to U_{n - 1}$ are in $P$. This follows from the aforementioned lemmas by turning $\mathcal{C}$ into a site whose coverings are $\{f : V \to U\}$ with $f \in P$ and taking $K$ given by $K_n = \{U_n \to X\}$.


hypercovering-remark-contractible-category

Note that the crux of the proof is to use Lemma \ref{lemma-add-simplices}. This lemma is completely general and does not care about the exact shape of the simplicial sets (as long as they have only finitely many nondegenerate simplices). It seems altogether reasonable to expect a result of the following kind: Given any morphism $a : K \times \partial \Delta[k] \to L$, with $K$ and $L$ hypercoverings, there exists a morphism of hypercoverings $c : K' \to K$ and a morphism  $g : K' \times \Delta[k] \to L$ such that $g|_{K' \times \partial \Delta[k]} = a \circ (c \times \text{id}_{\partial \Delta[k]})$. In other words, the category of hypercoverings is in a suitable sense contractible.


hypercovering-remark-not-covering-set

One feature of this description is that if one of the multiple intersections $U_{i_0} \cap \ldots \cap U_{i_{n + 1}}$ is empty then the covering on the right hand side may be the empty covering. Thus it is not automatically the case that the maps $I_{n + 1} \to (\text{cosk}_n\text{sk}_n I)_{n + 1}$ are surjective. This means that the geometric realization of $I$ may be an interesting (non-contractible) space. \medskip\noindent In fact, let $I'_n \subset I_n$ be the subset consisting of those simplices $i \in I_n$ such that $U_i \not = \emptyset$. It is easy to see that $I' \subset I$ is a subsimplicial set, and that $(I', \{U_i\})$ is a hypercovering. Hence we can always refine a hypercovering to a hypercovering where none of the opens $U_i$ is empty.


hypercovering-remark-repackage-into-simplicial-space

Let us repackage this information in yet another way. Namely, suppose that $(I, \{U_i\})$ is a hypercovering of the topological space $X$. Given this data we can construct a simplicial topological space $U_\bullet$ by setting $$ U_n = \coprod\nolimits_{i \in I_n} U_i, $$ and where for given $\varphi : [n] \to [m]$ we let morphisms $U(\varphi) : U_n \to U_m$ be the morphism coming from the inclusions $U_i \subset U_{\varphi(i)}$ for $i \in I_n$. This simplicial topological space comes with an augmentation $\epsilon : U_\bullet \to X$. With this morphism the simplicial space $U_\bullet$ becomes a hypercovering of $X$ along which one has cohomological descent in the sense of \cite[Expos\'e Vbis]{SGA4}. In other words, $H^n(U_\bullet, \epsilon^*\mathcal{F}) = H^n(X, \mathcal{F})$. (Insert future reference here to cohomology over simplicial spaces and cohomological descent formulated in those terms.) Suppose that $\mathcal{F}$ is an abelian sheaf on $X$. In this case the spectral sequence of Lemma \ref{lemma-cech-spectral-sequence} becomes the spectral sequence with $E_1$-term $$ E_1^{p, q} = H^q(U_p, \epsilon_q^*\mathcal{F}) \Rightarrow H^{p + q}(U_\bullet, \epsilon^*\mathcal{F}) = H^{p + q}(X, \mathcal{F}) $$ comparing the total cohomology of $\epsilon^*\mathcal{F}$ to the cohomology groups of $\mathcal{F}$ over the pieces of $U_\bullet$. (Insert future reference to this spectral sequence here.)


hypercovering-remark-taking-disjoint-unions

Let $\mathcal{C}$ be a site. Let $K$ and $L$ be objects of $\text{SR}(\mathcal{C})$. Write $K = \{U_i\}_{i \in I}$ and $L = \{V_j\}_{j \in J}$. Assume $U = \coprod_{i \in I} U_i$ and $V = \coprod_{j \in J} V_j$ exist. Then we get $$ \Mor_{\text{SR}(\mathcal{C})}(K, L) \longrightarrow \Mor_\mathcal{C}(U, V) $$ as follows. Given $f : K \to L$ given by $\alpha : I \to J$ and $f_i : U_i \to V_{\alpha(i)}$ we obtain a transformation of functors $$ \Mor_\mathcal{C}(V, -) = \prod\nolimits_{j \in J} \Mor_\mathcal{C}(V_j, -) \to \prod\nolimits_{i \in I} \Mor_\mathcal{C}(U_i, -) = \Mor_\mathcal{C}(U, -) $$ sending $(g_j)_{j \in J}$ to $(g_{\alpha(i)} \circ f_i)_{i \in I}$. Hence the Yoneda lemma produces the corresponding map $U \to V$. Of course, $U \to V$ maps the summand $U_i$ into the summand $V_{\alpha(i)}$ via the morphism $f_i$.


hypercovering-remark-take-unions-hypercovering

Let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has fibre products and equalizers and let $K$ be a hypercovering. Write $K_n = \{U_{n, i}\}_{i \in I_n}$. Suppose that \begin{enumerate} \item[(a)] $U_n = \coprod_{i \in I_n} U_{n, i}$ exists, and \item[(b)] $\coprod_{i \in I_n} h_{U_{n, i}} \to h_{U_n}$ induces an isomorphism on sheafifications. \end{enumerate} Then we get another simplicial object $L$ of $\text{SR}(\mathcal{C})$ with $L_n = \{U_n\}$, see Remark \ref{remark-taking-disjoint-unions}. Now we claim that $L$ is a hypercovering. To see this we check conditions (1), (2), (3) of Definition \ref{definition-hypercovering-variant}. Condition (1) follows from (b) and (1) for $K$. Condition (2) follows in exactly the same way. Condition (3) follows because \begin{align*} F((\text{cosk}_n \text{sk}_n L)_{n + 1})^\# & = ((\text{cosk}_n \text{sk}_n F(L)^\#)_{n + 1}) \\ & = ((\text{cosk}_n \text{sk}_n F(K)^\#)_{n + 1}) \\ & = F((\text{cosk}_n \text{sk}_n K)_{n + 1})^\# \end{align*} for $n \geq 1$ and hence the condition for $K$ implies the condition for $L$ exactly as in (1) and (2). Note that $F$ commutes with connected limits and sheafification is exact proving the first and last equality; the middle equality follows as $F(K)^\# = F(L)^\#$ by (b).


hypercovering-remark-take-unions-hypercovering-X

Let $\mathcal{C}$ be a site. Let $X \in \Ob(\mathcal{C})$. Assume $\mathcal{C}$ has fibre products and let $K$ be a hypercovering of $X$. Write $K_n = \{U_{n, i}\}_{i \in I_n}$. Suppose that \begin{enumerate} \item[(a)] $U_n = \coprod_{i \in I_n} U_{n, i}$ exists, \item[(b)] given morphisms $(\alpha, f_i) : \{U_i\}_{i \in I} \to \{V_j\}_{j \in J}$ and $(\beta, g_k) : \{W_k\}_{k \in K} \to \{V_j\}_{j \in J}$ in $\text{SR}(\mathcal{C})$ such that $U = \coprod U_i$, $V = \coprod V_j$, and $W = \coprod W_j$ exist, then $U \times_V W = \coprod_{(i, j, k), \alpha(i) = j = \beta(k)} U_i \times_{V_j} W_k$, \item[(c)] if $(\alpha, f_i) : \{U_i\}_{i \in I} \to \{V_j\}_{j \in J}$ is a covering in the sense of Definition \ref{definition-covering-SR} and $U = \coprod U_i$ and $V = \coprod V_j$ exist, then the corresponding morphism $U \to V$ of Remark \ref{remark-taking-disjoint-unions} is a covering of $\mathcal{C}$. \end{enumerate} Then we get another simplicial object $L$ of $\text{SR}(\mathcal{C})$ with $L_n = \{U_n\}$, see Remark \ref{remark-taking-disjoint-unions}. Now we claim that $L$ is a hypercovering of $X$. To see this we check conditions (1), (2) of Definition \ref{definition-hypercovering}. Condition (1) follows from (c) and (1) for $K$ because (1) for $K$ says $K_0 = \{U_{0, i}\}_{i \in I_0}$ is a covering of $\{X\}$ in the sense of Definition \ref{definition-covering-SR}. Condition (2) follows because $\mathcal{C}/X$ has all finite limits hence $\text{SR}(\mathcal{C}/X)$ has all finite limits, and condition (b) says the construction of ``taking disjoint unions'' commutes with these fimite limits. Thus the morphism $$ L_{n + 1} \longrightarrow (\text{cosk}_n \text{sk}_n L)_{n + 1} $$ is a covering as it is the consequence of applying our ``taking disjoint unions'' functor to the morphism $$ K_{n + 1} \longrightarrow (\text{cosk}_n \text{sk}_n K)_{n + 1} $$ which is assumed to be a covering in the sense of Definition \ref{definition-covering-SR} by (2) for $K$. This makes sense because property (b) in particular assures us that if we start with a finite diagram of semi-representable objects over $X$ for which we can take disjoint unions, then the limit of the diagram in $\text{SR}(\mathcal{C}/X)$ still is a semi-representable object over $X$ for which we can take disjoint unions.


algebraic-remark-flat-fp-presentation

If the morphism $f : \mathcal{S}_U \to \mathcal{X}$ of Lemma \ref{lemma-stack-presentation} is only assumed surjective, flat and locally of finite presentation, then it will still be the case that $f_{can} : [U/R] \to \mathcal{X}$ is an equivalence. In this case the morphisms $s$, $t$ will be flat and locally of finite presentation, but of course not smooth in general.


sites-remark-big-presheaves

As already pointed out we may consider the category of presheaves with values in any of the ``big'' categories listed in Categories, Remark \ref{categories-remark-big-categories}. These will be ``big'' categories as well and they will be listed in the above mentioned remark as we go along.


sites-remark-functoriality-presheaves-values

Suppose that $\mathcal{A}$ is a category such that any diagram $\mathcal{I}_Y \to \mathcal{A}$ has a colimit in $\mathcal{A}$. In this case it is clear that there are functors $u^p$ and $u_p$, defined in exactly the same way as above, on the categories of presheaves with values in $\mathcal{A}$. Moreover, the adjointness of the pair $u^p$ and $u_p$ continues to hold in this setting.


sites-remark-no-big-sites

(On set theoretic issues -- skip on a first reading.) The main reason for introducing sites is to study the category of sheaves on a site, because it is the generalization of the category of sheaves on a topological space that has been so important in algebraic geometry. In order to avoid thinking about things like ``classes of classes'' and so on, we will not allow sites to be ``big'' categories, in contrast to what we do for categories and $2$-categories. \medskip\noindent Suppose that $\mathcal{C}$ is a category and that $\text{Cov}(\mathcal{C})$ is a proper class of coverings satisfying (1), (2) and (3) above. We will not allow this as a site either, mainly because we are going to take limits over coverings. However, there are several natural ways to replace $\text{Cov}(\mathcal{C})$ by a set of coverings or a slightly different structure that give rise to the same category of sheaves. For example: \begin{enumerate} \item In Sets, Section \ref{sets-section-coverings-site} we show how to pick a suitable set of coverings that gives the same category of sheaves. \item Another thing we can do is to take the associated topology (see Definition \ref{definition-topology-associated-site}). The resulting topology on $\mathcal{C}$ has the same category of sheaves. Two topologies have the same categories of sheaves if and only if they are equal, see Theorem \ref{theorem-topology-and-topos}. A topology on a category is given by a choice of sieves on objects. The collection of all possible sieves and even all possible topologies on $\mathcal{C}$ is a set. \item We could also slightly modify the notion of a site, see Remark \ref{remark-shrink-coverings} below, and end up with a canonical set of coverings. \end{enumerate} Each of these solutions has some minor drawback. For the first, one has to check that constructions later on do not depend on the choice of the set of coverings. For the second, one has to learn about topologies and redo many of the arguments for sites. For the third, see the last sentence of Remark \ref{remark-shrink-coverings}. \medskip\noindent Our approach will be to work with sites as in Definition \ref{definition-site} above. Given a category $\mathcal{C}$ with a proper class of coverings as above, we will replace this by a set of coverings producing a site using Sets, Lemma \ref{sets-lemma-coverings-site}. It is shown in Lemma \ref{lemma-choice-set-coverings-immaterial} below that the resulting category of sheaves (the topos) is independent of this choice. We leave it to the reader to use one of the other two strategies to deal with these issues if he/she so desires.


sites-remark-sheaf-condition-empty-covering

If the covering $\{U_i \to U\}_{i \in I}$ is the empty family (this means that $I = \emptyset$), then the sheaf condition signifies that $\mathcal{F}(U) = \{*\}$ is a singleton set. This is because in (\ref{equation-sheaf-condition}) the second and third sets are empty products in the category of sets, which are final objects in the category of sets, hence singletons.


sites-remark-both-refine-same-H0

In particular this lemma shows that if $\mathcal{U}$ is a refinement of $\mathcal{V}$, and if $\mathcal{V}$ is a refinement of $\mathcal{U}$, then there is a canonical identification $H^0(\mathcal{U}, \mathcal{F}) = H^0(\mathcal{V}, \mathcal{F})$.


sites-remark-quasi-continuous

(Skip on first reading.) Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let us use the definition of tautologically equivalent families of maps, see Definition \ref{definition-combinatorial-tautological} to (slightly) weaken the conditions defining continuity. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Let us call $u$ {\it quasi-continuous} if for every $\mathcal{V} = \{V_i \to V\}_{i\in I} \in \text{Cov}(\mathcal{C})$ we have the following \begin{enumerate} \item[(1')] the family of maps $\{u(V_i) \to u(V)\}_{i\in I}$ is tautologically equivalent to an element of $\text{Cov}(\mathcal{D})$, and \item[(2)] for any morphism $T \to V$ in $\mathcal{C}$ the morphism $u(T \times_V V_i) \to u(T) \times_{u(V)} u(V_i)$ is an isomorphism. \end{enumerate} We are going to see that Lemmas \ref{lemma-pushforward-sheaf} and \ref{lemma-adjoint-sheaves} hold in case $u$ is quasi-continuous as well. \medskip\noindent We first remark that the morphisms $u(V_i) \to u(V)$ are representable, since they are isomorphic to representable morphisms (by the first condition). In particular, the family $u(\mathcal{V}) = \{u(V_i) \to u(V)\}_{i\in I}$ gives rise to a zeroth {\v C}ech cohomology group $H^0(u(\mathcal{V}), \mathcal{F})$ for any presheaf $\mathcal{F}$ on $\mathcal{D}$. Let $\mathcal{U} = \{U_j \to u(V)\}_{j \in J}$ be an element of $\text{Cov}(\mathcal{D})$ tautologically equivalent to $\{u(V_i) \to u(V)\}_{i \in I}$. Note that $u(\mathcal{V})$ is a refinement of $\mathcal{U}$ and vice versa. Hence by Remark \ref{remark-both-refine-same-H0} we see that $H^0(u(\mathcal{V}), \mathcal{F}) = H^0(\mathcal{U}, \mathcal{F})$. In particular, if $\mathcal{F}$ is a sheaf, then $\mathcal{F}(u(V)) = H^0(u(\mathcal{V}), \mathcal{F})$ because of the sheaf property expressed in terms of zeroth {\v C}ech cohomology groups. We conclude that $u^p\mathcal{F}$ is a sheaf if $\mathcal{F}$ is a sheaf, since $H^0(\mathcal{V}, u^p\mathcal{F}) = H^0(u(\mathcal{V}), \mathcal{F})$ which we just observed is equal to $\mathcal{F}(u(V)) = u^p\mathcal{F}(V)$. Thus Lemma \ref{lemma-pushforward-sheaf} holds. Lemma \ref{lemma-adjoint-sheaves} follows immediately.


sites-remark-explain-left-exact

The conditions of Proposition \ref{proposition-get-morphism} above are equivalent to saying that $u$ is left exact, i.e., commutes with finite limits. See Categories, Lemmas \ref{categories-lemma-finite-limits-exist} and \ref{categories-lemma-characterize-left-exact}. It seems more natural to phrase it in terms of final objects and fibre products since this seems to have more geometric meaning in the examples.


sites-remark-quasi-continuous-morphism-sites

(Skip on first reading.) Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Analogously to Definition \ref{definition-morphism-sites} we say that a {\it quasi-morphism of sites $f : \mathcal{D} \to \mathcal{C}$} is given by a quasi-continuous functor $u : \mathcal{C} \to \mathcal{D}$ (see Remark \ref{remark-quasi-continuous}) such that $u_s$ is exact. The analogue of Proposition \ref{proposition-get-morphism} in this setting is obtained by replacing the word ``continuous'' by the word ``quasi-continuous'', and replacing the word ``morphism'' by ``quasi-morphism''. The proof is literally the same.


sites-remark-pt-topos

There are many sites that give rise to the topos $\Sh(pt)$. A useful example is the following. Suppose that $S$ is a set (of sets) which contains at least one nonempty element. Let $\mathcal{S}$ be the category whose objects are elements of $S$ and whose morphisms are arbitrary set maps. Assume that $\mathcal{S}$ has fibre products. For example this will be the case if $S = \mathcal{P}(\text{infinite set})$ is the power set of any infinite set (exercise in set theory). Make $\mathcal{S}$ into a site by declaring surjective families of maps to be coverings (and choose a suitable sufficiently large set of covering families as in Sets, Section \ref{sets-section-coverings-site}). We claim that $\Sh(\mathcal{S})$ is equivalent to the category of sets. \medskip\noindent We first prove this in case $S$ contains $e \in S$ which is a singleton. In this case, there is an equivalence of topoi $i : \Sh(pt) \to \Sh(\mathcal{S})$ given by the functors \begin{equation} \label{equation-sheaves-pt-sets} i^{-1}\mathcal{F} = \mathcal{F}(e), \quad i_*E = (U \mapsto \Mor_{\textit{Sets}}(U, E)) \end{equation} Namely, suppose that $\mathcal{F}$ is a sheaf on $\mathcal{S}$. For any $U \in \Ob(\mathcal{S}) = S$ we can find a covering $\{\varphi_u : e \to U\}_{u \in U}$, where $\varphi_u$ maps the unique element of $e$ to $u \in U$. The sheaf condition implies in this case that $\mathcal{F}(U) = \prod_{u \in U} \mathcal{F}(e)$. In other words $\mathcal{F}(U) = \Mor_{\textit{Sets}}(U, \mathcal{F}(e))$. Moreover, this rule is compatible with restriction mappings. Hence the functor $$ i_* : \textit{Sets} = \Sh(pt) \longrightarrow \Sh(\mathcal{S}), \quad E \longmapsto (U \mapsto \Mor_{\textit{Sets}}(U, E)) $$ is an equivalence of categories, and its inverse is the functor $i^{-1}$ given above. \medskip\noindent If $\mathcal{S}$ does not contain a singleton, then the functor $i_*$ as defined above still makes sense. To show that it is still an equivalence in this case, choose any nonempty $\tilde e \in S$ and a map $\varphi : \tilde e \to \tilde e$ whose image is a singleton. For any sheaf $\mathcal{F}$ set $$ \mathcal{F}(e) := \Im( \mathcal{F}(\varphi) : \mathcal{F}(\tilde e) \longrightarrow \mathcal{F}(\tilde e) ) $$ and show that this is a quasi-inverse to $i_*$. Details omitted.


sites-remark-morphism-topoi-big

(Set theoretical issues related to morphisms of topoi. Skip on a first reading.) A morphism of topoi as defined above is not a set but a class. In other words it is given by a mathematical formula rather than a mathematical object. Although we may contemplate the collection of all morphisms between two given topoi, it is not a good idea to introduce it as a mathematical object. On the other hand, suppose $\mathcal{C}$ and $\mathcal{D}$ are given sites. Consider a functor $\Phi : \mathcal{C} \to \Sh(\mathcal{D})$. Such a thing is a set, in other words, it is a mathematical object. We may, in succession, ask the following questions on $\Phi$. \begin{enumerate} \item Is it true, given a sheaf $\mathcal{F}$ on $\mathcal{D}$, that the rule $U \mapsto \Mor_{\Sh(\mathcal{D})}(\Phi(U), \mathcal{F})$ defines a sheaf on $\mathcal{C}$? If so, this defines a functor $\Phi_* : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$. \item Is it true that $\Phi_*$ has a left adjoint? If so, write $\Phi^{-1}$ for this left adjoint. \item Is it true that $\Phi^{-1}$ is exact? \end{enumerate} If the last question still has the answer ``yes'', then we obtain a morphism of topoi $(\Phi_*, \Phi^{-1})$. Moreover, given any morphism of topoi $(f_*, f^{-1})$ we may set $\Phi(U) = f^{-1}(h_U^\#)$ and obtain a functor $\Phi$ as above with $f_* \cong \Phi_*$ and $f^{-1} \cong \Phi^{-1}$ (compatible with adjoint property). The upshot is that by working with the collection of $\Phi$ instead of morphisms of topoi, we (a) replaced the notion of a morphism of topoi by a mathematical object, and (b) the collection of $\Phi$ forms a class (and not a collection of classes). Of course, more can be said, for example one can work out more precisely the significance of conditions (2) and (3) above; we do this in the case of points of topoi in Section \ref{section-points}.


sites-remark-quasi-continuous-morphism-topoi

(Skip on first reading.) Let $\mathcal{C}$ and $\mathcal{D}$ be sites. A quasi-morphism of sites $f : \mathcal{D} \to \mathcal{C}$ (see Remark \ref{remark-quasi-continuous-morphism-sites}) gives rise to a morphism of topoi $f$ from $\Sh(\mathcal{D})$ to $\Sh(\mathcal{C})$ exactly as in Lemma \ref{lemma-morphism-sites-topoi}.


sites-remark-cartesian-cocontinuous

Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories. Given morphisms $g : u(U) \to V$ and $f : W \to V$ in $\mathcal{D}$ we can consider the functor $$ \mathcal{C}^{opp} \longrightarrow \textit{Sets},\quad T \longmapsto \Mor_\mathcal{C}(T, U) \times_{\Mor_\mathcal{D}(u(T), V)} \Mor_\mathcal{D}(u(T), W) $$ If this functor is representable, denote $U \times_{g, V, f} W$ the corresponding object of $\mathcal{C}$. Assume that $\mathcal{C}$ and $\mathcal{D}$ are sites. Consider the property $P$: for every covering $\{f_j : V_j \to V\}$ of $\mathcal{D}$ and any morphism $g : u(U) \to V$ we have \begin{enumerate} \item $U \times_{g, V, f_i} V_i$ exists for all $i$, and \item $\{U \times_{g, V, f_i} V_i \to U\}$ is a covering of $\mathcal{C}$. \end{enumerate} Please note the similarity with the definition of continuous functors. If $u$ has $P$ then $u$ is cocontinuous (details omitted). Many of the cocontinuous functors we will encounter satisfy $P$.


sites-remark-localize-presheaves

Localization and presheaves. Let $\mathcal{C}$ be a category. Let $U$ be an object of $\mathcal{C}$. Strictly speaking the functors $j_U^{-1}$, $j_{U*}$ and $j_{U!}$ have not been defined for presheaves. But of course, we can think of a presheaf as a sheaf for the chaotic topology on $\mathcal{C}$ (see Example \ref{example-indiscrete}). Hence we also obtain a functor $$ j_U^{-1} : \textit{PSh}(\mathcal{C}) \longrightarrow \textit{PSh}(\mathcal{C}/U) $$ and functors $$ j_{U*}, j_{U!} : \textit{PSh}(\mathcal{C}/U) \longrightarrow \textit{PSh}(\mathcal{C}) $$ which are right, left adjoint to $j_U^{-1}$. By Lemma \ref{lemma-describe-j-shriek} we see that $j_{U!}\mathcal{G}$ is the presheaf $$ V \longmapsto \coprod\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) $$ In addition the functor $j_{U!}$ commutes with fibre products and equalizers.


sites-remark-localization-cartesian-cocontinuous

Let $\mathcal{C}$ be a site. Let $U \to V$ be a morphism of $\mathcal{C}$. The cocontinuous functors $\mathcal{C}/U \to \mathcal{C}$ and $j : \mathcal{C}/U \to \mathcal{C}/V$ (Lemma \ref{lemma-relocalize}) satisfy property $P$ of Remark \ref{remark-cartesian-cocontinuous}. For example, if we have objects $(X/U)$, $(W/V)$, a morphism $g : j(X/U) \to (W/V)$, and a covering $\{f_i  : (W_i/V) \to (W/V)\}$ then $(X \times_W W_i/U)$ is an avatar of $(X/U) \times_{g, (W/V), f_i} (W_i/V)$ and the family $\{(X \times_W W_i/U) \to (X/U)\}$ is a covering of $\mathcal{C}/U$.


sites-remark-morphism-topoi-comes-from-morphism-sites

Notation and assumptions as in Lemma \ref{lemma-morphism-topoi-comes-from-morphism-sites}. If the site $\mathcal{D}$ has a final object and fibre products then the functor $u : \mathcal{D} \to \mathcal{C}'$ satisfies all the assumptions of Proposition \ref{proposition-get-morphism}. Namely, in addition to the properties mentioned in the lemma $u$ also transforms the final object of $\mathcal{D}$ into the final object of $\mathcal{C}'$. This is clear from the construction of $u$. Hence, if we first apply Lemmas \ref{lemma-topos-good-site} to $\mathcal{D}$ and then Lemma \ref{lemma-morphism-topoi-comes-from-morphism-sites} to the resulting morphism of topoi $\Sh(\mathcal{C}) \to \Sh(\mathcal{D}')$ we obtain the following statement: Any morphism of topoi $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ fits into a commutative diagram $$ \xymatrix{ \Sh(\mathcal{C}) \ar[d]_g \ar[r]_f & \Sh(\mathcal{D}) \ar[d]^e \\ \Sh(\mathcal{C}') \ar[r]^{f'} & \Sh(\mathcal{D}') } $$ where the following properties hold: \begin{enumerate} \item the morphisms $e$ and $g$ are equivalences given by special cocontinuous functors $\mathcal{C} \to \mathcal{C}'$ and $\mathcal{D} \to \mathcal{D}'$, \item the sites $\mathcal{C}'$ and $\mathcal{D}'$ have fibre products, final objects and have subcanonical topologies, \item the morphism $f' : \mathcal{C}' \to \mathcal{D}'$ comes from a morphism of sites corresponding to a functor $u : \mathcal{D}' \to \mathcal{C}'$ to which Proposition \ref{proposition-get-morphism} applies, and \item given any set of sheaves $\mathcal{F}_i$ (resp.\ $\mathcal{G}_j$) on $\mathcal{C}$ (resp.\ $\mathcal{D}$) we may assume each of these is a representable sheaf on $\mathcal{C}'$ (resp.\ $\mathcal{D}'$). \end{enumerate} It is often useful to replace $\mathcal{C}$ and $\mathcal{D}$ by $\mathcal{C}'$ and $\mathcal{D}'$.


sites-remark-equivalence-topoi-comes-from-morphism-sites

Notation and assumptions as in Lemma \ref{lemma-morphism-topoi-comes-from-morphism-sites}. Suppose that in addition the original morphism of topoi $\Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ is an equivalence. Then the construction in the proof of Lemma \ref{lemma-morphism-topoi-comes-from-morphism-sites} gives two functors $$ \mathcal{C} \rightarrow \mathcal{C}' \leftarrow \mathcal{D} $$ which are both special cocontinuous functors. Hence in this case we can actually factor the morphism of topoi as a composition $$ \Sh(\mathcal{C}) \rightarrow \Sh(\mathcal{C}') = \Sh(\mathcal{D}') \leftarrow \Sh(\mathcal{D}) $$ as in Remark \ref{remark-morphism-topoi-comes-from-morphism-sites}, but with the middle morphism an identity.


sites-remark-improve-proposition-points-limits

In fact, let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has a final object $X$  and fibre products. Let $p = u: \mathcal{C} \to \textit{Sets}$ be a functor such that \begin{enumerate} \item $u(X) = \{*\}$ a singleton, and \item for every pair of morphisms $U \to W$ and $V \to W$ with the same target the map $u(U \times_W V) \to u(U) \times_{u(W)} u(V)$ is surjective. \item for every covering $\{U_i \to U\}$ the map $\coprod u(U_i) \to u(U)$ is surjective. \end{enumerate} Then, in general, $p$ is {\bf not} a point of $\mathcal{C}$. An example is the category $\mathcal{C}$ with two objects $\{U, X\}$ and exactly one non-identity arrow, namely $U \to X$. We endow $\mathcal{C}$ with the trivial topology, i.e., the only coverings are $\{U \to U\}$ and $\{X \to X\}$. A sheaf $\mathcal{F}$ is the same thing as a presheaf and consists of a triple $(A, B, A \to B)$: namely $A = \mathcal{F}(X)$, $B = \mathcal{F}(U)$ and $A \to B$ is the restriction mapping corresponding to $U \to X$. Note that $U \times_X U = U$ so fibre products exist. Consider the functor $u = p$ with $u(X) = \{*\}$ and $u(U) = \{*_1, *_2\}$. This satisfies (1), (2), and (3), but the corresponding stalk functor (\ref{equation-stalk}) is the functor $$ (A, B, A \to B) \longmapsto B \amalg_A B $$ which isn't exact. Namely, consider $(\emptyset, \{1\}, \emptyset \to \{1\}) \to (\{1\}, \{1\}, \{1\} \to \{1\})$ which is an injective map of sheaves, but is transformed into the noninjective map of sets $$ \{1\} \amalg \{1\} \longrightarrow \{1\} \amalg_{\{1\}} \{1\} $$ by the stalk functor.


sites-remark-not-pushforward

Warning: The result of Lemma \ref{lemma-stalk-j-shriek} has no analogue for $j_{U, *}$.


sites-remark-no-pullback-presheaves

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{D} \to \mathcal{C}$ be a continuous functor which gives rise to a morphism of sites $\mathcal{C} \to \mathcal{D}$. Note that even in the case of abelian groups we have not defined a pullback functor for presheaves of abelian groups. Since all colimits are representable in the category of abelian groups, we certainly may define a functor $u_p^{ab}$ on abelian presheaves by the same colimits as we have used to define $u_p$ on presheaves of sets. It will also be the case that $u_p^{ab}$ is adjoint to $u^p$ on the categories of abelian presheaves. However, it will not always be the case that $u_p^{ab}$ agrees with $u_p$ on the underlying presheaves of sets.


sites-remark-compose-base-change

Consider a commutative diagram $$ \xymatrix{ \Sh(\mathcal{B}') \ar[r]_k \ar[d]_{f'} & \Sh(\mathcal{B}) \ar[d]^f \\ \Sh(\mathcal{C}') \ar[r]^l \ar[d]_{g'} & \Sh(\mathcal{C}) \ar[d]^g \\ \Sh(\mathcal{D}') \ar[r]^m & \Sh(\mathcal{D}) } $$ of topoi. Then the base change maps for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \begin{align*} m^{-1} \circ (g \circ f)_* & = m^{-1} \circ g_* \circ f_* \\ & \to g'_* \circ l^{-1} \circ f_* \\ & \to g'_* \circ f'_* \circ k^{-1} \\ & = (g' \circ f')_* \circ k^{-1} \end{align*} is the base change map for the rectangle. We omit the verification.


sites-remark-compose-base-change-horizontal

Consider a commutative diagram $$ \xymatrix{ \Sh(\mathcal{C}'') \ar[r]_{g'} \ar[d]_{f''} & \Sh(\mathcal{C}') \ar[r]_g \ar[d]_{f'} & \Sh(\mathcal{C}) \ar[d]^f \\ \Sh(\mathcal{D}'') \ar[r]^{h'} & \Sh(\mathcal{D}') \ar[r]^h & \Sh(\mathcal{D}) } $$ of ringed topoi. Then the base change maps for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \begin{align*} (h \circ h')^{-1} \circ f_* & = (h')^{-1} \circ h^{-1} \circ f_* \\ & \to (h')^{-1} \circ f'_* \circ g^{-1} \\ & \to f''_* \circ (g')^{-1} \circ g^{-1} \\ & = f''_* \circ (g \circ g')^{-1} \end{align*} is the base change map for the rectangle. We omit the verification.


sites-remark-enlarge-coverings

Enlarging the class of coverings. Clearly, if $\text{Cov}(\mathcal{C})$ defines the structure of a site on $\mathcal{C}$ then we may add to $\mathcal{C}$ any set of families of morphisms with fixed target tautologically equivalent (see Definition \ref{definition-combinatorial-tautological}) to elements of $\text{Cov}(\mathcal{C})$ without changing the topology.


sites-remark-shrink-coverings

Shrinking the class of coverings. Let $\mathcal{C}$ be a site. Consider the set $$ \mathcal{S} = P(\text{Arrows}(\mathcal{C})) \times \Ob(\mathcal{C}) $$ where $P(\text{Arrows}(\mathcal{C}))$ is the power set of the set of morphisms, i.e., the set of all sets of morphisms. Let $\mathcal{S}_\tau \subset \mathcal{S}$ be the subset consisting of those $(T, U) \in \mathcal{S}$ such that (a) all $\varphi \in T$ have target $U$, (b) the collection $\{\varphi\}_{\varphi \in T}$ is tautologically equivalent (see Definition \ref{definition-combinatorial-tautological}) to some covering in $\text{Cov}(\mathcal{C})$. Clearly, considering the elements of $\mathcal{S}_\tau$ as the coverings, we do not get exactly the notion of a site as defined in Definition \ref{definition-site}. The structure $(\mathcal{C}, \mathcal{S}_\tau)$ we get satisfies slightly modified conditions. The modified conditions are: \begin{enumerate} \item[(0')] $\text{Cov}(\mathcal{C}) \subset P(\text{Arrows}(\mathcal{C})) \times \Ob(\mathcal{C})$, \item[(1')] If $V \to U$ is an isomorphism then $(\{V \to U\}, U) \in \text{Cov}(\mathcal{C})$. \item[(2')] If $(T, U) \in \text{Cov}(\mathcal{C})$ and for $f : U' \to U$ in $T$ we are given $(T_f, U') \in \text{Cov}(\mathcal{C})$, then setting $T' = \{f \circ f' \mid f \in T,\ f' \in T_f\}$, we get $(T', U) \in \text{Cov}(\mathcal{C})$. \item[(3')] If $(T, U) \in \text{Cov}(\mathcal{C})$ and $g : V \to U$ is a morphism of $\mathcal{C}$ then \begin{enumerate} \item $U' \times_{f, U, g} V$ exists for $f : U' \to U$ in $T$, and \item setting $T' = \{\text{pr}_2 : U' \times_{f, U, g} V \to V \mid f : U' \to U \in T\}$ for some choice of fibre products we get $(T', V) \in \text{Cov}(\mathcal{C})$. \end{enumerate} \end{enumerate} And it is easy to verify that, given a structure satisfying (0') -- (3') above, then after suitably enlarging $\text{Cov}(\mathcal{C})$ (compare Sets, Section \ref{sets-section-coverings-site}) we get a site. Obviously there is little difference between this notion and the actual notion of a site, at least from the point of view of the topology. There are two benefits: because of condition (0') above the coverings automatically form a set, and because of (0') the totality of all structures of this type forms a set as well. The price you pay for this is that you have to keep writing ``tautologically equivalent'' everywhere.


sets-remark-how-to-use-reflection

The lemma above can also be proved using the reflection principle. However, one has to be careful. Namely, suppose the sentence $\phi_{scheme}(X)$ expresses the property ``$X$ is a scheme'', then what does the formula $\phi_{scheme}^{V_\alpha}(X)$ mean? It is true that the reflection principle says we can find $\alpha$ such that for all $X \in V_\alpha$ we have $\phi_{scheme}(X) \leftrightarrow \phi_{scheme}^{V_\alpha}(X)$ but this is entirely useless. It is only by combining two such statements that something interesting happens. For example suppose $\phi_{red}(X, Y)$ expresses the property ``$X$, $Y$ are schemes, and $Y$ is the reduction of $X$'' (see Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}). Suppose we apply the reflection principle to the pair of formulas $\phi_1(X, Y) = \phi_{red}(X, Y)$, $\phi_2(X) = \exists Y, \phi_1(X, Y)$. Then it is easy to see that any $\alpha$ produced by the reflection principle has the property that given $X \in \Ob(\Sch_\alpha)$ the reduction of $X$ is also an object of $\Sch_\alpha$ (left as an exercise).


sets-remark-what-is-not-in-it

Let $R$ be a ring. Suppose we consider the ring $\prod_{\mathfrak p \in \Spec(R)} \kappa(\mathfrak p)$. The cardinality of this ring is bounded by $|R|^{2^{|R|}}$, but is not bounded by $|R|^{\aleph_0}$ in general. For example if $R = \mathbf{C}[x]$ it is not bounded by $|R|^{\aleph_0}$ and if $R = \prod_{n \in \mathbf{N}} \mathbf{F}_2$ it is not bounded by $|R|^{|R|}$. Thus the ``And so on'' of Lemma \ref{lemma-what-is-in-it} above should be taken with a grain of salt. Of course, if it ever becomes necessary to consider these rings in arguments pertaining to fppf/\'etale cohomology, then we can change the function $Bound$ above into the function $\kappa \mapsto \kappa^{2^\kappa}$.


sets-remark-better

It is likely the case that, for some limit ordinal $\alpha$, the set of coverings $\text{Cov}(\mathcal{C})_\alpha$ satisfies the conditions of the lemma. This is after all what an application of the reflection principle would appear to give (modulo caveats as described at the end of Section \ref{section-reflection-principle} and in Remark \ref{remark-how-to-use-reflection}).


more-etale-remark-covariance-f-shriek-separated

Let $f : X \to Y$ be morphism of schemes which is separated and locally of finite type. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Let $X' \subset X$ be an open subscheme. Denote $f' : X' \to Y$ the restriction of $f$. There is a canonical injective map $$ f'_!(\mathcal{F}|_{X'}) \longrightarrow f_!\mathcal{F} $$ Namely, let $V \in Y_\etale$ and consider a section $s' \in f'_*(\mathcal{F}|_{X'})(V) = \mathcal{F}(X' \times_Y V)$ with support $Z'$ proper over $V$. Then $Z'$ is closed in $X \times_Y V$ as well, see Cohomology of Schemes, Lemma \ref{coherent-lemma-functoriality-closed-proper-over-base}. Thus there is a unique section $s \in \mathcal{F}(X \times_Y V) = f_*\mathcal{F}(V)$ whose restriction to $X' \times_Y V$ is $s'$ and whose restriction to $X \times_Y V \setminus Z'$ is zero, see Lemma \ref{lemma-section-support-in-locally-closed}. This construction is compatible with restriction maps and hence induces the desired map of sheaves $f'_!(\mathcal{F}|_{X'}) \to f_!\mathcal{F}$ which is clearly injective. By construction we obtain a commutative diagram $$ \xymatrix{ f'_!(\mathcal{F}|_{X'}) \ar[r] \ar[d] & f_!\mathcal{F} \ar[d] \\ f'_*(\mathcal{F}|_{X'}) & f_*\mathcal{F} \ar[l] } $$ functorial in $\mathcal{F}$. It is clear that for $X'' \subset X'$ open with $f'' = f|_{X''} : X'' \to Y$ the composition of the canonical maps $f''_!\mathcal{F}|_{X''} \to f'_!\mathcal{F}|_{X'} \to f_!\mathcal{F}$ just constructed is the canonical map $f''_!\mathcal{F}|_{X''} \to f_!\mathcal{F}$.


more-etale-remark-covariance-compact-support

Let $X$ be a separated scheme locally of finite type over a field $k$. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Exactly as in Remark \ref{remark-covariance-f-shriek-separated} there are injective maps $$ H^0_c(X', \mathcal{F}|_{X'}) \longrightarrow H^0_c(X, \mathcal{F}) $$ which turn $H^0_c$ into a ``cosheaf'' on the Zariski site of $X$.


more-etale-remark-f-shriek-base-change-composition

The isomorphisms between functors constructed above satisfy the following two properties: \begin{enumerate} \item Let $f : X \to Y$, $g : Y \to Z$, and $h : Z \to T$ be composable morphisms of schemes which are separated and locally of finite type. Then the diagram $$ \xymatrix{ (h \circ g \circ f)_! \ar[r] \ar[d] & (h \circ g)_! \circ f_! \ar[d] \\ h_! \circ (g \circ f)_! \ar[r] & h_! \circ g_! \circ f_! } $$ commutes where the arrows are those of Lemma \ref{lemma-f-shriek-composition}. \item Suppose that we have a diagram of schemes $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_c & X \ar[d]^f \\ Y' \ar[d]_{g'} \ar[r]_b & Y \ar[d]^g \\ Z' \ar[r]^a & Z } $$ with both squares cartesian and $f$ and $g$ separated and locally of finite type. Then the diagram $$ \xymatrix{ a^{-1} \circ (g \circ f)_! \ar[d] \ar[rr] & & (g' \circ f')_! \circ c^{-1} \ar[d] \\ a^{-1} \circ g_! \circ f_! \ar[r] & g'_! \circ b^{-1} \circ f_! \ar[r] & g'_! \circ f'_! \circ c^{-1} } $$ commutes where the horizontal arrows are those of Lemma \ref{lemma-base-change-f-shriek-separated} the arrows are those of Lemma \ref{lemma-f-shriek-composition}. \end{enumerate} Part (1) holds true because we have a similar commutative diagram for pushforwards. Part (2) holds by the very general compatibility of base change maps for pushforwards (Sites, Remark \ref{sites-remark-compose-base-change}) and the fact that the isomorphisms in Lemmas \ref{lemma-base-change-f-shriek-separated} and \ref{lemma-f-shriek-composition} are constructed using the corresponding maps fo pushforwards.


more-etale-remark-covariance-lqf-f-shriek

Let $f : X \to Y$ be locally quasi-finite morphism of schemes. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Let $X' \subset X$ be an open subscheme and denote $f' : X' \to Y$ the restriction of $f$. We claim there is a canonical map $$ f'_!(\mathcal{F}|_{X'}) \longrightarrow f_!\mathcal{F} $$ Namely, this map will be the sheafification of a canonical map $$ f'_{p!}(\mathcal{F}|_{X'}) \to f_{p!}\mathcal{F} $$ constructed as follows. Let $V \in Y_\etale$ and consider a section $s' = \sum_{i = 1, \ldots, n} (Z'_i, s'_i)$ as in (\ref{equation-formal-sum}) defining an element of $f'_{p!}(\mathcal{F}|_{X'})(V)$. Then $Z'_i \subset X'_V$ may also be viewed as a locally closed subscheme of $X_V$ and we have $H_{Z'_i}(\mathcal{F}|_{X'}) = H_{Z'_i}(\mathcal{F})$. We will map $s'$ to the exact same sum $s = \sum_{i = 1, \ldots, n} (Z'_i, s'_i)$ but now viewed as an element of $f_{p!}\mathcal{F}(V)$. We omit the verification that this construction is compatible with restriction mappings and functorial in $\mathcal{F}$. This construction has the following properties: \begin{enumerate} \item The maps $f'_{p!}\mathcal{F}' \to f_{p!}\mathcal{F}$ and $f'_!\mathcal{F}' \to f_!\mathcal{F}$ are compatible with the description of stalks given in Lemmas \ref{lemma-finite-support-stalk} and \ref{lemma-lqf-f-shriek-stalk}. \item If $f$ is separated, then the map $f'_{p!}\mathcal{F}' \to f_{p!}\mathcal{F}$ is the same as the map constructed in Remark \ref{remark-covariance-f-shriek-separated} via the isomorphism in Lemma \ref{lemma-finite-support-f-shriek-separated}. \item If $X'' \subset X'$ is another open, then the composition of $f''_{p!}(\mathcal{F}|_{X''}) \to f'_{p!}(\mathcal{F}|_{X'}) \to f_{p!}\mathcal{F}$ is the map $f''_{p!}(\mathcal{F}|_{X''}) \to f_{p!}\mathcal{F}$ for the inclusion $X'' \subset X$. Sheafifying we conclude the same holds true for $f''_!(\mathcal{F}|_{X''}) \to f'_!(\mathcal{F}|_{X'}) \to f_!\mathcal{F}$. \item The map $f'_!\mathcal{F}' \to f_!\mathcal{F}$ is injective because we can check this on stalks. \end{enumerate} All of these statements are easily proven by representing elements as finite sums as above and considering what happens to these elements.


more-etale-remark-alternative-lqf-f-shriek

Lemma \ref{lemma-lqf-colimit-f-shriek} gives an alternative construction of the functor $f_!$ for locally quasi-finite morphisms $f$. Namely, given a locally quasi-finite morphism $f : X \to Y$ of schemes we can choose an open covering $X = \bigcup_{i \in I} X_i$ such that each $f_i : X_i \to Y$ is separated. For example choose an affine open covering of $X$. Then we can define $f_!\mathcal{F}$ as the cokernel of the penultimate map of the complex of the lemma, i.e., $$ f_!\mathcal{F} = \Coker\left( \bigoplus\nolimits_{i_0, i_1} f_{i_0i_1, !} \mathcal{F}|_{X_{i_0i_1}} \to \bigoplus\nolimits_{i_0} f_{i_0, !} \mathcal{F}|_{X_{i_0}} \right) $$ where we can use the construction of $f_{i_0, !}$ and $f_{i_0i_1, !}$ in Section \ref{section-compact-support} because the morphisms $f_{i_0}$ and $f_{i_0 i_1}$ are separated. One can then compute the stalks of $f_!$ (using the separated case, namely Lemma \ref{lemma-lqf-f-shriek-separated-colimits}) and obtain the result of Lemma \ref{lemma-lqf-f-shriek-stalk}. Having done so all the other results of this section can be deduced from this as well.


more-etale-remark-construct-map-presheaves-downstairs

Let $g : Y' \to Y$ be a morphism of schemes. For an abelian presheaf $\mathcal{G}'$ on $Y'_\etale$ let us denote $g_*\mathcal{G}'$ the presheaf $V \mapsto \mathcal{G}'(Y' \times_Y V)$. If $\alpha : \mathcal{G} \to g_*\mathcal{G}'$ is a map of abelian presheaves on $Y_\etale$, then there is a unique map $\alpha^\# : \mathcal{G}^\# \to g_*((\mathcal{G}')^\#)$ of abelian sheaves on $Y_\etale$ such that the diagram $$ \xymatrix{ \mathcal{G} \ar[d] \ar[r]_\alpha & g_*\mathcal{G}' \ar[d] \\ \mathcal{G}^\# \ar[r]^-{\alpha^\#} & g_*((\mathcal{G}')^\#) } $$ is commutative where the vertical maps come from the canonical maps $\mathcal{G} \to \mathcal{G}^\#$ and $\mathcal{G}' \to (\mathcal{G}')^\#$. If $\alpha' : g^{-1}\mathcal{G}^\# \to (\mathcal{G}')^\#$ is the map adjoint to $\alpha^\#$, then for a geometric point $\overline{y}' : \Spec(k) \to Y'$ with image $\overline{y} = g \circ \overline{y}'$ in $Y$, the map $$ \alpha'_{\overline{y}'} : \mathcal{G}_{\overline{y}} = (\mathcal{G}^\#)_{\overline{y}} = (g^{-1}\mathcal{G}^\#)_{\overline{y}'} \longrightarrow (\mathcal{G}')^\#_{\overline{y}'} = \mathcal{G}'_{\overline{y}'} $$ is given by mapping the class in the stalk of a section $s$ of $\mathcal{G}$ over an \'etale neighbourhood $(V, \overline{v})$ to the class of the section $\alpha(s)$ in $g_*\mathcal{G}'(V) = \mathcal{G}'(Y' \times_Y V)$ over the \'etale neighbourhood $(Y' \times_Y V, (\overline{y}', \overline{v}))$ in the stalk of $\mathcal{G}'$ at $\overline{y}'$.


more-etale-remark-pointed-sets

The material in this section can be generalized to sheaves of pointed sets. Namely, for a site $\mathcal{C}$ denote $\Sh^*(\mathcal{C})$ the category of sheaves of pointed sets. The constructions in this and the preceding section apply, mutatis mutandis, to sheaves of pointed sets. Thus given a locally quasi-finite morphism $f : X \to Y$ of schemes we obtain an adjoint pair of functors $$ f_! : \Sh^*(X_\etale) \longrightarrow \Sh^*(Y_\etale) \quad\text{and}\quad f^! : \Sh^*(Y_\etale) \longrightarrow \Sh^*(X_\etale) $$ such that for every geometric point $\overline{y}$ of $Y$ there are isomorphisms $$ (f_!\mathcal{F})_{\overline{y}} = \coprod\nolimits_{f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} $$ (coproduct taken in the category of pointed sets) functorial in $\mathcal{F} \in \Sh^*(X_\etale)$ and isomorphisms $$ f^!(\overline{y}_*S) = \prod\nolimits_{f(\overline{x}) = \overline{y}} \overline{x}_*S $$ functorial in the pointed set $S$. If $F : \textit{Ab}(X_\etale) \to \Sh^*(X_\etale)$ and $F : \textit{Ab}(Y_\etale) \to \Sh^*(Y_\etale)$ denote the forgetful functors, compatibility between the constructions will guarantee the existence of canonical maps $$ f_!F(\mathcal{F}) \longrightarrow F(f_!\mathcal{F}) $$ functorial in $\mathcal{F} \in \textit{Ab}(X_\etale)$ and $$ F(f^!\mathcal{G}) \longrightarrow f^!F(\mathcal{G}) $$ functorial in $\mathcal{G} \in \textit{Ab}(Y_\etale)$ which produce the obvious maps on stalks, resp.\ skyscraper sheaves. In fact, the transformation $F \circ f^! \to f^! \circ F$ is an isomorphism (because $f^!$ commutes with products).


more-etale-remark-going-around

Consider a commutative diagram $$ \xymatrix{ X'' \ar[r]_{k'} \ar[d]_{f''} & X' \ar[r]_k \ar[d]_{f'} & X \ar[d]^f \\ Y'' \ar[r]^{l'} \ar[d]_{g''} & Y' \ar[r]^l \ar[d]_{g'} & Y \ar[d]^g \\ Z'' \ar[r]^{m'} & Z' \ar[r]^m & Z } $$ of schemes whose vertical arrows are proper and whose horizontal arrows are separated and locally quasi-finite. Let us label the squares of the diagram $A$, $B$, $C$, $D$ as follows $$ \begin{matrix} A & B \\ C & D \end{matrix} $$ Then the maps of Lemma \ref{lemma-shriek-proper-and-open} for the squares are (where we use $Rf_* = f_*$, etc) $$ \begin{matrix} \gamma_A : l'_! \circ f''_* \to f'_* \circ k'_! & \gamma_B : l_! \circ f'_* \to f_* \circ k_! \\ \gamma_C : m'_! \circ g''_* \to g'_* \circ l'_! & \gamma_D : m_! \circ g'_* \to g_* \circ l_! \end{matrix} $$ For the $2 \times 1$ and $1 \times 2$ rectangles we have four further maps $$ \begin{matrix} \gamma_{A + B} : (l \circ l')_! \circ f''_* \to f_* \circ (k \circ k')_*  \\ \gamma_{C + D} : (m \circ m')_! \circ g''_* \to g_* \circ (l \circ l')_! \\ \gamma_{A + C} : m'_! \circ (g'' \circ f'')_* \to (g' \circ f')_* \circ k'_! \\ \gamma_{B + D} : m_! \circ (g' \circ f')_* \to (g \circ f)_* \circ k_! \end{matrix} $$ By Lemma \ref{lemma-shriek-proper-and-open-compose-horizontal} we have $$ \gamma_{A + B} = \gamma_B \circ \gamma_A, \quad \gamma_{C + D} = \gamma_D \circ \gamma_C $$ and by Lemma \ref{lemma-shriek-proper-and-open-compose} we have $$ \gamma_{A + C} = \gamma_A \circ \gamma_C, \quad \gamma_{B + D} = \gamma_B \circ \gamma_D $$ Here it would be more correct to write $\gamma_{A + B} = (\gamma_B \star \text{id}_{k'_!}) \circ (\text{id}_{l_!} \star \gamma_A)$ with notation as in Categories, Section \ref{categories-section-formal-cat-cat} and similarly for the others. Having said all of this we find (a priori) two transformations $$ m_! \circ m'_! \circ g''_* \circ f''_* \longrightarrow g_* \circ f_* \circ k_! \circ k'_! $$ namely $$ \gamma_B \circ \gamma_D \circ \gamma_A \circ \gamma_C = \gamma_{B + D} \circ \gamma_{A + C} $$ and $$ \gamma_B \circ \gamma_A \circ \gamma_D \circ \gamma_C = \gamma_{A + B} \circ \gamma_{C + D} $$ The point of this remark is to point out that these transformations are equal. Namely, to see this it suffices to show that $$ \xymatrix{ m_! \circ g'_* \circ l'_! \circ f''_* \ar[r]_{\gamma_D} \ar[d]_{\gamma_A} & g_* \circ l_! \circ l'_! \circ f''_* \ar[d]^{\gamma_A} \\ m_! \circ g'_* \circ f'_* \circ k'_! \ar[r]^{\gamma_D} & g_* \circ l_! \circ f'_* \circ k'_! } $$ commutes. This is true because the squares $A$ and $D$ meet in only one point, more precisely by Categories, Lemma \ref{categories-lemma-properties-2-cat-cats} or more simply the discussion preceding Categories, Definition \ref{categories-definition-horizontal-composition}.


stacks-properties-remark-representable-over

Let $\mathcal{Y}$ be an algebraic stack. Consider the following $2$-category: \begin{enumerate} \item An object is a morphism $f : \mathcal{X} \to \mathcal{Y}$ which is representable by algebraic spaces, \item a $1$-morphism $(g, \beta) : (f_1 : \mathcal{X}_1 \to \mathcal{Y}) \to (f_2 : \mathcal{X}_2 \to \mathcal{Y})$ consists of a morphism $g : \mathcal{X}_1 \to \mathcal{X}_2$ and a $2$-morphism $\beta : f_1 \to f_2 \circ g$, and \item a $2$-morphism between $(g, \beta), (g', \beta') : (f_1 : \mathcal{X}_1 \to \mathcal{Y}) \to (f_2 : \mathcal{X}_2 \to \mathcal{Y})$ is a $2$-morphism $\alpha : g \to g'$ such that $(\text{id}_{f_2} \star \alpha) \circ \beta = \beta'$. \end{enumerate} Let us denote this $2$-category $\textit{Spaces}/\mathcal{Y}$ by analogy with the notation of Topologies on Spaces, Section \ref{spaces-topologies-section-procedure}. Now we claim that in this $2$-category the morphism categories $$ \Mor_{\textit{Spaces}/\mathcal{Y}}( (f_1 : \mathcal{X}_1 \to \mathcal{Y}), (f_2 : \mathcal{X}_2 \to \mathcal{Y})) $$ are all setoids. Namely, a $2$-morphism $\alpha$ is a rule which to each object $x_1$ of $\mathcal{X}_1$ assigns an isomorphism $\alpha_{x_1} : g(x_1) \longrightarrow g'(x_1)$ in the relevant fibre category of $\mathcal{X}_2$ such that the diagram $$ \xymatrix{ & f_2(x_1) \ar[ld]_{\beta_{x_1}} \ar[rd]^{\beta'_{x_1}} \\ f_2(g(x_1)) \ar[rr]^{f_2(\alpha_{x_1})} & & f_2(g'(x_1)) } $$ commutes. But since $f_2$ is faithful (see Algebraic Stacks, Lemma \ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}) this means that if $\alpha_{x_1}$ exists, then it is unique! In other words the $2$-category $\textit{Spaces}/\mathcal{Y}$ is very close to being a category. Namely, if we replace $1$-morphisms by isomorphism classes of $1$-morphisms we obtain a category. We will often perform this replacement without further mention.


stacks-properties-remark-more-general-presentation

The result of Lemma \ref{lemma-points-presentation} can be generalized as follows. Let $\mathcal{X}$ be an algebraic stack. Let $U$ be an algebraic space and let $f : U \to \mathcal{X}$ be a surjective morphism (which makes sense by Section \ref{section-properties-morphisms}). Let $R = U \times_\mathcal{X} U$, let $(U, R, s, t, c)$ be the groupoid in algebraic spaces, and let $f_{can} : [U/R] \to \mathcal{X}$ be the canonical morphism as constructed in Algebraic Stacks, Lemma \ref{algebraic-lemma-map-space-into-stack}. Then the image of $|R| \to |U| \times |U|$ is an equivalence relation and $|\mathcal{X}| = |U|/|R|$. The proof of Lemma \ref{lemma-points-presentation} works without change. (Of course in general $[U/R]$ is not an algebraic stack, and in general $f_{can}$ is not an isomorphism.)


stacks-properties-remark-list-properties-local-smooth-topology

Here is a list of properties which are local for the smooth topology (keep in mind that the fpqc, fppf, and syntomic topologies are stronger than the smooth topology): \begin{enumerate} \item locally Noetherian, see Descent, Lemma \ref{descent-lemma-Noetherian-local-fppf}, \item Jacobson, see Descent, Lemma \ref{descent-lemma-Jacobson-local-fppf}, \item locally Noetherian and $(S_k)$, see Descent, Lemma \ref{descent-lemma-Sk-local-syntomic}, \item Cohen-Macaulay, see Descent, Lemma \ref{descent-lemma-CM-local-syntomic}, \item reduced, see Descent, Lemma \ref{descent-lemma-reduced-local-smooth}, \item normal, see Descent, Lemma \ref{descent-lemma-normal-local-smooth}, \item locally Noetherian and $(R_k)$, see Descent, Lemma \ref{descent-lemma-Rk-local-smooth}, \item regular, see Descent, Lemma \ref{descent-lemma-regular-local-smooth}, \item Nagata, see Descent, Lemma \ref{descent-lemma-Nagata-local-smooth}. \end{enumerate}


stacks-properties-remark-local-source-warning

Warning: Lemma \ref{lemma-local-source} should be used with care. For example, it applies to $\mathcal{P}=$``flat'', $\mathcal{Q}=$``empty'', and $\mathcal{R}=$``flat and locally of finite presentation''. But given a morphism of algebraic spaces $f : X \to Y$ the largest open subspace $W \subset X$ such that $f|_W$ is flat is {\it not} the set of points where $f$ is flat!


stacks-properties-remark-local-source-apply

Notwithstanding the warning in Remark \ref{remark-local-source-warning} there are some cases where Lemma \ref{lemma-local-source} can be used without causing ambiguity. We give a list. In each case we omit the verification of assumptions (1) and (2) and we give references which imply (3) and (4). Here is the list: \begin{enumerate} \item \label{item-rel-dim-leq-d} $\mathcal{Q} = $``locally of finite type'', $\mathcal{R} = \emptyset$, and $\mathcal{P} =$``relative dimension $\leq d$''. See Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-dimension} and Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-openness-bounded-dimension-fibres} and \ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}. \item \label{item-loc-quasi-finite} $\mathcal{Q} =$``locally of finite type'', $\mathcal{R} = \emptyset$, and $\mathcal{P} =$``locally quasi-finite''. This is the case $d = 0$ of the previous item, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0}. On the other hand, properties (3) and (4) are spelled out in Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}. \item \label{item-unramified} $\mathcal{Q} = $``locally of finite type'', $\mathcal{R} = \emptyset$, and $\mathcal{P} =$``unramified''. This is Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-where-unramified}. \item \label{item-flat} $\mathcal{Q} =$``locally of finite presentation'', $\mathcal{R} =$``flat and locally of finite presentation'', and $\mathcal{P} =$``flat''. See More on Morphisms of Spaces, Theorem \ref{spaces-more-morphisms-theorem-openness-flatness} and Lemma \ref{spaces-more-morphisms-lemma-flat-locus-base-change}. Note that here $W(\mathcal{P}, f)$ is always exactly the set of points where the morphism $f$ is flat because we only consider this open when $f$ has $\mathcal{Q}$ (see loc.cit.). \item \label{item-etale} $\mathcal{Q} =$``locally of finite presentation'', $\mathcal{R} =$``flat and locally of finite presentation'', and $\mathcal{P}=$``\'etale''. This follows on combining (\ref{item-unramified}) and (\ref{item-flat}) because an unramified morphism which is flat and locally of finite presentation is \'etale, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-unramified-flat-lfp-etale}. \item Add more here as needed (compare with the longer list at More on Groupoids, Remark \ref{more-groupoids-remark-local-source-apply}). \end{enumerate}


stacks-properties-remark-stack-structure-locally-closed-subset

Let $X$ be an algebraic stack. Let $T \subset |\mathcal{X}|$ be a locally closed subset. Let $\partial T$ be the boundary of $T$ in the topological space $|\mathcal{X}|$. In a formula $$ \partial T = \overline{T} \setminus T. $$ Let $\mathcal{U} \subset \mathcal{X}$ be the open substack of $X$ with $|\mathcal{U}| = |\mathcal{X}| \setminus \partial T$, see Lemma \ref{lemma-open-substacks}. Let $\mathcal{Z}$ be the reduced closed substack of $\mathcal{U}$ with $|\mathcal{Z}| = T$ obtained by taking the reduced induced closed subspace structure, see Definition \ref{definition-reduced-induced-stack}. By construction $\mathcal{Z} \to \mathcal{U}$ is a closed immersion of algebraic stacks and $\mathcal{U} \to \mathcal{X}$ is an open immersion, hence $\mathcal{Z} \to \mathcal{X}$ is an immersion of algebraic stacks by Lemma \ref{lemma-composition-immersion}. Note that $\mathcal{Z}$ is a reduced algebraic stack and that $|\mathcal{Z}| = T$ as subsets of $|X|$. We sometimes say $\mathcal{Z}$ is the {\it reduced induced substack structure} on $T$.


stacks-properties-remark-dimension-empty-stack

If $\mathcal{X}$ is a nonempty stack of finite type over a field, then $\dim(\mathcal{X})$ is an integer. For an arbitrary locally Noetherian algebraic stack $\mathcal{X}$, $\dim(\mathcal{X})$ is in $Z\cup \{\pm \infty\}$, and $\dim(\mathcal{X}) = -\infty$ if and only if $\mathcal{X}$ is empty.


stacks-remark-alternative

Suppose that $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids. In this case we can prove Lemma \ref{lemma-painful} using Categories, Lemma \ref{categories-lemma-fibred-strict} which says that $\mathcal{S} \to \mathcal{C}$ is equivalent to the category associated to a contravariant functor $F : \mathcal{C} \to \textit{Groupoids}$. In the case of the fibred category associated to $F$ we have $g^* \circ f^* = (f \circ g)^*$ on the nose and there is no need to use the maps $\alpha_{g, f}$. In this case the lemma is (even more) trivial. Of course then one uses that the $\mathit{Mor}(x, y)$ presheaf is unchanged when passing to an equivalent fibred category which follows from Lemma \ref{lemma-presheaf-mor-map-fibred-categories}.