diff --git a/.cspell.json b/.cspell.json index 24ec3689f..ceb56302a 100644 --- a/.cspell.json +++ b/.cspell.json @@ -39,6 +39,7 @@ "anneaux", "Aronszajn", "Artin", + "Artinian", "Auslander", "Axiomatising", "axiomatization", @@ -217,6 +218,7 @@ "functionals", "functor", "functorial", + "functoriality", "functors", "Gillam", "Gille", @@ -312,6 +314,7 @@ "Neumann", "Neves", "Niefield", + "nilpotent", "nilradical", "nimbers", "nlab", @@ -377,6 +380,7 @@ "Sierpiński", "simplicial", "simplicially", + "Šmulian", "Specker", "Strecker", "subalgebra", @@ -414,7 +418,6 @@ "surjective", "surjectivity", "surjects", - "Šmulian", "Szamuely", "Tarski", "tensoring", @@ -429,6 +432,7 @@ "Tyrer", "Ulmer", "ultrafilters", + "ultrapower", "uncountably", "unital", "unitalization", diff --git a/CLAUDE.md b/CLAUDE.md index 8bfb1f015..a8285c799 100644 --- a/CLAUDE.md +++ b/CLAUDE.md @@ -120,7 +120,7 @@ Content: - **Easy to understand**: state the claim or key idea first, then the details ("We claim that ... To see this, ..."). Name objects and morphisms explicitly with source and target (`f : X \to Y`). For unsatisfied properties, give a concrete counterexample (e.g. "the embedding $C_2 \hookrightarrow S_3$") rather than an existence argument, whenever possible. - **Complete**: do not skip steps. Phrases like "It is easy to see" or "clearly" are only for exceptional cases, namely steps that are routine for the intended reader. One-line proofs such as "This is trivial." or "This holds by definition." are only for claims that follow immediately from the definitions. - **Classifications and equivalences**: when one direction is immediate, prove only the other one and say so ("For the non-trivial direction, ..."), as in most special-morphism proofs. In content pages, mark the two directions with `($\implies$)` and `($\impliedby$)`, and steps in a cycle of equivalences with `(1) $\implies$ (2)`. -- **Building on known facts**: a proof may use properties of the same structure that are assigned earlier in the file or deduced from them ("We already know that ..."), as well as its classification of special morphisms ("(see below)"). Results about other structures may be used with a link to them. If a proof depends on the proof of another assignment, give that assignment a `label` of the form `_` (e.g. `grp_no_cogenerator`) and list the label under `references`. +- **Building on known facts**: a proof may use properties of the same structure that are assigned earlier in the file or deduced from them ("We already know that ..."), as well as its classification of special morphisms ("(see below)"). Results about other structures may be used with a link to them. **Whenever a proof refers to the proof of another assignment**, give that assignment a `label` of the form `_` (e.g. `Grp_no_cogenerator`) and list the label under `references` of the citing assignment. This applies to every kind of reference, including loose ones such as "see the proof for $\LRS_R$" or "the idea is the same as in the proof for $\CAlg(R)$". It is essential to avoid dead references: assignments are regularly removed when they become redundant (e.g. after a new implication), and since `references` is a foreign key to the label, `db:seed` then fails instead of silently keeping a proof that refers to a deleted one. Without the reference, nothing detects this. Before removing a labeled assignment, rewrite the citing proofs. - **Reuse instead of repetition**: put general lemmas in `content/` pages instead of repeating an argument in several YAML files, and apply them explicitly, e.g. "apply the contrapositive of the dual of Lemma 2 here to the forgetful functor $\Ab \to \Grp$". - **Generalize concrete proofs**: once a proof for a specific structure works, check which features of the structure it really uses. If these are general properties (e.g. infinitary extensivity, existence of certain limits), state the result as an implication, or as a lemma in a `content/` page if it cannot be phrased with existing properties. An implication is much more powerful than a single assignment: the deduction system applies it, and its dual, to every structure, which decides further properties and makes existing direct proofs redundant. After adding one, run `pnpm db:update` and remove the redundant assignments (or mark them with `check_redundancy: false` if other proofs cite them). Example: the proof that $\LRS_R$ is not coaccessible only used that $\LRS_R$ is infinitary extensive, which led to the implication `infinitary_extensive_not_coaccessible`. It also decided `Met_c` and made the proofs for `LRS_R`, `Sch_R`, `Top`, and `Quiv_fc` redundant. diff --git a/content/Top-embeds-in-LRS.md b/content/Top-embeds-in-LRS.md deleted file mode 100644 index 3e10ad6fa..000000000 --- a/content/Top-embeds-in-LRS.md +++ /dev/null @@ -1,59 +0,0 @@ ---- -title: An embedding of the category of topological spaces in the category of locally ringed spaces -description: Describes a functor which makes the category of topological spaces a coreflective and "almost reflective" subcategory of the category of locally ringed spaces. From the properties of this embedding, we can rule out several properties for the category of locally ringed spaces, using the corresponding failures of these properties for the category of topological spaces. ---- - -# An embedding of the category of topological spaces in the category of locally ringed spaces - -For much of this development, we will be dealing with the case of $\LRS_k$ where $k$ is a field. We begin by describing $\Top$ as a reflective subcategory of $\LRS_k$. - -::: Lemma 1 -The forgetful functor $U : \LRS_k \to \Top$ has a right adjoint $K : \Top \to \LRS_k$ given by equipping a topological space $X$ with the constant sheaf $\underline{k}$. Furthermore, the functor $K$ is fully faithful, thus making $\Top$ into a reflective subcategory of $\LRS_k$. -::: - -::: Proof -In this adjunction, the counit $UK \to \id$ is just the identity. To describe the unit $\id \to KU$, we need to define a morphism $(X, \O_X) \to (X, \underline{k})$ for any locally ringed space $(X, \O_X)$ over $k$. This morphism will be the identity on topological spaces, and the pullback operation $\underline{k} \to \O_X$ will be the unique morphism of sheaves induced by the given structure of $\O_X$ as a sheaf of $k$-algebras. It is now straightforward to check that this indeed defines an adjunction; and since the counit is an isomorphism, that implies that $K$ is fully faithful. -::: - -We now show that this reflective subcategory is in fact also a coreflective subcategory. Recall that for $f \in \O_X(U)$ we have its vanishing set $V(f) \coloneqq \{x\in U : f \in \m_{X,x}\} = \{x \in U : f(x) = 0\}$, where $f(x) \in \kappa(x)$ is the image of $f_x \in \O_{X,x}$ in the residue field. - -::: Lemma 2 -For each object $X$ of $\LRS_k$, let $X_0$ be the set of points $x \in X$ such that the induced morphism from $k$ to the residue field $\kappa(x)$ is an isomorphism. We give $X_0$ the following strengthening of the subspace topology: it will be the topology where a neighborhood subbasis at $x \in X_0$ is the collection of sets of the form $X_0 \cap V(f)$ where $f \in \O_X(U)$ for some neighborhood $U$ of $x$ in $X$, and $x \in V(f)$. Then $X \mapsto X_0$ defines a functor $S : \LRS_k \to \Top$ that is right adjoint to $K$. -::: - -::: Proof -First, to see that $S$ is a functor, suppose we have a morphism $f : (X, \O_X) \to (Y, \O_Y)$. Then for $x \in X_0$, we have a sequence $k \to \kappa(f(x)) \to \kappa(x)$ where the composition is an isomorphism. Thus, $\kappa(f(x)) \to \kappa(x)$ is a surjective morphism of fields, and therefore an isomorphism. It follows that $k \to \kappa(f(x))$ is also an isomorphism of fields, so $f(x) \in Y_0$. To see that the restriction map $X_0 \to Y_0$ is continuous, suppose $g \in \O_Y(V)$ is such that $f(x) \in V(g)$. Then $x \in V(f^\sharp g)$, and -$$X_0 \cap f^{-1}(Y_0 \cap V(g)) = X_0 \cap V(f^\sharp g)$$ -where $f^\sharp g \in \O_X(f^{-1}(V))$. In other words, we have shown that the inverse image in $X_0$ of any subbasic neighborhood of $f(x)$ is a neighborhood of $x$. - -Now if we apply the functor $S$ to a space of the form $(X, \underline{k})$, then since by definition any section of the constant sheaf $\underline{k}$ is locally constant, we see that we recover exactly $X$ with its original topology. Thus, we can define the unit $\id \to SK$ of the adjunction to be the identity. - -As for the counit $KS \to \id$, for any locally ringed space $(X, \O_X)$ over $k$ we need to define a morphism $(X_0, \underline{k}) \to (X, \O_X)$. The map of topological spaces will be the inclusion map $i : X_0 \hookrightarrow X$, which is continuous since in particular for $U$ an open neighborhood of $x \in X_0$ we have $X_0 \cap U = X_0 \cap V(0_U)$, where $0_U \in \O_X(U)$ is the zero element. The pullback map $\O_X \to i_* \underline{k}$ takes $f \in \O_X(U)$ to the function $X_0 \cap U \to k$ where $x \in X_0 \cap U$ maps to the inverse image of $f(x) \in \kappa(x)$ under the isomorphism $k \to \kappa(x)$. An alternative description of this pullback is that $x \in X_0 \cap U$ maps to the unique $a\in k$ such that $x \in V(f-a)$. Since $X_0 \cap V(f-a)$ is a neighborhood of $x$ in $X_0$ by definition, this shows that we get a locally constant function to $k$ as required. - -From here, it is straightforward to show that this does in fact define an adjunction. -::: - -_Remark._ In the special case where $k$ is a finite field, we have -$$\textstyle X_0 \cap V(f) = \bigcap_{a \in k^\times} (X_0 \cap D(f-a)),$$ -which is already open in the subspace topology. Therefore, in this case, $X_0$ is given exactly the subspace topology. - -::: Corollary 3 -For any non-trivial commutative ring $R$, fix a quotient field $k$ of $R$. Then the functor $K_R : \Top \to \LRS_R$ given by equipping a topological space with the constant sheaf $\underline{k}$ is fully faithful; has a right adjoint; and preserves all inhabited limits. -::: - -::: Proof -The functor $K_R$ is the composition of $K : \Top \to \LRS_k$ and the forgetful functor $\LRS_k \to \LRS_R$. Since $\LRS_k$ is equivalent to the slice category of $\LRS_R$ over the subterminal object $\Spec k$, the forgetful functor is fully faithful; has right adjoint ${-} \times \Spec k$; and preserves all inhabited limits. Therefore, from the previously established results on $K$, the result follows. -::: - -::: Corollary 4 -Let $R$ be any non-trivial commutative ring. Then:
-(a) $\LRS_R$ is not cartesian closed.
-(b) $\LRS_R$ does not have cartesian filtered colimits.
-(c) $\LRS_R$ is not regular.
-(d) $\LRS_R$ does not have cofiltered-limit-stable epimorphisms.
-(e) $\LRS_R$ does not have effective cocongruences. -::: - -::: Proof -We already know that $\Top$ does not satisfy any of these properties. In order to conclude that $\LRS_R$ does not satisfy any of them either, we fix a quotient field of $R$ as above and consider the functor $K_R$. In each case, this is an easy application of the contrapositive of a result from [here](/content/subcategories) to the functor $K_R$. Namely, (a) follows from Lemma 5; (b) from Lemma 4; (c) from Lemma 7; (d) from the dual of Lemma 2 with the observation that $K_R$ preserves epimorphisms since it has a right adjoint; and (e) from the dual of Lemma 8. -::: diff --git a/content/local-ring-epimorphisms.md b/content/local-ring-epimorphisms.md new file mode 100644 index 000000000..951f1741b --- /dev/null +++ b/content/local-ring-epimorphisms.md @@ -0,0 +1,32 @@ +--- +title: Epimorphisms of local commutative rings +description: The codomain of an epimorphism in the category of local commutative rings is not much larger than its domain. This is used to prove that the category of locally ringed spaces is well-powered. +--- + +# Epimorphisms of local commutative rings + +We consider the category of local commutative rings and local homomorphisms. + +::: Lemma +Let $\varphi : A \to B$ be a local homomorphism of local commutative rings which is an epimorphism in the category of local commutative rings and local homomorphisms. Then $|B| \leq \max(|A|,\aleph_0)$. +::: + +::: Proof +The homomorphism $\varphi$ factors as a surjective homomorphism $A \to \varphi(A)$ followed by the inclusion $\varphi(A) \hookrightarrow B$. The ring $\varphi(A)$ is local since it is a non-zero quotient ring of $A$. The inclusion $\varphi(A) \hookrightarrow B$ is a local homomorphism and an epimorphism in the category of local commutative rings and local homomorphisms, and $|\varphi(A)| \leq |A|$. Hence, we may assume that $A$ is a subring of $B$ and that $\varphi$ is the inclusion. + +A _system_ over $A$ in the variables $Y_1,\dotsc,Y_n$ consists of finitely many polynomials $p_1,\dotsc,p_k \in A[Y_1,\dotsc,Y_n]$ and a subset $J \subseteq \{1,\dotsc,n\}$. A _solution_ of the system is a tuple $(c_1,\dotsc,c_n) \in B^n$ with $p_i(c_1,\dotsc,c_n) = 0$ for all $i$ and $c_j \in \m_B$ for all $j \in J$. We say that the system _determines_ $b \in B$ if it has a solution and every solution $(c_1,\dotsc,c_n)$ satisfies $c_1 = b$. Every system determines at most one element, and there are at most $\max(|A|,\aleph_0)$ systems over $A$. Hence, it suffices to prove that every $b \in B$ is determined by some system over $A$. + +Assume that $b \in B$ is not determined by any system over $A$. Let $\F$ be the set of finite subsets of $B$ containing $b$. For $F \in \F$ we consider the system $\Sigma_F$ over $A$ in the variables $Y_x$ for $x \in F$, enumerated such that $Y_b$ is the first variable, which consists of the polynomials + +- $Y_x + Y_{x'} - Y_{x''}$ for $x,x',x'' \in F$ with $x + x' = x''$, +- $Y_x Y_{x'} - Y_{x''}$ for $x,x',x'' \in F$ with $x x' = x''$, +- $Y_a - a$ for $a \in F \cap A$, + +and the subset $F \cap \m_B \subseteq F$. The tuple $(x)_{x \in F}$ is a solution of $\Sigma_F$. Since $\Sigma_F$ does not determine $b$, it has another solution $(g_F(x))_{x \in F}$ with $g_F(b) \neq b$. We extend $g_F$ to a map $g_F : B \to B$ by $g_F(x) \coloneqq 0$ for $x \notin F$. + +For a finite subset $E \subseteq B$ let $\F_{\supseteq E} \coloneqq \{F \in \F : E \subseteq F\}$. These sets are non-empty since $E \cup \{b\} \in \F_{\supseteq E}$, and they satisfy $\F_{\supseteq E} \cap \F_{\supseteq E'} = \F_{\supseteq E \cup E'}$. Hence, they have the finite intersection property, so that there is an ultrafilter $\U$ on $\F$ containing all of them. Let $T \coloneqq B^\F / \U$ be the ultrapower, i.e. the quotient of the ring $B^\F$ by the ideal of all families $(t_F)_{F \in \F}$ with $\{F \in \F : t_F = 0\} \in \U$. We write $[t_F]$ for the class of $(t_F)_{F \in \F}$. Then $T$ is a local commutative ring whose maximal ideal $\m_T$ consists of the classes $[t_F]$ with $\{F \in \F : t_F \in \m_B\} \in \U$. Indeed, these classes form an ideal not containing $1$, and every other class $[t_F]$ has $t_F \in B^\times$ for all $F$ in a set $M \in \U$, so that it is invertible with inverse $[u_F]$, where $u_F \coloneqq t_F^{-1}$ for $F \in M$ and $u_F \coloneqq 0$ otherwise. (Alternatively, this follows from [Łoś's theorem](https://en.wikipedia.org/wiki/Ultraproduct#%C5%81o%C5%9B's_theorem).) + +We define two maps $\delta, g : B \to T$ by $\delta(x) \coloneqq [x]$ and $g(x) \coloneqq [g_F(x)]$. The map $\delta$ is a local homomorphism since $x \in \m_B$ implies $[x] \in \m_T$. For $x,x' \in B$ and all $F \in \F_{\supseteq \{x,x',x+x',xx'\}}$ we have +$$g_F(x + x') = g_F(x) + g_F(x'), \quad g_F(x x') = g_F(x) \, g_F(x'),$$ +and for $a \in A$ and all $F \in \F_{\supseteq \{a\}}$ we have $g_F(a) = a$. Since $1 \in A$, this shows that $g$ is a ring homomorphism with $g(a) = \delta(a)$ for all $a \in A$. Moreover, $g$ is local since $g_F(x) \in \m_B$ for $x \in \m_B$ and all $F \in \F_{\supseteq \{x\}}$. Since the inclusion $A \hookrightarrow B$ is an epimorphism, we get $\delta = g$. But $g_F(b) \neq b$ for all $F \in \F$, so that $g(b) \neq \delta(b)$, a contradiction. +::: diff --git a/content/subcategories.md b/content/subcategories.md index 7c0066b5a..b93845a97 100644 --- a/content/subcategories.md +++ b/content/subcategories.md @@ -117,20 +117,16 @@ Finally, to prove the parenthetical remark, assume that $\C$ has binary products ::: ::: Lemma 9 -Let $L$ be a functor which is left adjoint to a faithful functor $U$. Then $L$ preserves generating collections. (Thus in particular, any reflective subcategory of a category with a generating collection also has a generating collection; and similarly for a single generator.) +Let $U : \C \to \D$ be a faithful functor with a left adjoint $L : \D \to \C$. Then $L$ preserves generating collections. If $U$ is also conservative, then $L$ preserves extremal generating collections. (Thus in particular, any reflective subcategory of a category with a generating collection also has a generating collection; and similarly for a single generator.) ::: ::: Proof -If $S$ is a generating collection, +Let $S$ be a collection of objects of $\D$. By the adjunction, the functor +$$(\Hom(L(G),-))_{G \in S} : \C \to (\Set^+)^S$$ +is isomorphic to the composition of $U : \C \to \D$ followed by +$$(\Hom(G,-))_{G \in S} : \D \to (\Set^+)^S.$$ -$$ -\begin{align*} -\prod_{G\in S} \Hom(L(G),-) & \cong \prod_{G\in S} \Hom(G,U(-)) \\ -& \cong \left( \prod_{G\in S} \Hom(G,-) \right) \circ U -\end{align*} -$$ - -is a composition of faithful functors, hence faithful. +If $S$ is a generating collection, this is a composition of faithful functors, hence faithful. If $S$ is an extremal generating collection and $U$ is conservative, this is a composition of faithful conservative functors, hence faithful and conservative. ::: ::: Lemma 10 diff --git a/database/data/categories/LRS_R.yaml b/database/data/categories/LRS_R.yaml index 8a6e4b17d..facb986c8 100644 --- a/database/data/categories/LRS_R.yaml +++ b/database/data/categories/LRS_R.yaml @@ -1,9 +1,9 @@ id: LRS_R name: category of locally ringed spaces notation: $\LRS_R$ -objects: locally ringed spaces over $R \neq 0$, i.e. pairs $(X,\O_X)$ consisting of a topological space and a sheaf of commutative $R$-algebras on it whose stalks are local rings -morphisms: 'A morphism $(f,f^\sharp) : (X,\O_X) \to (Y,\O_Y)$ consists of a continuous map $f : X \to Y$ and a homomorphism $f^\sharp : \O_Y \to f_* \O_X$ of sheaves of $R$-algebras such that for every $x \in X$ the induced map on stalks $f_x : \O_{Y,f(x)} \to \O_{X,x}$ is local.' -description: Here, $R$ is a fixed non-trivial commutative ring. The general properties of this category will not depend on the choice of $R$. For $R = \IZ$ we obtain the category $\LRS$ of locally ringed spaces. +objects: locally ringed spaces $X$ over $R \neq 0$, each consisting of a topological space $|X|$ and a sheaf $\O_X$ of commutative $R$-algebras on $|X|$ whose stalks are local rings +morphisms: 'A morphism $f : X \to Y$ consists of a continuous map $|f| : |X| \to |Y|$ and a homomorphism $f^\sharp : \O_Y \to |f|_* \O_X$ of sheaves of $R$-algebras such that for every $x \in |X|$ the induced map on stalks $f^\sharp_x : \O_{Y,|f|(x)} \to \O_{X,x}$ is local.' +description: Here, $R$ is a fixed non-trivial commutative ring. The general properties of this category will not depend on the choice of $R$. For $R = \IZ$ we obtain the category $\LRS$ of locally ringed spaces. We exclude $R=0$ because this yields the trivial category. nlab_link: https://ncatlab.org/nlab/show/locally+ringed+topological+space tags: @@ -14,58 +14,76 @@ related: - Top - CAlg(R) -comments: - - Monomorphisms $X \to Y$ can be formally described by the condition that the diagonal morphism $X \to X \times_Y X$ is an isomorphism. Since the fiber product of locally ringed spaces has a concrete description (MSE/1033675), we thus have a description of the monomorphisms, albeit a very complicated one. At least, we can say that every monomorphism is injective. - satisfied_properties: - property: locally small - proof: For two ringed spaces $(X,\O_X), (Y,\O_Y)$, the collection of morphisms $X \to Y$ of ringed spaces is the collection $\coprod_{f \in \Hom(X,Y)} \Hom(\O_Y,f_* \O_X)$, which is a set since $\Hom(X,Y)$ is a set and each $\Hom(\O_Y,f_* \O_X)$ is a set. + proof: For two ringed spaces $X,Y$, the collection of morphisms $X \to Y$ of ringed spaces is the collection $\coprod_{u \in \Hom(|X|,|Y|)} \Hom(\O_Y,u_* \O_X)$, where $u$ runs through the continuous maps $|X| \to |Y|$. This is a set since $\Hom(|X|,|Y|)$ is a set and each $\Hom(\O_Y,u_* \O_X)$ is a set. - property: complete proof: See Gillam. See also MSE/1033675. - property: cocomplete - proof: See Demazure-Gabriel's "Groupes algébriques", I. §1. 1.6. Specifically, the forgetful functor from locally ringed spaces to ringed spaces preserves colimits, and colimits of ringed spaces are built from colimits of topological spaces and limits of commutative rings, see MSE/1646202. - - - property: well-copowered - proof: This follows from the characterization of epimorphisms below. - - - property: concretizable proof: >- - Consider the functor $F : \LRS_R \to \Set \times \Set^{\op}$ defined on objects by - $$\textstyle F(X,\O_X) \coloneqq \bigl(|X|,\coprod_{U \subseteq X} |\O_X(U)|\bigr),$$ - where $U$ runs through the open subsets of $X$ and $|{-}|$ denotes the underlying set of a topological space or ring, respectively. If $(f,f^\sharp) : (X,\O_X) \to (Y,\O_Y)$ is a morphism of locally ringed spaces, then we get a map of sets $|f| : |X| \to |Y|$, and the ring homomorphisms $f^\sharp(V) : \O_Y(V) \to \O_X(f^*(V))$ for opens $V \subseteq Y$ induce a map of sets - $$\textstyle \alpha : \coprod_{V \subseteq Y} \O_Y(V) \to \coprod_{U \subseteq X} \O_X(U)$$ - defined by $\alpha \circ \iota_V = \iota_{f^*(V)} \circ f^\sharp(V)$. We let $F(f,f^\sharp) \coloneqq (|f|,\alpha)$. It is easy to verify that $F$ is a functor. - - To show that $F$ is faithful, assume that $(f,f^\sharp)$ and $(g,g^\sharp)$ are two morphisms with the same image under $F$. Then $|f| = |g|$, hence $f = g$. Also, we have - $$\iota_{f^*(V)} \circ f^\sharp(V) = \iota_{g^*(V)} \circ g^\sharp(V)$$ - for every open subset $V \subseteq Y$. Since $\iota_{f^*(V)} = \iota_{g^*(V)}$ is a monomorphism of sets, we conclude that $f^\sharp(V) = g^\sharp(V)$. Thus, $f^\sharp = g^\sharp$. We have shown $(f,f^\sharp) = (g,g^\sharp)$, as required. - - Since $\Set^{\op}$ is concretizable (say, via the contravariant power set functor), and $\Set \times \Set$ is also concretizable (because it is locally small and has a generating collection), it follows that $\LRS_R$ is concretizable. + Colimits are computed as in ringed spaces: for a diagram $(X_i)_{i \in I}$ in $\LRS_R$, let $X$ be the ringed space whose underlying space $|X| \coloneqq \colim_i |X_i|$ is the colimit of the underlying spaces, with colimit maps $u_i : |X_i| \to |X|$, and whose structure sheaf $\O_X \coloneqq \lim_i (u_i)_* \O_{X_i}$ is the limit of the push-forwards of the structure sheaves. Then $X$ is a locally ringed space, the morphisms $X_i \to X$ consisting of $u_i$ and the limit projections $\O_X \to (u_i)_* \O_{X_i}$ have local stalk maps, and they form a colimit in $\LRS_R$. See Demazure–Gabriel, Groupes algébriques, Tome I, Ch. I, §1, 1.6. See also MSE/1646202. + check_redundancy: false - property: infinitary extensive proof: >- The proof works for both ringed and locally ringed spaces and relies on the fact that $\Top$ is infinitary extensive. - First, recall some basic facts about open embeddings (often called open immersions in the context of schemes). If $(X,\O_X)$ is a locally ringed space and $U \subseteq X$ is an open subset, then, letting $\O_U \coloneqq (\O_X)|_U$, the pair $(U,\O_U)$ is again a locally ringed space. There is an inclusion morphism $(i,i^\sharp) : (U,\O_U) \to (X,\O_X)$, where $i$ is the inclusion map and $i^\sharp : \O_X \to i_* \O_U$ is defined via restriction maps. This is a monomorphism, and a morphism into $(X,\O_X)$ factors through it if and only if the image of its underlying map is contained in $U$. We will refer to any monomorphism isomorphic to one of this form as an open embedding. + First, recall some basic facts about open embeddings (often called open immersions in the context of schemes). If $X$ is a locally ringed space and $U \subseteq |X|$ is an open subset, then the space $U$ with the sheaf $(\O_X)|_U$ is again a locally ringed space, which we denote by $X_U$. There is an inclusion morphism $i : X_U \to X$, where $|i| : U \hookrightarrow |X|$ is the inclusion map and $i^\sharp : \O_X \to |i|_* (\O_X)|_U$ is defined via restriction maps. This is a monomorphism, and a morphism $g : T \to X$ factors through it if and only if $|g|(|T|) \subseteq U$. We will refer to any monomorphism isomorphic to one of this form as an open embedding. - Pullbacks along open embeddings exist and are themselves open embeddings. Indeed, if $(f,f^\sharp) : (X,\O_X) \to (Y,\O_Y)$ is a morphism of locally ringed spaces and $V \subseteq Y$ is an open subset, then $f^*(V) \subseteq X$ is an open subset, and $(f^*(V),\O_{f^*(V)}) \hookrightarrow (X,\O_X)$ is the pullback of $(f,f^\sharp)$ along $(V,\O_V) \hookrightarrow (Y,\O_Y)$. + Pullbacks along open embeddings exist and are themselves open embeddings. Indeed, if $f : X \to Y$ is a morphism of locally ringed spaces and $V \subseteq |Y|$ is an open subset, then $|f|^*(V) \subseteq |X|$ is an open subset, and the inclusion $X_{|f|^*(V)} \hookrightarrow X$ is the pullback of $f$ along the inclusion $Y_V \hookrightarrow Y$. - Since $\LRS_R$ is cocomplete, coproducts exist. Concretely, the coproduct of a family of locally ringed spaces $(X_i,\O_{X_i})$ is the locally ringed space $(X,\O_X)$, where $X \coloneqq \coprod_i X_i$ and $\O_X(U) \coloneqq \prod_i \O_{X_i}(U \cap X_i)$. Notice that the coproduct inclusions are open embeddings. Hence, pullbacks along coproduct inclusions exist. + Since $\LRS_R$ is cocomplete, coproducts exist. Concretely, the coproduct of a family of locally ringed spaces $X_i$ is the locally ringed space $X$ with $|X| \coloneqq \coprod_i |X_i|$ and $\O_X(U) \coloneqq \prod_i \O_{X_i}(U \cap |X_i|)$. Notice that the coproduct inclusions are open embeddings. Hence, pullbacks along coproduct inclusions exist. That coproducts are disjoint in $\LRS_R$ follows from the corresponding property of $\Top$ together with the following observations. By the descriptions above, the forgetful functor $\LRS_R \to \Top$ preserves coproducts and pullbacks along open embeddings (although it does not preserve pullbacks in general), and the empty topological space admits a unique structure sheaf, up to isomorphism, making it into a locally ringed space. - It remains to prove that coproducts are stable under pullbacks. To this end, let $(f,f^\sharp) : (T,\O_T) \to \coprod_i (X_i,\O_{X_i})$ be a morphism. Then $T_i \coloneqq f^*(X_i) \subseteq T$ is an open subset, and for $i \neq j$ we have $T_i \cap T_j = \varnothing$ because $X_i \cap X_j = \varnothing$ in the coproduct. We must show that the canonical morphism $(g,g^\sharp) : \coprod_i (T_i,\O_{T_i}) \to (T,\O_T)$ is an isomorphism. The underlying continuous map $g$ is a homeomorphism because coproducts in $\Top$ are stable under pullbacks. It therefore remains to show that $g^\sharp : \O_T \to g_* \O$, where $\O$ denotes the structure sheaf of the coproduct, is an isomorphism. Equivalently, for every open subset $U \subseteq T$, we must show that the canonical ring homomorphism - $$\textstyle \O_T(U) \to \prod_{i \in I} \O_{T_i}(U \cap T_i) = \prod_{i \in I} \O_T(U \cap T_i)$$ - is an isomorphism. This is precisely the sheaf axiom applied to the open cover of $U$ by the pairwise disjoint open subsets $U \cap T_i$. + It remains to prove that coproducts are stable under pullbacks. To this end, let $f : T \to \coprod_i X_i$ be a morphism. Then $U_i \coloneqq |f|^*(|X_i|) \subseteq |T|$ is an open subset, and for $i \neq j$ we have $U_i \cap U_j = \varnothing$ because $|X_i| \cap |X_j| = \varnothing$ in $\coprod_i |X_i|$. Let $T_i \coloneqq T_{U_i}$. We must show that the canonical morphism $g : \coprod_i T_i \to T$ is an isomorphism. The underlying continuous map $|g|$ is a homeomorphism because coproducts in $\Top$ are stable under pullbacks. It therefore remains to show that $g^\sharp : \O_T \to |g|_* \O$, where $\O$ denotes the structure sheaf of $\coprod_i T_i$, is an isomorphism. Equivalently, for every open subset $U \subseteq |T|$, we must show that the canonical ring homomorphism + $$\textstyle \O_T(U) \to \prod_{i \in I} \O_{T_i}(U \cap U_i) = \prod_{i \in I} \O_T(U \cap U_i)$$ + is an isomorphism. This is precisely the sheaf axiom applied to the open cover of $U$ by the pairwise disjoint open subsets $U \cap U_i$. + + - property: extremal cogenerator + proof: >- + Let $\RS_R$ be the category of ringed spaces over $R$, which is defined like $\LRS_R$, but without the conditions on stalks and stalk maps. By Gillam, Cor. 6 (stated there for $R = \IZ$, but the proof works verbatim for every $R$), the inclusion functor $\LRS_R \to \RS_R$ has a right adjoint $L : \RS_R \to \LRS_R$. The inclusion functor is faithful, and it is conservative since the inverse of an isomorphism of ringed spaces has bijective, hence local, stalk maps. By the dual of Lemma 9 here, $L$ maps extremal cogenerators of $\RS_R$ to extremal cogenerators of $\LRS_R$. So it suffices to find an extremal cogenerator of $\RS_R$. + + Let $Q$ be the ringed space over $R$ whose underlying space is $\{0,1\}^2$ with the open subsets + $$\varnothing,\quad W \coloneqq \{0,1\} \times \{1\},\quad \{0,1\}^2,$$ + and whose structure sheaf is given by + $$\begin{align*} + \O_Q(\{0,1\}^2) & \coloneqq R, \\ + \O_Q(W) & \coloneqq R[T], \\ + \O_Q(\varnothing) & \coloneqq 0, + \end{align*}$$ + with the inclusion $R \hookrightarrow R[T]$ as the restriction map. This is a sheaf since every open cover of a non-empty open subset contains this subset. We claim that $Q$ is an extremal cogenerator of $\RS_R$. + + A morphism $X \to Q$ in $\RS_R$ consists of a function $(h,k) : |X| \to \{0,1\}^2$ such that $U \coloneqq k^*(\{1\})$ is open, together with $R$-algebra homomorphisms $R \to \O_X(|X|)$ and $R[T] \to \O_X(U)$ that are compatible with the restriction maps. The first homomorphism is unique, the second one is determined by the image $s \in \O_X(U)$ of $T$, and the compatibility is automatic. Let $S(X)$ be the set of pairs $(U,s)$ with $U \subseteq |X|$ open and $s \in \O_X(U)$. We obtain a bijection + $$\Hom_{\RS_R}(X,Q) \cong \Hom_{\Set}(|X|,\{0,1\}) \times S(X).$$ + Under this bijection, precomposition with a morphism $f : X \to Y$ is the product of the maps + $$\alpha_f : \Hom_{\Set}(|Y|,\{0,1\}) \to \Hom_{\Set}(|X|,\{0,1\}), \quad h \mapsto h \circ |f|,$$ + $$\beta_f : S(Y) \to S(X), \quad (V,s) \mapsto (|f|^*(V), f^\sharp(s)).$$ + All four sets are non-empty, since they contain the constant function with value $0$ and the pair $(\varnothing,0)$, respectively. Hence, the product map determines $\alpha_f$ and $\beta_f$, and it is bijective if and only if $\alpha_f$ and $\beta_f$ are. + + To show that $Q$ is a cogenerator, let $f,g : X \rightrightarrows Y$ be two morphisms that induce the same map by precomposition, so that $\alpha_f = \alpha_g$ and $\beta_f = \beta_g$. Since $\{0,1\}$ is a cogenerator of $\Set$, the first equation implies $|f| = |g|$. The second equation means $f^\sharp(s) = g^\sharp(s)$ for all open subsets $V \subseteq |Y|$ and $s \in \O_Y(V)$, i.e. $f^\sharp = g^\sharp$. Hence, $f = g$. + + Now let $f : X \to Y$ be a morphism such that precomposition with it is bijective, so that $\alpha_f$ and $\beta_f$ are bijective. Since $\{0,1\}$ is an extremal cogenerator of $\Set$, $|f|$ is bijective. Since every pair $(U,0)$ lies in the image of $\beta_f$, every open subset of $|X|$ has the form $|f|^*(V)$ for an open subset $V \subseteq |Y|$, so $|f|$ is a homeomorphism. Finally, $\beta_f$ is the disjoint union of the maps $f^\sharp : \O_Y(V) \to \O_X(|f|^*(V))$, and $V \mapsto |f|^*(V)$ is a bijection between the open subsets of $|Y|$ and $|X|$. Hence, each of these maps is bijective. This shows that $f^\sharp : \O_Y \to |f|_* \O_X$ is an isomorphism, so that $f$ is an isomorphism of ringed spaces. + references: + - Set_extremal_cogenerator + + - property: well-copowered + proof: This follows from the characterization of epimorphisms below. + + - property: well-powered + proof: >- + Let $f : X \to Y$ be a monomorphism. By the classification of monomorphisms (see below), $|f| : |X| \to |Y|$ is injective, and the stalk maps $f^\sharp_x : \O_{Y,|f|(x)} \to \O_{X,x}$ are epimorphisms in the category of local commutative rings. By this lemma, every stalk of $\O_X$ has cardinality at most + $$\textstyle \lambda \coloneqq \max\bigl(\aleph_0, \sup_{y \in |Y|} \card(\O_{Y,y})\bigr).$$ + Fix a set $S$ of cardinality $\lambda$, and choose injective maps $\O_{X,x} \to S$. Since a section of $\O_X$ is determined by its germs, we obtain injective maps $\O_X(U) \to S^U$ for all open subsets $U \subseteq |X|$, which are compatible with the restriction maps. Transporting the structure along these maps and along $|f|$, we see that $X$ is isomorphic to a locally ringed space $X'$ with $|X'| \subseteq |Y|$ and $\O_{X'}(U) \subseteq S^U$ for all open subsets $U \subseteq |X'|$, such that the restriction maps are the projections. These locally ringed spaces form a set, and for each of them there is only a set of morphisms $X' \to Y$. Hence, $\Sub(Y)$ is essentially small. unsatisfied_properties: - property: skeletal proof: This is trivial. - property: balanced - proof: Choose a residue field $K$ of $R$. The canonical morphism $$\Spec(K[X]/X \times K[X,X^{-1}]) \longrightarrow \Spec(K[X])$$ is a mono- and an epimorphism, but no isomorphism. + proof: Choose a residue field $K$ of $R$. The canonical morphism $$\Spec(K[T]/T \times K[T,T^{-1}]) \longrightarrow \Spec(K[T])$$ is a mono- and an epimorphism, but no isomorphism. - property: semi-strongly connected proof: This is because already the full subcategory $\Sch_R$ of $R$-schemes is not semi-strongly connected. @@ -74,28 +92,58 @@ unsatisfied_properties: proof: This is because already the full subcategory $\Sch_R$ of $R$-schemes is not Malcev. - property: co-Malcev - proof: 'We can adjust the proof for $\Top$ (see MO/509548) as follows: Let $K$ be a residue field of $R$, let $X$ be a singleton and $C = \{u,v\}$ be the Sierpiński space where $\{u\}$ is open, but $\{v\}$ is not. Endow both with the sheaf of locally constant functions to $K$. Thus, $\O_X(X) = K$, $\O_C(C) = \O_C(\{u\}) = K$. Let $j_1, j_2 : X \rightrightarrows C$ be the two canonical morphisms with images $u$ and $v$, respectively. They form a coreflexive corelation that is not cosymmetric.' + proof: 'We can adjust the proof for $\Top$ (see MO/509548) as follows: Let $K$ be a residue field of $R$, let $X$ be the locally ringed space whose underlying space $|X|$ is a singleton, and let $C$ be the locally ringed space whose underlying space $|C| = \{u,v\}$ is the Sierpiński space where $\{u\}$ is open, but $\{v\}$ is not. Both have the sheaf of locally constant functions to $K$ as structure sheaf. Thus, $\O_X(|X|) = K$, $\O_C(|C|) = \O_C(\{u\}) = K$. Let $j_1, j_2 : X \rightrightarrows C$ be the two canonical morphisms such that $|j_1|$ and $|j_2|$ have the images $\{u\}$ and $\{v\}$, respectively. They form a coreflexive corelation that is not cosymmetric.' references: - Top_not_co-malcev - property: total proof: 'The adjunction between the global sections functor and the $\Spec$ functor (Grothendieck–Dieudonné, EGA I (1971), Ch. 1, Prop. 1.6.3) makes $\CAlg(R)^{\op}$ into a reflective subcategory of $\LRS_R$. Therefore, if $\LRS_R$ were total, then $\CAlg(R)$ would be cototal by Kelly, Cor. 6.2, which we know is not the case.' - - property: cartesian closed - proof: This is Corollary 4(a) here. - check_redundancy: false - - property: cartesian filtered colimits - proof: This is Corollary 4(b) here. + proof: >- + We embed $\Top$ into $\LRS_R$ and use that $\Top$ does not have cartesian filtered colimits. Choose a maximal ideal $\m$ of $R$ and let $K \coloneqq R/\m$. For a topological space $T$, let $\underline{K}_T$ be the constant sheaf with value $K$ on $T$, whose sections over an open subset $V \subseteq T$ are the locally constant functions $V \to K$, i.e. the continuous maps $V \to K$ for the discrete topology on $K$. Its stalks are isomorphic to $K$, so we obtain a locally ringed space $\Delta(T)$ with $|\Delta(T)| \coloneqq T$ and $\O_{\Delta(T)} \coloneqq \underline{K}_T$. A continuous map $u : T \to T'$ induces a morphism $\Delta(u) : \Delta(T) \to \Delta(T')$ with $|\Delta(u)| \coloneqq u$ whose sheaf map is given by $s \mapsto s \circ u$. This defines a functor + $$\Delta : \Top \to \LRS_R.$$ + We say that a locally ringed space $X$ is defined over $K$ if the unique morphism $X \to \Spec(R)$ factors through $\Spec(K)$, i.e. if $\m \, \O_X = 0$. For example, $\Delta(T)$ is defined over $K$ since $\m$ annihilates $\underline{K}_T$. + + Claim 1. Let $X$ be a locally ringed space and $T$ be a topological space. If $X$ is defined over $K$, then the map $\Hom(X,\Delta(T)) \to \Hom(|X|,T)$, $g \mapsto |g|$ is bijective. Otherwise, $\Hom(X,\Delta(T)) = \varnothing$. + + If there is a morphism $X \to \Delta(T)$, then $X$ is defined over $K$ since $\Delta(T)$ is. Conversely, assume that $X$ is defined over $K$, and let $u : |X| \to T$ be a continuous map. The morphisms $g : X \to \Delta(T)$ with $|g| = u$ correspond to the morphisms of sheaves of $R$-algebras $\underline{K}_T \to u_* \O_X$ with local stalk maps. By the adjunction $u^{-1} \dashv u_*$, morphisms $\underline{K}_T \to u_* \O_X$ correspond to morphisms $u^{-1} \underline{K}_T = \underline{K}_{|X|} \to \O_X$, and these correspond to $R$-algebra homomorphisms $K \to \O_X(|X|)$, since $\underline{K}_{|X|}$ is the constant sheaf with value $K$. There is exactly one such homomorphism, since $K = R/\m$ and $\m \, \O_X = 0$. The stalk maps $K \to \O_{X,x}$ of the resulting morphism $g$ are local since $K$ is a field. This proves Claim 1. + + Claim 2. The functor $\Delta$ is fully faithful and preserves non-empty limits. + + Let $T,T'$ be topological spaces. Since $\Delta(T')$ is defined over $K$, Claim 1 shows that the map $\Hom(\Delta(T'),\Delta(T)) \to \Hom(T',T)$, $g \mapsto |g|$ is bijective. Since $|\Delta(u)| = u$, its inverse is $u \mapsto \Delta(u)$. Hence, $\Delta$ is fully faithful. Now let $(T_i)_{i \in I}$ be a diagram in $\Top$ with $I \neq \varnothing$, with limit $T$ and limit projections $p_i : T \to T_i$. For a locally ringed space $X$ defined over $K$, Claim 1 identifies the map + $$\textstyle \Hom(X,\Delta(T)) \to \lim_{i \in I} \Hom(X,\Delta(T_i)), \quad g \mapsto (\Delta(p_i) \circ g)_{i \in I}$$ + with the bijection $\Hom(|X|,T) \to \lim_{i \in I} \Hom(|X|,T_i)$. For every other locally ringed space $X$, both sides are empty, the right side because $I \neq \varnothing$. Hence, the morphisms $\Delta(p_i)$ form a limit cone in $\LRS_R$. + + Claim 3. The functor $\Delta$ preserves colimits. + + Let $(T_i)_{i \in I}$ be a diagram in $\Top$ with colimit $T$ and colimit maps $u_i : T_i \to T$. Recall from the proof of cocompleteness that colimits in $\LRS_R$ are computed as in ringed spaces: the underlying space is the colimit of the underlying spaces, and the structure sheaf is the limit of the push-forwards of the structure sheaves. Hence, the colimit of the diagram $(\Delta(T_i))_{i \in I}$ has the underlying space $T$ and the structure sheaf $\O \coloneqq \lim_{i \in I} (u_i)_* \underline{K}_{T_i}$, and its colimit maps consist of the $u_i$ and the limit projections. Let $V \subseteq T$ be an open subset. Then $V$ is the colimit of the open subspaces $u_i^*(V) \subseteq T_i$ in $\Top$. Indeed, this holds for the underlying sets, since colimits in $\Set$ are stable under pullbacks. And if $W \subseteq V$ is a subset such that each $u_i^*(W)$ is open in $u_i^*(V)$, hence in $T_i$, then $W$ is open in $T$ since $T$ carries the final topology, hence $W$ is open in $V$. Since the sections of $\underline{K}_T$ and $\underline{K}_{T_i}$ are the continuous maps to the discrete space $K$, it follows that the map + $$\textstyle \underline{K}_T(V) \to \O(V) = \lim_{i \in I} \underline{K}_{T_i}(u_i^*(V)), \quad s \mapsto (s \circ u_i)_{i \in I}$$ + is bijective. These maps form an isomorphism of sheaves $\underline{K}_T \to \O$ under which the limit projections become the sheaf maps of the morphisms $\Delta(u_i)$. Hence, the morphisms $\Delta(u_i)$ form a colimit cocone in $\LRS_R$. + + In particular, $\Delta$ is fully faithful and preserves binary products and filtered colimits. Since $\Top$ does not have cartesian filtered colimits, the contrapositive of Lemma 4 here applied to $\Delta$ shows that $\LRS_R$ does not have cartesian filtered colimits either. + + Explicitly, if $S$ is a topological space and $(T_i)_{i \in I}$ is a filtered diagram in $\Top$ such that the canonical continuous map + $$\alpha : \colim_{i \in I} (S \times T_i) \to S \times \colim_{i \in I} T_i$$ + is not a homeomorphism, then, since $\Delta$ preserves binary products and colimits, the canonical morphism + $$\colim_{i \in I} \bigl(\Delta(S) \times \Delta(T_i)\bigr) \to \Delta(S) \times \colim_{i \in I} \Delta(T_i)$$ + in $\LRS_R$ identifies with $\Delta(\alpha)$, which is not an isomorphism since its underlying continuous map is $\alpha$. + label: LRS_R_embedding_of_Top - property: effective cocongruences - proof: This is Corollary 4(e) here. + proof: >- + Consider the functor $\Delta : \Top \to \LRS_R$ from the proof above. It is fully faithful and preserves non-empty limits and all colimits, in particular equalizers, pushouts, and binary coproducts. Since $\Top$ does not have effective cocongruences, the contrapositive of the dual of Lemma 8 here applied to $\Delta$ shows that $\LRS_R$ does not have effective cocongruences either. + + Concretely, this argument transports the counterexample from the proof for $\Top$ along $\Delta$: if $I$ denotes the indiscrete topological space with two points, then the two morphisms $\Spec(K) \rightrightarrows \Delta(I)$ corresponding to the two points of $I$ form a cocongruence in $\LRS_R$ that is not effective. Here, $\Delta(I)$ is the locally ringed space with underlying space $I$ and $\O_{\Delta(I)}(I) = K$. + references: + - LRS_R_embedding_of_Top + - Top_no_effective_cocongruences - property: pushout-stable regular monomorphisms proof: >- - Choose a maximal ideal $\m$ of $R$ and let $K \coloneqq R/\m$. We first reduce the claim to $\LRS_K$, which is the full subcategory of $\LRS_R$ consisting of those $X$ with $\m \O_X = 0$. Equivalently, $X$ admits a morphism $X \to \Spec(K)$ in $\LRS_R$, which is then unique. It follows that $\LRS_K$ is closed under pushouts in $\LRS_R$, and that every $T$ with a morphism $T \to X \in \LRS_K$ lies in $\LRS_K$. Hence, pushouts and equalizers in $\LRS_K$ are the same as in $\LRS_R$, and therefore also regular monomorphisms, since these are the equalizers of their cokernel pairs (by the dual of this lemma). So if $\LRS_R$ had pushout-stable regular monomorphisms, then so would $\LRS_K$. + Choose a maximal ideal $\m$ of $R$ and let $K \coloneqq R/\m$. We first reduce the claim to $\LRS_K$, which is the full subcategory of $\LRS_R$ consisting of those $X$ that are defined over $K$, i.e. for which the unique morphism $X \to \Spec(R)$ factors through $\Spec(K)$, i.e. $\m \, \O_X = 0$. In this case, the morphism $X \to \Spec(K)$ is unique. It follows that $\LRS_K$ is closed under pushouts in $\LRS_R$, and that every $T$ with a morphism $T \to X \in \LRS_K$ lies in $\LRS_K$. Hence, pushouts and equalizers in $\LRS_K$ are the same as in $\LRS_R$, and therefore also regular monomorphisms, since these are the equalizers of their cokernel pairs (by the dual of this lemma). So if $\LRS_R$ had pushout-stable regular monomorphisms, then so would $\LRS_K$. - So it suffices to construct, for every field $K$, a counterexample in $\LRS_K$. In fact, we will find a regular monomorphism $i : X \to Y$ which is the identity on underlying spaces, and a morphism $f : X \to Z$ to a one-point space such that $Z \to Y \sqcup_X Z$ is not a regular monomorphism. The idea is that $\O_Y$ has an infinitesimal direction over two open subsets which cannot be glued to a global section, whereas $\O_X$ has a global section in this direction. Collapsing the space to a point only keeps the global sections, so this direction gets lost. + So it suffices to construct, for every field $K$, a counterexample in $\LRS_K$. In fact, we will find a regular monomorphism $i : X \to Y$ which is the identity on underlying spaces, and a morphism $f : X \to Z$ to a locally ringed space $Z$ with a single point such that $Z \to Y \sqcup_X Z$ is not a regular monomorphism. The idea is that $\O_Y$ has an infinitesimal direction over two open subsets which cannot be glued to a global section, whereas $\O_X$ has a global section in this direction. Collapsing the space to a point only keeps the global sections, so this direction gets lost. Let $K[\varepsilon]$ be the commutative algebra of dual numbers, where $\varepsilon^2 = 0$, and let $L \coloneqq K[\varepsilon_1,\varepsilon_2]$ be the commutative algebra where $\varepsilon_1^2 = \varepsilon_1 \varepsilon_2 = \varepsilon_2^2 = 0$, so that $1,\varepsilon_1,\varepsilon_2$ is a basis of $L$ over $K$. @@ -105,60 +153,127 @@ unsatisfied_properties: $$F(U_1) \xrightarrow{\rho_1} F(\{x\}) \xleftarrow{\rho_2} F(U_2)$$ via $F(S) = F(U_1) \times_{F(\{x\})} F(U_2)$, and that a morphism of such sheaves is the same as a family of homomorphisms on $U_1$, $U_2$, $\{x\}$ that is compatible with the restriction maps. - Let $X \coloneqq (S,\O_X)$ and $Y \coloneqq (S,\O_Y)$, where $\O_X$ and $\O_Y$ are the sheaves given by the cospans + Let $X$ and $Y$ be the locally ringed spaces with $|X| = |Y| = S$ whose structure sheaves $\O_X$ and $\O_Y$ are given by the cospans $$K[\varepsilon] \xrightarrow{\id} K[\varepsilon] \xleftarrow{\id} K[\varepsilon], \qquad K[\varepsilon] \xrightarrow{\rho_1} L \xleftarrow{\rho_2} K[\varepsilon],$$ respectively, with $\rho_1(\varepsilon) \coloneqq \varepsilon_1$ and $\rho_2(\varepsilon) \coloneqq \varepsilon_2$. Both are locally ringed spaces since their stalks $K[\varepsilon]$ and $L$ are local rings. We have $\O_X(S) = K[\varepsilon]$. A pair $(\lambda_1 + \mu_1 \varepsilon, \lambda_2 + \mu_2 \varepsilon) \in K[\varepsilon]^2$ lies in $\O_Y(S) = K[\varepsilon] \times_L K[\varepsilon]$ iff $$\lambda_1 + \mu_1 \varepsilon_1 = \lambda_2 + \mu_2 \varepsilon_2 \quad \text{in } L,$$ i.e. iff $\lambda_1 = \lambda_2$ and $\mu_1 = \mu_2 = 0$. Hence, $\O_Y(S) \cong K$ consists only of the constants: unlike in $\O_X$, the sections $\varepsilon$ over $U_1$ and $U_2$ do not glue. - Define the morphism $i \coloneqq (\id_S,\pi) : X \to Y$, where $\pi : \O_Y \to \O_X$ is the identity on $U_1$ and $U_2$, and on $\{x\}$ it is the homomorphism + Define the morphism $i : X \to Y$ by $|i| \coloneqq \id_S$ and $i^\sharp \coloneqq \pi$, where $\pi : \O_Y \to \O_X$ is the identity on $U_1$ and $U_2$, and on $\{x\}$ it is the homomorphism $$\pi_x : L \to K[\varepsilon], \quad \pi_x(\varepsilon_1) \coloneqq \pi_x(\varepsilon_2) \coloneqq \varepsilon.$$ This is compatible with the restriction maps since $\pi_x \circ \rho_1 = \pi_x \circ \rho_2 = \id$. On global sections, $\pi$ is the inclusion $K \hookrightarrow K[\varepsilon]$. The stalk maps of $i$ are surjective homomorphisms of local rings, hence local. Since $i$ is the identity on underlying spaces and its stalk maps are surjective, $i$ is a regular monomorphism by the classification of regular monomorphisms (see below). - Let $Z \coloneqq \Spec(K[\varepsilon])$, and let $f : X \to Z$ be the morphism corresponding to the identity $K[\varepsilon] \to \O_X(S) = K[\varepsilon]$. Recall from the proof of cocompleteness that pushouts in $\LRS_K$ are computed as in ringed spaces: the underlying space is the pushout of the underlying spaces, and the structure sheaf is the fiber product of the push-forwards of the structure sheaves. Hence, the underlying space of $P \coloneqq Y \sqcup_X Z$ is the pushout of $\id_S$ and $S \to Z$, which is a point, and its algebra of global sections is - $$\O_Y(S) \times_{\O_X(S)} \O_Z(Z) = K \times_{K[\varepsilon]} K[\varepsilon] \cong K.$$ + Let $Z \coloneqq \Spec(K[\varepsilon])$, and let $f : X \to Z$ be the morphism corresponding to the identity $K[\varepsilon] \to \O_X(S) = K[\varepsilon]$. Recall from the proof of cocompleteness that pushouts in $\LRS_K$ are computed as in ringed spaces: the underlying space is the pushout of the underlying spaces, and the structure sheaf is the fiber product of the push-forwards of the structure sheaves. Hence, the underlying space of $P \coloneqq Y \sqcup_X Z$ is the pushout of $|i| = \id_S$ and $|f| : S \to |Z|$, which is a point, and its algebra of global sections is + $$\O_Y(S) \times_{\O_X(S)} \O_Z(|Z|) = K \times_{K[\varepsilon]} K[\varepsilon] \cong K.$$ So $P \cong \Spec(K)$, and the stalk map of the morphism $j : Z \to P$ is the inclusion $K \hookrightarrow K[\varepsilon]$. Since it is not surjective, $j$ is not a regular monomorphism by the classification of regular monomorphisms. + - property: filtered-colimit-stable monomorphisms + proof: >- + Choose a residue field $K$ of $R$. Let $V \coloneqq K^{(\IN)}$ with basis $(e_m)_{m \in \IN}$, and let $V_n \subseteq V$ be the subspace spanned by the $e_m$ with $m \geq n$. Then $V_0 \supseteq V_1 \supseteq \cdots$ and $\bigcap_{n \in \IN} V_n = 0$. Let $\sigma : V \to K$ be the linear map with $\sigma(e_m) = 1$ for all $m$, which sums the coefficients. + + For a $K$-vector space $W$, let $K \oplus W$ be its square-zero extension. It is a local ring with nilpotent maximal ideal $W$, and it is functorial in $W$: a linear map $f : W \to W'$ induces the homomorphism $K \oplus W \to K \oplus W'$, $(\lambda,w) \mapsto (\lambda,f(w))$. For example, $K \oplus K$ is isomorphic to the algebra $K[\varepsilon]$ of dual numbers, where $\varepsilon^2 = 0$, via $(\lambda,\mu) \mapsto \lambda + \mu \varepsilon$. + + Let $A_n \coloneqq K \oplus V_n$. Then $\Spec(A_n)$ has a single point, and its stalk at this point is $A_n$. The inclusions $V_{n+1} \hookrightarrow V_n$ induce inclusions $A_{n+1} \hookrightarrow A_n$, hence morphisms $\Spec(A_n) \to \Spec(A_{n+1})$, which form a filtered diagram indexed by $\IN$. The restriction $V_n \to K$ of $\sigma$ induces the homomorphism + $$p_n : A_n \to K[\varepsilon], \quad (\lambda,v) \mapsto \lambda + \sigma(v) \varepsilon.$$ + It is surjective since $\sigma(e_n) = 1$, and $p_{n+1}$ is the restriction of $p_n$ to $A_{n+1}$ by functoriality. Hence, the morphisms $\Spec(p_n) : \Spec(K[\varepsilon]) \to \Spec(A_n)$ form a morphism from the constant diagram $(\Spec(K[\varepsilon]))_n$ to the diagram $(\Spec(A_n))_n$. Each $\Spec(p_n)$ is a monomorphism by the classification of monomorphisms (see below): its underlying continuous map is a map between one-point spaces, and its stalk map $p_n$ is surjective, hence an epimorphism of local commutative rings. + + The colimit of the constant diagram is $\Spec(K[\varepsilon])$. Recall from the proof of cocompleteness that colimits in $\LRS_R$ are computed as in ringed spaces: the underlying space is the colimit of the underlying spaces, and the structure sheaf is the limit of the push-forwards of the structure sheaves. Hence, the colimit of the diagram $(\Spec(A_n))_n$ has a single point, and its stalk at this point is + $$\textstyle \lim_n A_n = \bigcap_{n \in \IN} A_n = K \oplus \bigcap_{n \in \IN} V_n = K,$$ + i.e. it is $\Spec(K)$, and the induced morphism $\Spec(K[\varepsilon]) \to \Spec(K)$ corresponds to the inclusion $K \hookrightarrow K[\varepsilon]$. This inclusion is not an epimorphism of local commutative rings, since the identity of $K[\varepsilon]$ and the local homomorphism $K[\varepsilon] \to K[\varepsilon]$, $\varepsilon \mapsto 0$, agree on $K$. Hence, the morphism $\Spec(K[\varepsilon]) \to \Spec(K)$ is not a monomorphism by the classification of monomorphisms. + + - property: regular subobject classifier + proof: >- + The idea is the same as for $\CAlg(R)$: a classifying morphism is invariant under automorphisms that preserve the subobject. Instead of the polynomial ring $K[X,Y]$ used there, we work with the local Artinian ring $K[X,Y]/\langle X,Y \rangle^2$, since its spectrum has a single point, and morphisms from such a locally ringed space into another one are described by a single point and a stalk map. + + Assume that $\top : \Spec(R) \to \Omega$ is a regular subobject classifier. Let $K$ be a residue field of $R$, and let $A \coloneqq K[X,Y]/\langle X,Y \rangle^2$, whose elements can be written uniquely as $a + bX + cY$ with $a,b,c \in K$. The projection $\pi : A \to K$, $a + bX + cY \mapsto a$ is surjective, hence a regular epimorphism in $\CAlg(R)$. Since $\Spec : \CAlg(R)^{\op} \to \LRS_R$ is fully faithful and a right adjoint, $\Spec(\pi) : \Spec(K) \to \Spec(A)$ is a regular monomorphism. So there is a unique morphism $\chi : \Spec(A) \to \Omega$ such that the square + $$\begin{CD} \Spec(K) @>{\Spec(\pi)}>> \Spec(A) \\ @V{!}VV @VV{\chi}V \\ \Spec(R) @>>{\top}> \Omega \end{CD}$$ + is a pullback. Let $p \in |\Omega|$ be the image of the unique point of $|\Spec(A)|$ under $|\chi|$, and let $\varphi : \O_{\Omega,p} \to A$ be the stalk map of $\chi$. + + Let $\sigma : A \to A$ be the automorphism that swaps $X$ and $Y$. Since $\pi \circ \sigma = \pi$, the square + $$\begin{CD} \Spec(K) @>{\Spec(\pi)}>> \Spec(A) \\ @V{\id}VV @VV{\Spec(\sigma)}V \\ \Spec(K) @>>{\Spec(\pi)}> \Spec(A) \end{CD}$$ + commutes, and it is a pullback since $\Spec(\sigma)$ is an isomorphism. Pasting it on top of the first square shows that $\chi \circ \Spec(\sigma)$ also classifies $\Spec(\pi)$, so that $\chi \circ \Spec(\sigma) = \chi$ by uniqueness. On stalks, this means $\sigma \circ \varphi = \varphi$, i.e. $\varphi$ takes values in the fixed ring + $$A^\sigma = \{a + b(X+Y) : a,b \in K\}.$$ + Now let $u : A \to A/\langle X+Y \rangle$ be the quotient map, and let $v : A \to A/\langle X+Y \rangle$ map $a + bX + cY$ to the class of $a$, so that $v$ factors through $\pi$. Both $u$ and $v$ map $a + b(X+Y)$ to the class of $a$, so they agree on $A^\sigma$. Since $\Spec(A/\langle X+Y \rangle)$ has a single point, a morphism from it to $\Omega$ is the same as a point of $|\Omega|$ together with a local homomorphism from its stalk to $A/\langle X+Y \rangle$. For $\chi \circ \Spec(u)$ and $\chi \circ \Spec(v)$, these are $p$ together with $u \circ \varphi$ and $v \circ \varphi$, which coincide since $\varphi$ takes values in $A^\sigma$. Since $\Spec(v)$ factors through $\Spec(\pi)$, it follows that + $$\chi \circ \Spec(u) = \chi \circ \Spec(v) = \top \circ {!}.$$ + Since the first square is a pullback, $\Spec(u)$ factors through $\Spec(\pi)$. Since $\Spec$ is fully faithful, $u$ factors through $\pi$. But then $X \in \ker(\pi) \subseteq \ker(u) = \langle X+Y \rangle$, which is a contradiction. + label: LRS_R_no_regular_subobject_classifier + references: + - CAlg(R)_no_regular_quotient_object_classifier + special_objects: initial object: - description: empty space + description: empty locally ringed space $X$, given by $|X| \coloneqq \varnothing$ and $\O_X(\varnothing) \coloneqq 0$ terminal object: - description: $\Spec(R)$ + description: affine scheme $\Spec(R)$ coproducts: - description: disjoint union with the product sheaf + description: The coproduct $X$ of a family $(X_i)_{i \in I}$ is given by $|X| \coloneqq \coprod_{i \in I} |X_i|$ and $\O_X(U) \coloneqq \prod_{i \in I} \O_{X_i}(U \cap |X_i|)$ for open subsets $U \subseteq |X|$. products: - description: See Gillam. See also MSE/1033675. + description: >- + Let $X$ be the product of a family $(X_i)_{i \in I}$. For a family $x = (x_i)_{i \in I}$ of points $x_i \in |X_i|$, let $A_x \coloneqq \bigotimes_{i \in I} \O_{X_i,x_i}$ be the tensor product of $R$-algebras. The points of $|X|$ are the pairs $(x,\p)$ of such a family $x$ and a prime ideal $\p \subseteq A_x$ whose preimage in each $\O_{X_i,x_i}$ is its maximal ideal. For a family $U = (U_i)_{i \in I}$ of open subsets $U_i \subseteq |X_i|$ with $U_i = |X_i|$ for all but finitely many $i$ and an element $s \in \bigotimes_{i \in I} \O_{X_i}(U_i)$, let $D(U,s)$ be the set of points $(x,\p)$ with $x_i \in U_i$ for all $i$ and $s_x \notin \p$, where $s_x \in A_x$ is the image of $s$. These sets form a basis of the topology. The stalk of $\O_X$ at $(x,\p)$ is the localization $(A_x)_\p$, and the sections over an open subset $W \subseteq |X|$ are the families in $\prod_{(x,\p) \in W} (A_x)_\p$ that are locally of the form $(x,\p) \mapsto a_x/s_x$ on some $D(U,s) \subseteq W$ with $a \in \bigotimes_{i \in I} \O_{X_i}(U_i)$. The projection $X \to X_i$ maps $(x,\p)$ to $x_i$, and its stalk maps are the canonical local homomorphisms $\O_{X_i,x_i} \to (A_x)_\p$. special_morphisms: isomorphisms: - description: 'A morphism $(f,f^\sharp) : (X, \O_X) \to (Y,\O_Y)$ is an isomorphism iff $f : X \to Y$ is a homeomorphism and $f^\sharp : \O_Y \to f_* \O_X$ is an isomorphism of sheaves.' + description: 'A morphism $f : X \to Y$ is an isomorphism iff $|f| : |X| \to |Y|$ is a homeomorphism and $f^\sharp : \O_Y \to |f|_* \O_X$ is an isomorphism of sheaves.' proof: This is easy. + monomorphisms: + description: 'A morphism $f : X \to Y$ is a monomorphism iff $|f| : |X| \to |Y|$ is injective and for every $x \in |X|$ the stalk map $f^\sharp_x : \O_{Y,|f|(x)} \to \O_{X,x}$ is an epimorphism in the category of local commutative rings and local homomorphisms. (This is weaker than being an epimorphism of commutative rings.)' + proof: >- + For a local commutative $R$-algebra $T$, let $P_T$ denote the locally ringed space with a single point whose stalk is $T$. A morphism $P_T \to X$ is the same as a point $x \in |X|$ together with a local homomorphism of $R$-algebras $\O_{X,x} \to T$, and its composite with $f$ corresponds to the point $|f|(x)$ together with the composite $\O_{Y,|f|(x)} \xrightarrow{f^\sharp_x} \O_{X,x} \to T$. + + If $x_1,x_2 \in |X|$ satisfy $|f|(x_1) = |f|(x_2) \eqqcolon y$, then there are a field $F$ and morphisms $\bar{x}_1,\bar{x}_2 : P_F \rightrightarrows X$ corresponding to the points $x_1,x_2$ that have the same composite with $f$. Indeed, the residue fields $\kappa(x_1)$ and $\kappa(x_2)$ are extensions of $\kappa(y)$. Since $\kappa(x_1) \otimes_{\kappa(y)} \kappa(x_2) \neq 0$, it has a quotient field $F$, which comes with homomorphisms $\kappa(x_i) \to F$ that agree on $\kappa(y)$. We regard $F$ as an $R$-algebra via $\kappa(y)$. For $i = 1,2$, let $\bar{x}_i$ be the morphism corresponding to the point $x_i$ and the local homomorphism $\O_{X,x_i} \to \kappa(x_i) \to F$. Its composite with $f$ corresponds to the point $y$ and the local homomorphism + $$\O_{Y,y} \to \O_{X,x_i} \to \kappa(x_i) \to F,$$ + which equals $\O_{Y,y} \to \kappa(y) \to \kappa(x_i) \to F$ and therefore does not depend on $i$. + + Assume that $f$ is a monomorphism. To show that $|f|$ is injective, let $x_1,x_2 \in |X|$ with $|f|(x_1) = |f|(x_2)$. Since $f$ is a monomorphism, the morphisms $\bar{x}_1,\bar{x}_2$ from above agree, so that $x_1 = x_2$. Thus, $|f|$ is injective. Next, let $x \in |X|$, $y \coloneqq |f|(x)$, and let $u,v : \O_{X,x} \rightrightarrows T$ be local homomorphisms into a local commutative ring $T$ with $u \circ f^\sharp_x = v \circ f^\sharp_x$. We regard $T$ as an $R$-algebra via $u$, which is the same as via $v$, since $R \to \O_{X,x}$ factors through $f^\sharp_x$. The morphisms $P_T \to X$ corresponding to $(x,u)$ and $(x,v)$ have the same composite with $f$, namely the one corresponding to $(y,u \circ f^\sharp_x)$. Since $f$ is a monomorphism, we get $u = v$. Thus, $f^\sharp_x$ is an epimorphism of local commutative rings. + + Conversely, assume that $f$ satisfies the two conditions, and let $g,h : Z \rightrightarrows X$ be two morphisms with $f \circ g = f \circ h$. Since $|f|$ is injective, $|f| \circ |g| = |f| \circ |h|$ implies $|g| = |h|$. For $z \in |Z|$ and $x \coloneqq |g|(z)$, the stalk maps $g^\sharp_z, h^\sharp_z : \O_{X,x} \rightrightarrows \O_{Z,z}$ are local homomorphisms with $g^\sharp_z \circ f^\sharp_x = h^\sharp_z \circ f^\sharp_x$, which implies $g^\sharp_z = h^\sharp_z$. So for every open subset $U \subseteq |X|$ and $s \in \O_X(U)$, the sections $g^\sharp(s), h^\sharp(s) \in \O_Z(|g|^*(U))$ have the same germs, which means $g^\sharp(s) = h^\sharp(s)$. Hence, $g = h$. + + Remark. The condition on the stalk maps cannot be replaced by the stronger condition that they are epimorphisms of commutative rings. To see this, let $K$ be a residue field of $R$, and let $\varphi : A \coloneqq K[X]_{(X)} \to B \coloneqq K[Y]_{(Y)}$ be the local homomorphism with $\varphi(X) \coloneqq Y(Y - 1)$. It is an epimorphism of local commutative rings: if $u,v : B \rightrightarrows T$ are local homomorphisms into a local commutative ring $T$ with $u \circ \varphi = v \circ \varphi$, then $a \coloneqq u(Y)$ and $b \coloneqq v(Y)$ satisfy $a(a - 1) = b(b - 1)$, i.e. + $$(a - b)(a + b - 1) = a^2 - a - b^2 + b = 0.$$ + Since $a,b \in \m_T$, the element $a + b - 1$ is a unit, so that $a = b$. Hence, $u$ and $v$ agree on $K[Y]$, and therefore on its localization $B$. But $\varphi$ is not an epimorphism of commutative rings: the inclusion $B \hookrightarrow K(Y)$ and its composite with the automorphism of $K(Y)$ that maps $Y$ to $1 - Y$ are different, but they agree on $\varphi(A)$ since + $$(1 - Y)\bigl((1 - Y) - 1\bigr) = (1 - Y)(-Y) = Y(Y - 1).$$ + So the morphism $P_B \to P_A$ with stalk map $\varphi$ is a monomorphism in $\LRS_R$, but its stalk map is not an epimorphism of commutative rings. In contrast, for morphisms of schemes, both conditions on the stalk maps are equivalent (see $\Sch_R$). epimorphisms: - description: 'A morphism $(f,f^\sharp) : (X, \O_X) \to (Y,\O_Y)$ is an epimorphism iff $f : X \to Y$ is surjective and $f^\sharp : \O_Y \to f_* \O_X$ is a monomorphism of sheaves.' + description: 'A morphism $f : X \to Y$ is an epimorphism iff $|f| : |X| \to |Y|$ is surjective and $f^\sharp : \O_Y \to |f|_* \O_X$ is a monomorphism of sheaves.' proof: >- - It is obvious that morphisms with this property are epimorphisms. For the opposite, assume that $(f,f^\sharp) : (X, \O_X) \to (Y,\O_Y)$ is an epimorphism. Then the codiagonal - $$(Y,\O_Y) \sqcup_{(X,\O_X)} (Y,\O_Y) \to (Y,\O_Y)$$ - is an isomorphism. By construction of the pushout, the underlying space of the pushout is $Y \sqcup_X Y$. Hence, $f$ is an epimorphism in $\Top$, i.e. it is surjective. For open subsets $V \subseteq Y$ the preimage in the pushout is $V \sqcup_{f^{-1}(V)} V$, and its ring of sections is the pullback $\O_Y(V) \times_{\O_X(f^{-1}(V))} \O_Y(V)$. Since the diagonal - $$\O_Y(V) \to \O_Y(V) \times_{\O_X(f^{-1}(V))} \O_Y(V)$$ - is an isomorphism, the map $f^\sharp(V) : \O_Y(V) \to \O_X(f^{-1}(V))$ is a monomorphism. Hence, $f^\sharp$ is a monomorphism of sheaves. + It is obvious that morphisms with this property are epimorphisms. For the opposite, assume that $f : X \to Y$ is an epimorphism. Then the codiagonal + $$Y \sqcup_X Y \to Y$$ + is an isomorphism. By construction of the pushout, its underlying space $|Y \sqcup_X Y|$ is the pushout $|Y| \sqcup_{|X|} |Y|$ of $|f|$ with itself in $\Top$. Hence, $|f|$ is an epimorphism in $\Top$, i.e. it is surjective. For open subsets $V \subseteq |Y|$ the preimage in $|Y| \sqcup_{|X|} |Y|$ is $V \sqcup_{|f|^*(V)} V$, and its ring of sections is the pullback $\O_Y(V) \times_{\O_X(|f|^*(V))} \O_Y(V)$. Since the diagonal + $$\O_Y(V) \to \O_Y(V) \times_{\O_X(|f|^*(V))} \O_Y(V)$$ + is an isomorphism, the map $f^\sharp : \O_Y(V) \to \O_X(|f|^*(V))$ is a monomorphism. Hence, $f^\sharp$ is a monomorphism of sheaves. regular monomorphisms: - description: 'A morphism $i : Z \to Y$ is a regular monomorphism iff its underlying continuous map is an embedding and its stalk maps $\O_{Y,i(z)} \to \O_{Z,z}$ are surjective for all $z \in Z$.' + description: 'A morphism $f : X \to Y$ is a regular monomorphism iff $|f| : |X| \to |Y|$ is an embedding and the stalk maps $f^\sharp_x : \O_{Y,|f|(x)} \to \O_{X,x}$ are surjective for all $x \in |X|$.' proof: >- - For a morphism $\varphi : T \to Y$ and $t \in T$, we write $\varphi_t : \O_{Y,\varphi(t)} \to \O_{T,t}$ for the stalk map, so that the germ of $\varphi^\sharp(b)$ at $t$ is $\varphi_t(b_{\varphi(t)})$ for every local section $b$ of $\O_Y$. We will use the following simple fact twice: if $\alpha : A \to B$ is a surjective homomorphism of local rings and $\beta : B \to C$ is a homomorphism into a local ring such that $\beta \circ \alpha$ is local, then $\beta$ is local, since every non-unit of $B$ is the image of a non-unit of $A$. + Recall that for a morphism $\varphi : T \to Y$, a point $t \in |T|$, and a local section $b$ of $\O_Y$, the germ of $\varphi^\sharp(b)$ at $t$ is $\varphi^\sharp_t(b_{|\varphi|(t)})$. We will use the following simple fact twice: if $\alpha : A \to B$ is a surjective homomorphism of local rings and $\beta : B \to C$ is a homomorphism into a local ring such that $\beta \circ \alpha$ is local, then $\beta$ is local, since every non-unit of $B$ is the image of a non-unit of $A$. - We first observe that every morphism $\varphi : T \to Y$ factors as $\varphi = j \circ e$, where $e$ is an epimorphism and $j$ is a morphism whose underlying continuous map is an embedding and whose stalk maps are surjective. Let $Y' \coloneqq \varphi(T) \subseteq Y$ carry the subspace topology, let $\iota : Y' \hookrightarrow Y$ be the inclusion, and let $\varphi' : T \to Y'$ be the continuous surjection with $\varphi = \iota \circ \varphi'$. The sheaf map $\varphi^\sharp : \O_Y \to \varphi_* \O_T = \iota_* \varphi'_* \O_T$ corresponds to a morphism of sheaves $\iota^{-1} \O_Y \to \varphi'_* \O_T$ on $Y'$, and we let $\O_{Y'} \subseteq \varphi'_* \O_T$ be its image. For $t \in T$ and $y \coloneqq \varphi(t)$, the stalk $\O_{Y',y}$ is the image of $\O_{Y,y}$ in $(\varphi'_* \O_T)_y$. It is a quotient of the local ring $\O_{Y,y}$, and it is non-zero since $\varphi_t$ factors through it and $\O_{T,t} \neq 0$. Hence, $\O_{Y',y}$ is local, and $(Y',\O_{Y'})$ is a locally ringed space. We obtain a factorization $\varphi = j \circ e$ with the morphisms - $$j \coloneqq (\iota, \O_Y \to \iota_* \iota^{-1} \O_Y \to \iota_* \O_{Y'}) : Y' \to Y,$$ - $$e \coloneqq (\varphi', \O_{Y'} \hookrightarrow \varphi'_* \O_T) : T \to Y'.$$ - Here, the stalk maps $j_y : \O_{Y,y} \to \O_{Y',y}$ are surjective, hence local, and since $e_t \circ j_y = \varphi_t$ is local, $e_t$ is local by the simple fact above. The underlying map of $e$ is surjective and its sheaf map is a monomorphism, so $e$ is an epimorphism by the classification of epimorphisms (see above). + We first observe that every morphism $\varphi : T \to Y$ factors as $\varphi = j \circ e$, where $e$ is an epimorphism and $j$ is a morphism such that $|j|$ is an embedding and the stalk maps of $j$ are surjective. Let $Y'$ be the ringed space whose underlying space $|Y'| \coloneqq |\varphi|(|T|) \subseteq |Y|$ carries the subspace topology, and whose structure sheaf is defined as follows. Let $\iota : |Y'| \hookrightarrow |Y|$ be the inclusion, and let $\varphi' : |T| \to |Y'|$ be the continuous surjection with $|\varphi| = \iota \circ \varphi'$. The sheaf map $\varphi^\sharp : \O_Y \to |\varphi|_* \O_T = \iota_* \varphi'_* \O_T$ corresponds to a morphism of sheaves $\iota^{-1} \O_Y \to \varphi'_* \O_T$ on $|Y'|$, and we let $\O_{Y'} \subseteq \varphi'_* \O_T$ be its image. For $t \in |T|$ and $y \coloneqq |\varphi|(t)$, the stalk $\O_{Y',y}$ is the image of $\O_{Y,y}$ in $(\varphi'_* \O_T)_y$. It is a quotient of the local ring $\O_{Y,y}$, and it is non-zero since $\varphi^\sharp_t$ factors through it and $\O_{T,t} \neq 0$. Hence, $\O_{Y',y}$ is local, and $Y'$ is a locally ringed space. Let $j : Y' \to Y$ be the morphism with $|j| \coloneqq \iota$ whose sheaf map is the composite $\O_Y \to \iota_* \iota^{-1} \O_Y \to \iota_* \O_{Y'}$, and let $e : T \to Y'$ be the morphism with $|e| \coloneqq \varphi'$ whose sheaf map is the inclusion $\O_{Y'} \hookrightarrow \varphi'_* \O_T$. These are morphisms of locally ringed spaces: the stalk maps $j^\sharp_y : \O_{Y,y} \to \O_{Y',y}$ are surjective, hence local, and since $e^\sharp_t \circ j^\sharp_y = \varphi^\sharp_t$ is local, $e^\sharp_t$ is local by the simple fact above. By construction, $\varphi = j \circ e$. Since $|e|$ is surjective and $e^\sharp$ is a monomorphism, $e$ is an epimorphism by the classification of epimorphisms (see above). - Now let $i : Z \to Y$ be a regular monomorphism, and let $i = j \circ e$ be the factorization from above. Since $i$ is an extremal monomorphism and $e$ is an epimorphism, $e$ is an isomorphism. Hence, the underlying map of $i$ is an embedding, and its stalk maps $i_z = e_z \circ j_{i(z)}$ are surjective. + Now let $i : Z \to Y$ be a regular monomorphism, and let $i = j \circ e$ be the factorization from above. Since $i$ is an extremal monomorphism and $e$ is an epimorphism, $e$ is an isomorphism. Hence, $|i| = \iota \circ |e|$ is an embedding, and the stalk maps $i^\sharp_z = e^\sharp_z \circ j^\sharp_{|i|(z)}$ are surjective. - For the converse direction, we first prove a factorization criterion. Let $i : Z \to Y$ be a morphism whose underlying map is an embedding and whose stalk maps $i_z : \O_{Y,i(z)} \to \O_{Z,z}$ are surjective. Let $\varphi : T \to Y$ be a morphism with $\varphi(T) \subseteq i(Z)$ such that $\varphi_t$ vanishes on $\ker(i_z)$ whenever $\varphi(t) = i(z)$. We claim that $\varphi$ factors uniquely through $i$. We may assume that $Z \subseteq Y$ is a subspace and that $i$ is the inclusion map. Then $\varphi = i \circ \varphi'$ for a unique continuous map $\varphi' : T \to Z$. Let $U \subseteq Z$ be open and $a \in \O_Z(U)$. Since the stalk maps of $i$ are surjective, every $z \in U$ has an open neighborhood $V \subseteq Y$ with $Z \cap V \subseteq U$ and a section $b \in \O_Y(V)$ such that $i^\sharp(b) = a$ on $Z \cap V$: first choose $b$ with the correct germ at $z$, then shrink $V$, using that $Z$ carries the subspace topology. We call such a section $b$ a local lift of $a$. If $b \in \O_Y(V)$ and $b' \in \O_Y(V')$ are local lifts of $a$, then for every $t \in \varphi^*(V \cap V')$ with $z \coloneqq \varphi(t)$ we have $b_z - b'_z \in \ker(i_z)$, hence $\varphi_t(b_z) = \varphi_t(b'_z)$. This means that $\varphi^\sharp(b)$ and $\varphi^\sharp(b')$ agree on $\varphi^*(V \cap V')$. Since the sets $\varphi^*(V) = (\varphi')^*(Z \cap V)$ for local lifts $b \in \O_Y(V)$ of $a$ cover $(\varphi')^*(U)$, the sections $\varphi^\sharp(b)$ glue to a section $\psi(a) \in \O_T((\varphi')^*(U))$. Since sums and products of local lifts are local lifts, this defines a morphism of sheaves $\psi : \O_Z \to \varphi'_* \O_T$. It satisfies $\psi(i^\sharp(b)) = \varphi^\sharp(b)$ for every local section $b$ of $\O_Y$, which means $i \circ (\varphi',\psi) = \varphi$. Its stalk maps satisfy $\psi_t \circ i_{\varphi'(t)} = \varphi_t$, so they are local by the simple fact above. Finally, if $\psi' : \O_Z \to \varphi'_* \O_T$ is another morphism with $i \circ (\varphi',\psi') = \varphi$, then $\psi'(a)$ restricts to $\psi'(i^\sharp(b)) = \varphi^\sharp(b)$ on $\varphi^*(V)$ for every local lift $b \in \O_Y(V)$ of $a$, so $\psi' = \psi$. This proves the claim. + For the converse direction, we first prove a factorization criterion. Let $i : Z \to Y$ be a morphism such that $|i|$ is an embedding and the stalk maps $i^\sharp_z : \O_{Y,|i|(z)} \to \O_{Z,z}$ are surjective. Let $\varphi : T \to Y$ be a morphism with $|\varphi|(|T|) \subseteq |i|(|Z|)$ such that $\varphi^\sharp_t$ vanishes on $\ker(i^\sharp_z)$ whenever $|\varphi|(t) = |i|(z)$. We claim that $\varphi$ factors uniquely through $i$. We may assume that $|Z| \subseteq |Y|$ is a subspace and that $|i|$ is the inclusion map. Then $|\varphi| = |i| \circ \varphi'$ for a unique continuous map $\varphi' : |T| \to |Z|$. Let $U \subseteq |Z|$ be open and $a \in \O_Z(U)$. Since the stalk maps of $i$ are surjective, every $z \in U$ has an open neighborhood $V \subseteq |Y|$ with $|Z| \cap V \subseteq U$ and a section $b \in \O_Y(V)$ such that $i^\sharp(b) = a$ on $|Z| \cap V$: first choose $b$ with the correct germ at $z$, then shrink $V$, using that $|Z|$ carries the subspace topology. We call such a section $b$ a local lift of $a$. If $b \in \O_Y(V)$ and $b' \in \O_Y(V')$ are local lifts of $a$, then for every $t \in |\varphi|^*(V \cap V')$ with $z \coloneqq |\varphi|(t)$ we have $b_z - b'_z \in \ker(i^\sharp_z)$, hence $\varphi^\sharp_t(b_z) = \varphi^\sharp_t(b'_z)$. This means that $\varphi^\sharp(b)$ and $\varphi^\sharp(b')$ agree on $|\varphi|^*(V \cap V')$. Since the sets $|\varphi|^*(V) = (\varphi')^*(|Z| \cap V)$ for local lifts $b \in \O_Y(V)$ of $a$ cover $(\varphi')^*(U)$, the sections $\varphi^\sharp(b)$ glue to a section $\psi(a) \in \O_T((\varphi')^*(U))$. Since sums and products of local lifts are local lifts, this defines a morphism of sheaves $\psi : \O_Z \to \varphi'_* \O_T$ with $\psi(i^\sharp(b)) = \varphi^\sharp(b)$ for every local section $b$ of $\O_Y$. Its stalk maps satisfy $\psi_t \circ i^\sharp_{\varphi'(t)} = \varphi^\sharp_t$, so they are local by the simple fact above. Hence, there is a morphism $g : T \to Z$ with $|g| \coloneqq \varphi'$ and $g^\sharp \coloneqq \psi$, and it satisfies $i \circ g = \varphi$. Finally, if $h : T \to Z$ is another morphism with $i \circ h = \varphi$, then $|h| = \varphi'$ since $|i|$ is injective, and $h^\sharp(a)$ restricts to $h^\sharp(i^\sharp(b)) = \varphi^\sharp(b)$ on $|\varphi|^*(V)$ for every local lift $b \in \O_Y(V)$ of $a$, so $h^\sharp = \psi$ and therefore $h = g$. This proves the claim. - Finally, let $i : Z \to Y$ be a morphism whose underlying map is an embedding and whose stalk maps are surjective. We show that $i$ is the equalizer of its cokernel pair. We may assume that $Z \subseteq Y$ is a subspace and that $i$ is the inclusion map. Recall from the proof of cocompleteness that pushouts in $\LRS_R$ are computed as in ringed spaces: the underlying space is the pushout of the underlying spaces, and the structure sheaf is the fiber product of the push-forwards of the structure sheaves. Hence, the cokernel pair $u_1,u_2 : Y \rightrightarrows W$ of $i$ looks as follows. The underlying space of $W$ consists of two copies of $Y$ glued along $Z$, and $u_1,u_2$ are the two inclusions. The open subsets of $W$ correspond to pairs $(V_1,V_2)$ of open subsets of $Y$ with $Z \cap V_1 = Z \cap V_2$, and - $$\O_W(V_1,V_2) = \O_Y(V_1) \times_{\O_Z(Z \cap V_1)} \O_Y(V_2),$$ + Finally, let $i : Z \to Y$ be a morphism such that $|i|$ is an embedding and the stalk maps of $i$ are surjective. We show that $i$ is the equalizer of its cokernel pair. We may assume that $|Z| \subseteq |Y|$ is a subspace and that $|i|$ is the inclusion map. Recall from the proof of cocompleteness that pushouts in $\LRS_R$ are computed as in ringed spaces: the underlying space is the pushout of the underlying spaces, and the structure sheaf is the fiber product of the push-forwards of the structure sheaves. Hence, the cokernel pair $u_1,u_2 : Y \rightrightarrows W$ of $i$ looks as follows. The underlying space $|W|$ consists of two copies of $|Y|$ glued along $|Z|$, and $|u_1|,|u_2|$ are the two inclusions. The open subsets of $|W|$ correspond to pairs $(V_1,V_2)$ of open subsets of $|Y|$ with $|Z| \cap V_1 = |Z| \cap V_2$, and + $$\O_W(V_1,V_2) = \O_Y(V_1) \times_{\O_Z(|Z| \cap V_1)} \O_Y(V_2),$$ where the sheaf maps of $u_1,u_2$ are the two projections. - Clearly, $u_1 \circ i = u_2 \circ i$. Now let $\varphi : T \to Y$ be a morphism with $u_1 \circ \varphi = u_2 \circ \varphi$. Since the forgetful functor $\Top \to \Set$ is a left adjoint, it preserves pushouts. Hence, the underlying set of $W$ is the quotient of $Y \sqcup Y$ that identifies the two copies of each $z \in Z$ and nothing else (this is already an equivalence relation). So $u_1(y) = u_2(y)$ only holds for $y \in Z$, and we get $\varphi(T) \subseteq Z$. Let $t \in T$, let $z \coloneqq \varphi(t)$, and let $a \in \ker(i_z)$. After shrinking, $a$ is the germ of a section $s \in \O_Y(V)$ over an open neighborhood $V$ of $z$ such that $i^\sharp(s) = 0$ in $\O_Z(Z \cap V)$. Thus, $(s,0)$ is a section of $\O_W$ over $(V,V)$. Since $u_1 \circ \varphi$ and $u_2 \circ \varphi$ have the same sheaf map, the two composites - $$\O_W(V,V) \rightrightarrows \O_Y(V) \xrightarrow{\varphi^\sharp} \O_T(\varphi^*(V))$$ - agree, so that $\varphi^\sharp(s) = \varphi^\sharp(0) = 0$. Taking germs at $t$, we get $\varphi_t(a) = \varphi_t(s_z) = \varphi^\sharp(s)_t = 0$. By the factorization criterion, $\varphi$ factors uniquely through $i$. + Clearly, $u_1 \circ i = u_2 \circ i$. Now let $\varphi : T \to Y$ be a morphism with $u_1 \circ \varphi = u_2 \circ \varphi$. Since the forgetful functor $\Top \to \Set$ is a left adjoint, it preserves pushouts. Hence, the underlying set of $|W|$ is the quotient of $|Y| \sqcup |Y|$ that identifies the two copies of each $z \in |Z|$ and nothing else (this is already an equivalence relation). So $|u_1|(y) = |u_2|(y)$ only holds for $y \in |Z|$, and we get $|\varphi|(|T|) \subseteq |Z|$. Let $t \in |T|$, let $z \coloneqq |\varphi|(t)$, and let $a \in \ker(i^\sharp_z)$. After shrinking, $a$ is the germ of a section $s \in \O_Y(V)$ over an open neighborhood $V$ of $z$ such that $i^\sharp(s) = 0$ in $\O_Z(|Z| \cap V)$. Thus, $(s,0)$ is a section of $\O_W$ over $(V,V)$. Since $u_1 \circ \varphi$ and $u_2 \circ \varphi$ have the same sheaf map, the two composites + $$\O_W(V,V) \rightrightarrows \O_Y(V) \xrightarrow{\varphi^\sharp} \O_T(|\varphi|^*(V))$$ + agree, so that $\varphi^\sharp(s) = \varphi^\sharp(0) = 0$. Taking germs at $t$, we get $\varphi^\sharp_t(a) = \varphi^\sharp_t(s_z) = \varphi^\sharp(s)_t = 0$. By the factorization criterion, $\varphi$ factors uniquely through $i$. + regular epimorphisms: + description: 'A morphism $f : X \to Y$ with kernel pair $p_1,p_2 : X \times_Y X \rightrightarrows X$ is a regular epimorphism iff $|f| : |X| \to |Y|$ is a surjective quotient map, $f^\sharp : \O_Y \to |f|_* \O_X$ is a monomorphism of sheaves, and for every open subset $V \subseteq |Y|$, every section $s \in \O_X(|f|^*(V))$ with $p_1^\sharp(s) = p_2^\sharp(s)$ lies in the image of $f^\sharp : \O_Y(V) \to \O_X(|f|^*(V))$. Moreover, a morphism $\Spec(\varphi) : \Spec(B) \to \Spec(A)$ of affine schemes is a regular epimorphism iff $\varphi : A \to B$ is a regular monomorphism in $\CAlg(R)$ and $|\Spec(\varphi)|$ is a surjective quotient map.' + proof: >- + Let $f : X \to Y$ be a morphism. Since $\LRS_R$ is complete, $f$ has a kernel pair $p_1,p_2 : X \times_Y X \rightrightarrows X$, and by this lemma, $f$ is a regular epimorphism iff it is the coequalizer of $p_1,p_2$. Recall from the proof of cocompleteness that colimits in $\LRS_R$ are computed as in ringed spaces: the underlying space is the colimit of the underlying spaces, and the structure sheaf is the limit of the push-forwards of the structure sheaves. Hence, the coequalizer $q : X \to Q$ of $p_1,p_2$ looks as follows. The underlying space $|Q|$ is the coequalizer of $|p_1|,|p_2|$ in $\Top$, the structure sheaf is given by + $$\O_Q(W) = \{s \in \O_X(|q|^*(W)) : p_1^\sharp(s) = p_2^\sharp(s)\}$$ + for open subsets $W \subseteq |Q|$, and $q^\sharp : \O_Q \to |q|_* \O_X$ is the inclusion. Let $c : Q \to Y$ be the unique morphism with $c \circ q = f$. Then $f$ is a regular epimorphism iff $c$ is an isomorphism. + + Since the forgetful functor $\LRS_R \to \Top$ preserves colimits, it preserves regular epimorphisms. So if $f$ is a regular epimorphism, then $|f|$ is a regular epimorphism of $\Top$, i.e. a surjective quotient map. Assume from now on that $|f|$ is a surjective quotient map. + + Let $x_1,x_2 \in |X|$ with $|f|(x_1) = |f|(x_2)$. By the proof for monomorphisms (see above), there are a field $F$ and morphisms $\bar{x}_1,\bar{x}_2 : P_F \rightrightarrows X$ corresponding to the points $x_1,x_2$ with $f \circ \bar{x}_1 = f \circ \bar{x}_2$, where $P_F$ is the locally ringed space with a single point whose stalk is $F$. The induced morphism $(\bar{x}_1,\bar{x}_2) : P_F \to X \times_Y X$ maps the unique point of $|P_F|$ to a point $t$ with $|p_1|(t) = x_1$ and $|p_2|(t) = x_2$. So every continuous map $g : |X| \to S$ into a topological space $S$ with $g \circ |p_1| = g \circ |p_2|$ satisfies $g(x_1) = g(x_2)$. Since $|f|$ is a surjective quotient map, every such $g$ factors uniquely through $|f|$. Hence, both $|f|$ and $|q|$ are coequalizers of $|p_1|,|p_2|$ in $\Top$, so $|c|$ is a homeomorphism. + + Consequently, $c$ is an isomorphism iff $c^\sharp : \O_Y \to |c|_* \O_Q$ is an isomorphism. For an open subset $V \subseteq |Y|$, we have $|q|^*(|c|^*(V)) = |f|^*(V)$ and therefore + $$(|c|_* \O_Q)(V) = \{s \in \O_X(|f|^*(V)) : p_1^\sharp(s) = p_2^\sharp(s)\}.$$ + Since $c \circ q = f$ and $q^\sharp$ is the inclusion, $c^\sharp$ maps a section $b \in \O_Y(V)$ to $f^\sharp(b)$. Hence, $f$ is a regular epimorphism iff for every open subset $V \subseteq |Y|$, the map $f^\sharp : \O_Y(V) \to \O_X(|f|^*(V))$ is injective and its image is $(|c|_* \O_Q)(V)$. Since $f \circ p_1 = f \circ p_2$, the image is always contained in $(|c|_* \O_Q)(V)$, which proves the first statement. + + Finally, let $\varphi : A \to B$ be a homomorphism of commutative $R$-algebras, and let $f \coloneqq \Spec(\varphi) : X \to Y$ with $X \coloneqq \Spec(B)$ and $Y \coloneqq \Spec(A)$. Recall that $\Spec : \CAlg(R)^{\op} \to \LRS_R$ is fully faithful and right adjoint to the global sections functor. If $f$ is a regular epimorphism, then $|f|$ is a surjective quotient map by the above, and $\varphi$ is a regular monomorphism in $\CAlg(R)$, since the global sections functor $\LRS_R \to \CAlg(R)^{\op}$ is a left adjoint, hence preserves regular epimorphisms, and maps $f$ to $\varphi$. Conversely, assume these two conditions. Since $\Spec$ is a right adjoint, it maps the cokernel pair $u_1, u_2 : B \rightrightarrows B \otimes_A B$ of $\varphi$ to a kernel pair of $f$, so we may take $X \times_Y X = \Spec(B \otimes_A B)$ and $p_i = \Spec(u_i)$. By the above, we need to show that $f^\sharp : \O_Y(V) \to (|c|_* \O_Q)(V)$ is bijective for every open subset $V \subseteq |Y|$. Since $\O_Y$ and $|c|_* \O_Q$ are sheaves on $|Y|$, it suffices to check this for the basic open subsets $V = D(g)$ with $g \in A$, where it means that + $$A_g \to B_g \rightrightarrows (B \otimes_A B)_g$$ + is an equalizer diagram. This is the localization at $g$ of the diagram $A \to B \rightrightarrows B \otimes_A B$, which is an equalizer diagram since $\varphi$ is the equalizer of its cokernel pair by the dual of this lemma. Since localization is exact, it preserves equalizers. diff --git a/database/data/categories/Sch_R.yaml b/database/data/categories/Sch_R.yaml index 5c1d6a37b..efd546cfe 100644 --- a/database/data/categories/Sch_R.yaml +++ b/database/data/categories/Sch_R.yaml @@ -16,9 +16,8 @@ related: - Z comments: - - Monomorphisms are discussed at MO/56591. At least the case of morphisms of locally finite type is understood. - Regular monomorphisms are discussed at MO/66279. - - Epimorphisms are discussed at MO/56564. Probably they cannot be classified. + - Epimorphisms are discussed at MO/56564. It is not clear if they can be classified. satisfied_properties: - property: locally small @@ -76,7 +75,7 @@ unsatisfied_properties: - property: pullback-stable regular epimorphisms proof: >- - Let $A \coloneqq R[t^2,t^3]$, $B \coloneqq R[t]$, so that $A \subseteq B$ consists of the polynomials $b = \sum_n b_n t^n$ with $b_1 = 0$. Let $X = \Spec(B)$, $Y = \Spec(A)$, and let $p : X \to Y$ be the morphism induced by the inclusion $A \hookrightarrow B$. For a field $R$, this is the normalization of the cuspidal cubic curve. We claim that $p$ is a regular epimorphism, but that its pullback along some closed immersion is not even an epimorphism. + Let $A \coloneqq R[t^2,t^3]$, $B \coloneqq R[t]$, so that $A \subseteq B$ consists of the polynomials $b = \sum_n b_n t^n$ with $b_1 = 0$. Let $X \coloneqq \Spec(B)$, $Y \coloneqq \Spec(A)$, and let $p : X \to Y$ be the morphism induced by the inclusion $A \hookrightarrow B$. For a field $R$, this is the normalization of the cuspidal cubic curve. We claim that $p$ is a regular epimorphism, but that its pullback along some closed immersion is not even an epimorphism. First, we show that $A$ is the equalizer of the two homomorphisms $B \rightrightarrows B \otimes_A B$, $b \mapsto b \otimes 1$ and $b \mapsto 1 \otimes b$. Let $R[\varepsilon]$ be the algebra of dual numbers, where $\varepsilon^2 = 0$. The two homomorphisms $\alpha_0,\alpha_1 : B \rightrightarrows R[\varepsilon]$ defined by $\alpha_0(t) \coloneqq 0$ and $\alpha_1(t) \coloneqq \varepsilon$ both map $t^2$ and $t^3$ to $0$, so they agree on $A$. Hence, they induce a homomorphism $\alpha : B \otimes_A B \to R[\varepsilon]$ with $\alpha(b \otimes b') = \alpha_0(b) \, \alpha_1(b')$. If $b \in B$ satisfies $b \otimes 1 = 1 \otimes b$, then $$b_0 = \alpha_0(b) = \alpha(b \otimes 1) = \alpha(1 \otimes b) = \alpha_1(b) = b_0 + b_1 \varepsilon,$$ @@ -84,25 +83,44 @@ unsatisfied_properties: Next, we show that $p$ is the coequalizer of its kernel pair $$p_1,p_2 : X \times_Y X = \Spec(B \otimes_A B) \rightrightarrows X$$ - in $\LRS_R$, and hence also in its full subcategory $\Sch_R$. Recall from the proof of cocompleteness of $\LRS_R$ that coequalizers in $\LRS_R$ are computed as in ringed spaces: the underlying space is the coequalizer of the underlying spaces, and the structure sheaf is the equalizer of the push-forwards of the structure sheaves. Since $B$ is integral over $A$, the continuous map $p$ is surjective and closed by the lying over and going up theorems (SP/00GQ, SP/00GU, SP/00HZ), hence a quotient map. If $x_1,x_2 \in X$ satisfy $p(x_1) = p(x_2)$, then there is some $z \in X \times_Y X$ with $p_1(z) = x_1$ and $p_2(z) = x_2$ by SP/01JT. Hence, $p$ is the coequalizer of $p_1,p_2$ in $\Top$. Let $q \coloneqq p \circ p_1 = p \circ p_2$. Since $p$ and $q$ are affine, the diagram of sheaves - $$\O_Y \to p_* \O_X \rightrightarrows q_* \O_{X \times_Y X}$$ + in $\LRS_R$, and hence also in its full subcategory $\Sch_R$. Recall from the proof of cocompleteness of $\LRS_R$ that coequalizers in $\LRS_R$ are computed as in ringed spaces: the underlying space is the coequalizer of the underlying spaces, and the structure sheaf is the equalizer of the push-forwards of the structure sheaves. Since $B$ is integral over $A$, the continuous map $|p|$ is surjective and closed by the lying over and going up theorems (SP/00GQ, SP/00GU, SP/00HZ), hence a quotient map. If $x_1,x_2 \in |X|$ satisfy $|p|(x_1) = |p|(x_2)$, then there is some $z \in |X \times_Y X|$ with $|p_1|(z) = x_1$ and $|p_2|(z) = x_2$ by SP/01JT. Hence, $|p|$ is the coequalizer of $|p_1|,|p_2|$ in $\Top$. Let $q \coloneqq p \circ p_1 = p \circ p_2$. Since $p$ and $q$ are affine, the diagram of sheaves + $$\O_Y \to |p|_* \O_X \rightrightarrows |q|_* \O_{X \times_Y X}$$ is the one associated with the diagram of $A$-modules $A \to B \rightrightarrows B \otimes_A B$. Since the functor $M \mapsto \widetilde{M}$ is exact, the first step implies that it is an equalizer diagram. This shows that $p$ is a regular epimorphism. Finally, let $I \coloneqq t^2 A$. The pullback of $p$ along the closed immersion $\Spec(A/I) \hookrightarrow Y$ is the morphism $p' : \Spec(B/IB) \to \Spec(A/I)$ induced by the canonical homomorphism $$j : A/I \to B/IB.$$ We claim that $p'$ is not an epimorphism. In fact, since the functor $\Spec : \CAlg(R)^{\op} \to \Sch_R$ is right adjoint to the global sections functor $\Gamma : \Sch_R \to \CAlg(R)^{\op}$, the latter preserves epimorphisms. So if $p'$ were an epimorphism, then $\Gamma(p') = j$ would be a monomorphism in $\CAlg(R)$, i.e. injective. But every element of $I$ has a vanishing coefficient at $t^3$, so $t^3 \notin I$, whereas $t^3 = t^2 \cdot t \in IB$. Thus, $[t^3] \in A/I$ a non-zero element in $\ker(j)$. + - property: regular subobject classifier + proof: The proof for $\LRS_R$ translates verbatim to $\Sch_R$. + references: + - LRS_R_no_regular_subobject_classifier + special_objects: initial object: - description: empty scheme + description: empty scheme $X \cong \Spec(0)$, given by $|X| \coloneqq \varnothing$ and $\O_X(\varnothing) \coloneqq 0$ terminal object: - description: $\Spec(R)$ + description: affine scheme $\Spec(R)$ coproducts: - description: disjoint union with the product sheaf + description: The coproduct $X$ of a family $(X_i)_{i \in I}$ is given by $|X| \coloneqq \coprod_{i \in I} |X_i|$ and $\O_X(U) \coloneqq \prod_{i \in I} \O_{X_i}(U \cap |X_i|)$ for open subsets $U \subseteq |X|$. products: - description: '[finite case] The idea is to use $\Spec(A) \times \Spec(B) = \Spec(A \otimes_R B)$ and then to glue affine pieces together. See EGA I, Chap. I, Thm. 3.2.1.' + description: '[finite case] There are two descriptions. (a) The product in $\LRS_R$ (see there) is a scheme. (b) The idea is to use $\Spec(A) \times \Spec(B) = \Spec(A \otimes_R B)$ and then to glue affine pieces together. See Grothendieck–Dieudonné, EGA I (1971), Ch. I, Thm. 3.2.1.' special_morphisms: isomorphisms: - description: pairs $(f,f^{\sharp})$ consisting of a homeomorphism $f$ and an isomorphism of sheaves $f^{\sharp}$ of $R$-algebras - proof: This is easy. + description: 'A morphism $f : X \to Y$ is an isomorphism iff $|f| : |X| \to |Y|$ is a homeomorphism and $f^\sharp : \O_Y \to |f|_* \O_X$ is an isomorphism of sheaves.' + proof: This description is inherited from $\LRS_R$. + monomorphisms: + description: 'A morphism $f : X \to Y$ is a monomorphism iff $|f| : |X| \to |Y|$ is injective and for every $x \in |X|$ the stalk map $f^\sharp_x : \O_{Y,|f|(x)} \to \O_{X,x}$ is an epimorphism of commutative rings.' + proof: >- + Assume that $f$ satisfies the two conditions. Then every stalk map $f^\sharp_x$ is also an epimorphism in the category of local commutative rings and local homomorphisms. By the classification of monomorphisms in $\LRS_R$, $f$ is a monomorphism in $\LRS_R$, hence also in its full subcategory $\Sch_R$. + + Conversely, assume that $f$ is a monomorphism. The proof for $\LRS_R$ shows that $|f|$ is injective, since the locally ringed spaces with a single point and a field $F$ as stalk used there are the schemes $\Spec(F)$. Now let $x \in |X|$ and $y \coloneqq |f|(x)$. Recall that there is a canonical morphism $\iota_x : \Spec(\O_{X,x}) \to X$, and that the square + $$\begin{CD} \Spec(\O_{X,x}) @>{\iota_x}>> X \\ @V{\Spec(f^\sharp_x)}VV @VV{f}V \\ \Spec(\O_{Y,y}) @>>{\iota_y}> Y \end{CD}$$ + commutes. We claim that $\iota_x$ is a monomorphism. Indeed, choose an affine open neighborhood $U \cong \Spec(A)$ of $x$, and let $\p \subseteq A$ be the prime ideal corresponding to $x$, so that $\O_{X,x} \cong A_\p$. Then $\iota_x$ is the composite of $\Spec(A_\p) \to \Spec(A)$ and the open embedding $U \hookrightarrow X$. The former is a monomorphism since the localization $A \to A_\p$ is an epimorphism of $R$-algebras and $\Spec : \CAlg(R)^{\op} \to \Sch_R$ is a right adjoint (of the global sections functor), and open embeddings are monomorphisms. + + Hence, $f \circ \iota_x = \iota_y \circ \Spec(f^\sharp_x)$ is a monomorphism, and therefore $\Spec(f^\sharp_x)$ is a monomorphism. Since $\Spec : \CAlg(R)^{\op} \to \Sch_R$ is fully faithful, $f^\sharp_x$ is an epimorphism in $\CAlg(R)$. This means that it is an epimorphism of commutative rings, since $\CAlg(R)$ is the coslice category of the category of commutative rings under $R$, and epimorphisms in a coslice category are the same as in the underlying category. + + The same classification is proven in MSE/1650634. For a classification of monomorphisms locally of finite type, see Grothendieck–Dieudonné, EGA IV4 (1967), Prop. 17.2.6. + + Alternatively, the inclusion functor $\Sch_R \to \LRS_R$ preserves pullbacks (SP/01JN), hence monomorphisms. Since it is also faithful, the monomorphisms in $\Sch_R$ are exactly the morphisms of schemes that are monomorphisms in $\LRS_R$, i.e. those $f$ for which $|f|$ is injective and the stalk maps are epimorphisms in the category of local commutative rings. This description only looks weaker than the one above: in general, an epimorphism of local commutative rings need not be an epimorphism of commutative rings (see the example in $\LRS_R$), but for the stalk maps of a morphism of schemes with injective underlying map, both notions agree by the proof above. diff --git a/database/data/macros.yaml b/database/data/macros.yaml index 07ac4bd8f..ebe3def24 100644 --- a/database/data/macros.yaml +++ b/database/data/macros.yaml @@ -26,6 +26,7 @@ \O: \mathcal{O} \S: \mathcal{S} \T: \mathcal{T} +\U: \mathcal{U} \Q: \mathcal{Q} # fraktur letters