8 Two-dimensional models
Fix a Grothendieck universe and call a category small when it is equivalent to one whose objects and morphisms form a set in that universe. All abelian categories below are assumed small in this sense, so that they, the exact functors and the natural transformations between them form a 2-category \(\mathit{AbCat}\). This is a size convention; it is preserved by every construction performed below, and no argument uses it otherwise.
8.2 Serre subcategories are the 2-kernels
The abelian category with one object is a strong bizero object of \(\mathit{AbCat}\).
Write \(0\) for that category. For every abelian category \(\mathsf{A}\) there is exactly one functor \(\mathsf{A}\to 0\), it is exact, and there is exactly one natural transformation between any two such; so \({\mathit{AbCat}}\left({\mathsf{A}},\, {0}\right)\) is the singleton category. An exact functor \(0\to \mathsf{A}\) is determined up to a unique natural isomorphism by its value on the unique object, which must be a zero object of \(\mathsf{A}\), so \({\mathit{AbCat}}\left({0},\, {\mathsf{A}}\right)\) is a contractible groupoid and in particular equivalent to \(\mathsf{1}\). Thus \(0\) is a bizero object, and a 1-cell is null exactly when it takes every object to a zero object. Given two parallel null functors \(F\), \(G\colon \mathsf{A}\to \mathsf{B}\), each component \(F(a)\to G(a)\) is the unique morphism between two zero objects, so there is exactly one natural transformation \(F\Rightarrow G\), and \(0\) is strong.
Let \(F\colon \mathsf{A}\to \mathsf{B}\) be an exact functor and let \(K\) be the full subcategory of \(\mathsf{A}\) on the objects that \(F\) annihilates. Then \(K\) is a Serre subcategory of \(\mathsf{A}\) and its inclusion \(k\colon K\to \mathsf{A}\) is a 2-kernel of \(F\).
That \(K\) is closed under subobjects, quotients and extensions is exactness of \(F\), the zero objects of \(\mathsf{B}\) having those three closure properties. By construction \(F\circ k\) takes every object to a zero object, so it is null and carries an invertible 2-cell \(\kappa \colon F\circ k\cong 0\); by Lemma 2.6 it is the only one.
For condition \((1)\) of Definition 2.18, let \(G\colon \mathsf{X}\to \mathsf{A}\) be exact with an invertible 2-cell \(F\circ G\cong 0\). Then \(F(G(x))\) is a zero object for every \(x\), so \(G\) takes its values in \(K\) and factors as \(k\circ u\) for a unique exact \(u\colon \mathsf{X}\to K\). Condition \((2)\) says that whiskering with \(k\) is a bijection on 2-cells, and this holds because \(k\) is fully faithful: a natural transformation between two functors landing in \(K\) is the same thing computed in \(K\) as computed in \(\mathsf{A}\).
Let \(F\colon \mathsf{A}\to \mathsf{B}\) be an exact functor and let \(S\) be the smallest Serre subcategory of \(\mathsf{B}\) containing the objects \(F(a)\). Then the projection \(q\colon \mathsf{B}\to \mathsf{B}/S\) is a 2-cokernel of \(F\).
The functor \(q\) annihilates \(S\) and hence every \(F(a)\), so \(q\circ F\) is null. Let \(z\colon \mathsf{B}\to \mathsf{Z}\) be exact with an invertible 2-cell \(z\circ F\cong 0\). Then \(z\) annihilates every \(F(a)\); the objects annihilated by \(z\) form a Serre subcategory, by Proposition 8.4, so \(z\) annihilates \(S\). By the universal property of the Serre quotient, \(z\) is naturally isomorphic to \(u\circ q\) for some exact \(u\colon \mathsf{B}/S\to \mathsf{Z}\), which is condition \((1)\) of Definition 2.19; and the same universal property says that composition with \(q\) is fully faithful on exact functors out of \(\mathsf{B}/S\), which is condition \((2)\).
In \(\mathit{AbCat}\), every normal 2-monomorphism is isomorphic to the inclusion of a Serre subcategory precomposed with an equivalence, and every normal 2-epimorphism is isomorphic to the projection onto a Serre quotient postcomposed with an equivalence.
8.7 The saturation, and the failure of (DI2)
Let \(K\) and \(S\) be Serre subcategories of an abelian category \(\mathsf{A}\). The \(S\)-saturation of \(K\), written \(K^{S}\), is the class of objects of \(\mathsf{A}\) that become isomorphic in \(\mathsf{A}/S\) to an object of \(K\); equivalently, the class of objects joined to an object of \(K\) by a span of morphisms all of whose kernels and cokernels lie in \(S\).
The class \(K^{S}\) contains \(K\) and \(S\), is closed under subobjects and under quotients, and is contained in every Serre subcategory of \(\mathsf{A}\) containing \(K\) and \(S\). It is therefore a Serre subcategory if and only if it is the join \(K\vee S\).
It contains \(K\) trivially, and it contains \(S\) because \(q\) sends an object of \(S\) to a zero object, which is \(q\) of the zero object of \(K\). Let \(X\in K^{S}\), say \(q(X)\cong q(a)\) with \(a\in K\), and let \(X'\subseteq X\). Then \(q(X')\) is a subobject of \(q(X)\cong q(a)\), and every subobject of \(q(a)\) is of the form \(q(a')\) for a subobject \(a'\subseteq a\); since \(K\) is closed under subobjects, \(a'\in K\) and \(X'\in K^{S}\). The argument for quotients is the same.
Let \(T\) be a Serre subcategory containing \(K\) and \(S\), and let \(X\xrightarrow []{g}D\xrightarrow []{f}a\) be a span with \(a\in K\) and with the kernels and cokernels of \(f\) and of \(g\) in \(S\). The image of \(f\) is a subobject of \(a\), hence lies in \(T\), and the kernel of \(f\) lies in \(S\subseteq T\); so \(D\), an extension of the one by the other, lies in \(T\). Running the same argument along \(g\) puts \(X\) in \(T\). Finally, \(K\vee S\) is the smallest Serre subcategory containing \(K\) and \(S\), so that \(K^{S}\subseteq K\vee S\) always, with equality precisely when \(K^{S}\) is a Serre subcategory.
Let \(K\) and \(S\) be Serre subcategories of an abelian category \(\mathsf{A}\). Then the exact functor \(K/(K\cap S)\to \mathsf{A}/S\) induced by the inclusion of \(K\) is fully faithful.
For \(a\) and \(b\) in \(K\), the hom-set \(\operatorname{Hom}_{\mathsf{A}/S}(\overline{a},\overline{b})\) is the filtered colimit 7, and \(\operatorname{Hom}_{K/(K\cap S)}(\overline{a},\overline{b})\) is the same colimit taken inside \(K\). Every subobject and every quotient of \(a\) or of \(b\) in \(\mathsf{A}\) already lies in \(K\), that subcategory being closed under subobjects and quotients, and such a subobject lies in \(S\) if and only if it lies in \(K\cap S\); so the two index categories coincide. The terms coincide as well, \(K\) being full. Hence the two colimits agree.
Condition (DI2) holds in \(\mathit{AbCat}\) if and only if for every abelian category \(\mathsf{A}\) and all Serre subcategories \(K\) and \(S\) of \(\mathsf{A}\), the \(S\)-saturation \(K^{S}\) is a Serre subcategory of \(\mathsf{A}\).
An antinormal 1-cell of \(\mathit{AbCat}\) is a composite \(e\circ m\) with \(m\) a normal 2-monomorphism into some abelian category \(\mathsf{A}\) and \(e\) a normal 2-epimorphism out of it. By Corollary 8.6 we may write \(m\cong k\circ u\) and \(e\cong v\circ q\), where \(k\colon K\to \mathsf{A}\) is the inclusion of a Serre subcategory, \(q\colon \mathsf{A}\to \mathsf{A}/S\) is the projection onto a Serre quotient, and \(u\) and \(v\) are equivalences; so \(e\circ m\cong v\circ w\circ u\) with \(w\mathrel {:=}q\circ k\), and by Corollary 3.8 the 1-cell \(e\circ m\) is normal if and only if \(w\) is. By Definition 8.8 the essential image of \(w\) is \(q(K^{S})\), and \(K^{S}\) contains \(S\) by Proposition 8.9; so Gabriel’s correspondence makes \(q(K^{S})\) a Serre subcategory of \(\mathsf{A}/S\) if and only if \(K^{S}\) is one of \(\mathsf{A}\).
Suppose \(K^{S}\) is a Serre subcategory. Then \(w\) factorises as
whose first factor is the 2-cokernel of the inclusion \(K\cap S\to K\), hence a normal 2-epimorphism. The second factor \(c\) is fully faithful by Lemma 8.10, hence a 2-monomorphism, and its essential image is the Serre subcategory \(q(K^{S})\); so \(c\) is an equivalence onto that subcategory, whose inclusion is a normal 2-monomorphism by Proposition 8.4, and Corollary 3.8 makes \(c\) a normal 2-monomorphism. Hence \(w\) is normal.
Conversely, suppose \(w\cong m'\circ e'\) with \(e'\) a normal 2-epimorphism and \(m'\) a normal 2-monomorphism. By Corollary 8.6 the functor \(e'\) is a Serre quotient projection composed with an equivalence, hence essentially surjective, and \(m'\) is the inclusion of a Serre subcategory \(T\) of \(\mathsf{A}/S\) composed with an equivalence. The essential image of \(w\) is then that of \(m'\), which is \(T\); so \(q(K^{S})=T\) is a Serre subcategory of \(\mathsf{A}/S\), and \(K^{S}\) is one of \(\mathsf{A}\).
The 2-category \(\mathit{AbCat}\) is not 2-di-exact.
Let \(\Lambda \) be the Nakayama algebra over a field \(\Bbbk \) on the cyclic quiver \(1\to 2\to 1\) with \(\operatorname {rad}^{3}=0\), let \(\mathsf{A}\) be its category of finite-dimensional modules and let \(S_{1}\), \(S_{2}\) be the two simple modules. Put
both plainly Serre subcategories. Let \(M\) be the uniserial module of length 3 with top \(S_{1}\), middle \(S_{2}\) and socle \(S_{1}\)—that is, the indecomposable projective \(\Lambda e_{1}\)—and let \(U\subset M\) be its submodule of length 2, with socle \(S_{1}\) and top \(S_{2}\).
Then \(U\in K^{S}\): the inclusion \(S_{1}\subseteq U\) has zero kernel and cokernel \(S_{2}\in S\), so \(\overline{U}\cong \overline{S_{1}}\) with \(S_{1}\in K\). Also \(M/U\cong S_{1}\) lies in \(K\). But \(M\notin K^{S}\), and this is a computation of hom-sets rather than an inspection of filtrations. The only submodule of \(M\) lying in \(S\) is \(0\), the only submodule \(M'\subseteq M\) with \(M/M'\in S\) is \(M\), and the only submodule \(X'\subseteq S_{1}\) with \(S_{1}/X'\in S\) is \(S_{1}\); so the colimits 7 collapse to a single term:
Hence \(\overline{M}\cong \overline{S_{1}}^{n}\) would force \(n=1\); but \(\operatorname{End}_{\mathsf{A}/S}(\overline{M})=\operatorname{End}_{\Lambda }(M)=e_{1}\Lambda e_{1}\) has dimension 2, being spanned by \(e_{1}\) and the path of length 2, whereas \(\operatorname{End}(\overline{S_{1}})=\Bbbk \). So \(\overline{M}\) is isomorphic to no object of \(K\). The short exact sequence \(0\to U\to M\to M/U\to 0\) therefore has both of its outer terms in \(K^{S}\) and its middle term outside, so \(K^{S}\) is not closed under extensions, and Proposition 8.11 applies.
The mechanism is that localisation creates \(\operatorname{Ext}\). Indeed \(\operatorname{Ext}^{1}_{\mathsf{A}}(S_{1},S_{1})=0\) in the example above, the quiver having no loop at the first vertex; the self-extension of \(\overline{S_{1}}\) witnessed by \(\overline{M}\) comes into being only once \(S_{2}\) has been killed, as the Yoneda composite of the two extensions between \(S_{1}\) and \(S_{2}\). Conversely, if for all \(a\), \(b\) in \(K\) the comparison
is surjective, then \(K^{S}\) is a Serre subcategory: an extension of \(q(b)\) by \(q(a)\) in \(\mathsf{A}/S\) is the image of one in \(\mathsf{A}\), whose middle term lies in \(K\). That happens for flat localisations, and whenever \(q\) admits a fully faithful right adjoint, so that the quotient sits inside \(\mathsf{A}\) as a full subcategory and can manufacture nothing. It also says where not to look: the categories of finite length, of artinian modules and of finite-dimensional modules over a finite-dimensional algebra are all ruled out, and ruled out for having too much \(\operatorname{Ext}\) rather than too little. That \(\mathit{AbCat}\) is not 2-di-exact is proved independently in [ 6 , Proposition 2.41 ] , by a Serre quotient of the finite-dimensional representations of the quiver \(1\to 2\to 3\); the obstruction isolated there [ 6 , Remark 2.42 ] —that the essential image of the quotient need not be closed under extensions—is the same one.
8.14 Saturated abelian categories
An abelian category \(\mathsf{A}\) is saturated when for all Serre subcategories \(K\) and \(S\) of \(\mathsf{A}\) the \(S\)-saturation \(K^{S}\) is again a Serre subcategory of \(\mathsf{A}\). We write \(\mathit{Sat}\) for the full sub-2-category of \(\mathit{AbCat}\) on the saturated abelian categories.
Saturation is inherited by Serre subcategories and by Serre quotients.
Let \(\mathsf{A}\) be saturated and let \(\mathsf{B}\) be a Serre subcategory of it. The Serre subcategories of \(\mathsf{B}\) are exactly the Serre subcategories of \(\mathsf{A}\) contained in \(\mathsf{B}\), that subcategory being closed in \(\mathsf{A}\) under subobjects, quotients and extensions; let \(K\) and \(S\) be two of them. By Lemma 8.10, applied to \(\mathsf{B}\) and \(S\) inside \(\mathsf{A}\), the induced functor \(\mathsf{B}/S\to \mathsf{A}/S\) is fully faithful, and a fully faithful functor reflects isomorphisms; so an object of \(\mathsf{B}\) becomes isomorphic to an object of \(K\) in \(\mathsf{B}/S\) exactly when it does in \(\mathsf{A}/S\). The \(S\)-saturation of \(K\) computed in \(\mathsf{B}\) is therefore the intersection with \(\mathsf{B}\) of the one computed in \(\mathsf{A}\). The latter is a Serre subcategory of \(\mathsf{A}\) by saturation, and it is contained in \(K\vee S\) by Proposition 8.9, hence in \(\mathsf{B}\); so the two coincide, and being a Serre subcategory of \(\mathsf{A}\) contained in \(\mathsf{B}\), it is a Serre subcategory of \(\mathsf{B}\).
Let now \(S\) be a Serre subcategory of \(\mathsf{A}\), write \(q\colon \mathsf{A}\to \mathsf{A}/S\), and let \(K'\) and \(T'\) be Serre subcategories of \(\mathsf{A}/S\). Put \(K\mathrel {:=}q^{-1}(K')\) and \(T\mathrel {:=}q^{-1}(T')\), which are Serre subcategories of \(\mathsf{A}\) containing \(S\), the functor \(q\) being exact and annihilating \(S\). By saturation the \(T\)-saturation \(K^{T}\) is a Serre subcategory of \(\mathsf{A}\); it contains \(T\) and hence \(S\), so by Gabriel’s correspondence \(q(K^{T})\) is a Serre subcategory of \(\mathsf{A}/S\). It contains \(K'\) and \(T'\), hence their join. It is also contained in \((K')^{T'}\): applying the exact functor \(q\) to a span witnessing membership in \(K^{T}\) produces a span witnessing membership in \((K')^{T'}\), the kernels and cokernels of the images being the images of the kernels and cokernels. So \((K')^{T'}\) contains \(K'\vee T'\); it is contained in that join by Proposition 8.9, and being equal to it, it is a Serre subcategory.
The 2-category \(\mathit{Sat}\) of saturated abelian categories, exact functors and natural transformations is 2-di-exact.
The abelian category with one object is saturated, and it is a strong bizero object of \(\mathit{Sat}\) by Proposition 8.3, whose proof takes place inside any full sub-2-category of \(\mathit{AbCat}\) containing it.
Condition (DI1). Let \(F\colon \mathsf{A}\to \mathsf{B}\) be an exact functor between saturated abelian categories. Its 2-kernel in \(\mathit{AbCat}\) is the inclusion of the Serre subcategory \(K\) of the objects that \(F\) annihilates, by Proposition 8.4, and its 2-cokernel is the projection onto \(\mathsf{B}/S\) for a Serre subcategory \(S\) of \(\mathsf{B}\), by Proposition 8.5. Both \(K\) and \(\mathsf{B}/S\) are saturated by Proposition 8.16; and since \(\mathit{Sat}\) is a full sub-2-category of \(\mathit{AbCat}\), a 2-kernel or a 2-cokernel computed in \(\mathit{AbCat}\) whose vertex lies in \(\mathit{Sat}\) is one computed in \(\mathit{Sat}\). The same remark makes Corollary 8.6 available in \(\mathit{Sat}\).
Condition (DI2). By the reduction carried out at the start of the proof of Proposition 8.11 it suffices to show that the composite \(w=q\circ k\) of the inclusion of a Serre subcategory \(K\) into a saturated abelian category \(\mathsf{A}\) with the projection onto a Serre quotient \(\mathsf{A}/S\) is normal. This is the first half of the proof of Proposition 8.11, whose hypothesis that \(K^{S}\) be a Serre subcategory is now supplied by saturation; and the intermediate category \(K/(K\cap S)\), being a Serre quotient of a Serre subcategory of \(\mathsf{A}\), is itself saturated by Proposition 8.16, so that the factorisation obtained there lies in \(\mathit{Sat}\).
8.18 A criterion on subobjects
Let \(K\) and \(S\) be Serre subcategories of an abelian category \(\mathsf{A}\). If every object of \(K\vee S\) admits a subobject in \(S\) with quotient in \(K\), or a subobject in \(K\) with quotient in \(S\), then \(K^{S}\) is a Serre subcategory.
Let \(M\) be an object of \(K\vee S\) and let \(M_{1}\subseteq M\) be a subobject as in the hypothesis. In the first case the projection \(M\twoheadrightarrow M/M_{1}\) has kernel \(M_{1}\in S\) and zero cokernel, so that \(\overline{M}\cong \overline{M/M_{1}}\) with \(M/M_{1}\in K\); in the second the inclusion \(M_{1}\to M\) has zero kernel and cokernel \(M/M_{1}\in S\), so that \(\overline{M}\cong \overline{M_{1}}\) with \(M_{1}\in K\). Either way \(M\in K^{S}\), so \(K\vee S\subseteq K^{S}\); the reverse containment is Proposition 8.9.
An abelian category \(\mathsf{A}\) satisfies \(\textup{(AS)}\) when for every Serre subcategory \(T\) of \(\mathsf{A}\) and every object \(X\): if every nonzero subobject of \(X\) has a nonzero subobject lying in \(T\), then \(X\) lies in \(T\).
A noetherian abelian category satisfying \(\textup{(AS)}\) is saturated.
Let \(K\) and \(S\) be Serre subcategories and let \(M\) lie in \(K\vee S\). By Proposition 8.19 it is enough to present \(M\) as an extension of an object of \(K\) by an object of \(S\).
Among the subobjects of \(M\) that lie in \(S\), choose one that is maximal; the zero subobject is available and \(M\) is noetherian. Call it \(M_{1}\) and put \(N\mathrel {:=}M/M_{1}\). Then \(N\) has no nonzero subobject lying in \(S\): the preimage in \(M\) of such a subobject is an extension of an object of \(S\) by \(M_{1}\), hence lies in \(S\), and it contains \(M_{1}\) strictly.
Every nonzero subobject \(W\) of \(N\) has a nonzero subobject lying in \(K\). Indeed \(W\), being a subobject of a quotient of \(M\), lies in \(K\vee S\); and \(K\vee S\) is the class of objects carrying a finite filtration whose successive subquotients lie in \(K\) or in \(S\), that class being closed under subobjects, quotients and extensions. The first nonzero step of such a filtration on \(W\) is a nonzero subobject of \(W\) lying in \(K\) or in \(S\), and the second case is excluded, since it would produce a nonzero subobject of \(N\) lying in \(S\).
By \(\textup{(AS)}\), applied with \(T=K\) to the object \(N\), we conclude that \(N\) lies in \(K\). The epimorphism \(M\twoheadrightarrow N\) has kernel \(M_{1}\) in \(S\), which is the presentation Proposition 8.19 asks for.
Condition \(\textup{(AS)}\) is the classical statement that the support of an object is the specialisation-closure of its associated points, written with neither supports nor associated points, and Kanda’s atom spectrum [ 13 , Definitions 2.1, 3.1, 3.4 and 3.7 ] is the dictionary. Call a nonzero object \(H\) monoform when no nonzero subobject of \(H\) embeds in a proper quotient of \(H\), and call two monoform objects atom-equivalent when they share a nonzero subobject; the atom spectrum \(\operatorname{ASpec}\mathsf{A}\) is the class of equivalence classes. Then \(\operatorname{ASupp}M\) is the class of atoms having a representative that is a subquotient of \(M\), and \(\operatorname{AAss}M\) the class of those having a representative that is a subobject of \(M\); and a subclass \(\Phi \) is open when every atom in it has a representative \(H\) with \(\operatorname{ASupp}H\subseteq \Phi \). For a noetherian and skeletally small \(\mathsf{A}\), the assignment \(T\mapsto \operatorname{ASupp}T\) is a bijection from the Serre subcategories onto the open subclasses, and every nonzero object has a monoform subobject [ 13 , Theorems 4.3 and 2.9 ] . Since every nonzero subobject of \(X\) then contains a monoform subobject, the hypothesis of Definition 8.20 says exactly that \(\operatorname{AAss}X\subseteq \operatorname{ASupp}T\) and its conclusion exactly that \(\operatorname{ASupp}X\subseteq \operatorname{ASupp}T\); so \(\textup{(AS)}\) says that \(\operatorname{ASupp}X\) is contained in every open subclass containing \(\operatorname{AAss}X\). For the finitely generated modules over a commutative Noetherian ring this returns \(\operatorname{ASpec}\mathsf{A}=\operatorname{Spec}R\), \(\operatorname{ASupp}=\operatorname{Supp}\), \(\operatorname{AAss}=\operatorname{Ass}\), and as open subclasses the specialisation-closed subsets.
Two traps make Definition 8.20 the better formulation. Kanda’s topology on \(\operatorname{ASpec}\mathsf{A}\) is the Hochster dual of the Zariski topology [ 13 , Remark 7.4 ] , so his “open” means “specialisation-closed” and his closure operator is generisation; and \(\operatorname{ASupp}H\) genuinely depends on the choice of monoform representative \(H\)—which is why openness has to quantify over representatives—so that the equality \(\operatorname{ASupp}X=\bigcup _{\alpha \in \operatorname{AAss}X}\overline{\{ \alpha \} }\) one first writes down is not even well posed. Nothing below uses this dictionary.
8.23 Modules and coherent sheaves
Let \(R\) be a commutative Noetherian ring. Then the category of finitely generated \(R\)-modules is a noetherian abelian category satisfying \(\textup{(AS)}\).
Only \(\textup{(AS)}\) is at issue. Let \(T\) be a Serre subcategory and let \(M\) be a module every nonzero submodule of which has a nonzero submodule in \(T\).
Let \(p\) be an associated prime of \(M\), so that \(R/p\) embeds in \(M\). By hypothesis \(R/p\) has a nonzero submodule \(J/p\) lying in \(T\); choose a nonzero \(x\) in \(J/p\). Multiplication by \(x\) is injective, \(R/p\) being a domain, so \(R/p\) is isomorphic to the submodule \(x\cdot (R/p)\) of \(J/p\) and therefore lies in \(T\).
Let now \(p\) be any prime in \(\operatorname{Supp}M\). Then \(M_{p}\neq 0\), so \(M_{p}\) has an associated prime, and pulling it back along \(R\to R_{p}\) gives an associated prime \(q\) of \(M\) with \(q\subseteq p\). By the previous paragraph \(R/q\) lies in \(T\); and \(R/p\) is a quotient of \(R/q\), so it lies in \(T\) as well.
Finally, \(M\) carries a filtration \(0=M_{0}\subset M_{1}\subset \dots \subset M_{n}=M\) whose successive quotients are of the form \(R/p_{i}\) for primes \(p_{i}\) [ 18 , Theorem 6.4 ] . Each \(R/p_{i}\) is a subquotient of \(M\), so that \(\operatorname{Supp}(R/p_{i})\subseteq \operatorname{Supp}M\) and in particular \(p_{i}\in \operatorname{Supp}M\). By the previous paragraph every \(R/p_{i}\) lies in \(T\), and \(T\) is closed under extensions, so \(M\) lies in \(T\).
Let \(X\) be a noetherian scheme, let \(\mathcal{F}\) be a coherent sheaf on \(X\) and let \(x\in \operatorname{Ass}\mathcal{F}\). Then \(\mathcal{F}\) has a coherent subsheaf \(\mathcal{G}\) with \(\operatorname{Ass}\mathcal{G}=\{ x\} \).
Write \(W=\overline{\{ x\} }\) and let \(\mathcal{G}_{0}\subseteq \mathcal{F}\) be the subsheaf of sections supported in \(W\), which is coherent because \(X\) is noetherian; it has \(x\) as an associated point and its support lies in \(W\). Let \(W'\) be the union of the closures of the associated points of \(\mathcal{G}_{0}\) other than \(x\). Each of these is a proper closed subset of \(W\), because \(x\) is the generic point of \(W\), so \(W'\) is a closed subset of \(X\) not containing \(x\). Let \(\mathcal{J}\) be the ideal sheaf of \(W'\) and let \(\mathcal{A}\subseteq \mathcal{G}_{0}\) be the subsheaf of sections supported in \(W'\); being coherent and supported in \(W'\), it is annihilated by a power of \(\mathcal{J}\). By the Artin–Rees lemma, applied on each member of a finite affine cover, \(\mathcal{J}^{n}\mathcal{G}_{0}\cap \mathcal{A}=0\) for \(n\) large; put \(\mathcal{G}=\mathcal{J}^{n}\mathcal{G}_{0}\) for such an \(n\). Then \(\mathcal{G}\) is nonzero, since \(\mathcal{J}_{x}=\mathcal{O}_{X,x}\), and it has no nonzero section supported in \(W'\). Its associated points are associated points of \(\mathcal{G}_{0}\), and one of them other than \(x\) would lie in \(W'\) and would give such a section.
For a noetherian scheme \(X\), the category \(\mathsf{Coh}(X)\) of coherent sheaves is a noetherian abelian category satisfying \(\textup{(AS)}\).
Only \(\textup{(AS)}\) is at issue. By Gabriel’s classification [ 9 , Proposition VI.2.4 ] the Serre subcategories of \(\mathsf{Coh}(X)\) are the \(T_{Z}=\{ \mathcal{F}\; :\; \operatorname{Supp}\mathcal{F}\subseteq Z\} \) for \(Z\subseteq X\) specialisation-closed. Let \(\mathcal{F}\) be coherent and suppose that every nonzero subsheaf of \(\mathcal{F}\) has a nonzero subsheaf lying in \(T_{Z}\).
Let \(x\in \operatorname{Ass}\mathcal{F}\) and take \(\mathcal{G}\subseteq \mathcal{F}\) as in Lemma 8.25. By hypothesis \(\mathcal{G}\) has a nonzero subsheaf \(\mathcal{H}\) with \(\operatorname{Supp}\mathcal{H}\subseteq Z\). Now \(\operatorname{Ass}\mathcal{H}\) is nonempty and contained in \(\operatorname{Ass}\mathcal{G}=\{ x\} \), so \(x\in \operatorname{Supp}\mathcal{H}\subseteq Z\). Hence \(\operatorname{Ass}\mathcal{F}\subseteq Z\); and since \(\operatorname{Supp}\mathcal{F}\) is the union of the closures of the points of \(\operatorname{Ass}\mathcal{F}\) [ 22 , Tag 05AL ] and \(Z\) is specialisation-closed, \(\operatorname{Supp}\mathcal{F}\subseteq Z\), which is to say that \(\mathcal{F}\) lies in \(T_{Z}\).
The 2-category \(\mathit{Sat}\) contains the category of finitely generated modules over every commutative Noetherian ring, and the category \(\mathsf{Coh}(X)\) of coherent sheaves on every noetherian scheme \(X\).
Condition \(\textup{(AS)}\) is exactly what the category of Proposition 8.12 violates. Take for \(T\) the Serre subcategory \(K\) of the \(S_{1}\)-modules and for \(X\) the module \(M=\Lambda e_{1}\). Every nonzero submodule of \(M\) contains the socle, which is \(S_{1}\) and lies in \(K\); but \(M\) does not lie in \(K\), having \(S_{2}\) among its composition factors. More generally, in a category of finite length the condition holds only for the semisimple ones, since it forces every simple subquotient of an object to be a subobject of it. That is the dividing line: the theory wants categories with room for specialisation, and finite length leaves none.
The class of saturated abelian categories is cut out by a property rather than exhibited by a construction, so Theorem 8.17 is an existence statement about a large 2-category rather than a small worked example; what Corollary 8.27 adds is that the 2-category is not one of the trivial ones. The classical inputs divide unevenly between the two families. The module case uses only the standard theory of associated primes and the filtration by primes. The geometric case uses Gabriel’s classification for \(\mathsf{Coh}(X)\), the description of the support of a coherent sheaf by its associated points, and the Artin–Rees lemma, which is the price of Lemma 8.25.
8.30 A locally ordered model: modular lattices
In a locally ordered 2-category the two-dimensional layer of the theory simplifies. An invertible 2-cell \(f\cong g\) consists of \(f\leq g\) and \(g\leq f\), so it is an identity; hence isomorphic 1-cells are equal, a factorisation up to an invertible 2-cell is a factorisation on the nose, and an equivalence is an isomorphism. The 2-cells themselves do not disappear: the hom-categories of \(\mathit{Sup}\) are far from discrete, and the two-dimensional universal properties of Definition 2.18 and Definition 2.19 quantify over them. Everything below therefore happens on the nose, and what remains two-dimensional is the order.
The one-element lattice is a strong bizero object of \(\mathit{Sup}\).
Write \(1\) for it. For every complete lattice \(L\) there is exactly one map \(L\to 1\), and a join-preserving map \(1\to L\) must send the empty join to the empty join, so the only one is the constant map at \(\bot \); both hom-posets are singleton categories, and \(1\) is a bizero object. A 1-cell \(L\to M\) is null precisely when it is the constant map at \(\bot \), so two parallel null 1-cells are equal and the unique 2-cell between them is their identity; thus \(1\) is strong.
The 2-category \(\mathit{Sup}\) is 2-z-exact. Explicitly, for a join-preserving map \(f\colon L\to M\), a 2-kernel of \(f\) is the inclusion of the down-segment \({\downarrow }a=\{ x\in L\mid x\leq a\} \) for \(a=\bigvee \{ x\in L\mid f(x)=\bot \} \), and a 2-cokernel of \(f\) is the map \(q_{s}\colon M\to {\uparrow }s=\{ y\in M\mid s\leq y\} \colon y\mapsto y\vee s\) for \(s=f(\top )\).
A down-segment is closed under all joins of \(L\), so it is a complete lattice and its inclusion \(k\) preserves them. An up-segment is a complete lattice whose nonempty joins are those of \(M\) and whose empty join is \(s\), and \(q_{s}\) preserves joins because \((\bigvee y_{i})\vee s=\bigvee (y_{i}\vee s)\).
Since \(f\) preserves joins, \(f(a)=\bigvee \{ f(x)\mid f(x)=\bot \} =\bot \), so \(f(x)=\bot \) for every \(x\leq a\), and \(f\circ k\) is the null 1-cell. For condition \((1)\) of Definition 2.18, let \(z\colon Z\to L\) carry an invertible 2-cell \(f\circ z\cong 0\), so that \(f\circ z=0\) by Remark 8.31. Then \(z(w)\leq a\) for every \(w\in Z\), and \(z\) corestricts to a join-preserving \(u\colon Z\to {\downarrow }a\) with \(k\circ u=z\). For condition \((2)\), a 2-cell \(k\circ u\Longrightarrow k\circ v\) says that \(u(w)\leq v(w)\) in \(L\) for every \(w\), which is the same statement in \({\downarrow }a\); whiskering with \(k\) is therefore a bijection on 2-cells, both sides having at most one element.
For the 2-cokernel, \(s=f(\top )=\bigvee _{x\in L}f(x)\) is the largest value of \(f\), so \(q_{s}\circ f\) is the constant map at \(s=\bot _{{\uparrow }s}\), the null 1-cell. Let \(z\colon M\to Z\) satisfy \(z\circ f=0\); then \(z(s)=z(f(\top ))=\bot \). The restriction \(u\colon {\uparrow }s\to Z\) of \(z\) preserves nonempty joins because \(z\) does, and the empty one because \(u(s)=\bot \); and \(u\circ q_{s}=z\) because \(z(y\vee s)=z(y)\vee z(s)=z(y)\). Condition \((2)\) holds because \(q_{s}\) is surjective: \(q_{s}(y)=y\) for \(y\in {\uparrow }s\), so a 2-cell \(u\circ q_{s}\Longrightarrow v\circ q_{s}\) gives \(u\leq v\) outright.
For every element \(a\) of a complete lattice \(L\), the inclusion \({\downarrow }a\to L\) is a 2-kernel of \(q_{a}\colon L\to {\uparrow }a\), and \(q_{a}\) is a 2-cokernel of that inclusion. Consequently, in \(\mathit{Sup}\) every normal 2-monomorphism is a down-segment inclusion precomposed with an isomorphism, every normal 2-epimorphism is a map \(q_{a}\) postcomposed with an isomorphism, and both classes are closed under composition.
The formulas of Proposition 8.33, applied to \(q_{a}\), return \(\bigvee \{ x\mid x\vee a=a\} =a\); applied to the inclusion, they return \(s=a\). A normal 2-monomorphism is a 2-kernel of some 1-cell, so by Proposition 8.33 and Corollary 2.22 it differs from a down-segment inclusion by an equivalence, which is an isomorphism by Remark 8.31; dually for normal 2-epimorphisms. For the closure, a down-segment of a down-segment is a down-segment, since \({\downarrow }b\) computed in \({\downarrow }a\) is \({\downarrow }b\) computed in \(L\) when \(b\leq a\); and for \(b\leq c\) the composite of \(q_{b}\colon L\to {\uparrow }b\) with \(q_{c}\colon {\uparrow }b\to {\uparrow }c\) is \(q_{c}\colon L\to {\uparrow }c\), since \((x\vee b)\vee c=x\vee c\). The general case follows by Corollary 3.8.
Up to composition with isomorphisms on either side, the antinormal morphisms of \(\mathit{Sup}\) are the maps
for elements \(a\), \(b\) of a complete lattice \(L\); and \(c_{a,b}\) is normal if and only if the transposition
is an isomorphism.
An antinormal morphism is a normal 2-monomorphism followed by a normal 2-epimorphism, so by Corollary 8.34 it is, up to isomorphisms on either side, of the form \(q_{b}\circ k\colon {\downarrow }a\to {\uparrow }b\) with \(k\) the inclusion, which is \(c_{a,b}\); and Corollary 3.8 makes normality insensitive to the isomorphisms.
Suppose that the transposition is an isomorphism \(\varphi \). The interval \([a\wedge b,a]\) is the up-segment \({\uparrow }(a\wedge b)\) of \({\downarrow }a\), and \([b,a\vee b]\) is the down-segment \({\downarrow }(a\vee b)\) of \({\uparrow }b\). Writing \(q\colon {\downarrow }a\to [a\wedge b,a]\) for the 2-cokernel \(q_{a\wedge b}\) and \(n\colon [b,a\vee b]\to {\uparrow }b\) for the 2-kernel inclusion, both provided by Corollary 8.34, we find \(n(\varphi (q(x)))=(x\vee (a\wedge b))\vee b=x\vee b=c_{a,b}(x)\), so that \(c_{a,b}=n\circ (\varphi \circ q)\) is a normal 2-epimorphism followed by a normal 2-monomorphism—Corollary 3.8 again—and hence normal.
Suppose conversely that \(c_{a,b}\) is normal, say \(c_{a,b}=m\circ e\) with \(e\) a normal 2-epimorphism and \(m\) a normal 2-monomorphism, the factorisation being on the nose by Remark 8.31. The 2-kernel of \(c_{a,b}\) is \({\downarrow }(a\wedge b)\), since for \(x\leq a\) we have \(x\vee b=b\) if and only if \(x\leq b\); by Proposition 3.9 it is also the 2-kernel of \(e\). By Corollary 8.34 we may write \(e=\psi \circ q_{t}\) with \(q_{t}\colon {\downarrow }a\to [t,a]\) and \(\psi \) an isomorphism; the 2-kernel of \(e\) is \({\downarrow }t\), so \(t=a\wedge b\). Replacing \(m\) by \(m\circ \psi \), a normal 2-monomorphism by Corollary 3.8, we may assume that \(c_{a,b}=m\circ q_{a\wedge b}\) with \(m\colon [a\wedge b,a]\to {\uparrow }b\) a normal 2-monomorphism; and by Corollary 8.34 again, \(m=n\circ \chi \) with \(\chi \colon [a\wedge b,a]\to [b,d]\) an isomorphism onto a down-segment \([b,d]\) of \({\uparrow }b\). Evaluating at \(y\in [a\wedge b,a]\) gives \(\chi (y)=c_{a,b}(y)=y\vee b\), so \(d=\chi (a)=a\vee b\), and \(\chi \) is the transposition.
For a complete lattice \(L\) the following conditions are equivalent:
\(L\) is modular;
for all \(a\), \(b\in L\), the antinormal morphism \(c_{a,b}\) is normal.
\((i)\Rightarrow (ii)\) In a modular lattice, the transposition \(y\mapsto y\vee b\colon [a\wedge b,a]\to [b,a\vee b]\) and the map \(z\mapsto z\wedge a\) in the other direction are mutually inverse: for \(y\in [a\wedge b,a]\) the modular law gives \((y\vee b)\wedge a=y\vee (b\wedge a)=y\), and for \(z\in [b,a\vee b]\) it gives \((z\wedge a)\vee b=z\wedge (a\vee b)=z\). Both maps are monotone, so the transposition is an order isomorphism, and an order isomorphism of complete lattices preserves all joins. This is Dedekind’s transposition principle [ 2 , Chapter I, Theorem 13 ] , and Proposition 8.35 concludes.
\((ii)\Rightarrow (i)\) If \(L\) is not modular then it contains a pentagon [ 2 , Chapter I, Theorem 12 ] : elements \(u{\lt}w\) and \(v\) with \(u\vee v=w\vee v\) and \(u\wedge v=w\wedge v\). Both \(u\) and \(w\) lie in the interval \([w\wedge v,w]\), and the transposition to \([v,w\vee v]\) carries both to \(w\vee v\), so it is not injective, and \(c_{w,v}\) is not normal by Proposition 8.35.
The pentagon \(N_{5}\) itself is a finite, hence complete, lattice, so \(\mathit{Sup}\) contains an object with a non-normal antinormal morphism, and condition (DI2) fails.
The locally ordered 2-category \(\mathit{Sup}_{\mathrm{mod}}\) of complete modular lattices and join-preserving maps is 2-di-exact.
Down-segments, up-segments and intervals of a lattice are sublattices, and modularity is inherited by sublattices. So the 2-kernels and 2-cokernels of Proposition 8.33, computed in \(\mathit{Sup}\) for a 1-cell of \(\mathit{Sup}_{\mathrm{mod}}\), have their vertices in \(\mathit{Sup}_{\mathrm{mod}}\); and since \(\mathit{Sup}_{\mathrm{mod}}\) is a full sub-2-category, a 2-kernel or 2-cokernel computed in \(\mathit{Sup}\) whose vertex lies in \(\mathit{Sup}_{\mathrm{mod}}\) is one computed in \(\mathit{Sup}_{\mathrm{mod}}\). This is condition (DI1), and the same remark transfers Corollary 8.34 and Proposition 8.35. For condition (DI2), an antinormal morphism of \(\mathit{Sup}_{\mathrm{mod}}\) is, up to isomorphisms, of the form \(c_{a,b}\) for elements \(a\), \(b\) of a modular lattice \(L\), hence normal by Proposition 8.36; and the middle vertex \([a\wedge b,a]\) of the factorisation constructed in Proposition 8.35 is an interval of \(L\), so the factorisation lies in \(\mathit{Sup}_{\mathrm{mod}}\).
The 2-category \(\mathit{Sup}\) is homologically self-dual, and its normal 2-monomorphisms and its normal 2-epimorphisms are closed under composition; but it does not satisfy condition \(\textup{(DPN)}\) of Definition 9.2.
Closure under composition is part of Corollary 8.34. For self-duality, let \((m,e)\) be an antinormal decomposition of the zero map in \(\mathit{Sup}\). Up to isomorphisms, which change nothing by Corollary 3.8 and Corollary 2.22, \(m\) is the inclusion \({\downarrow }a\to L\) and \(e=q_{b}\); and that \(e\circ m=c_{a,b}\) is null says that \(x\vee b=b\) for all \(x\leq a\), that is, \(a\leq b\). The dinverse pair is \((2\text{-}\mathrm{ker}(e),2\text{-}\mathrm{coker}(m))=({\downarrow }b\to L,q_{a})\), so the dinversion is \(c_{b,a}\colon {\downarrow }b\to {\uparrow }a\), whose transposition is \([b\wedge a,b]=[a,b]\to [a,b\vee a]=[a,b]\colon y\mapsto y\vee a\), the identity of \([a,b]\); by Proposition 8.35 the dinversion is normal, which is Definition 4.17.
For \(\textup{(DPN)}\), take the pentagon \(N_{5}=\{ \bot ,x,z,y,\top \} \) with \(\bot {\lt}x{\lt}z{\lt}\top \), the element \(y\) incomparable to \(x\) and \(z\), and \(x\vee y=z\vee y=\top \), \(x\wedge y=z\wedge y=\bot \). The antinormal pair \(({\downarrow }z\to N_{5},q_{y})\) has composite \(c_{z,y}\), whose transposition \([\bot ,z]\to [y,\top ]\) goes from a three-element chain to a two-element one, so it is no isomorphism and \(c_{z,y}\) is not normal. The dinversion is \(c_{y,z}\), whose transposition \([\bot ,y]\to [z,\top ]\) is an isomorphism of two-element chains, so \(c_{y,z}\) is normal, and the biconditional of Definition 9.2 fails.
Proposition 8.38 gives \(\mathit{Sup}\) the exact profile of \(\mathit{AbCat}\) established in Section 9.21—homologically self-dual, both composition properties, not \(\textup{(DPN)}\)—so each separation recorded in Corollary 9.26 has a five-element witness. The resonance with dimension one is no accident. In [ 1 , Example 4.2.1 ] the failure of \(\textup{(DPN)}\) in the category of commutative monoids is exhibited on a pentagonal semilattice, and [ 1 , Theorem 4.3.4 ] shows that in a di-exact category the lattice of normal subobjects of every object is modular. Proposition 8.36 closes the circle: read on the 2-category of complete lattices themselves, condition (DI2) is not merely implied by modularity but equal to it. The level in between is inhabited as well: Section 9.28 exhibits, in the Hilbert lattices, complete lattices that are not modular and yet support condition \(\textup{(DPN)}\).
Theorem 6.9, read in \(\mathit{Sup}_{\mathrm{mod}}\), is a Snake Lemma for modular lattices, and it is worth spelling out what it says. By Corollary 8.34, a 1-cell \(g\colon L\to M\) is normal precisely when, writing \(n\) for the largest element of \(L\) with \(g(n)=\bot \) and \(m=g(\top )\), the map \(g\) kills \({\downarrow }n\) and restricts to an isomorphism \([n,\top ]\to {\downarrow }m\). The rows of the ladder are, up to isomorphism, the sequences \({\downarrow }u\to L\to {\uparrow }u\) and \({\downarrow }v\to M\to {\uparrow }v\) of Corollary 8.34, for elements \(u\in L\) and \(v\in M\), and the squares commute on the nose by Remark 8.31, so the outer verticals are the restriction \(f\colon {\downarrow }u\to {\downarrow }v\) of \(g\), which forces \(g(u)\leq v\), and the induced map \(h\colon {\uparrow }u\to {\uparrow }v\colon y\mapsto g(y)\vee v\). Their normality comes for free: each factors as a normal 2-epimorphism, followed by a composite of one transposition with a restriction of the isomorphism above, followed by a normal 2-monomorphism, and the transpositions are invertible because \(L\) and \(M\) are modular. The input of the theorem thus reduces to a normal 1-cell \(g\) together with a pair of elements \(u\), \(v\) such that \(g(u)\leq v\). Writing \(s\) for the largest element of \(L\) with \(g(s)\leq v\), the snake sequence becomes
whose 1-cells are, in order, the inclusion, then \(x\mapsto x\vee u\), then the connecting 1-cell \(\partial \)—which may be taken to be \(g\) itself, restricted to \([u,s]\)—then \(y\mapsto y\vee m\), then \(z\mapsto z\vee v\). Every 1-cell of the sequence is normal, and 2-exactness at each of the four middle positions amounts to a single identity between an image element and a kernel element. At \(2\text{-}\mathrm{Ker}(g)\) and \(2\text{-}\mathrm{Cok}(g)\) the identity holds by construction; the two with content, at \(2\text{-}\mathrm{Ker}(h)\) and \(2\text{-}\mathrm{Cok}(f)\), read
and both are consequences of the isomorphism \([n,\top ]\to {\downarrow }m\).
The normal morphisms of \(\mathit{Sup}\) are, by Corollary 8.34, exactly the maps that project onto an up-segment and then embed it as a down-segment of the codomain. These are the exact Galois connections of Grandis’s lattice-theoretic homological algebra [ 10 , 11 ] —characterised in [ 11 , 1.2.5 ] by precisely this factorisation—and between modular lattices they are his modular connections [ 11 , 1.2.8 ] . The settings differ in two respects. Grandis’s lattices need not be complete, whereas on complete lattices a monotone map preserves all joins if and only if it is the covariant part of a Galois connection, so that his connections between complete lattices are exactly the 1-cells of \(\mathit{Sup}\); and his category \(\mathsf{Mlc}\) of modular lattices and modular connections keeps only the normal morphisms, where \(\mathit{Sup}_{\mathrm{mod}}\) keeps every join-preserving map and normality singles the connections out. His kernels, cokernels and short exact sequences agree with those of Proposition 8.33 and Corollary 8.34 [ 11 , 1.2.3 ] , and \(\mathsf{Mlc}\) is a p-exact category [ 11 , 1.2.8 ] , so the one-dimensional content of Remark 8.40—the six-term exact sequence of modular connections—is available in his framework: the connecting morphism exists in any of his homological categories [ 11 , Lemma 3.3.3 ] , and the six-term sequence is exact under modularity conditions on the diagram [ 11 , Proposition 3.3.4 ] which hold automatically for modular connections. Part (d) of the latter is the one-dimensional counterpart of the reduction in Remark 8.40: exactness of the middle vertical alone already gives exactness at the two central objects. What the two-dimensional reading adds is the derivation. The connections are not chosen but forced, as the normal 1-cells among all join-preserving maps, by universal properties whose 2-cells are the order; and modularity, which for Grandis delimits \(\mathsf{Mlc}\) by hypothesis, enters here as an instance of condition (DI2), by Proposition 8.36. The model is populated by the lattices on which classical homological algebra acts: submodule lattices of modules and, more generally, normal-subobject lattices in any semi-abelian category are complete modular lattices.
The two models of Section 8 meet over the same commutative algebra. Call a subset of \(\operatorname{Spec}R\) specialisation-closed when it contains, with every prime, all primes containing it. Such subsets are closed in the power set under arbitrary unions and arbitrary intersections, so they form a complete lattice \(\mathrm{Sc}(R)\) whose joins are unions; both operations being computed in the power set, that lattice is distributive, hence modular, and \(\mathrm{Sc}(R)\) is an object of \(\mathit{Sup}_{\mathrm{mod}}\). Its 1-cells come from geometry. A ring homomorphism \(\varphi \colon R\to S\) makes \(\operatorname{Spec}\varphi \colon \operatorname{Spec}S\to \operatorname{Spec}R\) continuous, and a continuous map preserves specialisation, so that \((\operatorname{Spec}\varphi )^{-1}\) carries specialisation-closed subsets to specialisation-closed subsets and preserves arbitrary unions: it is a 1-cell \(\mathrm{Sc}(R)\to \mathrm{Sc}(S)\) of \(\mathit{Sup}_{\mathrm{mod}}\), and a 2-cell between two such is an inclusion of subsets.
These are the lattices of Section 8.23, read one dimension down. By the classification invoked in the proof of Proposition 8.26, the Serre subcategories of the category of finitely generated \(R\)-modules are the \(T_{Z}=\{ M\; :\; \operatorname{Supp}M\subseteq Z\} \) for \(Z\subseteq \operatorname{Spec}R\) specialisation-closed [ 9 , Proposition VI.2.4 ] , so \(\mathrm{Sc}(R)\) is the lattice of Serre subcategories of that object of \(\mathit{Sat}\), which by Proposition 8.4 is its lattice of 2-kernels. The same noetherian spectrum is therefore read by Corollary 8.27 as an object of \(\mathit{Sat}\) and by the present subsection as an object of \(\mathit{Sup}_{\mathrm{mod}}\): the two models do not merely coexist, they overlap, and no reference beyond those already used in Section 8.23 is needed to see it.
The family is honest but undemanding. Distributivity gives modularity for nothing, so these lattices satisfy (DI2) for a reason without content, and by Remark 8.40 the Snake Lemma reads over them as two identities between specialisation-closed subsets. What makes Proposition 8.36 say something are the modular lattices that are not distributive, of which the submodule lattices named at the end of Remark 8.41 are the standard supply; the spectra show instead that the objects Section 8.23 produces are not foreign to the present one.
In the locally discrete abelian reading, the hypothesis of Theorem 6.9 that the three verticals be normal is invisible: in an abelian category every morphism is normal. In \(\mathit{Sup}_{\mathrm{mod}}\) it has bite, and it cannot be dropped. Let \(L=\{ \bot {\lt}x{\lt}\top \} \) and \(M=\{ \bot {\lt}\top \} \) be chains—distributive, so certainly modular—and let \(g\colon L\to M\) send \(\bot \) to \(\bot \) and the other two elements to \(\top \): a join-preserving map with trivial 2-kernel which is not a 2-monomorphism, the freedom that abelian categories lack, and which normality of \(g\) would remove. Take the rows determined by \(u=x\) and \(v=\top \). Both squares commute on the nose, and the outer verticals are normal: \(f\colon {\downarrow }x\to {\downarrow }\top =M\) is an isomorphism of two-element chains, and \(h\colon {\uparrow }x\to {\uparrow }\top \) is null, hence normal by Proposition 4.13. Yet no 1-cell \(\partial \) makes the snake sequence 2-exact at \(2\text{-}\mathrm{Ker}(h)\): the lattice \(2\text{-}\mathrm{Cok}(f)\) is trivial, so \(\partial \) is null and its 2-kernel is all of \(2\text{-}\mathrm{Ker}(h)=[x,\top ]\), while \(\overline{b}\colon 2\text{-}\mathrm{Ker}(g)={\downarrow }\bot \to [x,\top ]\) is null, with trivial 2-image. Indeed the first exactness identity displayed in Remark 8.40 fails: here \(s=\top \), so its left-hand side is \(\top \) while \(u\vee n=x\). In the form \(\bigvee \{ x'\in L\mid g(x')\leq g(x)\} =x\vee n\), valid for normal \(g\) and violated here at \(x\), this is exactly the identity which Grandis isolates as the lattice-theoretic substitute for the subtraction step of the classical diagram chase [ 11 , 2.3.5 ] .
The two models are complementary. In \(\mathit{Sup}_{\mathrm{mod}}\) every invertible 2-cell is an identity, so the coherence layer of the theory is invisible there, while the order-theoretic 2-cells over which the two-dimensional universal properties quantify are everywhere; in \(\mathit{Sat}\) it is the invertible 2-cells that do the work. Neither model is locally discrete, and each trivialises exactly what the other exercises.