The Zariski spectrum of the category of finitely presented modules

Mike Prest Department of Mathematics University of Manchester Manchester M13 9PL England

October 23, 1998

The Ziegler spectrum [Zie] of a ring R is a topological space, the points of which are the isomorphism classes of indecomposable pure-injective right Rmodules. It is well-known that the points of this space may be identified with the isomorphism classes of indecomposable injective objects in the functor category (R−mod, Ab).

In [Pre5] the underlying set of the the Ziegler spectrum was retopologised with the ”dual Ziegler” topology by taking, for a basis of open sets, the complements of the compact open sets in the Ziegler topology. It was then pointed out how, with the above functorial interpretation of the points of the space, one may identify this new topology as the ”Gabriel-Zariski spectrum” of the locally coherent category (R−mod, Ab).

We begin by extending this to more general categories. We define the Ziegler and Zariski spectra for arbitrary locally finitely presented additive categories, we describe the particular form these topologies take when restricted to the indecomposable injectives in a locally coherent category and we show that the Zariski spectrum of a locally finitely presented abelian category is homeomorphic to the Gabriel-Zariski spectrum of an associated locally coherent category.

Then, in the second part of the paper, we present some elementary facts about these topologies and their inter-relationship. We also describe both topologies explicitly for the case of Mod-R with R a Dedekind domain (for the Ziegler topology this is already in [Zie]) and we say a little about the case where R is an artin algebra.

Finally we examine the topologies induced on the set of indecomposable injective modules over a ring. We obtain a rather complete picture in the case that R is a commutative coherent ring: we describe the (irreducible) closed subsets and determine when the space is noetherian. In contrast we are able to say very little in the case that R is right noetherian.

There is a sheaf of rings defined over the ”dual Ziegler” topology, but we explore this elsewhere [Pre7], [Pre8].

At the beginning of the second part of the paper we summarise what has been done, in terms of modules, for the reader who wishes to skip the rather general development in the first part.

1 Part 1

1.1 Locally finitely presented additive categories

Throughout the paper we deal with additive categories (and all functors are to be additive even though we do not say this again). How much could be done in more general categories we do not know (but see [MaPa], [AdRo]).

This first part is devoted to describing the Ziegler and Zariski spectra of certain categories. The Ziegler spectrum of a locally coherent category has been treated extensively elsewhere ([Her2], [Kra2]). Nevertheless, we believe that there is some value in our turning over this ground again since our approach is somewhat different, being somewhat more model-theoretic.

We begin by describing some results from [C-B3] and discussing their relevance to the spectra that we will define.

An additive category C is locally finitely presented (lfp) if it has direct limits, its full subcategory fp − C of finitely presented objects is skeletally small and every object of C is a direct limit of finitely presented objects (an object C of C is finitely presented (fp ) if the functor (C, −) commutes with direct limits). The category C is locally coherent (lc) if it is lfp abelian (hence [C-B3, 2.4] Grothendieck) and every finitely generated subobject of a finitely presented object is finitely presented (an object D is finitely generated (fg) if there is an epimorphism C ⟶ D with C finitely presented).

If C is lfp (all categories will be additive) then a sequence L ⟶ M ⟶ N in C with composition 0 is pure-exact if for every fp C in C the induced sequence 0 ⟶ (C, L) ⟶ (C, M) ⟶ (C, N) ⟶ 0 is exact. The object L of C is pure- injective if every pure-exact sequence starting with L is split.

In [C-B3] a construction (one of two) is given which embeds C as a full subcategory of an lfp abelian category and which has the property that a sequence in C is pure-exact iff its image in the larger category is exact.

Define, for C lfp, DC = Flat(fp − (fp − C, Ab), Ab) - the category of flat functors from fp − (fp − C, Ab) to Ab. We suppose now that C also has products: then the following is proved in [C-B3, 3.3].

∗ DC is locally coherent (in particular is Grothendieck abelian and so has injective hulls).

∗ There is a canonical functor D : C ⟶ DC which is full and faithful, preserves direct limits and products and takes fp − C into fp − DC.

∗ A sequence 0 ⟶ L ⟶ M ⟶ N ⟶ 0 in C is pure-exact iff its image in DC is exact.

∗ The absolutely pure objects of DC are the exact functors from fp − (fp − C, Ab) to Ab and also are precisely the functors in the image of D (an object M is absolutely pure = fp − injective if Ext^{1}(D, M) = 0 for every finitely presented object D).

∗ The functor D restricts to an identification of the pure-injective objects of C with the injective objects of DC. Thus the category DC which, in the case C = Mod−R, is equivalent [C-B3, 3.3] to (R−mod, Ab) (note the change in side) retains many of the properties of this special case.

We may begin to define the Ziegler spectrum ZgC of the category C (lfp with products) by taking for its points the (iso classes of) indecomposable pure-injective objects of C (noting that these correspond to the (iso classes of) indecomposable injectives in the locally coherent category DC). The Ziegler topology for Mod-R was originally defined in terms of pp formulas, equivalently in terms of the pp-definable subgroups (=subgroups of finite definition) that such formulas define (these subgroups also have more ”algebraic” definitions). If we wish to make sense of pp formulas in this generality we must set up an appropriate language for C: this will be done below. Before doing that we observe that there is already a notion of pp- definable subgroup in [C-B3]. We recall this and will show that there is no conflict between these routes to the notion of ”pp-definable subgroup”. C continues to be an lfp category with products.

Let α : G ⟶ H be a morphism in fp − C. Then for any M in C we have the induced map (α, M) : (H, M) ⟶ (G, M) given by composition with α. We write Mα (Crawley-Boevey writes M_{α}) for the image of (α, M) and refer to this as a subgroup of finite definition of M of sort G (the reference to sorts is our addition). For instance, with C = Mod−R and α being the map from R to R which is left multiplication by r (for some r ∈ R) then Mα = Mr.

Define an order on such functors (M ↦ Mα) by setting α ≤ β, where α : G ⟶ H and β : G ⟶ K are in fp − C, if for every M in C we have Mα ≤ Mβ (it is enough to check for M in fp − C since every object of C is a limit of objects of fp − C and since G,H, K are finitely presented). Taking M = K we obtain the following (which is contained in [C-B3, 3.4]).

Lemma 1.1 Given α : G ⟶ H and β : G ⟶ K in fp − C, we have α ≤ β iff there is ν : H ⟶ K with β = να.

(In the case C = Mod−R, G = R^{n} this is a basic property of ”free realisations” [Pre4, 8.12].)

We will show that these subgroups of finite definition coincide with the pp- definable subgroups in the natural language for C, which language we now describe (for C lfp abelian this is in [Pre2]).

1.2 The language of a locally finitely presented category

Let C be lfp. Given any generating set Γ for C contained in fp − C we define a corresponding (multi-sorted) language.

For each object G in Γ we introduce a sort and we say, for M in C, that the elements of M of sort G are the elements of (G, M). We equip each sort with a symbol for addition and a symbol for 0. Any given variable ranges over the elements of a single sort and we use a subscript to denote that sort: thus v_{G}, when interpreted in M, is a variable for elements of (G, M).

For each morphism α : G ⟶ H in (the full subcategory on) Γ we introduce a function symbol which, when interpreted, will be the map, which we have seen just above, ”composition with α” from sort H to sort G. We will use the same symbol for the function symbol as for the function (as usual).

Denote the resulting language by L^{Γ}.

Let Ax(Γ) be a set of axioms expressing the positive atomic diagram of Γ, including the specification that all functions are additive (in the case C = Mod−R, Γ = {R} the set Ax(Γ) is the usual set of axioms for the theory of R-modules). For example if α : G ⟶ H and β : H ⟶ K are in Γ and if γ = βα then we put the axiom ∀v_{K}(v_{K}γ = v_{K}β.α) into Ax(Γ). Then it is easy to see that the category of models for Ax(Γ) (where the morphisms are the L^{Γ}-structure-preserving maps) is the category of right Γ-modules - that is, the category of functors (Γ^{op}, Ab). We regard the category C as a subcategory of (Γ^{op}, Ab) via the full faithful embedding which takes C to (−, C) : Γ^{op} ⟶ Ab and thus we regard the objects of C as structures for this language (we will see later that if C is abelian then it is an elementary class of L^{Γ}-structures).

We have a multi-sorted first order language and so all the usual definitions and theorems of model theory apply; we just occasionally have to take care because of there being possibly infinitely many sorts (most model theory texts deal with just the single-sorted case but one can find these details worked out in some places, e.g. [Fis]). In particular we define pp formulas in the usual way (as formulas in the closure of the set of equations under conjunction and existential quantification) and so obtain a notion of ”ppdefinable subgroup” of an object of C. Before we show that this coincides with the subgroups of finite definition in [C-B3] we consider the relationship between the various languages that we obtain as we vary the choice of Γ.

There is a canonical choice of Γ, namely (a skeletal version of) fp − C (except, of course when we consider R-modules, where we take Γ = {R} and so obtain the usual 1-sorted language since (R, M) ≃ M canonically). It is clear that every L^{Γ}- formula is also an L^{fp−C}-formula. The converse - that every L^{fp−C}-formula is equivalent to an L^{Γ}-formula - was shown to be true in the case that C is abelian in [Pre2]. In fact, the argument there works provided C has pullbacks and Γ satisfies the condition: given any object C in fp − C there are β : K ⟶ G and α : G ⟶ C with G,K finite direct sums of objects in Γ and with α epi and α a pseudocokernel of β. We say that α is a pseudocokernel of β (α,β as above) if αβ = 0 and if for every γ : G ⟶ H with γβ = 0 there is δ : C ⟶ H with δα = γ (see [C-B3]).

For the rest of this discussion we will work with Γ = fp − C with the above as some justification. We will call the corresponding language the canonical language of C.

Note that the subgroups of finite definition in the sense of [C-B3] are exactly those which are definable by a pp formula of the form ”α | v_{G}” that is, of the form ”∃v_{H}(v_{G} = v_{H}α)” where α : G ⟶ H is in fp − C.

Proposition 1.2 Suppose that C is lfp with products. Every pp formula in the language of C is equivalent to one of the form α | v.

Proof First we show that every quantifier-free formula v_{G}β = 0 is equivalent to one of the given form (since fp − C is closed under finite direct sums we may always suppose that a formula has at most a single free variable). The key fact is the following: an lfp category C has products iff fp − C has pseudocokernels [C-B3, 2.1].

Suppose that β : K ⟶ G is in fp − C and let α : G ⟶ H be a pseudocokernel of β in fp − C. Then v_{G}β = 0 is equivalent to α | v_{G}. For if α | v_{G} (read v_{G} now as a typical element of sort (G, −)), say v_{G} = v_{H}α, then v_{G}β = v_{H}αβ = 0 and, conversely, if v_{G}β = 0 then, by definition of pseudocokernel, we have that v_{G} factors through α, as required.

Now take a general pp formula with free variable v_{G}: ∃v_{G′}(v_{G}β +v_{G′}β′ = 0). This is equivalent to the formula ∃v_{G′}(v_{G}, v_{G′})(β, β′)^{T} = 0 where (v_{G}, v_{G′}) is regarded as a variable of type G⊕G′ and where (β, β′)^{T} is the obvious map to G ⊕ G′. By the above (v_{G}−, v_{G′})(β, β′)^{T} = 0 is equivalent to a formula of the form α_{1} | (v_{G}, v_{G′}) for some morphism α_{1} = (α, α′) : G ⊕ G′ ⟶ H in fp − C. So our original formula is equivalent to ∃v_{G′}∃v_{H}(v_{H}α_{1} = (v_{G}, v_{G′})) that is, to ∃v_{H}∃v_{G′}(v_{H}α = v_{G} ∧ v_{H}α′ = v_{G′}) which, in turn, is equivalent to ∃v_{H}(v_{H}α = v_{G}), as required.

It is easily seen that if C has kernels then every divisibility formula α|v is equivalent to an annihilation formula (one of the form vβ = 0). Thus one obtains the result one might have expected - elimination of quantifiers in the canonical language - in the case of an lfp abelian category.

In summary, if C is lfp with products then the indecomposable pureinjectives in C correspond bijectively to the indecomposable injectives of a locally coherent category DC. The isomorphism classes of these objects are the points of the Ziegler spectrum ZgC of C. We have given two ways of defining the subgroups of finite definition of objects of C and we have seen these agree. So we could try to mimick the usual definition of the Ziegler topology by taking as basic open sets the (ϕ/ψ) = {N ∈ ZgC : ϕ(N) > ψ(N)}. It is not clear, however, that in this generality we do get a basis for a topology: given N ∈ (ϕ/ψ) ∩ (ϕ′/ψ′) is there (ϕ′′/ψ′′) with N ∈ (ϕ′′/ψ′′) ⊆ (ϕ/ψ) ∩ (ϕ′/ψ′) ?

We will resolve this problem by regarding C as embedded in DC, considering DC as an elementary localisation of the functor category ((fp − DC)^{op}, Ab) and then inducing the relative topology on ZgC by considering it as a (closed) subset of the Ziegler spectrum of ((fp−DC)^{op}, Ab), the point being that we can show that the above basis property holds when we have a functor category.

1.3 Functor categories

Throughout this section let C = Mod−Γ = (Γ^{op}, Ab) be the category of ”right Γ-modules” where Γ is any small preadditive category. The canonical embedding Γ ⟶ Mod−Γ given by sending G to (−, G) identifies Γ with a full generating subcategory of finitely generated projectives and from now on we will identify Γ with this subcategory.

Since C = Mod−Γ is lfp abelian the language of C is as described before and, as we saw, the language which is based on Γ ⊆ fp − C as generating set is inter-interpretable with that based on fp − C.

Notice that for Γ = {R} with R a ring and so C = Mod−R the interinterpretability amounts to the fact that if C is in mod-R then ”there is a morphism from C to (arbitrary) M such that ...” is expressible in the usual language of R-modules, plus the fact that morphisms in mod-R lift to morphisms between finitely generated free modules.

Now suppose that ϕ is any pp formula (in whichever language we have chosen to use). Then the assignment to M of ϕ(M) defines a functor from mod-Γ to Ab. This functor is determined, among those functors which commute with direct limits, by its restriction F_{ϕ} to mod − Γ (see [Pre4, 12.3] for the proof for R-modules which applies equally well here). Indeed the proof [Bur] that for C = Mod−R we have an equivalence between fp − (mod−Γ, Ab) and (Mod−Γ)^{eq+} also works here and so, in particular, we may identify pp formulas in (Mod−Γ)^{eq+} with the finitely presented functors from mod−Γ to Ab. (For the model-theoretic eq+ construction see [KP]: what is there done for Mod-R may as well be done for Mod-Γ.)

We also have, just as with R-modules, a duality δ : fp − (mod−Γ, Ab) ≃ (fp−(Γ−mod, Ab))^{op} and hence (see [C-B3, 3.3]) the identification of D(Mod−Γ) with (Γ−mod, Ab) where the map D : Mod−Γ ⟶ D(Mod−Γ) takes M to M ⊗_{Γ} −. Furthermore [C-B3, 3.4] (δF, M ⊗ −) ≃ F (M) for any F in fp − (mod−Γ, Ab) (where F (M) should be read as F̃(M), F̃ being that extension of F to Mod-Γ which commutes with direct limits). Let us write D_{Γ} for D(Mod−Γ).

Denote by Zg_{Γ} the right Ziegler spectrum Zg(Mod−Γ) of Γ (we would use _{Γ}Zg to denote the left Ziegler spectrum Zg(Γ−Mod) of Γ). From the discussion above we have the following alternative functorial description of Zg_{Γ}, where we use inj(C) to denote the set of isomorphism classes of indecomposable injective objects of C.

The points of Zg_{Γ} are the points of inj(D_{Γ}).

The basic open sets are of the form (F) = {E ∈ inj(D_{Γ}) : (F, E) ≠ 0} where F is any object in fp − (D_{Γ}). We now prove that this does give a basis for a topology.

Theorem 1.3 ([Zie, 4.9] for the case Γ = {R}) Let Γ be a small preadditive category and suppose that N ∈ (ϕ/ψ)∩(ϕ′/ψ′) in Zg_{Γ}. Then there is a basic open set (ϕ′′/ψ′′) with N ∈ (ϕ′′/ψ′′) ⊆ (ϕ/ψ) ∩ (ϕ′/ψ′).

Proof (following [Bur, 4.1.4]) Let F and F′ be finitely presented functors in (mod−Γ, Ab) such that (F) = (ϕ/ψ) and (F′) = (ϕ′/ψ′) under the identification D : N ↦ N ⊗− of Zg_{Γ} with inj(D_{Γ}). By hypothesis, (F, dN) ≠ 0 and (F′, dN) ≠ 0, say f : F ⟶ dN and f′ : F′ ⟶ dN are non-zero morphisms in D_{Γ}. Since dN is an indecomposable injective, imf ∩ imf′ ≠ 0 and so this intersection contains a non-zero finitely generated subfunctor, Z say. Choose subfunctors H of F and H′ of F′ which map under f, respectively f′, onto Z.

Since D_{Γ} is a functor category there is a finitely generated projective functor P and an epimorphism g : P ⟶ Z. By projectivity this lifts to h : P ⟶ H and h′ : P ⟶ H′ - so fh = g = f′h′. The images of h and h′ are finitely generated and hence, since D_{Γ} is locally coherent, are finitely presented. So K = kerh and K′ = kerh′ are finitely generated subfunctors of P and so, therefore, is their sum K + K′. Thus g factorises through the finitely presented object F′′ = P/(K + K′).

By construction dN ∈ (F′′). We claim that (F′′) ⊆ (F) ∩ (F′). So suppose that dN′ ∈ (F′′), say f′′ : F′′ ⟶ dN′ is non-zero. Composing with the canonical epimorphism imh ⟶ F′′ we get a non-zero morphism from a subfunctor of F to dN′. By injectivity of dN′ this extends to a morphism from F to dN′, as required.

Theorem 1.4 ([Zie, 4.9] for the case Γ = {R}) Let Γ be a small preadditive category. Then every basic open set (ϕ/ψ) of Zg_{Γ} is compact.

Proof Ziegler’s argument uses the compactness theorem of model theory and works just as well in this context, as follows. Suppose that (ϕ/ψ) = ⋃_{λ}(ϕ_{λ}/ψ_{λ}). Consider the set of sentences {∃v(ϕ(v)∧¬ψ(v)}∪{∀v_{λ}(ϕ_{λ}(v_{λ}) → ψ_{λ}(v_{λ}))}_{λ} (note that the various formulas may have (sequences of) variables belonging to different sorts). If there is no finite subcover of the above cover of (ϕ/ψ) then this set is consistent and hence has a model M. Then, just as in modules (see [Zie] (4.5) or [Pre4, 4.35] ) any maximal pp-type in the theory of M containing ϕ(v) and omitting ψ(v) has as its hull an indecomposable pure-injective object in the difference (ϕ/ψ) \ ⋃_{λ}(ϕ_{λ}/ψ_{λ}) - contradiction.

Example 1.5 Zg_{Γ} need not be compact - notice that, in contrast to Rmodules, the whole space need not be basic open (if there are infinitely many sorts). Let Q be a quiver with (countably) infinitely many vertices and no arrows and let C be the category of k- representations of Q where k is any field - so C = Mod−Γ where Γ is the category which has one object G(i) for each element of Q and (G(i), G(j)) = k if i = j and = 0 if i ≠ j.

For each i, we have the open set (v_{i} = v_{i}/v_{i} = 0) where v_{i} denotes a variable of sort G(i). Clearly N ∈ (v_{i} = v_{i}/v_{i} = 0) iff (G(i), N) ≠ 0. Let e_{i} denote the identity of G(i). Then any object M of C decomposes in C as ⨁_{i} Me_{i} and so ZgC is the disjoint union of the infinitely many non-empty open sets (v_{i} = v_{i}/v_{i} = 0) and hence is not compact.

1.4 Elementary localisation

Now let C be any lfp abelian category, with generating set Γ of finitely presented objects. Then [Pre3] there is a hereditary torsion theory τ on Mod-Γ which is of finite type such that C is equivalent to the localisation (Mod−Γ)_{τ} of Mod-Γ at τ. Furthermore [Pre3] C forms a definable subclass of Mod-Γ. That is, membership of C may be axiomatised in the language for Mod-Γ based on Γ. But that language may be identified with the language L^{Γ} of C since Γ, as a full subcategory of Mod-Γ is isomorphic to Γ as a full subcategory of C. So it follows that the topology on ZgC may be defined in C or equally well in Mod-Γ and then relativised to C. So we see that our tentative basis for the Ziegler topology does, for C lfp abelian, indeed define a basis for a topology: given the sets (ϕ/ψ) and (ϕ′/ψ′) of ZgC and N in their intersection we read these as basic open subsets of Zg_{Γ} and a point of Zg_{Γ} so that, by 1.3 above, there are pp formulas ψ′′ ≤ ϕ′′ in the language of Mod-Γ based on Γ with N ∈ (ϕ′′/ψ′′) ⊆ (ϕ/ψ) ∩ (ϕ′/ψ′); now intersect with ZgC to obtain the desired conclusion.

We are now in the position to treat any lfp abelian category C in an entirely analogous way to a module category over a ring: we have a first order language for the model theory of objects in C (the basics of model theory in this context are developed in [Pre2]); we have the Ziegler spectrum defined as for modules (and just have to remember that the whole space may not be compact); we have the interpretation of the Ziegler spectrum as a topology on the indecomposable injectives of a related functor category (and hence the functorial approach to the model theory of modules can be taken here as well). This means that we can generalise ”the model theory of modules” to ”the model theory of objects in lfp abelian categories” by making rather minor modifications to proofs. To actually perform this generalisation would be a rather large, though routine, task (a small part was done in [Pre2] where the results of [Pre1] were generalised to just this context) but we believe that by the above development we have made it easy to check the generalisation of any particular group of results.

1.5 Definability of Abs(C) when C is locally coherent

A result originally proved by Eklof and Sabbagh for module categories [ES] is the following.

Theorem 1.6 [Pre2] Let C be lfp abelian. Then the class Abs(C) of absolutely pure objects of C is elementary (that is, axiomatisable in the language of C) iff C is locally coherent.

We note the following short proof of one direction in the case that C is a functor category (then the general case is immediate from 1.4 above). (As a general, point the proofs in [Pre2] are close generalisations of the 1-sorted proofs and do not take full advantage of the possibilities made available by working in the categorical context.) Recall that M in C is absolutely pure iff Ext^{1}(C, M) = 0 for all C in fp − C. Given C in fp − C take any presentation 0 ⟶ A ⟶ B ⟶ C ⟶ 0 in fp − C (using that fp − C is abelian) where we may assume if C is a functor category that B is projective. Then we obtain the exact sequence 0 ⟶ (C, M) ⟶ (B, M) ⟶ (A, M) ⟶ Ext^{1}(C, M) ⟶ Ext^{1}(B, M) = 0. So to say that M is absolutely pure is just to say that (B, M) ⟶ (A, M) is epi (over all C in fp − C). But that is just the sentence ∀v_{A}∃v_{B}(v_{A} = v_{B}f) where f : A ⟶ B is the morphism above.

In particular, if C is locally coherent then the set inj(C) of isomorphism classes of indecomposable injectives of C is a closed subset of the Ziegler spectrum (one may check that closed subsets of the Ziegler spectrum correspond to definable classes of objects just as in the case of module categories [Zie, 4.10], [Pre4, 4.67]).

The following result (elimination of quantifiers) also has a proof which generalises that for Mod-R (generalise [ES, 3.2] or follow [Pre4, 15.40] and use that C is locally coherent).

Theorem 1.7 Let C be locally coherent, let Γ be any generating set of finitely presented objects and let ϕ(\overline{v}) be a pp formula with free variables \overline{v} in the language L^{Γ} for C. Then there is a morphism f in ⨁ Γ (the closure of Γ under finite direct sums) such that, in every absolutely pure object M of C the formula ϕ(\overline{v}) defines the same subgroup as does the (quantifierfree) formula (\overline{v})f = 0 (if Γ ≠ ⨁ Γ then read the latter is a convenient abbreviation for a formula in L^{Γ} which has the form ⋀ Σv_{i}f_{ij} = 0).

1.6 The dual-Ziegler topology

Just as for modules [Pre5] (also cf. [Pre4, Section 4.7 Example 2] ) we define the dual-Ziegler or Zariski topology ZarC (Zar_{Γ} for C = Mod−Γ) to have the same underlying set as ZgC but to have, as a basis of open sets the complements [ϕ/ψ] = (ϕ/ψ)^{c} of the compact Ziegler-open sets.

Recall that, even for C = Mod−R, this space is not in general compact: although we are using Hochster’s definition [Ho] which was made for spectral spaces neither the Ziegler nor dual-Ziegler spectrum is spectral! See [Pre5] and Part 2 below for some discussion of the extent to which these satisfy the requirements for being a spectral space.

1.7 The Gabriel-Zariski spectrum of a locally coherent category

Now we recall how the definition of the Zariski spectrum of a commutative noetherian ring may be cast, following [Gab], [Roo], into a form which makes sense in much more general contexts. All this is ”folklore”.

1.7a The ”classical” case Suppose that R is a commutative ring. The (Zariski) spectrum, spec(R), of R is the set of all prime ideals of R, equipped with the topology which has, as a basis of open sets, those of the form D(R) = {P ∈ spec(R) : r ∉ P } for r ∈ R.

We set V (R) = D(R)^{c} = {P ∈ spec(R) : r ∈ P } and the typical closed set has the form V (I) = {P ∈ spec(R) : I ≤ P } where I is any ideal of R.

Recall also that a closed set of a topological space is said to be irreducible if it cannot be expressed as the union of two proper closed subsets. A generic point of a closed set C is a point whose closure is C.

Every irreducible closed set of spec(R) has a generic point. Such a generic point is unique since spec(R) is a T_{0} space (that is, contains no topologically indistinguishable points).

1.7b spec(R)) defined module-theoretically

Now suppose that R is commutative and noetherian.

To the prime ideal P we associate the injective hull E(R/P) of the factor module R/P: since R is noetherian this is a bijection between the prime ideals of R and the isomorphism classes of indecomposable injective R-modules and so allows us to identify spec(R) with inj_{R} (=inj(Mod−R)).

With this identification V (R) = {E(R/Q) : Hom_{R}(R/rR, E(R/Q)) ≠ 0}. For certainly, if r ∈ Q then we have the canonical surjection from R/rR to R/Q and hence a non-zero map to E(R/Q). Conversely, if we have a nonzero map f from R/rR to E(R/Q), let M be the annihilator in E(R/Q) of Q and let s ∈ R be such that f(s) ∈ M (M is essential in E(R/Q)). Since sr ∈ rR and since the annihilator in R of f(s) is easily seen to be Q, we must have r ∈ Q, as required.

We now claim that if we replace the modules R/rR by arbitrary finitely presented (=finitely generated) R-modules then we do not enrich the topology on inj_{R}. We must show that the sets of the form V (M) = {E(R/Q) : Hom_{R}(M, E(R/Q)) ≠ 0} for M a finitely presented R-module are already closed. We show this first if M is cyclic (and finitely presented).

Suppose then, that M = R/Σ_{i}a_{i}R for some a_{1}, ..., a_{n} ∈ R. Certainly each D(a_{i}) is contained in D(M) = V (M)^{c} and, conversely, if E(R/Q) is not contained in the union of the D(a_{i}) then as argued above, each a_{i} is in Q and hence Σ_{i}a_{i}R ≤ Q so, as before, E(R/Q) is contained in V (M). Hence D(M) = D(a_{1}) ∪ ... ∪ D(a_{n}), as required.

Now suppose that M is finitely presented: in particular, M is n-generated for some n. Just since R is coherent, there is a sequence M = M_{n} > ... > M_{1} > M_{0} = 0 of finitely generated submodules of M such that all the factor modules C_{i} = M_{i}/M_{i−1} are cyclic and finitely presented. Then V (M) = V (C_{1}) ∪ ... ∪ V (C_{n}). For, by injectivity, any non-zero morphism from C_{i} to E(R/Q) extends to a morphism from M/M_{i−1} to E(R/Q) and hence induces a non-zero morphism from M to E(R/Q). Conversely, given a non-zero morphism f from M to E(R/Q), let i be minimal such that the restriction of f to M_{i} is non-zero: then f induces a non-zero morphism from C_{i} to E(R/Q), as desired.

We have thus obtained the following reformulated definition of the Zariski spectrum of a commutative noetherian ring.

The points of the spectrum are the isomorphism classes of indecomposable injective modules and the topology has, for a basis of open sets, those of the form D(F) = [F] = {E : Hom(F, E) = 0} where F is a finitely presented module.

This definition makes sense in any additive category (and will be a reasonable definition in any locally coherent category).

1.7c The general case

Let C be a locally coherent category. Following the above we denote by GZ(C) the Gabriel-Zariski spectrum of C: the set of isomorphism classes of indecomposable injective objects of C equipped with the topology which has, as a basis of open sets, the [F] = {E ∈ inj(C) : Hom(F, E) = 0} where F is in fp − C. Notice that [F] ∩ [F′] = [F ⊕ F′] is open.

We have already seen in Section 1.3 that if C is lfp abelian then ZgC is homeomorphic to inj(DC) equipped with the relative topology inherited from Zg(DC) (and hence the same for the dual-Ziegler topologies ZarC and Zar(DC)).

Theorem 1.8 ([Pre5] for C = Mod−R) Let C be lfp abelian. Then the identification of ZarC(=ZgC as a set) and inj(DC) already described is a homeomorphism ZarC ≃ GZ(DC).

Proof The basic open sets D(F) of GZ(DC) are open sets in ZarC (just by definition of the latter). Conversely we know that ZarC can be regarded as the dual Ziegler topology on Zg(DC) relativised to inj(DC). By 1.7 a typical basic Ziegler-open set of the locally coherent category DC when intersected with inj(DC) is equal to the intersection with a set of the form (vf = 0/vg = 0) where f : K ⟶ G, g : L ⟶ G are in fp − DC (so v = v_{G}) with vg = 0 ≤ vf = 0 and hence (1.1) such that there is a morphism k : K ⟶ L with f = gk. Let f′ : G ⟶ C and g′ : G ⟶ D be the cokernels of f and g respectively and let k′ : C ⟶ D be the morphism induced by k. Let h : H ⟶ C be the kernel of k′. Note that all these are in fp − DC. Then we claim that (v_{G}f = 0/v_{G}g = 0) = (v_{H} = v_{H}/v_{H} = 0) (that is [vf = 0/vg = 0] = [H] which is what we want).

For suppose that N ∈ (vf = 0/vg = 0): that is, there is a morphism a : G ⟶ N such that af = 0 but ag ≠ 0. Then there is a morphism b from C to N which does not factor through k′, that is, such that bh ≠ 0. Thus (take v_{H} = bh) N ∈ (H). For the converse, if there is a non-zero morphism c from H to N then, by injectivity of N, there is a morphism from C to N which does not factor through D and so there is a morphism from G to N which annihilates f but does not annihilate g, as claimed.

2 Part 2

We now give various topological properties of the Ziegler and Zariski spectra. We do this for module categories but we trust that anyone who requires these results in more general contexts will encounter no difficulty in lifting the arguments, using the ideas and results above.

When dealing with R-modules, one normally uses the language based on {R} as generating set Γ for Mod-R, rather than that based on mod-R. We begin by summarising some of what we have already discussed as applied to Mod-R.

2.1 The Ziegler and Dual Ziegler Spectra

2.1a The Ziegler spectrum

Let R be any ring. By Zg_{R} we denote the right Ziegler spectrum of the ring R: the set of (isomorphism classes of) indecomposable pureinjective (=algebraically compact) right R-modules, endowed with the topology which has as a basis of open sets the (ϕ/ψ) = {N ∈ Zg_{R} : |ϕ(N)/ψ(N)| > 1} where ϕ > ψ are pp formulas. Ziegler showed [Zie, 4.9] that these sets are compact. In this definition one may use just the pp formulas in one free variable but, if one allows pp formulas in any finite number of free variables then the topology is unchanged and one obtains all the compact open sets (since (ϕ(\overline{v})/ψ(\overline{v})) ∪ (ϕ′(\overline{v})/ψ′(\overline{v})) = ((ϕ(\overline{v}) ∧ ϕ′(\overline{v}′))/(ψ(\overline{v}) ∧ ψ′(\overline{v}′)))).

An alternative way of describing the basic open sets is as the supports of those functors in (Mod−R, Ab) which commute with direct limits and whose restrictions to (mod−R, Ab) are finitely presented.

Yet again, this space is homeomorphic (via N ↦ N ⊗_{R} −) to the set inj(D_{R}), where D_{R} = (R−mod, Ab), equipped with the topology which has as a basis of open sets the (F) = {E ∈ inj(D_{R}) : (F, E) ≠ 0} as F ranges over the finitely presented functors in D_{R} - we use the notation (F) rather than D(F) above.

The Ziegler spectrum is not in general T_{0}. One may make it so by identifying topologically indistinguishable points

Lemma 2.1 Points N and N′ belong to exactly the same open sets of the Ziegler spectrum iff they are elementarily equivalent.

Proof The direction ”⇐” is immediate from the definition of the topology. For ”⇒”, just from the definition of the topology, the infinite powers of N and N′ are elementarily equivalent since they have the same (Baur-Garavaglia-Monk) invariants (see [Pre4, Section 2.4] e.g.). So it is enough to show that each invariant of N is infinite if greater than 1. If we have ϕ(N)/ψ(N) finite but greater than 1 then clearly N has a minimal pair (modulo its own theory) so by [Zie, 7.10] any indecomposable pure-injective elementarily equivalent to N must be isomorphic to N, as required.

We therefore denote by Zg_{R}/ ≡ the Ziegler spectrum with topologically equivalent points identified and endowed with the quotient topology. There are rings (other than simple artinian rings) for which Zg_{R}/ ≡ is a single point (simple regular rings [Tyu] and some other examples [PRZ]) but for many rings this collapsing is quite mild (for instance, over an artin algebra each finite-length point is equivalent to no other point [Pre4, 13.1]).

The condition in the definition [Ho] of spectral space which is far from being satisfied is that the collection of compact open sets should be closed under finite intersection.

Example 2.2 Consider the Ziegler spectrum Zg_{ℤ} over the ring ℤ of integers (this already is a T_{0} space). The basic open set (2v = 0/v = 0) consists of all indecomposable pure-injectives which have an element of order 2, that is it consists of the prüfer group ℤ_{⊭^{∞}} together with all the groups ℤ_{⊭^{⋉}}. Dually, the basic open set (v = v/2|v) consists of the 2-adic integers \overline{ℤ_{(⊭)}} together with all the ℤ_{⊭^{⋉}}. Thus the intersection of these sets consists of all the groups ℤ_{⊭^{⋉}}. This set is, however, not compact since each point ℤ_{⊭^{⋉}} is isolated in Zg_{ℤ}: specifically ℤ_{⊭^{⋉}} is the only indecomposable pure-injective that lies in the open set (2^{n−1}|v ∧ 2v = 0/2^{n}|v ∧ 2v = 0).

2.1b The dual Ziegler spectrum

The dual Ziegler, or Zariski, topology, Zar_{R}, has the same underlying set as Zg_{R} but has, for a basis of open neighbourhoods, the sets [ϕ/ψ] = (ϕ/ψ)^{c}.

The Ziegler topology (and hence the Zariski topology) is, in general, nonspectral and it is not clear whether one can reverse the duality to recover the Ziegler space from the topological structure of the Zariski space. In any case, we do have a duality, if not a purely topological one, between these spaces.

We have seen that the identification of the underlying set of Zg_{R} with inj(D_{R}) is a homeomorphism of Zg_{R} with the topology induced on inj(D_{R}) from Zg(D(R)) (Section 1.3) and we also saw that Zar_{R} and GZ(D_{R}) are, under this identification of points, homeomorphic (1.8).

2.2 Open and closed sets

2.2a Ziegler topology A typical Ziegler-open set has the form ⋃{(ϕ_{i}/ψ_{i}) : i ∈ I} and so a typical Ziegler-closed set has the form ⋂{[ϕ_{i}/ψ_{i}] : i ∈ I}. Every Ziegler-closed subset is compact (for module categories! - cf. 1.5 above). A clopen set, being closed, must be compact and hence must be a basic open set. Over an artin algebra any finite set of indecomposable finitely presented modules (and so the complement of such a set) is clopen. Any decomposition of a ring as a product gives a disconnection of the space.

The model-theoretic importance of this topology is well-expressed by the correspondence [Zie, 4.10] between Ziegler-closed sets and theories of modules. The support of a module M is the subset supp(M) = {N ∈ Zg_{R} : N is a direct summand of a module elementarily equivalent to M} of Zg_{R}. It is a closed subset which depends only on the theory of M, it is non-empty if M is non-zero and every closed subset has this form. The support of a point of Zg_{R} coincides with its Ziegler- closure. Conversely, to any closed set C we associate the set of all modules with support contained in C: this is an elementary class - with theory T_{C}, say, - whose class of models is closed under pure submodules and direct products (a ”definable subcategory” in Crawley-Boevey’s terminology [C-B4]). Moreover there are modules M such that supp(M) = C and for such M the complete theory of M^{ℵ_{0}} depends only on C and is denoted T_{C}^{∗}.

2.2b Zariski topology A typical Zariski-open set has the form ⋃_{i}[ϕ_{i}/ψ_{i}] and so a typical Zariski-closed set has the form ⋂_{i}(ϕ_{i}/ψ_{i}).

2.3 Closures

Lemma 2.3 Let N, N′ be indecomposable pure-injectives. Then N′ ∈ Zar − cl(N) iff N ∈ Zg − cl(N′).

Proof N′ ∈ Zar − cl(N) iff every (basic) Zariski-open set [ϕ/ψ] containing N′ also contains N, which is so iff every basic Ziegler-open set (ϕ/ψ) containing N also contains N′.

2.3a Ziegler-closure Let us say that two points of a topological space are unseparated if every open neighbourhood of the one intersects every open neighbourhood of the other.

Proposition 2.4 N ∈ Zg − cl(ϕ/ψ) iff there is N′ ∈ (ϕ/ψ) such that N and N′ are unseparated.

Proof The condition is sufficient since it implies that every open neighbourhood of N intersects the open neighbourhood (ϕ/ψ) of N′. For the converse, if there is no such point N′ then, for each N′ ∈ (ϕ/ψ) choose (ϕ_{N′}/ψ_{N′}) ⊆ (ϕ/ψ) such that N′ ∈ (ϕ_{N′}/ψ_{N′}) and such that there is a neighbourhood (σ_{N′}/τ_{N′}) of N such that (ϕ_{N′}/ψ_{N} ∩ (σ_{N′}/τ_{N′}) = ∅. By compactness of (ϕ/ψ) there are N_{1}, ...N_{k} ∈ (ϕ/ψ) such that (ϕ/ψ) = (ϕ_{N_{1}}/ψ_{N_{1}})∪...∪ (ϕ_{N_{k}}/ψ_{N_{k}}). But then the open neighbourhood (σ_{N_{1}}/τ_{N_{1}}) ∩ ... ∩ (σ_{N_{k}}/τ_{N_{k}}) of N has empty intersection with (ϕ/ψ) and so N ∉ Zg − cl(ϕ/ψ).

Example 2.5 R = ℤ: Zg − cl(2x = 0/x = 0) = (2x = 0/x = 0) ∪ \{\overline{ℤ_{(⊭)}}\} ∪ {ℚ}. The module ℚ is in the closure since every open neighbourhood of ℚ contains ℤ_{⊭^{∞}} ∈ (2x = 0/x = 0) and \overline{ℤ_{(⊭)}} is in the closure because it is unseparated from ℤ_{⊭^{∞}} (although neither is in the closure of the other).

Corollary 2.6 The Ziegler-interior of a basic Ziegler-closed set [ϕ/ψ] consists of all points which are separated from each point in the complement (ϕ/ψ): [ϕ/ψ]^{o} = {N ∈ [ϕ/ψ] : ∀N′ ∈ (ϕ/ψ) ∃ open U, U′ with U ∩ U′ = ∅ and N ∈ U, N′ ∈ U′}.

Proof This is immediate from the above since [ϕ/ψ]^{o} = (Zg − cl(ϕ/ψ))^{c}.

Observation 2.7 Suppose that Zg − cl{N} (= supp(N)) is Zariski-open. Then it has the form [ϕ/ψ] for some ϕ,ψ.

Proof By assumption Zg − cl{N} has the form ⋃_{λ}[ϕ_{λ}/ψ_{λ}]. Say N ∈ [ϕ_{λ_{0}}/ψ_{λ_{0}}] - a Ziegler-closed set which must therefore equal Zg − cl{N}.

In particular an irreducible Ziegler-closed subset over a countable ring is Zariski-open iff it has the form [ϕ/ψ] (by Herzog’s result 2.13 below).

2.3b Zariski-closure

Proposition 2.8 Suppose that X ⊆ Zg_{R}. If N ∈ Zar − cl(X) then there is N′ ∈ Zg − cl(X) such that N ∈ Zar − cl(N′).

Proof Let F be the filter of basic Zariski-open neighbourhoods of N and, for each [ϕ/ψ] ∈ F, choose N_{ϕ/ψ} ∈ X ∩ [ϕ/ψ]. Consider F as a downwards-directed (by inclusion) set and let U be any ultrafilter on F extending the filter base of final sets (those of the form {[ϕ′/ψ′] : [ϕ′/ψ′] ⊆ [ϕ/ψ]} for some [ϕ/ψ] ∈ F). Let M be the ultraproduct ∏_{F} N_{ϕ/ψ}/U. By Los’ theorem and choice of filter base, for each [ϕ/ψ] ∈ F we have M ⊧ |ϕ/ψ| = 1 so supp(M) ⊆ supp(N). Also, since each N_{ϕ/ψ} lies in X we have supp(M) ⊆ Zg − cl(X) (essentially by [Zie, 4.10]). Take N′ ∈ supp(M): then N′ ∈ Zg − cl(N) and so N ∈ Zar − cl(N′) , as required.

Example 2.9 The result above does not characterise Zariski-closure: to see this, take R = ℤ, X = \{ℤ_{⊭^{∞}}\} and N = \overline{ℤ_{(⊭)}}. So N ∉ Zar − cl(X) but ℚ ∈ Zg − cl(𝕏) and N ∈ Zar − cl(ℚ).

Corollary 2.10 Suppose that X is Ziegler-closed. Then Zar − cl(X) = ⋃{Zar − cl(N′) : N′ ∈ X}.

2.4 Closed points

2.4a Ziegler-closed points

Proposition 2.11 If an indecomposable pure-injective N is of finite endolength then it is a closed point of the Ziegler spectrum. If R is countable then the converse is true.

Proof The first part is a result of Garavaglia [Gar]. Briefly, if N is of finite endolength then the theory of its infinite power has no non-isolated irreducible types and hence the Ziegler closure of {N} is simply {N} - or use Ziegler’s structure theory [Zie] (see [Pre4, Section 10.4]).

Suppose, conversely, that R is countable and that N is Ziegler- closed. Then, by [Zie, 8.3], the theory of the infinite power of N has m- dimension and then by the proof of [Pre4, 10.17] for instance, the fact that {N} is closed implies that N must be of finite endolength.

Open question Are Ziegler-closed points of finite endolength, without any assumption on the ring?

2.4b Zariski-closed points

Observation 2.12 An indecomposable pure-injective module N is Zariskiclosed iff N ∈ Zg − clN′ implies N′ ≃ N. The image of N in Zar_{R}/ ≡ is closed iff N ∈ Zg − clN′ implies N′ ≡ N. (Note that we may identify Zar_{R}/ ≡ with the dual space of Zg_{R}/ ≡.)

Proof Immediate from the relation between Zariski- and Zieglerclosure of points and the characterisation (2.1) of topological equivalence as elementary equivalence.

2.5 Irreducible closed sets

Note that the closure of any point is irreducible.

2.5a Ziegler topology

Theorem 2.13 [Her1, 4.7] If Zg_{R} has a countable basis of open sets (in particular, if the ring R is countable) then every irreducible Ziegler-closed set has a generic point (which is unique up to elementary equivalence).

Open question Is the above true for arbitrary rings? That is, is every irreducible Ziegler-closed set the closure of a point?

Corollary 2.14 If R is countable and if [ϕ/ψ] is an irreducible Zieglerclosed set then [ϕ/ψ] is compact as a Zariski- open set.

Proof By Herzog’s result there is N ∈ Zg_{R} with [ϕ/ψ] = Zg − cl(N). Given a Zariski-open cover [ϕ/ψ] = ⋃_{λ}[ϕ_{λ}/ψ_{λ}] there is λ with N ∈ [ϕ_{λ}/ψ_{λ}] and so [ϕ_{λ}/ψ_{λ}] = [ϕ/ψ].

2.5b Zariski topology

Proposition 2.15 Every irreducible Zariski-closed set of the form (ϕ/ψ) has a Zariski-generic point.

Proof Suppose, for a contradiction, that (ϕ/ψ) does not have a Zariski-generic point. Then for each N ∈ (ϕ/ψ) there is N′ ∈ (ϕ/ψ) such that N′ ∉ Zar − clN, that is, such that N ∉ Zg − clN′ and so there is a Ziegler-open neighbourhood (ϕ_{N}/ψ_{N}) of N which is strictly contained in (ϕ/ψ). Thus we obtain a representation of (ϕ/ψ) as a union of proper Ziegler-open subsets and so, since (ϕ/ψ) is Ziegler-compact, there is a finite subcover. But this gives a covering of (ϕ/ψ) by finitely many Zariski-closed sets so, by Zariski-irreducibility of (ϕ/ψ), there is a single one of these, (ϕ_{N}/ψ_{N}) say, which is equal to (ϕ/ψ) - contradiction, as required.

Question Does every irreducible finite (resp. arbitrary) intersection of basic Zariski-closed sets have a generic point?

Observation 2.16 relating to the above question: Suppose that V is an irreducible Zariski-closed set. Let F = {[ϕ/ψ] : [ϕ/ψ] ∩ V ≠ ∅}. Then F has the finite intersection property since V is irreducible. So, by compactness of the Ziegler spectrum, ⋂ F ≠ ∅. Then V ⊆ Zar − clN for any N ∈ ⋂ F: indeed ⋂ F = ⋂{Zg − cl(N′) : N′ ∈ V }. The question, then, is whether ⋂ F contains a point of V. Certainly it may strictly contain V: take R = ℤ, V = {ℤ_{⊭^{∞}}} and then ⋂ F = {ℤ_{⊭^{∞}}, ℚ}.

2.6 Generic points

We will say that a point in a space is weakly generic if it is not in the closure of any other point. For an example, if a countable ring is ”indiscrete” in the sense of [PRZ] and not artinian (for example if it is simple von Neumann regular non-artinian) then the Ziegler spectrum is uncountable but carries the indiscrete topology, hence Zg_{R}/ ≡ is a singleton but no point of Zg_{R} is weakly generic.

Proposition 2.17 (a) A point N is weakly generic in Zar_{R}/ ≡ iff Zg − clN carries the indiscrete topology.

(b) A point is weakly Zariski-generic iff it is Ziegler-closed

(c) Any point of finite endolength is weakly Zariski-generic and, if R is countable, these are exactly the weakly Zariski-generic points.

Proof (a) N is weakly generic in Zar_{R}/ ≡ iff N ∈ Zar − clN′ implies N ≡ N′ iff N′ ∈ Zg − clN implies N ≡ N′ and hence implies Zg − clN′ = Zg − clN.

(b) N is weakly Zariski-generic iff N ∈ Zar − clN′ implies N ≃ N′ iff N′ ∈ Zg − clN implies N ≃ N′ iff Zg − clN = {N}.

(c) By (b) and 2.11.

Let R be an artin algebra. Crawley-Boevey in [C-B1] has called a point generic if it is of finite endolength but not of finite length: we will follow Krause in extending this terminology to arbitrary rings, by taking the exclusion clause to be ”but not finitely presented”. If R is a countable artin algebra then by the above and [Pre4, 13.4] we may characterise these points as those which are weakly Zariski-generic and which are not isolated. For general rings this is not a characterisation: if R is any Dedekind domain with only finitely many primes then the fraction field of R is an isolated point of Zar_{R} (see 2.8b below).

Question For which rings is it true that if the Ziegler spectrum is not discrete (i.e. if the ring is not of finite representation type) then there is a weakly Zariski- generic point which is not finitely presented? If this were true of artinian rings then one would have a positive answer to the conjecture that pure-semisimple rings are of finite representation type.

2.7 Isolated points

Recall that a point N is isolated if {N} is a (necessarily basic) open set.

2.7a Ziegler-isolation N is Ziegler-isolated iff there is a pair ϕ > ψ of pp formulas such that N is the unique indecomposable pure-injective N′ with |ϕ(N′)/ψ(N′)| > 1. We can also express this condition functorially: there exists a finitely presented functor F ∈ (R−mod, Ab) such that N ⊗ − is the unique indecomposable injective solution E to (F, E) ≠ 0.

It is an open question whether or not every Ziegler-isolated point must have a minimal pair (that is, belong to ϕ′/ψ′) with ϕ′ > ψ′ such that there is no θ with ϕ′ > θ > ψ′). It is equivalent to ask whether N being Zieglerisolated entails N ⊗ − being the injective hull of a finitely presented simple functor. If the ring is countable or if there are no continuous pure-injectives then the answer is positive:

(a) [Zie, 8.3] if R is countable and if C is a closed subset of the Ziegler spectrum and if N is an isolated point of C (that is, {N} = C ∩ (ϕ/ψ) for some ϕ/ψ) then N has a C-minimal pair (ϕ/ψ is C-minimal iff the localisation of the functor F_{Dψ/Dϕ} at the torsion theory on D_{R} = (R−mod, Ab) corresponding to C is simple);

(b) [Zie, 8.11] If C is a closed subset of the Ziegler spectrum and if T_{C} has m-dimension (and hence C has Cantor-Bendixson rank) or, more generally, if there is no continuous pure-injective module with support contained in C, and if N is an isolated point of C then N has a C-minimal pair.

2.7b Zariski-isolation

Proposition 2.18 If a point N is isolated in Zar_{R}/ ≡ then the theory of N^{ℵ_{0}} is finitely axiomatisable, given the theory of R-modules, modulo being infinite.

Proof If N is isolated in Zar_{R}/ ≡ then there is a basic Zariskiopen neighbourhood [ϕ/ψ] containing only points which are elementarily equivalent to N. In that case, the theory of N^{ℵ_{0}} is axiomatised modulo being infinite by |ϕ/ψ| = 1.

The converse is false.

Example 2.19 Take R to be the localisation of ℤ at the prime p. Then the theory of R^{ℵ_{0}} is finitely axiomatised (modulo the theory of R-modules!), modulo being infinite, by |px = 0/x = 0| = 1 but \overline{R} is not Zariski-isolated since it is not Ziegler-closed (ℚ is in its Ziegler-closure).

2.8 The Ziegler and Zariski spectra of a Dedekind domain

Let R be a Dedekind domain (we mean commutative) which is not a field. Then the indecomposable pure-injective R-modules are those of finite length R/P^{n}, the prüfer modules R_{P^{∞}}, the P -adic modules \overline{R_{(P)}} where P is a nonzero prime, and the fraction field K of R. Since R is noetherian and so its ideals are finitely generated, one may write down pp formulas which express conditions such as P^{n}|x (i.e. x lies in P^{n}M) and P^{n}x = 0. Below, ”P ” will always be used for a non-zero prime.

2.8a Neighbourhood bases in the Ziegler topology [Zie, 9.5] R/P^{n} : {R/P^{n}} = ((P^{n−1}|x ∧ P x = 0)/(P^{n}|x ∧ P x = 0))

”There is a non-zero element annihilated by P (this excludes all but R_{P^{∞}} : and the various R/P^{m}) which is divisible by P^{n−1} but not by P^{n}.”

\overline{R_{(P)}} : \{\overline{R_{(P)}}\} ∪ \{R/P^{m} : m ≥ n + 1\} = (P^{n}|x/P^{n+1}|x) for n ≥ 0 ”There is an element divisible by P^{n} but not by P^{n+1}.” (P acts surjectively on all but R_{P} and the various R/P^{m} and (R/P^{k})P^{n} = 0 for k ≤ n)

R_{P^{∞}}: {R_{P^{∞}}} ∪ {R/P^{m} : m ≥ n + 1} = (P^{n+1}x = 0/P^{n}x = 0) for n ≥ 0 (dual to \overline{R_{(P)}}) ”There is an element annihilated by P^{n+1} but not by P^{n}.” (P acts injectively on all but R_{P^{∞}} and the various R/P^{m} and (R/P^{k})P^{n} = 0 for k ≤ n)

K : \{K\} ∪ \{\overline{R_{(P)}} : P prime \} ∪ \{R_{P^{∞}} : P prime \} ∪ \{R/P^{n} : n ≥ 0, P ≠ P_{1}, ..., P_{k}} ∪ {R/Q^{n} : n ≥ m, Q = P_{1}, ..., P_{k}} = (J|x/x = 0) for P_{1}, ..., P_{k} primes and m ≥ 1, where J = (P_{1}...P_{k})^{m−1} (this open set is also defined by the dual (x = x/Jx = 0)) ”There is a non-zero element divisible by J.” equivalently, ”Not every element is annihilated by J.” (This excludes only the finitely many finite points R/P_{i}^{t} with i = 1, ..., k and t < m.)

Note that: the finite-dimensional points are both open and closed; the Ziegler- closed points are exactly those of finite endolength, namely the various R/P^{n} and K.

2.8b Neighbourhood bases in the Zariski topology

R/P^{n} : {R/P^{n}} = [P^{n}|x/x = 0] ∩ {[(P^{m−1}|x ∧ P x = 0)/(P^{m}|x ∧ P x = 0)] : m = 1, ..., n − 1} = [P^{n}|x/x = 0] ∩ [P x = 0/(P^{n−1}|x ∧ P x = 0)] (The second term of the intersection excludes just the R/P^{m} for m < n and the first term includes only the R/P^{k} for k ≤ n.)

\overline{R_{(P)}} : \{K\} ∪ \{\overline{R_{(P)}}\} ∪ \{\overline{R_{(Q)}}, R_{Q^{∞}}, R/Q^{n} : n ≥ 1, Q ≠ Q_{1}, ..., Q_{k}\} = [P x = 0/x = 0] ∩ {[x = x/Q_{i}|x] ∩ [Q_{i}x = 0/x = 0] : i = 1, ..., k} for Q_{1}, ..., Q_{k} any primes different from P.

Thus we can totally remove finitely many primes different from P as well as the prüfer and all the finite points associated to P. Notice that if R has only finitely many primes then we obtain a minimal neighbourhood of the form \{\overline{R_{(P)}}, K\}.

R_{P^{∞}} : \{K\} ∪ \{R_{P^{∞}}\} ∪ \{\overline{R_{(Q)}}, R_{Q^{∞}}, R/Q^{n} : n ≥ 1, Q ≠ Q_{1}, ..., Q_{k}\} = [x = x/P |x] ∩ {[x = x/Q_{i}|x] ∩ [Q_{i}x = 0/x = 0] : i = 1, ..., k} for Q_{1}, ..., Q_{k} any primes different from P.

Thus we can totally remove finitely many primes different from P as well as the P -adic and all the finite points associated to P. If R has only finitely many primes then we obtain a minimal neighbourhood of the form {R_{P^{∞}}, K}.

K : \{K\}∪\{\overline{R_{(Q)}}, R_{Q^{∞}}, R/Q^{n} : n ≥ 1, P ≠ P_{1}, ..., P_{k}\} = ∩\{[x = x/P_{i}|x]∩ [P_{i}x = 0/x = 0] : i = 1..., k} for P_{1}, ..., P_{k} any primes .

Thus we can totally remove finitely many primes.

Observe that these Zariski-open sets, apart from the finite ones, are not compact: e.g. X = ⋂{[x = x/P_{i}|x]∩[P_{i}x = 0/x = 0] : i = 1, ..., k} = ⋂{[x = x/P_{i}|x]∩ [P_{i}x = 0/x = 0] : i = 1, ..., k + 1} ∪ (X ∩ [x = x/P_{k+1}|x]) ∪ (X ∩ [P_{k+1}x = 0/x = 0]) ∪ {[(P_{k+1})^{n}|x/x = 0] ∩ [P_{k+1}x = 0/(P_{k+1})^{n−1}|x ∧ P_{k+1}x = 0] : n ≥ 1} is an open cover with no finite subcover; e.g. [x = x/x = 0] = [x = x/P |x] ∪ [P x = 0/x = 0] ∪ {[P^{n}|x/x = 0] ∩ [P x = 0/(P^{n−1}|x ∧ P x = 0)] : n ≥ 1} is an open cover of the whole space with no finite subcover.

Thus, Zar_{R} may fail to have a basis of compact open sets.

Observe also that the Zariski-closed points are the various R/P^{n}, the R_{P} and the prüfer points R_{P^{∞}} for P a prime.

Finally note that in the Zariski spectrum of a Dedekind domain with spec(R) finite each infinite-dimensional point has a minimal neighbourhood consisting of itself together with the generic point and, in particular, the generic point is isolated.

2.9 Artin algebras

2.9a Ziegler topology Recall the following [Pre2, 13.1-13.4] and 2.11 above.

Theorem 2.20 Let N be an indecomposable finitely generated (hence pureinjective) module over the artin algebra R. Then N is both open (i.e. isolated) and closed in the Ziegler topology. The set of isolated points is dense in Zg_{R} and hence is exactly the set of finitely generated points. Every point of finite endolength is closed and, if R is countable, then there are no further closed points.

Question If R is an artin algebra and N is a closed point of the Ziegler spectrum, must N be of finite endolength?

2.9b Zariski topology

Proposition 2.21 Let N be an indecomposable finitely generated module over the artin algebra R. Then N is both open and closed in the Zariski topology.

Proof N is Ziegler-isolated - say {N} = (ϕ/ψ) for some pp formulas ψ ≤ ϕ. This is, by definition, a Zariski-closed set.

With the same notation, {N}^{c} is, by the above theorem, both Zariskiclosed (hence compact) and Zariski open and hence {N}^{c} = (ϕ′/ψ′)for some pp formulas ψ′ ≤ ϕ′. Therefore {N} = [ϕ′/ψ′], as required.*

* Thanks to Henning Krause for pointing out this very short argument for the second point.

Lemma 2.22 Let R be any ring and let C be any closed subset of Zg_{R} such that the complete theory T = T_{C}^{∗} with support C has m-dimension. Then every point of C is in the Zariski-closure of a point of finite endolength.

Proof Since T has m-dimension C, and hence any closed subset of C, has Cantor-Bendixson rank (see [Pre4, 10.19]). Let N ∈ C and let C′ be the Ziegler-closure of N. Let N′ be a point of maximal C-B rank in C′: then N′ has finite endolength [Pre4, 10.19] and since N′ ∈ Zg − cl(N) we have N ∈ Zar − cl(N′).

Proposition 2.23 Let R be a countable artin algebra and let N ∈ Zar_{R} be Zariski-isolated. Then N is of finite endolength. Let R be a finitedimensional algebra over a countable algebraically closed field and let N ∈ Zar_{R} be Zariski-isolated. Then N is of finite length.

Proof Say {N} = [ϕ/ψ]. Let T be the theory of N^{ℵ_{0}}: then T must have m-dimension - otherwise, by [Zie, 8.3], there would be 2^{ℵ_{0}} points in supp(T) ⊆ [ϕ/ψ] (this is where we use the assumption that R is countable). Since T has m-dimension supp(T) contains a point of finite-endolength (by the Lemma) - which must be N, as required.

For the second statement, if N is not of finite length then it is a generic point in the sense of [C-B1] and so, by [C-B1], there is a Dedekind domain D with infinitely many primes and a representation embedding from Mod-D to Mod-R with N being the image of the generic D-module Q (=the quotient field of D). By [Pre6] this induces a homeomorphism of the Zariski spectrum of D into that of R. Since Q is not isolated in Zar_{D}, N cannot be isolated in Zar_{R}.

Question Can we remove the countability hypothesis from the above? Question Is the second statement true for artin algebras (using [C-B2])? Question Is the set of finite-dimensional points in Zar_{R} for R an artin algebra Zariski-dense?

3 Part 3

3.1 inj_{R} over a right coherent ring

Let R be a right coherent ring. Recall that inj_{R} denotes the subset of Zg_{R} consisting of the indecomposable injective right R-modules. Since R is right coherent, inj_{R} is a closed set in the Ziegler topology (1.6).

Examples 3.1 As a subset in the Zariski topology inj_{R} need not be open (consider R = ℤ - the Zariski-open neighbourhoods of the injective points can be found in 2.8b above) although it may be open (e.g. over an artin algebra). Nor need it be closed: by 2.10 the Zariski-closure of inj_{ℤ} consists of all infinite points of Zar_{ℤ}. Nor need it be dense: any finite point over ℤ is Zariski-closed.

If E is any injective module we denote by cogE the torsionfree class cogenerated by E: the class of all modules which embed in some power of E. More generally if E is a set of injective modules then put cogE = cog(⨁ E).

Theorem 3.2 Let R be right coherent. A subset C of inj_{R} is Ziegler-closed iff C = F ∩ inj_{R} for some torsionfree class F for a hereditary torsion theory on Mod-R of finite type.

Proof ⇒ The assumption on F implies [Pre1, 2.4] that F is elementary (= axiomatisable) and hence the intersection with inj_{R} is Ziegler-closed.

⇐ If C is Ziegler-closed then cogC, being the class of models of the universal sentences in T_{C}^{∗}, is an elementary class. So by [Pre1, 2.4] cogC is a hereditary torsionfree class of finite type and hence, since C is Ziegler-closed, we must have inj_{R} ∩ cogC = C.

Lemma 3.3 (e.g. [LM]) For any ring R, right ideal J and ∩ -irreducible right ideal I the following pairs of conditions are equivalent. Furthermore (a) ⇒ (b) ⇒ (c).

(a) (i) E(R/J) ∈ cogE(R/I) (ii) ∀r ∈ R\J∃r′ ∈ R∃s ∈ R\I with rr′ ∉ J and (J : rr′) ≤ (I : s)

(b) (i) Hom(R/J, E(R/I)) ≠ 0 (ii) ∃r ∈ R\J∃s ∈ R\I with (J : R) ≤ (I : s)

(c) Suppose that J also is ∩-irreducible.

(i) Hom(E(R/J), E(R/I)) ≠ 0 (ii) ∃J′ ∼ J∃s ∈ R\I with J′ ≤ (I : s) (we write J ∼ J′ if there are R,r′ with (J′ : r′) = (J : R) ≠ R)

Proof (a) (i)⇒(ii) Since E(R/J) embeds in a power of E(R/I), given r ∈ R\J there is f : E(R/J) ⟶ E(R/I) with f(R + J) ≠ 0 and so, by uniformity of E(R/I), with f(rr′ + J) = s + I ∈ R/I ≤ E(R/I) for some r′and some s ∉ I. Since ann_{R}(rr′ + J) ≤ ann_{R}(s + I) we obtain (ii). (ii)⇒(i) Given r ∈ R\J and r′,s as in (ii) we have a well-defined morphism f : R/(J : rr′) ⟶ E(R/I) given by 1 + (J : rr′) ↦ 1 + (I : s). Since R/(J : rr′) ≃ (rr′R + J)/J and since E(R/I) is injective we obtain a morphism f_{r} : E(R/J) ⟶ E(R/I) with f_{r}(R) ≠ 0. The product of these morphisms over all r ∈ R\I then gives an embedding of E(R/J) into E(R/I)as required.

(b) (i)⇒(ii) Given non-zero f : R/J ⟶ E(R/I) there are r,s such that f(r + J) = s + I ≠ 0. (ii)⇒ (i) With r,s as given, there is an epimorphism R/(J : r) ⟶ R/(I : s) which, using R/(J : r) ≃ (rR + J)/J and injectivity of E(R/I), gives (i).

(c) (i)⇒(ii) Let f : E(R/J) ⟶ E(R/I) be non- zero with, say, fe = s + I ≠ 0. Then [LM] J′ = ann_{R}e satisfies the conditions. (ii)⇒(i) This follows from (b) since J ∼ J′ with J ∩ -irreducible implies E(R/J) ≃ E(R/J′). The implications (a)⇒(b)⇒(c) are trivial.

Denote by U_{E} the Gabriel filter consisting of those right ideals I of R such that Hom(R/I, E) = 0. If E = E(R/P) then write U_{P} for U_{E}.

3.2 inj_{R} over a commutative coherent ring

Suppose now that R is commutative and coherent

Let E be an indecomposable injective module. Let P = P (E) = Σ{J : eJ = 0 for some e ∈ E, e ≠ 0}. Since R is commutative, P = Σ{J : MJ = 0 for some M ≤ E, M ≠ 0}. Note also that, since E is uniform, the annihilators of non-zero submodules of E form a directed set with P as the union.

Lemma 3.4 P is a prime ideal and if ann_{E}P ≠ 0 then E = E(R/P).

Proof If ab ∈ P then eab = 0 for some non-zero e ∈ E. So either ea = 0, in which case a ∈ P or ea ≠ 0 in which case b ∈ P. If e ∈ E, e ≠ 0 and eP = 0 then, by construction, P = ann_{R}e and so E = E(eR) ≃ E(R/P).

Theorem 3.5 Let R be commutative coherent, let E be an indecomposable injective module and let P (E) be the prime ideal defined above. Then E and E(R/P) are topologically indistinguishable in Zg_{R} and hence also in Zar_{R}. Therefore E ≡ E(R/P).

Proof Let p(x) be the (partial but, in fact, complete) type in T = Th(E^{ℵ_{0}}) which says ”x ≠ 0” and ”xP = 0”. By construction this is consistent with T and also, by construction of P, implies ”xr ≠ 0” for every r ∈ R \ P (it is realised in E if P is finitely generated). Therefore p(x) is realised, by an element a say, in some model M of T which, without loss of generality, we may take to be injective (since every model of T is absolutely pure - for R is coherent - every pure-injective model is injective). Since ann_{R}a = P we have that E(R/P) ∈ Zg − cl(E).

Conversely, suppose that E = E(R/I). Let r ∈ R\I and set e = r + I ∈ E. Then ann_{R}e = (I : r) and so (I : r) ≤ P by construction. So Hom(R/(I : r), E(R/P)) ≠ 0, that is, Hom((rR + I)/I, E(R/P)) ≠ 0 and so, by injectivity of E(R/P), there is a non-zero morphism f_{r} : R/I ⟶ E with f_{r}(R) ≠ 0. Therefore the morphism from R/I to E(R/P)^{R\I} given by s ↦ (f_{r}(s))_{r} is an embedding. Hence E = E(R/I) embeds in a product of copies of E(R/P) and hence E ∈ Zg − cl(E(R/P)) as required. The last statement follows by 2.1.

So in the commutative coherent case the Ziegler and Zariski topologies on inj_{R} are essentially topologies on spec(R). If R is commutative von Neumann regular then, since any ∩ -irreducible ideal must be maximal, hence prime, every indecomposable injective has the form E(R/P) with P prime. The following example, which was pointed out to me by T. Kucera, shows that this is not so for arbitrary commutative coherent rings.

Example 3.6 Let R = k[X_{n}(n ∈ ω)] be a polynomial ring over a field k in infinitely many commuting indeterminates. Then R is coherent (immediate from the characterisation in [Ch]). Let I = (X_{n}^{n+1} : n ∈ ω). Then I is not prime (since X_{1} ∉ I but X_{1}^{2} ∈ I) but E = E(R/I) is an indecomposable injective. To see this, it is enough to show that R/I is uniform and this may be shown as follows. First note that a polynomial Σa_{ν}X^{ν} (each multi-index ν occurring at most once) is in I iff each of its monomial factors is in I. Let x_{i} denote the image in R/I of X_{i}. Let p ∈ R\I. A short inductive argument shows that there is a multiple of p whose image in R/I has the form x_{1}x_{2}^{2}...x_{n}^{n} (which, note, is non-zero) for some n. Hence any two non-zero elements of R/I have a common multiple of this form.

On the other hand, E does not have the form E(R/P) for any prime P. Indeed, it is easy to see that P (E) is the maximal ideal (X_{n} : n ∈ ω). Now E(R/(X_{n} : n ∈ ω)) has non-zero socle whereas E(R/I) has zero socle (if p ∈ R\I, say p ∈ k[X_{0}, ...X_{n}], then (p + I)X_{n+1} generates a non-zero proper submodule of the submodule generated by p + I).

Theorem 3.5 above shows that E is elementarily equivalent to E(R/(X_{n} : n ∈ ω)).

The next result shows that an indecomposable injective E is an elementary cogenerator in the sense of [Pre4, Section 9.4] (that is E ⊗ − is the injective hull of a simple functor in (R−mod, Ab)) iff it has the form E(R/P) for some prime P.

Proposition 3.7 Let E be an indecomposable injective module over the commutative coherent ring R. Then cogE is of finite type iff E = E(R/P) for some prime P. In particular E(R/Q) ∈ Zg − clE(R/P) iff E(R/Q) ∈ cogE(R/P).

Proof ⇐ By 3.3(b) we have I ∈ U_{P} iff for all r ∈ R, (I : R) is not contained in P. Choose I ∈ U_{P} - so I is not contained in P. Choose c ∈ I \ P. Let r ∈ R and suppose that (cR : r) ≤ P. In particular, since rc = cr and so c ∈ (cR : r), we contradict c ∉ P. Hence cR ∈ U_{P} and cR ≤ I, as required.

⇒ If cogE is of finite type then it is an elementary class [Pre1, 2.4] and so, by 3.5, E(R/P) ∈ cogE where P = P (E). So there is an embedding R/P ⟶ E^{κ} for some κ , taking 1 + P to (e_{i})_{i<κ} say. Some e_{i(0)} is non-zero and so, since e_{i(0)}P = 0 we have ann_{E}P ≠ 0 and hence E = E(R/P) by 3.5. The last statement is by 3.2.

Proposition 3.8 Let R be commutative coherent. Then the correspondence P ↦ cog(E(R/P) between spec(R) and the set of prime torsion theories of finite type is 1-1.

Proof Let E = E(R/I) and E′ = E(R/P) be indecomposable injectives with P prime and suppose that cogE = cogE′. Then E is a direct summand of a power of E′ and vice versa, so E ≡ E′. Let r ∈ I. Then E ⊧ ∃x(x ≠ 0 ∧ xr = 0) and so E′ also satisfies this sentence - say e ∈ E′, e ≠ 0 with er = 0. Since P is the maximal annihilator of non-zero elements of E′, we obtain r ∈ P and hence I ≤ P. If I = Q is also prime then, by symmetry, we conclude Q = P, as claimed.

It also follows that for a commutative coherent ring R the correspondence P ↦ E(R/P) between spec(R) and inj_{R} is a bijection iff the Ziegler topology coincides with the topology introduced by Burke in [Bur].

Theorem 3.9 Let R be commutative coherent and let C ⊆ inj_{R} be Zieglerclosed. Then C is irreducible iff C = cog(E(R/P)) ∩ inj_{R} for some prime ideal P of R.

Proof The direction ”⇐” is immediate: we prove ”⇒”. So assume that C is irreducible. Let F = cogC - an elementary class since C is closed (proof of 3.2). Let P be the set of maximal annihilators of elements of members of F (by the Compactness Theorem every annihilator of an element of a member of F is contained in a member of P). Choose P_{0} ∈ P and set E_{0} = E(R/P_{0}), E′ = ⨁{E(R/P) : P ∈ P, P ≠ P_{0}}. Let C_{0} = suppE_{0} and C′ = suppE′.

We show that C = C_{0} ∪ C′. If E ∈ C then E ≡ E(R/P (E)) and so, since C is closed, it will be sufficient to show that every point of C of the form E(R/P) with P prime is in C_{0} ∪ C′. So, given such E = E(R/P) choose P_{1} ∈ P with P ≤ P_{1}. Then E ∈ cogE(R/P_{1}) since, if r ∈ R \ P then, since P is prime (P : r) ≤ P ≤ P_{1} and so we have the criterion of 3.3(a)(ii) above. Hence E ∈ C_{0} ∪ C′.

Since C is irreducible we have C = C_{0} as required or C = C′. In the latter case we would have E_{0} ∈ cogE′ and hence there would be an embedding of the form f : R/P_{0} ⟶ ∏{E(R/P)^{κ(P)} : P ∈ P, P ≠ P_{0}} with, say (1 + P_{0}) ⟶ (e_{λ})_{λ}. Some e_{λ} is non-zero and so P_{0} would be (properly!) contained in ann_{R}e_{λ} - contradicting P_{0} ∈ P.

Lemma 3.10 Let P, Q be prime ideals of the commutative ring R. Then E(R/P) ∈ Zar − cl(E(R/Q)) iff E(R/Q) ∈ Zg − cl(E(R/P)) iff Q ≤ P.

Proof Immediate from 2.3, 3.3(a) and 3.7.

Proposition 3.11 Let R be commutative coherent. A subset V of inj_{R} is Zariski-closed and irreducible iff there is a prime ideal Q of R such that V = {E : P (E) ≥ Q}.

Proof By the arguments of section 1.7 above, the bijection P ↦ E(R/P) between spec(R) and its image in inj_{R} is a homeomorphism where spec(R) carries its usual topology and inj_{R} has the Gabriel-Zariski topology (defined with respect to Mod-R). The arguments there also show that this coincides with the Zariski(=dual-Ziegler) topology on inj_{R}. Since, up to topological equivalence, the points E(R/P) are typical in inj_{R} (3.5) we obtain the above (the characterisation of irreducible closed sets is easy to see and well-known for spec(R)).

Theorem 3.12 Let R be commutative coherent. Then there are bijections between the following: the set of irreducible Ziegler-closed subsets of inj_{R}; spec(R); the set of irreducible Zariski-closed subsets of inj_{R}; given by {E : P (E) ≤ Q} ∼ Q ∼ {E : P (E) ≥ Q}.

Hence the dcc on irreducible Ziegler-closed sets corresponds to the dcc on primes and the dcc on irreducible Zariski-closed sets corresponds to the acc on primes (both of which fail for the coherent ring k[X_{n} : n ∈ ω]).

3.3 Inj_{R} over a right noetherian ring

Throughout this section R will be right noetherian (we repeat this in the statements of results where the implicit hypothesis is not immediately obvious). Compared with the commutative coherent situation we know rather little about inj_{R}.

Corollary 3.13 Let R be right notherian. A subset C of inj_{R} is Zieglerclosed iff C = F ∩ inj_{R} for some torsionfree class F for a hereditary torsion theory on Mod-R.

The corollary is immediate from 3.2. The following facts which are collected here are also immediate from what has gone before. Recall that a basis of open sets for the Ziegler (respectively Zariski) topology on inj_{R} is the set of all (xI = 0/x = 0) (resp. [xI = 0/x = 0]) for I a right ideal of R.

Observations 3.14 For any E ⊆ inj_{R} and E ∈ inj_{R} we have

∗ E ∈ Zg − cl(E) iff E ∈ cogE.

∗ E ∈ (xI = 0/x = 0) iff Hom(R/I, E) ≠ 0 iff I ∉ U_{E} and hence

∗ a basis of open neighbourhoods of E within inj_{R} is {(xI = 0/x = 0)∩inj_{R} : I ∉ U_{E}}

∗ E ∈ Zg − cl(E) iff ⋂{U_{E′} : E′ ∈ E} ⊆ U_{E}

∗ E ∈ [xI = 0/x = 0] iff Hom(R/I, E) = 0 iff I ∈ U_{E} and hence

∗ a basis of open neighbourhoods of E within inj_{R} is {[xI = 0/x = 0] ∩ inj_{R} : I ∈ U_{E}}

∗ a sub-basis of the above is obtained by restricting I to be ∩- irreducible (since if I = I_{1} ∩ ... ∩ I_{n} then it is easy to check that [xI = 0/x = 0] ∩ inj_{R} = ⋂{[xI_{i} = 0/x = 0] ∩ inj_{R} : i = 1, ..., n}

∗ For any E ⊆ inj_{R} and E ∈ inj_{R} we have E ∈ Zar − cl(E) iff U_{E} ⊆ ⋃{U_{E′} : E′ ∈ E}

Lemma 3.15 Let E = E(R/I) ∈ inj_{R}. The following sets form a neighbourhood system of Ziegler-open sets for E within inj_{R}: (xI = 0/x(I +rR) = 0) where R ranges over R \ I.

Proof Let e ∈ E be any element with annihilator I. By [Zie, 4.9] a neighbourhood system of open sets of E is given by any system of sets (ϕ(x)/ψ(x)) with the pairs ϕ/ψ cofinal in the ± pp- type of e (where the type of e is measured relative to the theory of injectives). Hence the set of all (xI = 0/xJ = 0) with J > I forms such a system. But, by ∩-irreducibility of I, the set of these with J 1-generated over I is cofinal, as claimed.

Proposition 3.16 Let R be right noetherian. If C ⊆ inj_{R} is Ziegler-closed then C is irreducible iff C = inj_{R} ∩ cog_{E} = Zg − cl(E) for some E ∈ inj_{R}.

Proof The direction ”⇐” is obvious: we prove ”⇒”. Since R is right noetherian the theory of injectives is totally-transcendental (=Σ-pure-injective). In particular [Zie], the Cantor-Bendixson rank of inj_{R} and hence of C is defined. Therefore the set C_{0} of isolated points is dense (see below for the proof). If C_{0} = {E} is a singleton then we are finished. Otherwise (and for a contradiction) choose E ∈ C_{0} and set C′_{0} = C_{0} \ {E}. Then C_{1} = Zg − cl(E) and C_{2} = Zg − cl(C′_{0}) are closed subsets of C. If E were in C_{2} we would contradict the fact that E is isolated in C_{0}. Similarly C_{1} is a proper subset of C. Finally, C = Zg − cl({E} ∪ C′_{0}) = Zg − cl(E) ∪ Zg − cl(C′_{0}) = C_{1} ∪ C_{2} - contradiction, as required.

We recall here, for the reader’s convenience, the proof that if the space C has Cantor-Bendixson rank then the set of isolated points is dense. So let U ⊆ C be open and non-empty. Choose c ∈ U of minimal CB-rank, α , say. By definition of the rank of a point there is V ⊆ C open such that V ⊆ C_{β} ∪ {c} where C_{β} is the set of points of CB-rank≤ β. By choice of α, U ∩ C_{β} = ∅ and hence U ∩ V = {c} and c is isolated (i.e. α = 0).

Example 3.17 A right noetherian ring need not have dcc on irreducible Ziegler- closed sets. Jategaonkar [Jat] gives examples of right principal ideal rings with arbitrarily long (any ordinal) chains of primes. By 3.3(a) above if Q < P are prime ideals then E_{Q} ∈ cogE_{P} but not conversely and so we can obtain arbitrarily long descending chains of irreducible closed sets in Zg_{R} for R right noetherian.

Next recall that, associated to any indecomposable injective E, is the prime ideal P (E) = ass(E) which is the maximal annihilator of a non- zero submodule of E. Recall [LM] that E(R/P) is a direct sum of finitely many copies of an indecomposable injective, denoted E_{P}, which is the unique P - torsionfree indecomposable injective with associated prime P. Here we have one half of 3.5.

Proposition 3.18 If E is an indecomposable injective over the right noetherian ring R and if P = ass(E) then E ∈ Zar − cl(E(R/P).

Proof It will be enough to show that ”ann_{R}x = P ” is consistent with Th(E) (for them E(R/P) will be in Zg − cl(E)). So, given a finitely generated right ideal J ≤ P and an element s ∈ R \P (it suffices to consider one s since we are dealing with a theory closed under products), we must show that there is an element of E annihilated by J and not annihilated by s. By construction there is a non-zero submodule M of E with MJ = 0. Let e ∈ M, e ≠ 0: so eRJ = 0. Since s ∈ R \ P there is r ∈ R with ers ≠ 0 and so er witnesses the consistency of xJ = 0 ∧ xs ≠ 0.

Example 3.19 We do not in general have E(R/P) ∈ Zar − cl(E). Let R be the first Weyl algebra over a field of characteristic 0. So (e.g. [McR]) R is a simple noetherian domain and has infinitely many simple modules. The injective hulls of these provide infinitely many Ziegler-closed points of inj_{R} each of which, by the above, is in the Zariski closure of the unique non-closed point (the injective hull of R).

Each injective hull E = E(S) of a simple module, S = R/I say, is isolated in the Ziegler topology on inj_{R} (and hence is Zariski-closed in inj_{R})

- by the Ziegler-open set (xI = 0/x = 0). In particular, E ∉ Zg − cl(E(R)) and so E(R) ∉ Zar − cl(E).

On the other hand it is easy to check that the generic point is in the Zariski- closure of any infinite set. For (3.14) a basis of neighbourhoods of E = E(R) is given by the [xI = 0/x = 0] as I ranges over the non-zero right ideals of R. Now E′ ∈ [xI = 0/x = 0] iff Hom(R/I, E′) = 0 iff E′ is not the injective hull of one of the finitely many simple composition factors of R/I. So each open neighbourhood of E excludes only finitely many of the Zariski-closed points.

Hence the Zariski topology on inj_{R} has its closed points parametrised by the simple modules and has one more point, which is a generic point for the space. (The Weyl algebra in characteristic greater than 0 is fully bounded noetherian and so (as is pointed out in [Kra3]) the correspondence between primes and indecomposable injectives is a homeomorphism between spec(R) and inj_{R} equipped with the Zariski topology.)

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