Tensor-triangular fields: Ruminations

We examine the concept of field in tensor-triangular geometry. We gather examples and discuss possible approaches, while highlighting open problems. As the construction of residue tt-fields remains elusive, we instead produce suitable homological tensor-functors to Grothendieck categories.


Introduction to tt-fields and statement of results
Let us start by recalling an elementary pattern of commutative algebra, namely the reduction of problems to local rings, and then to residue fields; schematically: The method for recovering global information from local data is generically called descent. On the other hand, a useful way to recover local information from the residue field is Nakayama's Lemma, at least under suitable finiteness conditions. We would like to assemble a similar toolkit in general tt-geometry. Localization of tensor-triangulated categories is well understood (see Remark 4.4) and descent has been extensively studied. The present article focusses on the righthand part of the above picture. Given a local tt-category T (Definition 4.3) we want to find a tt-category F together with a tt-functor F : T → F such that F is a 'tt-field'. Clarifying the latter notion is one first difficulty. Such a tt-functor F should satisfy some form of Nakayama, a property which most probably means that F is conservative (detects isomorphisms) on compact-rigids. This conservativity on T c matches the behavior in examples and implies that x ∈ T c F (x) = 0 is the unique closed point (0) in the spectrum Spc(T c ). So what should 'tt-fields' be? In colloquial terms, a tt-field F should be 'an end to the tt-road': It should not admit non-trivial localizations, nor non-trivial 'quotients'. In particular, we should not be able to 'mod out' any non-zero object nor any non-zero morphism by applying a tt-functor going out of F.
Although vague, this preliminary intuition is sufficient to convince ourselves that some well-known tt-categories should be recognized as tt-fields. Of course, in commutative algebra, the derived category of a good old commutative field k should be a tt-field. Also, the module category over Morava K-theory (in a structured enough sense) should be a tt-field in topology. In these examples, the homotopy groups are 'graded fields' k[t, t −1 ] with k a field and t in even degree, and all modules are sums of suspensions of the trivial module. (See some warnings in Remark 1.4.) Consideration of so-called 'cyclic shifted subgroups', or 'π-points', in modular representation theory of finite groups lead us to another class of candidates: Proposition. Let p be a prime, G = C p the cyclic group of order p and k a field of characteristic p. Let F = Stab(kG) be the stable module category, i.e. the additive quotient of the category of kG-modules by the subcategory of projective modules.
Then every non-zero coproduct-preserving tt-functor F : F → S is faithful.
This is Proposition 5.1, where further 'field-like' properties of Stab(kC p ) are isolated. This result tells us that one should probably accept F = Stab(kC p ) as a tt-field although it is quite different from 'classical' fields. Granted, for p = 2, the above category F ∼ = Mod-k is very close to a 'good old field'. However, for p > 2, the homotopy groups in F form the Tate cohomology ring π − * (1) = Hom F (1, Σ * 1) ∼ = H * (C p , k), which is the graded ring k[t ±1 , s]/s 2 with t in degree 2 and s in degree 1. In particular, s is a nilpotent element in π * which cannot be killed off by any non-trivial tt-functor out of F, because of the above proposition.
In other words, one should renounce some traditional definitions of fields. Topologists sometimes call fields those (nice enough 1 ) rings over which every module is a sum of suspensions of the ring itself. This property still holds in the example of F = Stab(kC p ) for p = 2 or p = 3. However, for every p ≥ 5 there are objects in F which are not direct sums of suspensions of the ⊗-unit 1. A generalization of the old topological definition has been used in the motivic setting in recent work of Heller and Ormsby [HO16]; they call 'field' a (nice enough) ring whose modules are sums of ⊗-invertibles. This generalized definition is motivated by the existence of additional 'spheres' in the motivic setting, namely G m or P 1 . However, for p ≥ 5 our F = Stab(kC p ) is stubbornly not a field in the Heller-Ormsby sense either, for its only ⊗-invertibles are the 'usual' spheres Σ * 1. See Proposition 5.1.
In summary, because of simple examples in modular representation theory, we cannot define tt-fields as those tt-categories whose homotopy groups are graded fields, nor as those tt-categories in which every object is a sum of spheres, or even a sum of ⊗-invertibles. We need something more flexible.
Dwelling on the instructive example of F = Stab(kC p ) for a moment longer, we see that every object of F is a coproduct of finite-dimensional indecomposable objects whose dimension is invertible in k. See Proposition 5.1 again. It follows that F is a tt-field in the sense of the following tentative generalization: 1.1. Definition. A non-trivial 'big' tt-category F will be called a tt-field if every object X of F is a coproduct X ≃ i∈I x i of compact-rigid objects x i ∈ F c and if every non-zero X is ⊗-faithful, meaning that the functor X ⊗ − : F → F is faithful (if X ⊗ f = 0 for some morphism f in F then f = 0). [Bal10b,§ 4.3] by only asking that every non-zero object of F c be ⊗-faithful. Under the assumption that every object of F is a coproduct of compacts, the two versions are evidently the same. We do not know any example of F in which every non-zero (compact) object is ⊗-faithful without having the other property. So it is possible that Definition 1.1 contains some redundancy.

Remark. A similar definition was suggested in
We establish basic properties of such tt-fields in Section 5. For instance, we show in Theorem 5.21 that being a field is equivalent to the internal hom functor hom F (−, X) being faithful for every non-zero object X -a very simple formulation.
A tt-field F must have a minimal spectrum: Spc(F c ) = { * } (Proposition 5.15). In other words, every non-zero object generates the whole category. This matches the intuition that a field should be very small. It also explains how, for T local, the expected 'Nakayama property' of the residue tt-functor F : T → F, namely the conservativity of F : T c → F c , simply means that the induced map Spc(F ) : Spc(F c ) → Spc(T c ) sends the unique point of Spc(F c ) to the closed point of Spc(T c ).
We are thus led to the quest for tt-residue fields.
1.3. Question. Given a local tt-category T, is there a coproduct-preserving ttfunctor F : T → F to a tt-field F (Definition 1.1), with F conservative on T c ?
In this generality, Question 1.3 remains an open problem, and most probably a difficult one. We propose here a palliative approach via abelian categories, which works unconditionally. Before explaining this, let us add some warnings.
1.4. Remark. In topology, the Morava K-theories at the prime 2 do not admit a homotopy-commutative ring structure. So it is not even clear that their modules form a tensor -triangulated category. In the same vein, in representation theory, each point of the support variety of a finite group G is detected by a 'π-point', which comes with an exact functor from the stable category of G to a tt-field Stab(kC p ). Again, this restriction is not always a tensor -functor, unless one tinkers with the tensor in Stab(kG). Both examples point to the possibility that one might need to adjust the role of the tensor product in the construction of tt-residue fields.
As mentioned in the foreword, it is possible that the theory of ∞-categories, or that of model categories, could help us solve Question 1.3. Mathew shows in [Mat17] how to produce (topological) residue fields when working over a field of characteristic zero and when T is a stable ∞-category with only even homotopy groups. Even this very special case seems remarkably difficult. * * * Let us now say a word about the announced approach to residue fields via abelian categories. Recall that there exists a restricted-Yoneda functor (1.5) h : We attack the problems presented above from the angle of the module category A when T is local. There are two related facets to this idea. First, we want to produce a 'residue abelian category'Ā together with a coproduct-preserving homological tensor-functorh : T →Ā which is conservative on T c and in such a way thatĀ is 'very small'. Second, in case there miraculously exists a tt-residue field F : T → F at the triangular level, we would like to relate the corresponding categories of modules, Mod-T c and Mod-F c , and theĀ constructed above.
Our first series of results establishes the unconditional existence of such anĀ.  Our second collection of results can be summarized as follows: if T admits a tt-residue field then it produces, in a natural way, a 'residue abelian category' as above and this abelian category is very close to the category of modules over the tt-field. 1.7. Theorem. Let T be a local 'big' tt-category and F : T → F be a coproductpreserving tt-functor into a tt-field F (Definition 1.1). Suppose that F is conservative on T c and surjective up to direct summands. (See Hypothesis 6.1.) Then there exists a ring-object E in T satisfying all the following properties:  Proof. Again, this is a summary of Section 6. See specifically Corollaries 6.9 and 6.12, Propositions 6.15, 6.17 and 6.21.

Modules, Serre ideals and quotients
Recall the Grothendieck category A = Mod-T c of Section 1.
2.1. Notation. The finitely presented objects A fp are denoted by mod-T c . The latter is the Freyd envelope of T c [Nee01, Chap. 5] and the (usual) Yoneda embedding h : T c ֒→ mod-T c , x →x = Hom T c (−, x), identifies T c with the projective (and injective) objects in mod-T c . Together with restricted-Yoneda of (1.5), we have the following commutative diagram The objectsx remain projective but usually not injective in the whole module category Mod-T c . See Remark 2.6 for injectives in A.

Remark. The category
In other words, for any coproduct-preserving homological functor G : T → C to a Grothendieck category C there exists a unique exact colimit-preserving functorĜ : Mod-T c → C making the following diagram commute: 2.4. Remark. The Day convolution product on Mod-T c is the unique symmetric monoidal ⊗ : Mod-T c × Mod-T c → Mod-T c such that M ⊗ − and − ⊗ N commute with colimits and such thatx ⊗ŷ = x ⊗ y for every x, y ∈ T c . Then restricted-Yoneda h : T → Mod-T c is a monoidal functor andX is flat in Mod-T c for every X ∈ T, even non-compact (see [BKS17,Prop. A.14]). Also,x is rigid when x ∈ T c , as every monoidal functor preserves rigid objects.
In fact the category A = Mod-T c is moreover closed, i.e. it admits an internal hom which we denote by hom A . This follows from a general fact about existence of adjoints in Grothendieck categories (Proposition A.2) or can be seen by another Day convolution argument. This internal hom functor hom A is characterized by the fact that hom A (−, M ) sends colimits to limits and the fact that hom A (x, −) = hom T (x, −) for every x ∈ T c , in the sense of Remark 2.3. Note that h : T → A needs not be a closed functor, outside of T c .
2.5. Lemma. Let x be an object of T c , with dual x ∨ = hom(x, 1), and M an object of A = Mod-T c . Then there is a natural isomorphism of functors (T c ) op → Ab: In particular, we have a natural isomorphism Σ1 ⊗ M ∼ = ΣM in A.
Proof. This is an immediate consequence of the description of the Day convolution product on Mod-T c ; the key observation is that it holds for representable functorŝ 2.6. Remark. Although restricted-Yoneda h : T → A = Mod-T c is in general neither full nor faithful outside of T c , the functor h restricts to an equivalence between the pure-injective objects Y ∈ T and the injective objects of A. See [Kra00, Cor. 1.9]. By definition, Y is pure-injective if every pure mono Y → Z in T splits, and a morphism Y → Z is a pure mono if the induced morphismŶ →Ẑ is a monomorphism. The interesting point is what happens on morphisms. We even have slightly more than full-faithfulness. Indeed, the functor h induces an isomorphism Injective objects in Mod-T c are also injective with respect to the internal hom: 2.9. Remark. The content of the first part of the proof of the lemma is that thê c for c ∈ T c are, up to choosing a skeleton, a set of (finitely presented) projective generators for A.
(a) For every X ∈ T, the morphism X ⊗ f is a phantom.
Proof. By Remark 2.4, we have X ⊗ f =X ⊗f = 0, hence (a). This implies that X ⊗ − preserves the class of exact triangles whose third map is a phantom. In words, this means that every X ⊗ − is pure-exact. Consequently, the functor is the composite of two pure-exact functors if Y is pure-injective. This gives (c). Finally, to check (b), for every X ∈ T, we have where the second isomorphism holds by (2.7) and the vanishing by (a). This shows that hom(f, Y ) = 0 by Yoneda.

* * *
We are interested in the Serre subcategories B of mod-T c and Mod-T c . We focus on the ⊗-ideals meaning of course M ⊗ B ⊆ B for every M in the ambient category.
2.11. Convention. All Serre subcategories of Mod-T c that we consider are assumed stable under suspension (Remark 2.2). For ⊗-ideals it follows from Lemma 2.5 that we can safely omit this condition and we shall do so from now on.
The ⊗-ideal condition can be tested just using the finitely presented projectives: 2.12. Lemma. Let B be a Serre subcategory of mod-T c which is closed under tensoring with finitely presented representable functors, i.e. closed under the action of T c under the Yoneda embedding. Then B is a Serre ⊗-ideal in mod-T c .
Proof. For every M ∈ mod-T c , there is an epimorphismx ։ M with x ∈ T c . Tensoring with any N ∈ B we getx ⊗ N ։ M ⊗ N andx ⊗ N belongs to B by assumption, hence so does M ⊗ N since B is Serre.
We collect a few facts about the quotients of A = Mod-T c by a Serre ⊗-ideal. (2.14) Proof. All this is standard abelian category theory. See Appendix A.
2.16. Notation. When the category B ⊆ mod-T c is clear from the context, we shall often denote the compositeh := Q • h : T →Ā by the simple notation for every object X and morphism f in T.
The above proposition holds for any locally coherent Grothendieck category with a tensor. In the particular case of A = Mod-T c , we have the following consequences.
Moreover, for every object X ∈ T, the functorh induces an isomorphism (d) Any injective object in the categoryĀ is flat.
Proof. We use everywhere the results of Proposition 2.13. Part (a) holds since Q is exact and coproduct-preserving and monoidal. Part (b) holds since Q preserves flat objects. Combining Remark 2.6 and QR ∼ = Id, we obtain the first sentence of (c). It remains to prove (2.19). It is the following composite of isomorphisms: using (2.7), the defining relationÊ ∼ = RĒ and the Q ⊣ R adjunction. Finally (d) results from (c) and (b).
2.20. Remark. The analogue of Lemma 2.8 also holds inĀ, namely, if J is an injective object ofĀ then it is also injective for the internal hom i.e. the functor homĀ(−, J) is exact. To see this, as homĀ(−, J) is a right adjoint, we only need to prove that it is right exact; keep in mind that it is contravariant. Since R preserves injectives (Proposition 2.13), the object RJ is injective in A. By (2.15) and QR ∼ = Id, our functor homĀ(−, J) is the following composition of functors: The first one, R, is left exact (Proposition 2.13). The second one is contravariant and exact by Lemma 2.8 since RJ is injective. The third one is exact. So the composite is right exact, as desired.

Constructing pure-injectives
For this section, we fix a Serre ⊗-ideal B ⊂ A fp = mod-T c of finitely presented T c -modules. As in Section 2, we denote by A = Mod-T c the whole category of T c -modules and by Q : A →Ā = Mod-T c / − → B the corresponding quotient. We writē h : T →Ā, or X →X, for the composed functor Q • h, see (2.17).
Let us investigate the properties of the pure-injective object E = E(B) of Construction 3.1. We already know by Corollary 2.18 (d) thatĒ is flat and clearlyĒ is non-zero inĀ as soon as B ⊂ A is proper, since1 is a subobject ofĒ.
3.3. Proposition. Consider an exact triangle ∆ in T on the morphism η : 1 → E: It satisfies the following properties: Then Proof. Part (a) is immediate from the fact thath : T →Ā is homological and the fact thath(η) =η is a monomorphism (3.2). To prove (b), let f : X → 1 in T such thatf = 0. By exactness of ∆, we only need to prove that the composite η f : X → E vanishes in T. By pure-injectivity of E, this follows from (2.19) and the vanishing η f =ηf = 0 inĀ. We also deduce (c) from the same isomorphism (2.19) in Corollary 2.18 applied to the morphism ξ ⊗ id E from X := Σ −1 W ⊗ E to E and the already establishedξ = 0 in (a). For (d), consider the morphism g ∈ Hom T (c ∨ ⊗ X, 1) ∼ = Hom T (X, c) ∋ f corresponding to f under the adjunction. This uses rigidity of c. Explicitly, g is the following composite: In particular, sinceh : T −→Ā is monoidal, we see thatḡ = 0. By (b) applied to g, we see that g factors through ξ and thus by (c), we have g ⊗ E = 0. We can then recover f from g as the composite The following lemma formalizes an elementary argument that we will use numerous times.
3.4. Lemma. Suppose we are given maps Proof. We can arrange the various maps between tensor products in a square which commutes by bifunctoriality of the tensor product. By hypothesis the bottom horizontal map vanishes, and thus so do both composites. As the rightmost vertical map is a monomorphism this forces f ⊗ X ′ = 0 as claimed.
3.5. Theorem. With the notation of Construction 3.1 for There exists a morphism f : X → Y in T such that M is the image off . Indeed, we may take a projective presentation of M using objects in the image of Add(T c ) and then take the cone on the map giving the presentation. The assumption thatÊ ⊗ M = 0 impliesĒ ⊗f = 0 inĀ. Applying Lemma 3.4 tof :X →Ȳ andη :1 →Ē, which is reasonable sinceη is a monomorphism andȲ is flat, we seef = 0. Therefore, by exactness of Q, we have Then there exists a morphism f : x → y in T c such that M is the image off . By Proposition 3.3 (d), we know thatf = 0 inĀ forces E ⊗ f = 0 in T and thereforê E ⊗ M = 0 in A as wanted.
Proof. This follows from the previous corollary for f = 1 : X → X since restricted-Yoneda X →X is conservative:Ê ⊗X = 0 =⇒ E ⊗ X = 0.
3.8. Remark. It follows from the above discussion that a Serre ⊗-ideal B in mod-T c is determined by each of the following: (1) The morphisms f : c → 1 between compact objects in T such thatf = 0.
(2) The morphisms f : c → d between compact objects in T such thatf = 0.
In the spirit of [BKS17, Theorem 3.10] we can also give a generator for − → B . Proof. By Construction 3.1, we know that Q(η) =η :1 Ē is a monomorphism.
3.13. Remark. Unfortunately, we do not know how to upgrade µ : E ⊗ E → E to an associative, or commutative, ring structure.
Let us end the section with a rather amusing application of Proposition 3.3; we will show that one can characterize phantoms ending in a compact object in terms of tensoring with pure-injectives. First, a preparatory lemma which could be of independent interest. 3.14. Lemma. Suppose f : x → c is a morphism between compacts and let E(c) denote the pure-injective envelope of c. Then Proof. We are in the situation of Lemma 3.4: we have mapsf :x →ĉ and η :ĉ E(c) and we know thatĉ ⊗η is a monomorphism, sinceĉ is flat. Thus iff ⊗ E(c) vanishes so doesf ⊗ĉ. It follows that f ⊗ c is trivial as x ⊗ c is compact. Now suppose f ⊗ c is zero. Then we can apply Lemma 3.4 again to the maps f : x → c and coev c : 1 → c ∨ ⊗ c, since c⊗ coev c is a split monomorphism, to deduce that f = 0. Proof. We first show that if f is phantom then E(c) witnesses this. Consider Proposition 3.3 for B = 0. We have an exact triangle where c → E(c) is a pure-injective envelope of c and p is the (weakly) universal phantom with target c. The proposition tells us that E(c) ⊗ p = 0. The statement then follows as any phantom f : is zero then f is a phantom. To this end, suppose f ⊗ E(c) = 0 but f is not phantom. Then there is an x ∈ T c and a non-vanishing composite we see that f g⊗E(c) = 0, since f ⊗E(c) = 0, but this contradicts Lemma 3.14.

Maximal Serre tensor ideals
We now want to isolate interesting Serre ⊗-ideals of B ⊂ mod-T c = A fp , to which we can apply the constructions of the previous sections.
Proof. This is immediate by Zorn since the union of a tower of Serre ⊗-ideals which do not meet a given class of objects still has this property.
Conceptually, it means that the spectrum Spc(T c ) is a local topological space. It has a unique closed point, namely m = (0). By extension, we shall say that T is local when T c is. Note that T itself will almost never satisfy the property that X ⊗ Y = 0 =⇒ X = 0 or Y = 0, as one can easily see with Rickard idempotents for instance.

Remark.
To every point P ∈ Spc(T c ) of the tt-spectrum of T c we can associate a local category T P := T/ Loc(P). Its rigid-compacts (T P ) c coincide, by a result of Neeman [Nee92], with the idempotent-completion of the Verdier localization T c /P. This construction extends the algebro-geometric one, in the sense that if we start with the derived category T = D(X) := D Qcoh (O X ) of a quasi-compact and quasi-separated scheme, so that T c = D perf (X) is the derived category of perfect complexes, and if P = P(x) is the prime corresponding to a point x ∈ X under the homeomorphism X ≃ Spc(D perf (X)), then the local category T P is naturally equivalent to the derived category D(O X,x ) over the local ring O X,x at x. (a) The homological ⊗-functorh : T →Ā = Mod-T c / − → B is conservative on compacts, that is, every non-zero c ∈ T c has non-zero imagec = 0 inĀ fp , or equivalently it detects isomorphisms between compact objects. (b) For every non-zero c ∈ T c , the Serre ⊗-ideal c generated byc inĀ fp is the wholeĀ fp .
Proof. Part (a) is immediate fromĉ / ∈ B for all non-zero c ∈ T c . Detection of isomorphism then follows by applying the homological functor c →c to the cone of a morphism in T c . Part (b) is immediate from Lemma 4.7 and so is (c) by an application of Lemma 3.4 tof and coev c : 1 → c ∨ ⊗ c, using that coev c ⊗ 1 is a monomorphism by Lemma 4.7 and flatness ofȲ .
From this we deduce that although we only assumed B maximal among those subcategories meeting T c trivially, it is automatically plain maximal in A fp . (c) Every non-zero object ofĀ fp generatesĀ fp as a Serre ⊗-ideal, and generates A as a localizing ⊗-ideal.
Proof. Let B C ⊆ A fp be a Serre ⊗-ideal andC = Q(C) = 0 the corresponding Serre ⊗-ideal of the quotientĀ fp . By maximality of B among the Serre ⊗-ideals of A fp avoiding ĉ c ∈ T c , c = 0 (Definition 4.2), there exists c ∈ T c non-zero such thatĉ ∈ C, that is,c ∈C. We conclude by Proposition 4.8 (b) thatC is the whole categoryĀ fp . Hence C = Q −1 (C) is the whole of A fp . Therefore − → C is the whole of −→ A fp =Ā and all the statements follow.
We now have the following consequences: 4.10. Corollary. Under Hypothesis 4.5, let X ∈ T and c ∈ T c non-zero such that Another upshot is that if there is more than one choice for B then the corresponding pure-injectives interact in the way one would expect of field objects. 4.11. Corollary. Suppose we are in the situation of Hypothesis 4.5, with B 1 and Proof. Recall from Theorem 3.5 that we have equalities is Serre, and it contains both B 1 and B 2 . But the subcategories B i are maximal Serre ⊗-ideals by Corollary 4.9. By the assumption that B 1 and B 2 are distinct we deduce that Ker(Ê 1 ⊗Ê 2 ⊗ −) cannot be a proper Serre ⊗-ideal of A fp . Thus it contains1 forcing E 1 ⊗E 2 ∼ = 0. 4.12. Remark. The corollary tells us that if B is a T c -maximal Serre ⊗-ideal with associated pure-injective E then Ker(E ⊗ −) seems quite large, at least relatively speaking; it contains the pure-injective associated to any other choice of T c -maximal Serre ⊗-ideal. It is then natural to wonder if Ker(E ⊗ −) satisfies some sort of maximality property itself. 4.13. Corollary. Under Hypothesis 4.5, let Y ⊆ Spc(T c ) be a Thomason subset (e.g. a closed subset with quasi-compact complement) and assume Y non-empty. Consider the idempotent triangle e Y → 1 → f Y → Σe Y associated to Y ; see [BF11].
We conclude by Corollary 4.10 for X = f Y . 4.14. Remark. The last corollary shows thath : X →X is only conservative on compacts, as in Proposition 4.8 (a), but not on all objects. It also shows thath can send non-compact objects of T to finitely presented ones inĀ. The following lemma allows us to convert this into a problem about Serre ⊗ideals, which we are by now well equipped to handle.
4.17. Lemma. Let C be an abelian ⊗-category, with ⊗ right exact. Let f : X → Y be a morphism in C with Y flat. Then Proof. It is easy to check that Nil(f ) is ⊗-ideal and stable by quotients and subobjects (the latter uses that Y is flat). Consider an exact sequence Replacing f by f ⊗n for n large enough, we can as well assume that M ′ ⊗ f = 0 and M ′′ ⊗ f = 0. It suffices to show that M ⊗ f ⊗2 = 0. Use flatness of Y to obtain the exact rows of the following commutative (plain) diagram, in which we replace M ′ ⊗ f and M ′′ ⊗ f by zero: The vanishing of the diagonal in the left-hand square gives the existence of a morphism ℓ such that 1 ⊗ f = ℓ • (h ⊗ 1) and similarly, the right-hand square gives the existence of k such that 1 ⊗ f = (g ⊗ 1) • k. Then we can tensor the first relation by Y on the right and the second one by X "in the middle" (meaning on the right and then swap the last two factors) to get the following two commuting triangles: As one of them is non-zero and they both belong toĀ fp , we know by Corollary 4.9 that they will generate the whole categoryĀ fp as a Serre ⊗-ideal. Consequently Nil(f ) =Ā fp ∋1 which means that f ⊗n = 0 for some n > 0. As1 is the unit in the symmetric monoidal categoryĀ, composition in EndĀ(1) and tensoring endomorphisms coincide, which shows every non-invertible element is nilpotent as promised.
4.18. Remark. The ring EndĀ(1) is local and receives the local ring End T (1) via a ring homomorphism, which is a local ring homomorphism since c →c is conservative (Proposition 4.8 (a)). Consequently, the image of the unique prime in EndĀ (1)

contains no non-zero pure-injective, for instance if it is trivial, then B is the unique
Proof. Suppose B ′ were another, distinct, T c -maximal Serre ⊗-ideal with associated pure-injective E ′ . By Lemma 4.21 we know that E ′ lies in T m and by Corollary 4.11 we know that E ⊗ E ′ = 0. But this would contradict our hypothesis and so such a B ′ cannot exist.
We now exhibit a situation that (likely due to our lack of general knowledge) seems to occur frequently and in which we can apply the theorem. Recall that T m is said to be minimal if it contains no proper non-trivial localizing ⊗-ideal. In other words every non-zero object of T m generates it as a localizing ⊗-ideal.

4.23.
Example. When the localizing ⊗-ideals of T are classified by the subsets of Spc(T c ), then it is clear that T m is minimal, since the corresponding subset of Spc(T c ) is the one-point closed subset {m}.

4.24.
Remark. Minimality of T m implies that if E 1 , E 2 ∈ T m satisfy E 1 ⊗ E 2 = 0 then E 1 = 0 or E 2 = 0. Indeed, if E 2 = 0, we see that T m ∩ Ker(E 1 ⊗ −) is a non-zero (it contains E 2 ) localizing ⊗-ideal contained in T m , hence it must be equal to it by minimality. This gives e m ∈ T m = Ker(E 1 ⊗ −) and E 1 ≃ e m ⊗ E 1 ≃ 0.

Corollary. Suppose that T is local as in Hypothesis 4.19, with minimal localizing ⊗-ideal T m . Then there is a unique
Proof. By Proposition 4.1 we know such a B exists. By Remark 4.24 we know tensoring with the corresponding pure-injective kills no non-zero object of T m . Combining these two facts we can apply Theorem 4.22 to conclude that B is unique as claimed.
4.26. Remark. All of this suggests that, in situations where the closed point m ∈ Spc(T c ) has quasi-compact complement but T m is not minimal, it could be instructive to examine the collection of all T c -maximal Serre ideals. One could, somewhat speculatively, hope to augment the information coming from the spectrum by this set of subcategories, i.e. to consider Spc T c together with some additional data at each visible point, and use this to try to classify localizing ideals.

Examples and properties of tt-fields
This section should help readers build an intuition of what tt-fields should be. We point out why some naive guesses are not appropriate solutions.
We start with an example in modular representation theory. As explained in Section 1, we want to count this category among tt-fields. Moreover, every indecomposable object of F is endofinite.
Proof. Part (a) is well-known, see [CJ64] or [War69]. Part (c) will follow from (b) by a general argument given below in Corollary 5.6. Similarly, (a) implies F pure-semisimple by a general result of [Kra00, Bel00] (see Theorem 5.7 below). Endofiniteness then follows from finite-dimensionality over k of the homomorphism spaces in F c .
5.2. Example. With the notation of the proposition, if σ ∈ C p is a generator and we let t = σ − 1 then kC p ≃ k[t]/t p . The indecomposable non-projective modules are [i] = k[t]/t i , for i = 1, . . . , p − 1. One can show that for i ≤ j: 5.4. Proposition. Let F : T → S be a coproduct-preserving tt-functor between 'big' tt-categories (Hypothesis 0.1). Then F admits a right adjoint U : S → T satisfying the projection formula for every X ∈ T and Y ∈ S:

Corollary. Let F : T → S be a coproduct-preserving tt-functor between 'big' ttcategories and suppose that every non-zero object of T is ⊗-faithful (e.g. this holds if T is a tt-field, Definition 1.1). Then F is faithful.
Proof. We use the notation of Proposition 5.4. From We shall see in Remarks 5.13 and 5.20 that the hypotheses of Corollary 5.6 are necessary. We also want to explain why being a pure-semisimple triangulated category is not sufficient to be a reasonable candidate for fieldness. Recall the (non-tensor) notion of pure-semisimplicity: 5.7. Theorem. Let F be a compactly generated triangulated category. The following are equivalent: Another illustration of the smallness of tt-fields is the following: The assumption about F being pure-semisimple removes any ambiguity about the meaning of 'every non-zero object is ⊗-faithful'. Suppose the weakest form, namely that the functor x ⊗ − : F c → F c is faithful for every nonzero compact x ∈ F c . Then it immediately follows from Theorem 5.7 (iii) that x ⊗ − : F → F is faithful as well and, then any non-zero X ⊗ − : F → F is faithful. 5.11. Example. Let C p n denote the cyclic group with p n elements and let k be a field of characteristic p. The stable category Stab kC p n is pure-semisimple. This follows, for instance, from the fact that kC p n has finite representation type (see [Bel00,§ 12]). Moreover, the compact part stab kC p n is local and its spectrum is a single point. However, Stab kC p n should morally not be a tt-field for n ≥ 2. Indeed, restriction along the inclusion C p → C p n gives a functor Stab kC p n → Stab kC p which should be regarded as a residue field. In other words, Stab kC p n can be made 'smaller'. Indeed, the tt-category Stab kC p n is not a tt-field in the sense of Definition 1.1 as the p n−1 -dimensional module k(C p n /C p ) is not ⊗-faithful.

5.12.
Example. Consider the group of quaternions Q 8 . As its center C 2 is the maximal elementary abelian subgroup, Spc(stab kQ 8 ) is a point. In this case, the residue field functor is probably given by restriction from Q 8 to C 2 . It is another amusing case of something like an artinian local tt-category whose residue field is still (probably) anétale extension in the sense of [BDS15]. 5.13. Remark. The faithfulness of any coproduct-preserving tt-functor F : F → S out of a tt-field F (Corollary 5.6) cannot hold if F is merely pure-semisimple. Indeed, as in Example 5.11, let G = C p n with n ≥ 2 and C p < G the maximal elementary abelian. Then Res G Cp : Stab(kG) → Stab(kC p ) is a non-faithful tt-functor and Stab(kG) is pure-semisimple.
The following example illustrates that it is not sufficient for an object of a 'big' tt-category T to not kill any compacts under tensoring to justify that the object survives in the residue field. Compare Corollary 4.10. 5.14. Example. Let T = SH (p) denote the p-local stable homotopy category and IS the Brown-Comenetz dual of the sphere spectrum. This object is characterized by the existence of natural isomorphisms Hom T (?, IS) ∼ = Hom Z (π 0 (?), Q/Z). The functor IS ∧− has no kernel on finite spectra, so one might suspect it should survive in some residue field. However, IS ∧ IS ∼ = 0 which suggests that in fact IS must become zero in any residue field. Details and references for the facts in this example can be found in [HP99, Section 7]. * * * We now discuss further properties of tt-fields in the sense of our Definition 1.1. 5.15. Proposition. Let F be a tt-category such that every object of F c is ⊗-faithful (for instance, a tt-field). Then Spc(F c ) is a point.
Proof. We need to show that any non-zero object x ∈ F c generates the whole category as a ⊗-ideal i.e. x = F c . We know that x ⊗ coev x is a split monomorphism, where coev x : 1 → x ∨ ⊗ x is the unit of the x ⊗ − ⊣ x ∨ ⊗ − adjunction. It follows from faithfulness of x ⊗ − that coev x is a split monomorphism, hence 1 ∈ x . (c) Let X ∈ F be a non-zero object and f : Y → Z a morphism in F such that Hom F (c ⊗ f, X) = 0 for all c ∈ F c . Then f = 0.
Proof. For (a), let E = E(B) ∈ F be the pure-injective associated to B as in Construction 3.1. As every M ∈ mod-F c is the image of f : x → y in F c , Corollary 3.6 and ⊗-faithfulness of E imply that B = 0 as wanted. For (b), we have from Theorem 5.7 (iii) that X ≃ i∈I x i with x i ∈ F c compact. Hence hom(X, −) ≃ i∈I hom(x i , −) and hom(x i , −) ∼ = x ∨ i ⊗ − is faithful for x i non-zero. On the other hand, since each x i is a summand of X, it follows that hom(−, x i ) is a summand of hom(−, X) and it suffices to prove that hom(−, x i ) is faithful. We have hom(−, x i ) ∼ = hom(−, 1) ⊗ x i . Indeed, this holds when the source is compact, which is sufficient up to pulling out coproducts as products since the category is pure-semisimple. We are left to show that (−) ∨ = hom(−, 1) is faithful on the whole of F. This is immediate from Theorem 5.7 (iii) again: Let f : Y → Z be a morphism in F and Y ≃ j∈J y j and Z ≃ k∈K z k with y j , z k ∈ F c . Then f is characterized by f kj : y j → z k (using compactness of the y j ). Let j ∈ J and showing that f ∨ = 0. For (c), it now suffices to show that the morphism hom(f, X) is zero in F. As F is phantom-free (Theorem 5.7), it suffices to show that this morphism is a phantom, i.e. that it maps to zero under every Hom F (c, −) for c ∈ F c . The result now follows from adjunction: Hom F (c, hom(f, X)) ∼ = Hom F (c ⊗ f, X) = 0. 5.18. Example. It follows easily from Theorem 5.17 (a) that for every non-zero X ∈ F and M ∈ mod-F c , we haveX ⊗ M = 0 only if M = 0. In fact the argument given in the proof, replacing E by X, shows this. 5.19. Example. One cannot conclude from Theorem 5.17 that every proper Serre subcategory of mod-F c is zero, without the ⊗-ideal assumption. In fact, F can even additively decompose itself, as the following simple example shows. Let F 0 be a tt-field in the sense of Definition 1.1, for instance the derived category of a field k. Let F 1 := F 0 another copy of the same triangulated category and F = F 0 × F 1 with component-wise morphisms. We define the tensor product in a Z/2-graded way: This makes F into a tt-category; for instance 1 F = (1, 0). It is easy to verify that F remains a tt-field in the sense of Definition 1.1. Moreover, (0, 1) is invertible of order two, so every object of F is a direct sum of invertible objects.
On the other hand, F displays some behaviour that is, at least at first glance, not desirable from a provincial point of view on fields. There is a natural collapsing functor π : F → F 0 defined by (x 0 , x 1 ) −→ x 0 ⊕ x 1 . One easily checks that π is strong monoidal and faithful; our field F faithfully (but not fully) embeds via a tensor functor into a 'smaller' field F 0 . This reflects the situation at the level of abelian categories: the category of Z/2Z-graded kvector spaces (of which F is the derived category) exhibits the same embedding into ungraded k-vector spaces. 5.20. Remark. The above example also shows that one cannot simply define a field by requesting that every triangulated functor out of F be faithful. Indeed, the projection on the first factor F ։ F 0 is not faithful. Note that it is not a tt-functor (compare Corollary 5.6). Example 5.19 also shows that Corollary 5.6 cannot hold for the adjoints of tt-functors. Indeed, the projection F = F 0 × F 1 ։ F 0 is (twosided) adjoint to the coproduct-preserving tt-functor F 0 ֒→ F given by inclusion.
Here is a very short characterization of being a tt-field (Definition 1.1). Proof. We have already seen in Theorem 5.17 (b) that the condition is necessary. Conversely, choose a non-zero pure-injective X ∈ F. By Proposition 2.10 (b), the functor hom(−, X) vanishes on phantom maps; but we are also assuming that this functor is faithful. It follows that F is phantomless and we conclude by Theorem 5.7 that every object is a coproduct of compacts. It follows that for every x ∈ T c , the functor hom(−, x) ≃ (−) ∨ ⊗ x is faithful, and in particular on compacts − ⊗ x becomes faithful, which we know suffices by Remark 5.10.

Theorem. A 'big' tt-category F is a tt-field if and only if
We note for future use that the tt-subcategories of tt-fields are fields themselves: 5.22. Proposition. Let F be a tt-field and F : G ֒→ F be a fully-faithful coproductpreserving tt-functor, where G is rigidly-compactly generated. Then G is a tt-field.
Proof. As F is fully-faithful and G is idempotent-complete, for every X ∈ G, every summand of F (X) in F must come from G and it follows that if F (X) ≃ i∈I y i in F then each y i ≃ F (x i ) for x i ∈ G and X ≃ i∈I x i in G. As F is monoidal, it preserves rigids, hence compacts. Conversely, if F (x) is compact, it is clear from F being fully-faithful and coproduct-preserving that Hom G (x, −) ∼ = Hom F (F (x), F (−)) will commute with coproducts; hence such x ∈ G is compact. We have shown that every X ∈ G is a coproduct of compacts. It is clear from F being faithful and a ⊗-functor that every non-zero X ∈ G is ⊗-faithful.

Abelian residues of tt-residue fields
In this section we explore some consequences of the hypothetical existence of a tt-residue field. We emphasize those phenomena which match the results we proved on module categories in Sections 2-4.
Let T be a 'big' tt-category (Hypothesis 0.1) which is local (Definition 4.3). Let us then clarify what we tentatively mean by the existence of a tt-residue field.
6.1. Hypothesis. Assume we are given a coproduct preserving tt-functor F : T −→ F satisfying the following three conditions: (1) The tt-category F is a tt-field in the sense of Definition 1.1. See also Section 5.
(2) The functor F is conservative on the compact-rigid objects, i.e. the preimage of zero x ∈ T c F (x) = 0 is the unique closed point (0) of Spc(T c ).
(3) Every object Y of F is a summand of the image F (X) of some X ∈ T.
6.2. Remark. Any strong monoidal functor preserves rigid objects-having a dual is defined in terms of identity maps, composition, and tensor products, all of which are preserved by a strong monoidal functor. Thus it is automatic from the hypotheses (those above together with the always standing Hypothesis 0.1) that F sends compact objects to compact objects.
6.3. Remark. We have already motivated Condition (2) in Section 1. It means that the induced map Spc(F ) : Spc(F c ) → Spc(T c ) sends the unique point of Spc(F c ) (Proposition 5.15) to the unique closed point of Spc(T c ), as one expects with residue fields. This property can be seen as a tt-generalization of Nakayama for local rings. Condition (3) is a tentative and imperfect way to express that F is some sort of 'quotient' of T, i.e. that F is not too far from the image subcategory F (T). This is inadequate for several reasons, in particular because it does not prevent F : T → F from 'overshooting the mark'. Indeed, even in commutative algebra, such an F could be a field extension of the actual residue field.
However, Condition (3) should be easy to verify in practice since F should be very small. For instance, it clearly holds if every object of F is a coproduct of suspensions of 1 F = F (1 T ), or a summand thereof, as in the topologists' definition.
An interesting open problem would be to replace Condition (3) by a more restrictive one which would capture the idea that F is the 'smallest' field into which T maps, while still satisfying Condition (2). This minimality could be stated by saying that any factorization of F as T → F ′ → F via another tt-field F ′ must be trivial: F ′ ∼ → F. In view of Proposition 5.22, such 'minimality' of F easily implies that F is generated by F (T) as a localizing subcategory. On the other hand, it is not clear how much fullness of F could be obtained from minimality, nor whether Condition (3) would follow. Finally, it would be interesting to prove that any ttfunctor to a field factors via a minimal one, but this also remains elusive. For these reasons, we do not include minimality among our hypotheses. 6.4. Notation. We denote by U : F → T the right adjoint to F , which exists by Proposition 5.4 and satisfies the projection formula (5.5).
6.5. Remark. The rather mild Condition (3) already forces faithfulness of the right adjoint U : F → T, as is well-known. Indeed, Condition (3) implies that the counit ǫ Y : F U (Y ) → Y is a split epimorphism and thus every morphism f : Another immediate consequence of Condition (3) is the following: 6.6. Proposition. Under Hypothesis 6.1, the adjunction F ⊣ U induces an equivalence between the idempotent-completion of the Kleisli category of free modules over the monad U F on the category T and the residue field F, in such a way that our functors F : T → F and U : F → T become respectively the free U F -module functor T → (U F -Free T ) ♮ and the underlying-object functor (U F -Free T ) ♮ −→ T.
Proof. By general category theory [ML98, § VI.5], the Kleisli comparison functor K : U F -Free T → F associated to the (F, U )-adjunction is fully-faithful. Our Condition (3) immediately implies that this functor K is surjective up to direct summand, hence an equivalence on idempotent-completions. 6.7. Remark. Given an exact monad M, like the above M = U F , on a triangulated category T, there is no guarantee that the Kleisli category or its idempotentcompletion (M-Free T ) ♮ be triangulated. In the above result, we heavily use that the monad is already realized by an exact adjunction of triangulated categories.
6.8. Notation. Let E = U (1 F ) the image of the ⊗-unit of F in T.
6.9. Corollary. Under Hypotheses 6.1, the object E = U (1 F ) of T is equipped with the structure of a commutative ring object in T, coming from the fact that U is laxmonoidal. We have an isomorphism of monads U F (−) ∼ = E ⊗ − where the latter is a monad via the above ring structure. We have an equivalence of categories between the idempotent-completion of the Kleisli category of free E-modules in T and the tt-field F, in such a way that the functors F : T → F and U : F → T become the free E-module and underlying-object functors respectively.
Proof. The right-adjoint U to the monoidal functor F is lax-monoidal, hence preserves (commutative) ring objects. This explains the ring structure on E = U (1). The isomorphism of monads is a general consequence of the projection formula; see [BDS15, Lemma 2.8]. The statement now follows from Proposition 6.6.
We now want to discuss the module-category analogue of the above.
6.10. Construction. Our basic framework is outlined in the following diagram (6.11) where the two vertical pairs of functors are adjoint. The functorF : Mod-T c → Mod-F c is the exact colimit-preserving functor that F induces by the universal property (Remark 2.3), that is, the left Kan extension (F c ) ! along the restriction F c of F to compacts. This functorF is strong monoidal. Its right adjoint U : Mod-F c → Mod-T c is the restriction (F c ) * along F c : T c → F c or, equivalently, the exact colimit-preserving functor induced by the universal property of the module category but now applied to U . Exactness of the functorF implies in particular thatÛ preserves injectives or, equivalently, that U preserves pure-injectives.
6.12. Corollary. The ring-object Proof. Every object of F is pure-injective and U preserves pure-injectives.
The following lemma is immediate from the above discussion of the functorÛ . However, we state it and indicate the proof for psychological reasons.
Proof. There is a chain of isomorphismsÛ (ĉ) = F(F −, c) ∼ = T(−, U c) = U (c), the first by the definition ofÛ as the restriction along F c , the second by adjunction, and the third again by definition.
Proof. ProvingÛ is faithful is equivalent to showing that the counit ǫ :FÛ → Id of the adjunction betweenF andÛ is an epimorphism. For finitely presented projectives this is clear from the fact that U is faithful: if c ∈ F c we haveFÛ (ĉ) ∼ = F U (c) and under this identification ǫ isǫ. Given this, we can use naturality ofǫ, the fact thatF andÛ are coproduct-preserving, and the fact that every object M ∈ Mod-T c is a quotient of a coproduct of finitely-presented projectives to deduce that everyǫ M is an epimorphism. 6.15. Proposition. The objectÊ =Û (1 F ) of A = Mod-T c is a commutative ringobject, which is injective. Moreover,Û is monadic and identifies Mod-F c with the Eilenberg-Moore categoryÊ-Mod A of modules in A over this ring-object.
Proof. The functorÛ is faithful by Lemma 6.14 and so we can apply Proposition A.7 to deduce that it is monadic. All that remains is to identify the monadÛF with the monad associated toÛ (1 F ) ∼ =Ê. This is a consequence of Corollary 6.9. 6.16. Remark. It follows, more or less, from Remark 5.8 that the object E is not just pure-injective, but actually endofinite. Indeed, by said remark the object 1 F is endofinite, by virtue of the pure-semisimplicity of F. To conclude that E is also endofinite it is enough to note that U preserves endofiniteness. This follows from the fact that U preserves products (as a right adjoint) and coproducts (its left adjoint preserves compacts cf. Remark 6.2), see for instance [KR01, Corollary 3.8].

* * *
We now wish to construct, in the spirit of Sections 2 and 3, a quotient of A = Mod-T c which is closer to Mod-F c .
6.17. Proposition. Under Hypothesis 6.1, consider the localizing ⊗-ideal Ker(F ) of A = Mod-T c and B = Ker(F ) ∩ A fp , its Serre ⊗-ideal subcategory of finitely presented objects. Then Ker(F ) is locally finitely presented, that is, Proof. This follows from the fact thatF preserves finitely presented objects and a general Grothendieck category argument. See Proposition A.6.
6.18. Construction. We can extend our diagram (6.11) by factoringF as follows Here Q is the Gabriel quotient and R its right adjoint as in Proposition 2.13. The functorF is characterized byF • Q =F and is therefore exact. By uniqueness of right adjoints, we must haveÛ ∼ = R •Ū , and therefore, applying Q(−), we get 6.20. Lemma. In the above diagram, the functorF :Ā → Mod-F c is faithful and strong monoidal, andŪ is faithful. In particular,Ū is monadic and we can identify Mod-F c with the Eilenberg-Moore categoryĒ-ModĀ ofĒ-modules inĀ over the ring-objectĒ =Ū (1 F ) ∼ = Q h(E).
Proof. It is clear that the functorF is faithful since it is conservative and exact (between abelian categories) and it is strong monoidal as is a factorization of the strong monoidal functorF through the monoidal localization corresponding to its kernel. We have seen in Lemma 6.14 thatÛ is faithful and so it follows from the isomorphismÛ ∼ = RŪ thatŪ is faithful. By Beck's monadicity theorem (Proposition A.7) this showsŪ is monadic. Hence Mod-F c is the category of modules inĀ over the monadŪF . Now,ŪF Q ∼ = QÛF ∼ = Q U F ∼ = Q(Ê ⊗ −) ∼ =Ē ⊗ Q and therefore the monadŪF ∼ =Ē ⊗ − is again 'monoidal' given by the ringĒ. Proof. SinceF is monoidal its kernel Ker(F ) is a ⊗-ideal. Thus B = (KerF ) fp is a Serre ⊗-ideal in A fp . We have assumed in Condition (2) that F has no non-zero compact objects in its kernel so B does not meet ĉ c ∈ T c , c = 0 as wanted. Thus there exists a T c -maximal Serre ⊗-ideal B ′ containing B and we claim it is equal to B. Suppose ab absurdo that there exists M ∈ B ′ \ B. This implies thatF (M ) = 0 in mod-F c . By Theorem 5.17 (a), we know that this non-zero object must generate the whole of mod-F c as a Serre ⊗-ideal. By Lemma 2.12, mod-F c is generated as a Serre (non-ideal) subcategory by ŷ ⊗F (M ) y ∈ F c . Hence Mod-F c is generated as a localizing (non-ideal) subcategory by the same objects. By Condition (3), we can replace the collection of y ∈ F c by F (X) for X ∈ T. So, we have shown that Mod-F c is generated as a localizing subcategory by F (X ⊗ M ) X ∈ T . In particular,1 F belongs to that localizing subcategory. ApplyingÛ , we see thatÊ =Û (1 F ) belongs to the localizing subcategory of A In summary, we have proved that if T has a tt-residue field F as in Hypothesis 6.1 then it gives rise to a T c -maximal Serre ⊗-ideal B in mod-T c . Moreover, the abelian shadow Mod-F c of the tt-field F can be reconstructed from the abelian residue field A that we have constructed. Thus if there is an honest tt-residue field it gives rise to the structure we have been considering in Mod-T c .
We conclude with an example.
6.23. Example. Let R = (R, m, k) be a discrete valuation ring. Then D(R) is local and π * : D(R) → D(k) verifies Hypothesis 6.1. Moreover, we know that D m (R) is minimal. Thus we can apply Corollary 4.25 to deduce that there is a unique D perf (R)-maximal Serre ⊗-ideal B of mod-D perf (R). By Proposition 6.21 it is none other than the kernel of the induced functor π * : mod-D perf (R) → mod-D perf (k).
The subcategory B can be described a bit more explicitly as follows. Let t ∈ m be a uniformizer and consider the corresponding triangle R t / / R π / / k / / ΣR giving rise to k. The object I = Im(t :R →R) in mod-D perf (R) is none other than the I from Proposition 3.9 and thus one can describe B as the Serre ⊗-ideal generated by I. The corresponding E is just the residue field k, which is endofinite and thus pure-injective as required.

Appendix A. Generalities on Grothendieck categories
We record some general facts and constructions concerning Grothendieck categories which we used throughout. Everything in this section is standard, although there do not necessarily exist convenient references. As a result we indicate some of the proofs. We tacitly assume that all functors are additive unless explicitly mentioned otherwise. We begin with a well-known fact. Proof. For each a ∈ A the existence of hom(a, −) follows from Theorem A.1 applied to the colimit-preserving functor a ⊗ −. Given a morphism f ∈ A(a, a ′ ) there is a corresponding natural transformation a⊗(−) → a ′ ⊗(−), which induces a morphism hom(a ′ , −) → hom(a, −) between right adjoints. It is routine to verify that these transformations assemble to give a bifunctor hom which is right adjoint to ⊗ in both variables.
A.3. Remark. The Grothendieck categories we use in the text are locally coherent, meaning the finitely presented objects form an abelian subcategory A fp ⊂ A and every object of A is a filtered colimit of finitely presented ones (i.e. A is locally finitely presented ).
We fix a locally coherent Grothendieck category A with colimit-preserving tensor ⊗ and enough flats with respect to ⊗. We now want to justify the various claims made in Proposition 2.13, most of which can be found in [Gro57,Gab62].
Given B ⊆ A fp a Serre ⊗-ideal, we can consider (Assuming P finitely generated projective does not change the above definition in the case of A = Mod-T c , i.e. we can test the condition with P =x for x ∈ T c .) These are the objects M such that every finitely generated subobject of M is a quotient of an object of B, i.e. the filtered colimits (in A) of objects of B. The above category − → B is a localizing (i.e. Serre and closed under coproducts) ⊗-ideal of A. For the ⊗-ideal property, note that −→ A fp = A and that B is ⊗-ideal in A fp . Moreover, the subcategory of finitely presented objects of − → B exactly coincides with B = ( − → B ) fp = − → B ∩ A fp . The Grothendieck-Gabriel quotientĀ = A/L by the Serre subcategory L is the localization with respect to all morphisms s : M → M ′ whose kernel and cokernel belong to L. We have a localization functor Q which admits a right adjoint R when L is localizing. To check that the tensor descends to the quotient ⊗ :Ā ×Ā →Ā in such a way that Q : A →Ā is monoidal, it suffices to check that the collection of morphisms s : M → M ′ with kernel and cokernel in L are preserved by tensoring with any N ∈ A. For this, we use of course that L is a ⊗-ideal. Decomposing s, we can treat separately the case where s is a monomorphism with cokernel in L, and the case where s is an epimorphism with kernel in L; the latter is easy since Ker(s ⊗ N ) is a quotient of Ker(s) ⊗ N . For the former, it is enough to convince oneself that for any L ∈ L the object Tor 1 (N, L) lies in L. This follows by computing with a flat resolution of N and using that L is both Serre and a ⊗-ideal. Now, for every flat object M ∈ A, we have a commutative diagram / /Ā whose top-right composition is exact and therefore so is the bottom functor, since Q is universal among exact functors. Therefore Q(M ) remains flat.
The induced monoidal structure onĀ is compatible with colimits, i.e. for every Y ∈Ā the functor Y ⊗(−) preserves colimits. Indeed, suppose J is a small category and F : J →Ā is a functor. We can assume Y is of the form QM , for instance by using the natural isomorphism IdĀ ∼ = QR. Colimit preservation is a consequence of the following string of natural isomorphisms Hence by Proposition A.2, the symmetric monoidal Grothendieck categoryĀ is closed. We denote the internal hom on A andĀ by hom A and homĀ respectively. The final statement of the lemma then comes down to noting that for Y, Y ′ ∈Ā R(homĀ(Y, Y ′ )) ∼ = R(homĀ(QRY, Y ′ )) ∼ = hom A (RY, RY ′ ).
A.6. Proposition. Let A and C be locally coherent Grothendieck categories (Remark A.3) and F : A → C an exact colimit-preserving functor which sends finitely presented objects to finitely presented objects. Then Ker F is generated by finitely presented objects of A, i.e. Ker F is the filtered colimit closure of Ker F ∩ A fp .
Proof. Replacing A by A modulo the localising subcategory generated by Ker F ∩ A fp and F by the induced functor to C we can reduce to showing that if Ker F ∩ A fp is trivial then F has no kernel. Suppose then that this is the case. We first note that F is faithful on A fp . This is true for any exact conservative functor on an abelian category, which one sees by testing vanishing via the image object. As the functor F preserves colimits, by Theorem A.1, it has a right adjoint G which must preserve filtered colimits because F preserves finitely presented objects. As shown above F is faithful when restricted to A fp . Thus the components of the unit η of this adjunction at finitely presented objects are monomorphisms. Since A is locally coherent it is, in particular, locally finitely presented -every object X of A is a filtered colimit of finitely presented objects. As both F and G preserve filtered colimits we thus see that η X can be written as a filtered colimit of components of η at finitely presented objects. So we see η X is a filtered colimit of monomorphisms and hence, since A is Grothendieck, is itself a monomorphism. This proves F is faithful and completes the argument.