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S A mdls u o nall in not but -modules, uhta h truncated the that such e [ per Gro57 / ≃ C sascae to associated ds e,ieptn monad, idempotent ies, X A A ⊂ C A L / Ser55 stectgr of category the as and d ≃ C ≥ ,adsnethen since and ], Coh d 0 C H ,equivalent ], slocalizing is X Coh stethick the is 0 ( nwhich in F X Coh ( d but , )) X is 2 MOHAMEDBARAKATANDMARKUSLANGE-HEGERMANN in the following sense a distinguished representative within this class; it is a so-called saturated object with respect to C. The appropriate categorical setup for a Serre quotient A/C of an A modulo a thick subcategory C ⊂A was introduced by Grothendieck [Gro57, Chap. 1.11] and then more elaborately in Gabriel’s thesis [Gab62]. Later Gabriel and Zisman developed in [GZ67] a localization theory of categories in which a Serre quotient A/C is an outcome of a special localization A→A[Σ−1] ≃ A/C. Their theory is also general enough to enclose Verdier’s localization of triangulated categories, which he used in his 1967 thesis (cf. [Ver96]) to define derived categories. Thanks to Simpson’s work [Sim06] the Gabriel-Zisman localization is now completely formalized in the proof assistant Coq [Coq04]. In many applications, as assumed in [GZ67], the localization A[Σ−1] is equivalent to a full subcategory of A, the subcategory of all Σ-local objects of A. This favorable situation (which in our context means that C is a localizing subcategory of A) is characterized by the existence of an idempotent monad associated to the localization. For a further overview on localizations we refer to the arXiv version of [Tho11]. In our application to Coh X we are in the setup of Serre quotient categories A/C, which are the outcome of an exact localization having an associated idempotent monad. We call this monad the Gabriel monad (cf. Definition 3.3). Gabriel monads satisfy a set of properties which we use as a simple set of axioms to define what we call a C-saturating monad. The goal of this paper is to characterize Gabriel monads conversely as C-saturating monads (Theorem 4.6). Such a characterization enables us in [BLH14b] to show that several known algorithmically computable in the context of coherent sheaves on a projective scheme1 are C-saturating2. This yields a constructive, unified, and simple proof that those functors are equivalent to the M 7→ L H0(M(d)), and hence compute the truncated module d≥d0 f of twisted global sections. Among those are the functors computing the graded ideal transform and the graded module given by the 0-th strand of the Tate resolution. The proof there relies on checking the defining set of axioms of a C-saturating monad, which turns out to be a relatively easy task. In particular, the proof does not rely on the (full) BGG correspondence [BGG78] of triangulated categories, the Serre-Grothendieck correspondence [BS98, 20.3.15], or the local duality, as used in [EFS03]. A stronger computability notion is that of the Ext-computability. Furthermore, in [BLH14d] we use the Gabriel monad of a localizing Serre quotient A/C to show that the so-called Ext- computability of A/C follows from that of A. In particular, the Abelian category Coh X is Ext-computable (cf. [BLH14b]).

2. PRELIMINARIES In this short section we collect some standard categorical preliminaries, which should make this paper self-contained for the readers interested in our applications to coherent sheaves [BLH14b]. For details we refer to [Bor94a, Bor94b] or the active nLab wiki [nLa12].

2.1. and monads.

1This technique applies to other classes of varieties admitting a finitely generated Cox ring S. 2In this context the Σ-local objects are Gabriel’s C-saturated objects. MONADSOFREFLECTIVELOCALIZATIONSOFABELIANCATEGORIES 3

Definition 2.1. Two functors F : A → B and G : B→A are called a pair of adjoint functors and denoted by F ⊣ (G : B→A) if there exists a binatural of Hom functors HomB(F (−), −) ≃ HomA(−, G (−)). F is called a left adjoint of G and G is a right adjoint of F . Remark 2.2. Two right adjoints (resp. two left adjoints) are naturally isomorphic. Proposition 2.3. An adjunction F ⊣ (G : B→A) is characterized by the existence of two natural transformations

δ : F ◦ G → IdB and η : IdA → G ◦ F , called the counit3 and unit4 of the adjunction, respectively, such that the two zig-zag (or - counit) identities hold, i.e., the compositions of the natural transformations F η F ηG G (zz) F −−→ F ◦ G ◦ F −−→δ F and G −→ G ◦ F ◦ G −→δ G must be the identity on functors. Definition 2.4. A monad on a category A is an endofunctor W : A→A together with two natural transformations 2 η : IdA → W and µ : W → W , called the unit and multiplication of the monad, respectively, such that following two coher- ence conditions hold:

µ ◦ W µ = µ ◦ µW and µ ◦ W η = µ ◦ ηW = IdW . Remark 2.5. Let F ⊣ (G : B→A) be an adjoint pair with unit η and counit δ. The composed endofunctor W := G ◦ F : A→A together with the unit of the adjunction η : IdA → G ◦ F as the monad unit η : IdA → W and µ := G δF : W 2 → W as the monad multiplication is a monad. It is called the monad associated to the adjoint pair. 2.2. Idempotent monads and reflective localizations. Definition 2.6. A monad (W ,η,µ) is called an idempotent monad if the multiplication µ : W 2 → W is a natural isomorphism. Definition 2.7. A full subcategory B⊂A is called a reflective subcategory if the inclusion functor ι : B→A has a left adjoint, called the reflector. Proposition 2.8 ([Bor94b, §4.2]). Let (W , η,µ) be a monad on the category A. Then the following statements are equivalent: e (a) The monad (W , η,µ) is idempotent. (b) The two natural transformationse W η, ηW : W → W 2 are equal. (c) The (essential) image of W in A is ae reflectivee subcategory.

3 δb corresponds to 1G b under the natural isomorphism HomB((F ◦ G )b,b) =∼ HomA(G b, G b). 4 ηa corresponds to 1Fa under the natural isomorphism HomA(a, (G ◦ F )a) =∼ HomB(F a, F a). 4 MOHAMEDBARAKATANDMARKUSLANGE-HEGERMANN

(d) The corestriction G W W of W to its (essential) image W is left adjoint b := co-res (A) (A) to the inclusion functor of W into : G W . ι (A) A b ⊣ (ι : (A) →A) (e) There exists an adjunction F ⊣ (G : B→A) with unit η and counit δ where G is fully faithful such that its associated monad (G ◦ F , η, G δF ) is isomorphic to (W , η,µ). e A localization of a category A at collection of Σ is roughly speaking the process of adjoining formal inverses of the morphisms in Σ (cf. [Wei94, Definition 10.3.1]). Definition 2.9 ([GZ67, §I.1.1]). A localization A→A[Σ−1] (at a subset of morphisms Σ of A) is called reflective localization if it admits a fully faithful right adjoint G : A[Σ−1] →A. The following proposition states, among other things, that all functors having fully faithful right adjoints are in fact reflective localizations. Proposition 2.10 ([GZ67, Proposition I.1.3]). Let F ⊣ (G : B→A) be a pair of adjoint functors with unit η and counit δ. Further let (W ,η,µ) := (G ◦ F , η, G δF ) be the associated monad. The following statements are equivalent: (a) G is fully faithful (inducing an equivalence between B and its (essential) image, a re- flective subcategory of A). (b) The associated monad (W ,η,µ) is idempotent5, F is essentially surjective, and G is conservative6. 7 (c) The counit δ : F ◦ G → IdB is a natural isomorphism . (d) If Σ is the set of morphisms ϕ ∈A such that F (ϕ) is invertible in B, then F : A → B realizes the reflective localization A→A[Σ−1]. In particular, W (A) ≃ G (B) ≃B≃A[Σ−1].

3. GABRIEL LOCALIZATIONS In this section we recall some results about Serre quotients arising from reflective localiza- tions. From now on A is an Abelian category. Definition 3.1 ([Wei94, Exer. 10.3.2]). A non-empty full subcategory C of an Abelian cate- gory A is called thick8 if it is closed under passing to subobjects, factor objects, and extensions. In other words, for every short exact sequence 0 → M ′ → M → M ′′ → 0 in A the object M lies in C if and only if M ′ and M ′′ lie in C. Pierre Gabriel defines in his thesis [Gab62, §III.1] the (Serre) A/C with the same objects as A and with Hom-groups defined by Hom (M, N) = lim Hom (M ′, N/N ′), A/C −→ A M ′,N ′

5In this case one says that the adjunction F ⊣ (G : B → A) is idempotent. 6i.e., G reflects : G (α) isomorphism =⇒ α isomorphism. 7In particular, G is a right inverse of F and F is essentially surjective. 8Gabriel uses the notion épaisse. Thick subcategories are automatically replete. MONADSOFREFLECTIVELOCALIZATIONSOFABELIANCATEGORIES 5 where the direct is taken over all subobjects M ′ ≤ M and N ′ ≤ N such that M/M ′ and N ′ belong to C. Further he defines the canonical functor Q : A→A/C which is the identity 9 on objects and which maps a ϕ ∈ HomA(M, N) to its image in HomA/C (M, N) under the maps ′ ′ ′ ′ .( Hom(M ֒→ M, N ։ N/N ) : HomA(M, N) → HomA(M , N/N Gabriel proves in [Gab62, Proposition III.1.1] that A/C is an Abelian category and that the canonical functor Q : A→A/C is exact. The canonical functor Q : A→A/C fulfills the following universal property. Proposition 3.2 ([Gab62, Cor. III.1.2]). Let C be a thick subcategory of A and G : A→D an into the Abelian category D. If G (C) is zero then there exists a unique functor H : A/C→D such that G = H ◦ Q. Definition 3.3. The thick subcategory C of the Abelian category A is called a localizing subcategory if the canonical functor Q : A→A/C admits a right adjoint S : A/C→A, i.e., a functor together with a binatural isomorphism

HomA/C(Q(−), −) ≃ HomA(−, S (−)). S is called the10 section functor of Q. Denote by

δ : Q ◦ S → IdA/C and η : IdA → S ◦ Q the counit and unit of the adjunction S ⊣ Q, respectively. We call such a canonical functor Q a Gabriel localization and the associated monad (S ◦ Q,η,µ = S δQ) the Gabriel monad. The following definition describes the Σ-local objects in the sense of Gabriel-Zisman, where Σ is the collection of all morphisms in A with both and cokernel in C. Definition 3.4. Let C be a thick subcategory of A. We call an object M in A C-saturated (or C-closed11) if it has no non-trivial subobjects in C and every extension of M by an object C ∈ C is trivial, i.e., each short exact sequence 0 → M → E → C → 0 in A is split. Define

SatC(A) := full (replete) subcategory of C-saturated objects. The following proposition together with Proposition 2.10 imply that A/C is a reflective local- −1 ization; more precisely A/C≃A[Σ ] ≃ SatC(A), where Σ is the collection of all morphisms in A with both kernel and cokernel in C. Proposition 3.5 ([Gab62, Proposition III.2.3 and its corollary]). Let C⊂A be a localizing subcategory of the Abelian category A with section functor S : A/C→A. (a) S : A/C→A is left exact and preserves products12. ∼ (b) The counit of the adjunction δ : Q ◦ S −→ IdA/C is a natural isomorphism.

9So Q is surjective (on objects) but in general neither faithful nor full. 10Cf. Remark 2.2. 11Gabriel uses fermé. 12Right adjoint functors preserve kernels and products [HS97, Theorem II.7.7]. 6 MOHAMEDBARAKATANDMARKUSLANGE-HEGERMANN

(c) An object M in A is C-saturated if and only if ηM : M → (S ◦ Q)(M) is an isomor- phism, where η is the unit of the adjunction. Corollary 3.6. Let C ⊂A be a localizing subcategory of the Abelian category A. Then S (A/C) ≃ SatC(A) are reflective subcategories of A, in particular, the following holds: (a) The section functor S is fully faithful. (b) S is conservative. (c) η(S ◦ Q)=(S ◦ Q)η. Proof. Proposition 3.5.(b) establishes the context of Propositions 2.10 and 2.8. 

The existence of enough C-saturated objects turns SatC(A) into an “orthogonal complement” of C. Definition 3.7. Let C be a thick subcategory of A. We say that A has enough C-saturated objects if for each M ∈ A there exists a C-saturated object N and a morphism ηM : M → N such that HC(M) := ker ηM ∈ C. 13 It follows from the fundamental theorem on homomorphisms that HC(M) is the maximal subobject of M in C. Proposition 3.8 ([Gab62, Proposition III.2.4]). The thick subcategory C⊂A is localizing iff A has enough C-saturated objects. Remark 3.9. The existence of the maximal subobject in C simplifies the description of Hom- groups in A/C, namely Hom (M, N) = lim Hom (M ′,N/H (N)). A/C −→ A C M ′֒→M M/M ′∈C The existence of the section functor S as the right adjoint of Q reduces the computability of Hom-groups in A/C ≃ SatC(A) even further to their computability in A ∼ HomA/C(M, N) = HomA/C (Q(M), Q(N)) = HomA(M, (S ◦ Q)(N)), avoiding the direct limit in the definition of HomA/C completely.

Corollary 3.10. The image S (A/C) of S is a subcategory of SatC(A) and the inclusion functor S (A/C) ֒→ SatC(A) is an equivalence of categories with the restricted-corestricted Gabriel monad S ◦ Q : SatC(A) → S (A/C) as a quasi-inverse. In other words, SatC(A) is the essential image of S and, hence, of the Gabriel monad S ◦ Q.

Proof. The inclusion of S (Q(A)) = S (A/C) in SatC(A) will follow from Proposition 3.5.(c) as soon as we can show that ηS , or equivalently η(S ◦ Q), is a natural isomorphism. Propo- sition 2.10 applied to Proposition 3.5.(b) states that the multiplication µ of the Gabriel monad S ◦Q is a natural isomorphism. Finally, the second coherence condition µ◦η(S ◦Q) = IdS ◦Q implies that η(S ◦ Q)= µ−1 is also a natural isomorphism. 

13 The notation HC is motivated by the notation for local cohomology. MONADSOFREFLECTIVELOCALIZATIONSOFABELIANCATEGORIES 7

Corollary 3.11. The restricted canonical functor Q : SatC(A) →A/C and the corestricted section functor S : A/C → SatC(A) are quasi-inverse equivalences of categories. In particu- lar, SatC(A) ≃ S (A/C) ≃A/C is an Abelian category. Q) S ◦ A ) ( Q (A C Sat ι S S ◦ Q := cores Qb Q SatC(A) S (A/C)= S (Q(A)) A/C S Q S ◦ | SatC (A)

Define the reflector Q := co-res (S ◦ Q): A → Sat (A). The adjunction Q ⊣ (ι : b SatC(A) C b → SatC(A) ֒→A) corresponds under the above equivalence to the adjunction Q ⊣ (S : A/C . They both share the same adjunction monad S Q Q . In particular, the A) ◦ = ι ◦ b : A→A reflector Q is exact14 and is left exact. b ι Let C ⊂A be a localizing subcategory of the Abelian category A. S (A/C) ≃ SatC(A) are not in general Abelian subcategories of A in the sense of [Wei94, p. 7], as short exact sequences in SatC(A) are not necessarily exact in A. This is due to the fact that the left exact section functor S (cf. Proposition 3.5.(a)) is in general not exact; the cokernel in SatC(A) ⊂A differs from the cokernel in A (cf. [BLH14c]). The full subcategory AC ⊂ A consisting of all objects with no nontrivial subobject in C is 15 a pre-Abelian category . The kernel of a morphisms in AC is its kernel in A. The cokernel of a morphism in AC is isomorphic to its cokernel as a morphism in A modulo the maximal subobject in C, in case C is a localizing subcategory (cf. Proposition 3.8). An in that case SatC(A) ⊂ AC is the completion of AC with respect to the property that every extension by an object in C is trivial. This completion is given by the Gabriel monad S ◦ Q restricted to AC.

4. CHARACTERIZING GABRIELLOCALIZATIONSAND GABRIEL MONADS The next proposition states that in fact all exact reflective localizations in the setup Abelian categories are (reflective) Gabriel localizations. Proposition 4.1 ([Gab62, Proposition III.2.5], [GZ67, Chap. 1.2.5.d]). Let Q S e ⊣ ( f: B → be a pair of adjoint functors of Abelian categories. Assume, that Q is exact and the counit A) e Q S of the adjunction is a natural isomorphism. Then Q is a localizing δ : e ◦ f→ IdB C := ker e

14 SatC(A) is then a Giraud subcategory. 15although not an Abelian category, in general, as monomorphisms need not be kernels of their cokernels. For example, the monomorphism 2 : Z → Z in AC = {f.g. torsion-free Abelian groups} is not kernel of its cokernel (which is zero), where A := {f.g. Abelian groups}⊃C := {f.g. torsion Abelian groups}. This gives an example of a morphism which is monic and epic in AC but not an isomorphism. 8 MOHAMEDBARAKATANDMARKUSLANGE-HEGERMANN subcategory of and the adjunction Q S induces an adjoint equivalence from A e ⊣ ( f: B→A) B to A/C. Now, we approach the central definition of this paper which collects some properties of Gabriel monads. Definition 4.2. Let C ⊂A be a localizing subcategory of the Abelian category A and ι : SatC(A) ֒→A the full embedding. We call an endofunctor W : A→A together with a natural transformation η : IdA → W C-saturating if the following holds: e (a) C ⊂ ker W , (b) W (A) ⊂ SatC(A), G W (c) := co-resSatC (A) is exact, (d) ηW = W η, and e e W 16 (e) ηι : IdA|SatC(A) → | SatC(A) is a natural isomorphism . e Let H be the unique functor from Proposition 3.2 such that G = H ◦Q. We call the composed functor H H the colift of W along Q, since W G H Q. f:= ι ◦ = ι ◦ = f◦ Lemma 4.3. Let Q : A→A/C be a Gabriel localization with section functor S . Then each C-saturating endofunctor W of A is naturally isomorphic to S ◦ Q. Furthermore, the colift H of W along Q is also a section functor naturally isomorphic to S . f G W H Proof. For := co-resSatC(A) let be the unique functor from Proposition 3.2 (using Definition 4.2.(a)) such that G = H ◦ Q. G = H ◦ Q

≃ H ◦ Q ◦ S ◦ Q (IdA/C ≃ Q ◦ S by Proposition 3.5.(b)) = G ◦ S ◦ Q S Q S Q ≃ IdSatC (A) ◦ ◦ (using ( (A)) ⊂ SatC(A) and 4.2.(e)) S Q = co-resSatC(A)( ◦ ). Then, using the notation of Definition 4.2,

W H Q G S Q. = f◦ = ι ◦ ≃ ◦ This also proves the equivalence H S , as Q is surjective.  f≃ Proposition 4.4. Let Q : A→A/C be a Gabriel localization and (W , η) be a C-saturating e endofunctor of with colift H along Q. Then there exists a natural transformation Q A f δe : ◦ H → IdA/C such that Q and H form an adjoint pair Q ⊣ H with unit η and counit δ. f f f e e Definition 4.5. Hence, each C-saturating endofunctor (W , η) is the monad (W , η, H δQ) e e fe associated to the adjunction Q H . We call it a -saturating monad. ⊣ f C

16 In particular, W (A) is an essentially wide subcategory of SatC(A). MONADSOFREFLECTIVELOCALIZATIONSOFABELIANCATEGORIES 9

Proof of Proposition 4.4. We define a natural transformation Q H and check δe : ◦ f → IdA/C the two zig-zag identities (zz), i.e., that the compositions of natural transformations

Qe eQ eHf Hfe Q η Q H Q δ Q and H η H Q H δ H −→ ◦ f◦ −→ f−−→ f◦ ◦ f−−→ f 4.2.(d) are the identity of functors. By 4.2.(e) we know that (H ◦Q)η = W η = ηW =(ηι)G is an f e e e e isomorphism. Hence, also Q is a natural isomorphism, because the functor H is equivalent to η f S by Lemma 4.3 and, thus, reflectse isomorphisms by Lemma 3.6.(b). This allows us to define in such a way to satisfy the first zig-zag identity, i.e., set Q Q −1. This defines as Q δe δe := ( η) δe is surjective (on objects). The second zig-zag identity is equivalent, againe due to the surjectivity of Q, to the second zig-zag identity applied to Q, i.e., (H δQ) ◦ η(H ◦ Q) being the identity fe e f transformation of the functor H Q W . Now f◦ = (H δQ) ◦ η(H ◦ Q)=(H (Qη)−1) ◦ η(H ◦ Q) (by the definition δQ := (Qη)−1) fe e f f e e f e e = ((H ◦ Q)η)−1 ◦ η(H ◦ Q) f e e f 4.2.(d) =(η(H ◦ Q))−1 ◦ η(H ◦ Q) (using η(H ◦ Q) = (H ◦ Q)η) e f e f e f f e f = IdH ◦Q .  We now approach our main result. Theorem 4.6 (Characterization of Gabriel monads). Each Gabriel monad is a C-saturating monad. Conversely, each C-saturating monad is equivalent to a Gabriel monad. Proof. The conditions in Definition 4.2 clearly apply to a Gabriel monad S ◦ Q by definition of the canonical functor Q, Corollary 3.10, Corollary 3.11, Corollary 3.6.(c), and Proposi- tion 3.5.(c), respectively. The converse follows directly from Lemma 4.3 and Proposition 4.4 which prove that the two adjunctions Q H and Q S are equivalent, and so are their associated monads.  ⊣ f ⊣ REFERENCES [BGG78] I. N. Bernšte˘ın, I. M. Gel′fand, and S. I. Gel′fand, Algebraic vector bundles on Pn and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67. MR MR509387 (80c:14010a) 2 [BLH11] Mohamed Barakat and Markus Lange-Hegermann, An axiomatic setup for algorithmic homological algebra and an alternative approach to localization, J. Algebra Appl. 10 (2011), no. 2, 269–293, (arXiv:1003.1943). MR 2795737 (2012f:18022) 1 [BLH14a] Mohamed Barakat and Markus Lange-Hegermann, Characterizing Serre quotients with no section functor and applications to coherent sheaves, Appl. Categ. Structures 22 (2014), no. 3, 457–466, (arXiv:1210.1425). MR 3200455 1 [BLH14b] Mohamed Barakat and Markus Lange-Hegermann, A constructive approach to coherent sheaves via Gabriel monads,(arXiv:1409.6100), 2014. 2 [BLH14c] Mohamed Barakat and Markus Lange-Hegermann, Gabriel morphisms and the computability of Serre quotients with applications to coherent sheaves,(arXiv:1409.2028), 2014. 7 [BLH14d] Mohamed Barakat and Markus Lange-Hegermann, On the Ext-computability of Serre quotient cate- gories, Journal of Algebra 420 (2014), no. 0, 333–349, (arXiv:1212.4068). 2 10 MOHAMEDBARAKATANDMARKUSLANGE-HEGERMANN

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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KAISERSLAUTERN, 67653 KAISERSLAUTERN, GER- MANY E-mail address: [email protected]

LEHRSTUHL B FÜR MATHEMATIK, RWTH AACHEN UNIVERSITY, 52062 AACHEN,GERMANY E-mail address: [email protected]