A resolution theorem for extriangulated categories with applications to the index

Quillen's Resolution Theorem in algebraic $K$-theory provides a powerful computational tool for calculating $K$-groups of exact categories. At the level of $K_0$, this result goes back to Grothendieck. In this article, we first establish an extriangulated version of Grothendieck's Resolution Theorem. Second, we use this Extriangulated Resolution Theorem to gain new insight into the index theory of triangulated categories. Indeed, we propose an index with respect to an extension-closed subcategory $\mathscr{N}$ of a triangulated category $\mathscr{C}$ and we prove an additivity formula with error term. Our index recovers the index with respect to a contravariantly finite, rigid subcategory $\mathscr{X}$ defined by J{\o}rgensen and the second author, as well as an isomorphism between $K_0^{\mathsf{sp}}(\mathscr{X})$ and the Grothendieck group of a relative extriangulated structure $\mathscr{C}_{R}^{\mathscr{X}}$ on $\mathscr{C}$ when $\mathscr{X}$ is $n$-cluster tilting. In addition, we generalize and enhance some results of Fedele. Our perspective allows us to remove certain restrictions and simplify some arguments. Third, as another application of our Extriangulated Resolution Theorem, we show that if $\mathscr{X}$ is $n$-cluster tilting in an abelian category, then the index introduced by Reid gives an isomorphism $K_0(\mathscr{C}_R^{\mathscr{X}}) \cong K_0^{\mathsf{sp}}(\mathscr{X})$.


Introduction
Algebraic K-theory has its roots in geometry, drawing its name from Grothendieck's proof of what is now known as the Grothendieck-Riemann-Roch theorem [BS58], [SGA6].The base case of this theory focuses on the Grothendieck group K 0 (C ) of a, classically, (skeletally small) abelian or exact category C , which is defined in a purely algebraic fashion.The abelian group K 0 (C ) is the free group generated on the set of isomorphism classes [A], for A ∈ C , modulo the relations [A] − [B] + [C] for each admissible short exact sequence A B ։ C in C .However, a priori, computing K 0 (C ) may be quite difficult.For example, if R is a ring and mod R is the exact category of finitely generated (left) R-modules, then determining K 0 (mod R) has led to a rich literature, e.g.[HR65,HMW92,Hol15].
A Resolution Theorem, attributed to Grothendieck (see Bass [Bas68,Thm. VIII.8.2], or Theorem 4.4), shows that if we can identify a suitable subcategory D of C , then K 0 (C ) ∼ = K 0 (D).Here, 'suitable' means each object in C admits a finite resolution by objects in D (see Definition 4.2), and D is an extension-closed subcategory of C that is also closed under taking kernels of admissible deflations.In the case C = mod R, if every M ∈ mod R has finite projective dimension, then we may choose D = proj R, the subcategory of finitely generated projective R-modules.Moreover, since all short exact sequences in proj R split, the group K 0 (proj R) is in principle simpler to compute.
The first aim of this article is to establish an extriangulated version of the Resolution Theorem (see Theorem A below).The notion of an extriangulated category was introduced by Nakaoka-Palu [NP19] and is a unification of Quillen's exact categories and Grothendieck-Verdier's triangulated categories.Extriangulated category theory serves as a convenient framework in which to write down proofs that apply to both exact and triangulated categories, and more generally to their substructures.Like the theory of exact categories, but in contrast to the triangulated category theory, each extension-closed subcategory of an extriangulated category inherits an extriangulated structure.We recall the necessary preliminaries on extriangulated categories in §2.
Since the theory of extriangulated categories was introduced in 2019, it has allowed many notions and constructions to be streamlined, such as cotorsion pairs [LN19] and Auslander-Reiten theory [INP24].In this same flow, our Extriangulated Resolution Theorem is indeed a generalization of Grothendieck's Resolution Theorem.The Grothendieck group of an extriangulated category is defined in a similar fashion to that of an exact category; see Definition 4.1.
Theorem A (Theorem 4.5).Assume that X is a full additive subcategory of a skeletally small extriangulated category (C , E, s), and that X is extension-closed and closed under taking cocones of s-deflations g : B → C where B, C ∈ X .If each object C ∈ C admits a finite X -resolution, then we have a group isomorphism K 0 (C ) K 0 (X ).

∼ =
We also mention here that Quillen founded higher algebraic K-theory for exact categories in [Qui73], wherein he defined K-groups K i (C ) for C exact and all integers i ≥ 0. In addition, Quillen generalized the Resolution Theorem to higher K-groups (see [Qui73,Thm. 3]) and this result is a fundamental theorem in K-theory, providing powerful computational machinery.There is also an ever-growing number of results on the K-theory of triangulated categories, e.g.[TT90,Nee05,Sch06,CX12].
The second aim of this article is to augment the index theory of triangulated categories by exploiting that Theorem A offers new insight into this theory.The index was introduced by Palu [Pal08] in order to better understand the Caldero-Chapoton map, and it continues to find beneficial applications in various contexts.For example, in [GNP23] it was revealed that the index is closely related to 0-Auslander extriangulated categories in connection with various mutations in representation theory, such as cluster tilting mutation [BMRRT,IY08] and silting mutation [AI12,Kim24].
For a suitable skeletally small 2-Calabi-Yau triangulated category C with suspension functor [1] and a 2-cluster tilting subcategory X ⊆ C (see Example 4.3), Palu defined the index of an object C ∈ C with respect to X as a certain element in the split Grothendieck group K sp 0 (X ) of X .The group K sp 0 (X ) is the abelian group freely generated on isomorphism classes [X] sp of objects X ∈ X , modulo the relations [X ′ ] sp − [X] sp + [X ′′ ] sp for each split exact sequence X ′ → X → X ′′ in X .In this setup, the index of C ∈ C is defined as the element ind Using extriangulated categories, Padrol-Palu-Pilaud-Plamondon [PPPP23] observed that this triangle can be viewed as a projective presentation of C. Indeed, in a certain extriangulated category (C , E X R , s X R ) relative to the triangulated structure on C (see §2.2), the subcategory X is precisely the subcategory of all E X R -projective objects (see [NP19,Def. 3.23]).Moreover, it follows from [PPPP23, Prop.4.11] that the index induces the following isomorphism (see also Corollary 5.3).
The isomorphism (1.1) prompted Jørgensen and the second author to define an index with respect to a more general class of subcategory in [JS24b].We note here that the index has been defined for any contravariantly finite, additive, direct-summand-closed subcategory of a skeletally small, idempotent complete triangulated category in [FJS24a].
Assume the following for the remainder of §1.
Setup B. Let C be a skeletally small, idempotent complete, triangulated category with suspension [1].In addition, suppose X ⊆ C is a full, additive subcategory that is contravariantly finite, rigid and closed under direct summands.
One can set which is an extension-closed subcategory of C , and define an extriangulated category (C , E R N , s R N ) (see Proposition 3.11) relative again to the triangulated structure on C .We prove in Lemma 3.15 that, as extriangulated categories, we have In particular, we have , and hence we make the following definition.
Definition C (Definition 5.7).The (right) index with respect to N of an object C ∈ C is its class We remark that Definition C makes sense for any extension-closed subcategory N of C , not just X ⊥ 0 , and hence it is a strict generalization of the index [−] X R of [JS24b].The upshot of defining an index with respect to an extension-closed subcategory is that we can utilize the localization theory of extriangulated categories as established by Nakaoka, Sakai and the first author [NOS22] (see §3).In [NOS22] a unification of the Serre quotient and Verdier quotient constructions is given.The extriangulated localization theory for triangulated categories is explored further in [Oga22b] and, in particular, there is a certain localization, or quotient, C /N of C with respect to N that is abelian; see (3.1) and Theorem 3.16.We denote the localization functor by Q : C → C /N .Our results in §5.2 culminate in the following additivity formula with error term for our index Theorem D. There is a well-defined group homomorphism θ R N : We recover the additivity formula with error term for the index [−] X R (see [JS24b, Thm.A]) as a special case of Theorem D; see Remark 5.8.Moreover, the theory of [NOS22] allows us to simplify some arguments.
The homomorphism θ R N captures the kernel of the canonical surjection π R N : as the next main result shows.This is part of Proposition 5.15.
Various indices have also appeared recently in higher homological algebra, e.g.[Rei20a,Rei20b,Fed21,Jør21,JS24a,FJS24b].In §5.3 we add our index to this landscape.We focus again on the n-cluster tilting situation, extending and strengthening some results of Fedele [Fed21, Thm.C, Prop.3.5], and also a result of Palu [Pal09, Lem.9] when n = 2.That we can use the right exact sequence in Proposition E to improve and/or strengthen these results adds to the mounting evidence that index theory for triangulated categories should be viewed through the lens of extriangulated categories.
In the last section, we turn our attention to the abelian setting and, using a straightforward application of Theorem A, deduce an analogue of (1.1) for an n-cluster tilting subcategory in an abelian category; see Theorem 6.5.This isomorphism is induced by the index defined by Reid in [Rei20b, Sec.1]; see Remark 6.6.
Notation and conventions.All categories and functors in this article are always assumed to be additive, and subcategories will always be full and closed under isomorphisms in the ambient category.For a category C , we denote the class of all morphisms in C by Mor C , and mod C is the category of finitely presented contravariant functors from C to the abelian category Ab of abelian groups.In addition, if A ⊆ C is an additive subcategory, then we denote by [A ] the (two-sided) ideal of morphisms in C that factor through an object in A .The canonical additive quotient functor is denoted (−) :

Extriangulated categories
An extriangulated category is defined to be an additive category C equipped with (1) a biadditive functor E : C op × C → Ab, where Ab is the category of abelian groups, and (2) a correspondence s that associates an equivalence class s where the triplet (C , E, s) satisfies the axioms laid out in [NP19, Def.2.12].We only recall some terminology and basic properties here, referring the reader to [NP19] for an in-depth treatment.For An extriangulated category (C , E, s) is simply denoted by C if there is no confusion.For the remainder of §2, let C = (C , E, s) be an extriangulated category.
Definition 2.1.We recall the following terminology.( ] is called an s-conflation, and in addition f is called an s-inflation and g an s-deflation.The pair δ and we call it an s-triangle. (3) A morphism of s-triangles from b, c) of morphisms in C with a * δ = c * δ ′ and so that the following diagram commutes.
is an s-conflation, then f is a weak kernel of g and g is a weak cokernel of f (see [NP19,Prop. 3.3]).Recall that a weak kernel of g is a morphism K k −→ B with gk = 0, and such that any morphism x ∈ C (X, B) with gx = 0 factors (not necessarily uniquely) through k.A weak cokernel is defined dually.Weak (co)kernels are not necessarily uniquely determined up to isomorphism, unlike (co)kernels.
be an s-conflation.Then we call C a cone of f and put Cone(f ) := C. Similarly, we denote the object A by CoCone(g) and call it a cocone of g.We note that this notation is justified since a cone of f (resp.cocone of g) is uniquely determined up to isomorphism (see [NP19,Rem. 3.10]).For any subcategories U and V in C , we denote by Cone(V, U ) the subcategory consisting of objects X appearing in an s-conflation V −→ U −→ X with U ∈ U and V ∈ V.The subcategory CoCone(V, U ) is defined similarly.
Note that if U and V are additive, then so are Cone(V, U ) and CoCone(V, U ).However, the subcategories Cone(V, U ) and CoCone(V, U ) are not necessarily closed under direct summands in general.
Definition 2.3.A subcategory X of C is said to be closed under taking cones of s-inflations if is an s-conflation with A, B ∈ X , then we have C ∈ X .Being closed under taking cocones of s-deflations is defined dually.Triangulated (resp.exact) structures on an additive category C naturally give rise to extriangulated structures on C .
In this case, we say that the extriangulated category/structure (C , E, s) corresponds to a triangulated category.
Example 2.5.[NP19, Exam.2.13] Suppose (A , E) is an exact category.Consider the collection δ is an s-triangle if and only if In this case, we say that the extriangulated category/structure (A , E, s) corresponds to an exact category.In addition, if (A , E) is in fact abelian, then we say (A , E, s) corresponds to an abelian category.Note that the set-theoretic assumption above is satisfied if A is skeletally small, or if A has enough projectives or enough injectives.
One of the advantages of revealing an extriangulated structure lies in the fact that it is closed under certain key operations: taking extension-closed subcategories; passing to a substructure using relative theory; taking certain ideal quotients; and localization.We now describe the parts of the first two of these operations relevant for our intentions, and localization is treated separately in §3.

Extension-closed subcategories
If the extriangulated structure on C is understood and no confusion may arise, we say that N is an extension-closed subcategory of C .An extension-closed subcategory of an extriangulated category inherits an extriangulated structure in a canonical way.
Proposition 2.6.[NP19, Rem.2.18] Suppose N is an extension-closed subcategory of C .Define E| N to be the restriction of E to N op × N , and similarly s| N := s| E| N .Then (N , E| N , s| N ) is an extriangulated category.
2.2.Relative theory.By a relative extriangulated structure, we mean a "coarser" or "less refined" structure, analogously to what is usually meant in topology.Relative theory for triangulated (resp.exact) categories has been considered in various contexts, e.g.[Bel00, Kra00] (resp.[AS93,DRSS99]).More recently, relative theory for n-exangulated categories (n ≥ 1 an integer) was introduced in Proposition 2.7.The following conditions are equivalent for an additive subfunctor F ⊆ E.
(1) (C , F, s| F ) forms an extriangulated category, where s| F is the restriction of s to F.
If an additive subfunctor F ⊆ E satisfies the equivalent conditions of Proposition 2.7, then it is called closed.Furthermore, we say that the extriangulated structure/category (C , F, s| F ) is relative to or a relative theory of (C , E, s).
For the remainder of §2.2, we assume (C , E, s) corresponds to a triangulated category with suspension [1].We now recall how a subcategory X ⊆ C determines relative extriangulated structures on C .Definition 2.8.For objects A, C ∈ C , we define subsets of E(C, A) as follows: These give rise to closed subfunctors E X L and E X R of E by [HLN21, Prop.3.19].Actually, they coincide with the closed subfunctors E X [1] and E X in the notation [HLN21, Def.3.18], respectively.In particular, putting E X := E X L ∩ E X R , we have three extriangulated substructures and C X := (C , E X , s X ) on C , which are relative to (C , E, s).In fact, by [JS24a, Thm.2.12], these are extriangulated subcategories of (C , E, s) in the sense of [Hau21,Def. 3.7].

Localization of extriangulated categories
In the pursuit of unifying Verdier [Ver96] and Serre quotients [Gab62], the localization theory of extriangulated categories with respect to suitable classes of morphisms was introduced in [NOS22].Since we will not need the theory of [NOS22] in full generality, we only provide Theorem 3.8, which follows from the main results of [NOS22].Furthermore, in §3.1 and §3.2 we specialize to the case of the localization of a triangulated category by an extension-closed subcategory as investigated in [Oga22b].
The following notion of an exact functor generalizes the classical ones when both (C , E, s) and (C ′ , E ′ , s ′ ) correspond to exact or triangulated categories.
δ is an s-triangle.
One can compose exact functors in the obvious way to obtain another exact functor; see [NOS22, Def.2.11], also [BTHSS23,Lem. 3.19].Extension-closure and relative theory provide typical examples of exact functors.
(2) For a closed subfunctor F ⊆ E and the relative extriangulated category (C , F, s| F ), the identity id C and inclusion In these situations, both (N , E| N , s| N ) and (C , F, s| F ) are extriangulated subcategories of (C , E, s).
To avoid any set-theoretic problem, we will work under the following setup when considering the localization in the sense of [GZ67].
Setup 3.3.We let (C , E, s) denote a skeletally small extriangulated category.Theorem 3.8 recalls sufficient conditions on the pair (C , N ), where N is a subcategory of C , to give rise to an extriangulated "quotient" category of C by N .We now lay out the terminology and notation necessary to state this result.In contrast to the triangulated and abelian cases, it is not clear if the localization C [S −1 N ] is equipped with a natural extriangulated structure in general.However, sufficient conditions for this are identified in [NOS22, p. 343], namely, conditions (MR1)-(MR4) concerning S N .Since we will not need these conditions explicitly, we omit recalling them here.
Consider the localization The class S N is said to be saturated if, for any f ∈ Mor C , we have L(f ) is an isomorphism if and only if f ∈ S N .Let (−) : C → C denote the quotient functor, where C := C /[N ] is the additive ideal quotient.We put S N := f f ∈ S N accordingly.Note that f = 0 if and only if f factors through an object in N .The localization of C at S N is denoted by and the localization functor by One can then check that N is an inverse of the functor M from (3.2).
Thus, if C is weakly idempotent complete (e.g. a triangulated category), we may think of the functor Q : C → C /N as the localization of C at S N .
3.1.Localization of triangulated categories.We now specialize to the case when (C , E, s) corresponds to a triangulated category and recall the relevant localization theory.
Setup 3.10.We fix a skeletally small, triangulated category C with suspension [1] and an extensionclosed subcategory N ⊆ C that is closed under direct summands.We denote by (C , E, s) the extriangulated category corresponding to the triangulated category C .
Since N is extension-closed in (C , E, s), it is immediate that it is extension-closed in any extriangulated substructure (C , F, s| F ) of (C , E, s).In particular, it is extension-closed in the following relative extriangulated structures defined using N .These differ to those defined in Definition 2.8, but we make a comparison of these structures in a special case in Lemma 3.15.
We remark that the above structures are generalized in [Che23, Prop.A.4] but from the viewpoint of constructing exact substructures of an exact category.With respect to the relative structure C N , the pair (C , N ) yields a saturated class S N of morphisms in C with S N satisfying the needed conditions (MR1)-(MR4) to obtain an extriangulated localization.

Abelian localization of triangulated categories.
Here, we review localizations of triangulated categories that are abelian.Abelian localizations of triangulated categories can arguably be traced back to hearts of t-structures in the sense of [BBD82].Since then, abelian localizations have been found using cluster tilting subcategories [BMR07, KR07, KZ08].These constructions were unified in [Nak11,AN12] and placed in an extriangulated context in [LN19].A generalization from cluster tilting to rigid subcategories was initiated in [BM12,BM13], and has been further developed in [Bel13,Nak13,HS20].See Example 3.20 for some details.Lemma 3.14.We have the following identities.
Relative structures like the above can be obtained from rigid subcategories.The next result clarifies how these structures relate to those in Definition 2.8 in case N is the kernel of C (X , −) for a contravariantly finite, rigid subcategory X ⊆ C (see [Oga22b, Exam.2.4]).We will use this in §5.2 to produce a generalization of the index defined in [JS24b].Recall that Lemma 3.15.Let X ⊆ C be a subcategory and consider the extension-closed subcategory M := Proof.We only check the first equation.Note that CoCone(M , M ) = C holds.Indeed, any object C ∈ C admits a right X -approximation x : X → C which yields a triangle A) and consider the triangle (3.4) as above.Then the composite hx vanishes, whence h factors through y, and so h ∈ [M ].The claim follows from the second identity of Lemma 3.14.
The main result we recall in §3.2 is the following, which establishes the abelian localization of a triangulated category with respect to an extension-closed subcategory.
Theorem 3.16.[Oga22b, Thm.4.2, Cor.4.3] The subcategory N is Serre in C N and the localization (C /N , E N , s N ) corresponds to an abelian category.Furthermore, the functor We include a simple example to demonstrate that even under very nice assumptions, we cannot hope that N is a Serre subcategory of C R N .
Example 3.17.Let C denote the cluster category associated to the quiver 1 → 2 (in the sense of [BMRRT]) and consider the cluster tilting subcategory N and so certainly cannot be Serre.
Despite this example, we need to understand how Q acts on s R N -conflations in order to connect this viewpoint to the aforementioned index.It turns out that Q sends s R N -conflations to right exact sequences in C /N .Definition 3.18.[Oga21, Def.2.7] Suppose (D, F, t) is an extriangulated category and A is an abelian category.A covariant additive functor F : D → A is right exact if, for every t-conflation is exact in A .Left exact functors are defined dually.
In the setup of Definition 3.18, suppose that Ext 1 A (C, A) is a set for all A, C ∈ A , and equip A with its canonical extriangulated structure (A , Ext 1 A , u).Then the functor F is both left and right exact (as just defined), if and only if F forms part of an exact functor (D, F, t) → (A , Ext 1 A , u) in the sense of Definition 3.1.In this language we thus have: We end this subsection by giving examples of such abelian localizations.
Example 3.20.Using the theory developed in [Oga22b], we can obtain more information about the localization functor in the situations considered in [BBD82] and [BM12,BM13].The triplet (C , E, s) still denotes a skeletally small triangulated category (see Setup 3.10).

An extriangulated resolution theorem
The aim of this section is to prove Theorem A(=Theorem 4.5), which is an extriangulated version of a resolution theorem for exact categories (see Theorem 4.4).We will use Theorem 4.5 in §5 to investigate the relationship between Grothendieck groups arising from a triangulated category (C , E, s) and an n-cluster tilting subcategory X ⊆ C .In this case, we know each object in C has a finite X -resolution (see Example 4.3), but it is not necessarily true that X is closed under taking cocones of s-deflations since any morphism is an s-deflation.Thus, we must pass to a relative extriangulated structure on C ; see Corollary 5.6 for details.
Although Quillen produced a resolution theorem in the framework of higher algebraic K-theory in [Qui73, §4], the idea was first established by Grothendieck at the level of K 0 (see [Bas68, , E, s) be a skeletally small extriangulated category.The Grothendieck group of (C , E, s) is defined to be If it will cause no confusion, we will abbreviate K 0 (C , E, s) as K 0 (C ).
To recall Grothendieck's Resolution Theorem, we need the following notion.
Definition 4.2.Let (C , E, s) be an extriangulated category, let X ⊆ C be a subcategory and fix an object where In this case, we say that the X -resolution (4.1) is of length n.In particular, any object X ∈ X has an X -resolution of length 0. The notion of finite X -coresolution is defined dually.
Typical examples of such resolutions arise from cluster tilting theory.
Example 4.3.Suppose C is an idempotent complete triangulated category with suspension [1] and let n ≥ 2 be an integer.Recall from [IY08, §3] that a subcategory X ⊆ C is called an n-cluster tilting subcategory of C if X is functorially finite in C and  A typical example of the resolution theorem is as follows.For an abelian category C with enough projectives, if each object in C has finite projective dimension, then we have K 0 (P) ∼ = K 0 (C ) for P ⊆ C the subcategory of projectives.Note that K 0 (P) is same as the split Grothendieck group K sp 0 (P) of P.
The following is an extriangulated version of the classical resolution theorem.The dual of Theorem 4.5 also holds.
Theorem 4.5 (Extriangulated Resolution Theorem).Let (C , E, s) be a skeletally small extriangulated category.Suppose X is an extension-closed subcategory of (C , E, s), such that X is closed under taking cocones of s-deflations.If any object C ∈ C admits a finite X -resolution, then we have an isomorphism where (4.1) is an X -resolution of C ∈ C .4.1.The proof of Theorem 4.5.In this subsection, we work under the hypotheses of Theorem 4.5.
Setup 4.6.Suppose (C , E, s) is a skeletally small extriangulated category and X ⊆ C is an extension-closed subcategory that is closed under taking cocones of s-deflations, such that every object in C has a finite X -resolution.Put X 0 := X and, for any i > 0, we denote by X i the subcategory of C consisting objects which admit finite X -resolutions of length at most i.Note that X i = Cone(X i−1 , X ) and it is additive and closed under isomorphisms.Furthermore, we have an ascending chain X 0 ⊆ X 1 ⊆ X 2 ⊆ • • • , and the union i≥0 X i coincides with C because each object in C has a finite X -resolution.In other words, we have colim − −−−− →i X i = C , where the left-hand side is a colimit in the category of additive categories and functors.
Proposition 4.7.For all i ≥ 0, the subcategory X i is extension-closed in (C , E, s), and hence inherits an extriangulated structure.
Proof.We prove this by induction on i ≥ 0. First, note that X 0 = X is extension-closed by assumption.Thus, suppose X i is extension-closed in C for some i ≥ 0, so that we may show X i+1 is extension-closed.To this end, let A B C f g δ be an s-triangle in C with A, C ∈ X i+1 = Cone(X i , X ).Then we know there are s-triangles where the rows and columns are s-triangles.In particular, we see that h is an s-deflation.
As each object in C is assumed to admit a finite X -resolution, there is an s-triangle of the form Y X B ′ q p γ with X ∈ X (and Y ∈ X r for some r ≥ 0).By (ET4 op ) applied to the where the rows and columns are s-triangles.Since X is closed under taking cocones of s-deflations, we see that Q = CoCone(g ′ p) ∈ X .By [NP19, Cor.3.16], the morphism ( f ′ a, p ) : X A ⊕X → B ′ is an s-deflation.Let us recall how the corresponding s-triangle is found.We have the s-triangle Y X B ′ q p γ from the middle column of (4.4), and the morphism f ′ a : X A → B ′ .We realise the E-extension (f ′ a) * γ as follows: Then [LN19, dual of Prop.1.20] yields an s-triangle (see (4.2)) and where the rows and columns are s-triangles.We use here that s(( ] as in (4.5).Since Y A ∈ X i by assumption and Q ∈ X ⊆ X i , we see that D ∈ X i because X i is extension-closed by the inductive hypothesis.Now we are in position to show B ∈ X i+1 as needed.Applying axiom (ET4 op ) to the composition where the rows and columns are s-triangles.
and we are done.
We denote the inherited extriangulated structure on X i by (X i , E| X i , s| X i ) (see §2.1).Now we investigate the Grothendieck groups K 0 (X i ) := K 0 (X i , E| X i , s| X i ) of these extriangulated subcategories of (C , E, s).
Proposition 4.8.The ascending chain Proof.Fix an integer i ≥ 0. We shall show that the natural group homomorphism φ : K 0 (X i ) → K 0 (X i+1 ) given by φ([C]) = [C] is an isomorphism by constructing an inverse as follows.Let C ∈ X i+1 = Cone(X i , X ) be arbitrary.Then there is an s-conflation P 1 p ′ −→ P 0 p −→ C with P 0 ∈ X ⊆ X i and P 1 ∈ X i .We claim that the assignment ψ : [C] → [P 0 ] − [P 1 ] gives rise to a group homomorphism ψ : K 0 (X i+1 ) → K 0 (X i ).
To see this, let Q 1 where all rows and columns are s-conflations.Since X i is extension-closed by Proposition 4.7, we see that P lies in X i .Hence, in K 0 (X i ) we have It is straightforward to check that ψ induces a group homomorphism as claimed, and that φ and ψ are mutually inverse.
We are now in position to prove Theorem 4.5.
Proof of Theorem 4.5.We will show that K 0 (C ) = K 0 (C , E, s) satisfies the universal property of the filtered colimit colim − −−−− →i K 0 (X i ).First, for each integer i ≥ 0, there are the canonical group homomorphisms α i : K 0 (X i ) → K 0 (C ) and On the other hand, let C ∈ C be arbitrary and suppose (4.1) is an X -resolution of C. Define To see that this is independent of the chosen is another X -resolution of C. Set N := max{n, m}.Then all the s-conflations involved in (4.1) and (4.6) are s| X N -conflations.In particular, in K 0 (X N ) and hence also in colim − −−−− →i K 0 (X i ), we have that It is clear that δ is constant on isoclasses of objects.Furthermore, any s-conflation A −→ B −→ C in (C , E, s) lies in X j for some sufficiently large integer j.Thus, by definition, [A] − [B] + [C] = 0 occurs in K 0 (X j ) and also in colim − −−−− →i K 0 (X i ).Hence, δ induces a group homomorphism δ : K 0 (C ) → colim − −−−− →i K 0 (X i ).Finally, it is clear that γ and δ are mutually inverse.As the canonical morphism K 0 (X i ) ) is an isomorphism for any i ≥ 0 by Proposition 4.8, the canonical morphism K 0 (X ) → colim − −−−− →i K 0 (X i ) is also an isomorphism and we are done.

Applications to the index in triangulated categories
As an application of our Extriangulated Resolution Theorem (Theorem 4.5), in §5.1 we recover some index isomorphisms that have recently appeared in the literature.This allows us to propose an index with respect to an extension-closed subcategory in §5.2, as well as establish an additivity formula for this index.Lastly, we prove a generalization of Fedele's [Fed21, Thm.C] in §5.3.5.1.The PPPP and JS index isomorphisms.Throughout §5.1, we fix the following setup.
Setup 5.1.Suppose (C , E, s) corresponds to a skeletally small, idempotent complete, triangulated category C .We also assume X ⊆ C is an additive subcategory that is contravariantly finite, rigid and closed under direct summands.
Recall from Example 4.3 that if X is n-cluster tilting (n ≥ 2), then any object C ∈ C admits a finite X -resolution It is straightforward to check that the following isomorphism is a direct consequence of [PPPP23,Prop. 4.11].We call it the PPPP index isomorphism; it is a special case of Corollary 5.5.
Corollary 5.3 (PPPP index isomorphism).Suppose X is 2-cluster tilting in C Then the Palu index ind X yields the following isomorphism of abelian groups.
The PPPP index isomorphism shows that the class [C] X R in the Grothendieck group K 0 (C X R ) of an object C ∈ C can be interpreted as the Palu index of C with respect to X .This suggests that one can define an index using the relative extriangulated structure C X R , even without the 2-cluster tilting assumption.This is what is done in [JS24b].
, which is more natural in our framework (see Proposition 5.13).However, in the setup of [JS24b,§3], one can pass between θ X and θ R N using the equivalence G is the induced isomorphism, then θ X G * = θ R N .Just like in [JS24b], we determine additivity formulae for our left and right indices in this subsection; see Theorem 5.14.Although the approach we take is similar to [JS24b,§3], some arguments are simplified by utilizing the localization theory of extriangulated categories.Furthermore, we also show that the image of θ R N captures the kernel of the natural surjection K 0 (C R N ) ։ K 0 (C ) (see Proposition 5.15).Note that our right index is a strict generalization of the Jørgensen-Shah index since there are extension-closed subcategories of triangulated categories that do not arise as X ⊥ 0 for any contravariantly finite, rigid subcategory X .
Example 5.9.Consider the quiver 1 → 2, its path algebra Λ over a field k and the bounded derived category C of finite-dimensional Λ-modules.The module category N := mod Λ is extension-closed in C , but a straightforward argument shows it cannot be of the form X ⊥ 0 for any rigid subcategory X ⊆ C .Setup 5.10.In the remainder of §5.2, we suppose (C , E, s) corresponds to a skeletally small, idempotent complete, triangulated category C with suspension [1].We also assume N is an extensionclosed subcategory of C , such that Cone(N , N ) = C .We remind the reader that under Setup 5.10 all the results from §3 apply.In particular, S N is a multiplicative system in C := C /[N ] (this is condition (MR2) from [NOS22] and follows from Theorem 3.12(2)), the localization ).In the sequel we focus on the right index and state, but do not prove, the corresponding assertions for the left index.
Our goal is to establish Theorem D. In order to define θ R N , we need two preliminary results.Note that the assumption Cone(N , N ) = C is not needed to prove Lemma 5.11.
Proof.We show the first equation.Since S N is a multiplicative system in C (i.e.admits a calculus of left and right fractions in C ), an isomorphism α : Q(B) (5.5) with f and h factoring through objects in N .Since The second column is an s R Following Construction 5.20, we can complete the morphism X 1 X 0 g1 to an (n + 2)-angle (5.16).Thus, due to Remark 5.21, we have In particular, the morphism s : C ′ 0 → C 0 lies in S N , and hence We end this section by making a remark on a class of examples where Theorem 5.22 applies but [Fed21, Thm.C] may not.
Remark 5.23.As a benefit of our generalization, n-cluster tilting subcategories closed under nshifts in singularity categories fall within the of Theorem 5.22.We recall that, for a Noetherian k-algebra Λ over a field k, the singularity category D sg (Λ) is defined to be the Verdier quotient of the bounded derived category D b (Λ) by the subcategory perf(Λ) of perfect complexes.Some examples and constructions of n-cluster tilting subcategories in D sg (Λ) are given in e.g.[Iya18,Kva21].It is known that D sg (Λ) is not Hom-finite in general; indeed, the Hom-finiteness of D sg (Λ) implies that Λ is Gorenstein in some cases (see [Kal21, Thm.2.1]).Thus, Theorem 5.22 is effective even in the case of singularity categories by passing to the idempotent completion.

Application to the index in abelian categories
In this section we will use Theorem 4.5 to prove an analogue of Corollary 5.5 for abelian categories.Since we will not need the details of the definition of an n-cluster tilting subcategory of an abelian category, we refer the reader to [Jas16, Def.3.14] for the precise formulation.Instead we just recall the needed consequences of the definition below.We fix the following setup for the remainder of the article.Setup 6.1.Suppose (C , E, s) is a skeletally small abelian category and that X ⊆ C is an n-cluster tilting subcategory (n ≥ 1).
By the dual of [Jas16, Prop.3.17], for each C ∈ C , there is a diagram (1) X i ∈ X for each 0 ≤ i ≤ n − 1; (2) 0 C i+1 X i C i 0 fi gi is a short exact sequence for each 0 ≤ i ≤ n − 2; and (3) the morphism g i : X i → C i is a right X -approximation for each 0 ≤ i ≤ n − 2.
In particular, we see that each object in C has a finite X -resolution (of length at most n − 1) in (C , E, s) in the sense of Definition 4.2.Although X is also extension-closed and closed under direct summands in C , we cannot apply Theorem 4.5 yet.Indeed, it easy to find examples where X is not closed under cocones of s-deflations.We will need to pass to a relative structure on C first.R , s X R ) which is relative to (C , E, s).In fact, C X R is an exact category by [NP19, Cor.3.18], because each s X R -inflation (resp.s X R -deflation) is a monomorphism (resp.an epimorphism).Lemma 6.3.X is closed under cocones of s X R -deflations.
Proof.Suppose A X 1 X 0 δ is an s X R -triangle with X i ∈ X .Consider the identity morphism id X 0 : X 0 → X 0 and notice that δ = id * X 0 δ = 0 as X 0 ∈ X .In particular, A is a direct summand of X 1 and hence A ∈ X .Now that we have passed to a relative structure, we must show the X -resolution of C in (C , E, s) arising from (6.1) is still an X -resolution in C X R .
Lemma 6.4.Each s-triangle C i+1 X i C i fi gi δi arising in (6.1) is in fact an s X R -triangle.Proof.Let x : X → C i be a morphism in C with X ∈ X .Since g i is a right X -approximation of C i , we have that x factors through g i .This implies x * δ i = 0 by [NP19, Cor.3.5], so δ i ∈ E X R .Putting all this together, we immediately have the following by Theorem 4.5.Theorem 6.5.There is an isomorphism of abelian groups R of length n − 1.We now explain how the above relates to the index defined by Reid [Rei20b].
Remark 6.6.The sum n−1 i=0 (−1) i [X i ] sp was defined to be the index index X (C) of C with respect to X ; see [Rei20b, Sec.1].Since C is idempotent complete, we can always find a minimal X -resolution of C by removing trivial summands (i.e. In particular, the value index X (C) in K sp 0 (X ) does not change when we remove such summands and, since a minimal X -resolution is unique up to isomorphism, the index is well-defined; see [Rei20b, Rem.2.1].
Definition 3.4.[NOS22,Def.4.1] An additive subcategory N of (C , E, s) is called thick if it is closed under direct summands, and N satisfies the 2-out-of-3 property for s-conflations, that is, for any s-conflation A −→ B −→ C, if any two of A, B or C belong to N , then so does the third.Notice that any thick subcategory N of C is automatically extension-closed by definition.In the case that (C , E, s) corresponds to a triangulated category, the notion of a thick subcategory as in Definition 3.4 coincides with the usual one for triangulated categories.In addition, a thick subcategory N ⊆ C is said to be Serre if wheneverA −→ B −→ C is an s-conflation with B ∈ N ,then we have A, C ∈ N (see [Oga22b, Def.1.17]).This generalizes the notion of a Serre subcategory for exact categories introduced in [CE98, 4.0.35].Definition 3.5.[NOS22, Def.4.3] We associate the following classes of morphisms to a thick subcategory N ⊆ C : (1) L := { f ∈ Mor C | f is an s-inflation with Cone(f ) ∈ N }; (2) R := { g ∈ Mor C | g is an s-deflation with CoCone(g) ∈ N }; and (3) S N is the smallest subclass of Mor C closed under compositions containing both L and R. By [NOS22, p. 374], the class S N consists of all finite compositions of morphisms in L and R and, moreover, it satisfies the following condition.(M0) S N contains all isomorphisms in C , and is closed under compositions and taking finite direct sums.If N ⊆ C is thick, then prototypical examples of localizing C at the class S N are Verdier and Serre quotients of triangulated and abelian categories, respectively; see [NOS22, §4.2].Example 3.6.(1) (Verdier quotient.)[NOS22, Exam.4.8] Let (C , E, s) be a triangulated category and N a thick subcategory of C .The class S N satisfies S N = L = R and the localization C [S −1 N ] is just the usual Verdier quotient of C with respect to N .(2) (Serre quotient.)[NOS22, Exam.4.9] Let N be a Serre subcategory of an abelian category (C , E, s).Then we have S N = L • R and C [S −1 N ] is the usual Serre quotient of C with respect to N .

( 1 )
The subcategory N ⊆ C is closed under taking cones of s R N -inflations.Dually, N is closed under taking cocones of s L N -deflations.(2) The subcategory N is thick in C N .The corresponding class S N ⊆ Mor C (see Definition 3.5) is saturated and S N satisfies (MR1)-(MR4) with respect to C N = (C , E N , s N ).Moreover, S N = R ret • L = R • L sec holds, where L sec denotes the class of sections belonging to L and R ret denotes the class of retractions belonging to R. (3) There exists an extriangulated localization (Q, µ) : C N → (C /N , E N , s N ) with Ker Q = N .Proof.(1): Suppose A B C f g h is an s R N -triangle with A, B ∈ N .It follows that h ∈ [N ] by Proposition 3.11.Then [Oga22b, Lem.2.5(2)] implies C ∈ N .The second assertion is proved dually.

Setup 3. 13 .
In addition to Setup 3.10, we assume further that Cone(N , N ) = C holds in §3.2.Note that when (C , E, s) corresponds to a triangulated category (as we are currently assuming), we have Cone(N , N ) = C if and only if CoCone(N , N ) = C .In this case, the bifunctors E L N and E R N can be described more simply.The next observation follows from the proof of [Oga22b, Lem.4.1].

F
2.1]) with U [1] ⊆ U .There is a cohomological functor H : C → H to the abelian heart of (U , V) and we put N := Ker H. Then [Oga22b, Thm.5.8] and Corollary 3.19 tell us that H induces a right exact functor H : C R N → C /N and a left exact functor H : C L N → C /N .We remark that the assertion still holds for the general heart construction of Abe-Nakaoka [Nak11, AN12].(2) Let X ⊆ C be an additive, contravariantly finite, rigid subcategory that is closed under isomorphisms and direct summands, and put N := X ⊥ 0 .Then Cone(N , N ) = C holds by Lemma 3.15, and we have a natural exact equivalence G : C /N ≃ −→ mod X of abelian categories by universality as below; see [Oga22b, Sec.5.3.2].(−) := C (?, −)| X ≃ G The functor F is the restriction to X of the Yoneda embedding, namely, for C ∈ C we put F (C) := C (?, C)| X ∈ mod X .In this case, by Corollary 3.19, the exact functor Q induces a right exact functor Q : C R N → C /N and a left exact functor Q : C L N → C /N .
Ch. VIII, Thm.4.2], or [Wei13, Ch.II, Thm.7.6]).Let us begin by recalling the definition of the Grothendieck group for an extriangulated category.Recall that, for a skeletally small additive category C , the split Grothendieck group K sp 0 (C ) of C is the free abelian group generated on the set of isomorphism classes [A] for A ∈ C , modulo the relations [A]−[B]+[C] for each split exact sequence A B C in C .Definition 4.1.[ZZ21, §4] Let (C Thm. 5.3].This implies that any C ∈ C admits an X -resolution in the relative extriangulated category C X R (= C R N by Lemma 3.15 where N = X ⊥ 0 ) of length at most n − 1 (see [IY08, Cor.3.3], or [Bel15, Prop.3.2, Thm.3.4]).

Theorem 4. 4 (
Resolution Theorem).[Bas68, Ch.VIII, Thm.4.2] Let C be a skeletally small exact category.Assume that X ⊆ C is extension-closed and closed under taking kernels of admissible deflations in C .If any object C ∈ C admits a finite X -resolution, then we have K 0 (X ) ∼ = K 0 (C ).
an abelian category and the localization functor Q : C → C /N is cohomological (see Lemma 3.7 and Theorem 3.16), and Q induces a right exact functor Q : C R N → C /N (see Corollary 3.19 Lemma 5.11.(cf.[JS24b, Lem.3.8]) Let B, C ∈ C be objects with an isomorphism Q Definition 6.2.(cf.Definition 2.8) For objects A, C ∈ C , we define:E X R (C, A) := δ = [A f −→ B g −→ C] ∈ E(C, A) x * δ = 0 for all x : X → C with X ∈ X .Note that x * δ = E(x, A)(δ) is the pullback of the short exact sequence A f −→ B g −→ C along x : X → C. Just like in the triangulated case, E X R (which is denoted E X in [HLN21, Def.3.18]) is a closed subfunctor of E by[HLN21, Prop.3.19].This yields the extriangulated category C X R := (C , E X