Matrix factorizations for self-orthogonal categories of modules

MATRIX FACTORIZATIONS FOR SELF-ORTHOGONAL CATEGORIES OF MODULES

PETTER ANDREAS BERGH AND PEDER THOMPSON

Abstract. For a commutative ringS and self-orthogonal subcategory Cof Mod(S), we consider matrix factorizations whose modules belong toC. Let f∈Sbe a regular element. Iff isM-regular for everyM∈C, we show there is a natural embedding of the homotopy category ofC-factorizations off into a corresponding homotopy category of totally acyclic complexes. Moreover, we prove this is an equivalence ifCis the category of projective or flat-cotorsion S-modules. Dually, using divisibility in place of regularity, we observe there is a parallel equivalence whenCis the category of injectiveS-modules.

Introduction

Matrix factorizations of a nonzero element f in a regular local ring Q were in- troduced by Eisenbud [12] and shown to correspond to maximal Cohen-Macaulay Q/(f)-modules; in turn Buchweitz [5] gave a relation between these and totally acyclic complexes of finitely generated projectiveQ/(f)-modules. Indeed, this cor- respondence can be described as an equivalence of triangulated categories,

HMF(Q, f) ^' //K_tac(prj(Q/(f))),

where HMF(Q, f) is the homotopy category of matrix factorizations of f, and K_tac(prj(Q/(f))) is the homotopy category of totally acyclic complexes of finitely generated projective Q/(f)-modules. In part, our goal is to develop the notion of matrix factorizations more generally—relative to a self-orthogonal category of modules—with an emphasis on extending this equivalence.

LetS be a commutative ring, letf ∈S, and letCbe an additive subcategory of Mod(S), the category ofS-modules. A linear factorization off, defined by Dycker- hoff and Murfet [11], is a pair ofS-modulesM0andM1along with homomorphisms d1 :M1 →M0 andd0 :M0 →M1 satisfyingd1d0 =f1^M0 and d0d1 =f1^M1. We define aC-factorization off to be a linear factorization off such thatM0, M1∈C.

The homotopy category ofC-factorizations off, denotedHF(C, f), is the category whose objects areC-factorizations off and whose morphisms are homotopy classes of the natural maps betweenC-factorizations; see Section 2. TakingCto be the cat- egory of finitely generated projective modules over a regular local ring, one obtains the usual notion of matrix factorizations in [12].

SetR=S/(f). To relate aC-factorization offto a suitable type of totally acyclic complex of R-modules, a natural setting to consider is whenCis self-orthogonal, that is, Extⁱ_S(M, M⁰) = 0 for everyM, M⁰ ∈Cand i ≥1. If Cis self-orthogonal

Date: December 3, 2019.

2010Mathematics Subject Classification. Primary 13D02. Secondary 18E30, 13D09.

Key words and phrases. Matrix factorization, totally acyclic complex, flat-cotorsion module.

1

andf ∈S isS-regular andM-regular for everyM ∈C, then the category R⊗SC is self-orthogonal—see Proposition 1.8—in which case there is a natural notion of total acyclicity. Proposition 2.5 thus relatesC-factorizations off toR⊗SC-totally acyclic complexes. Here, for a self-orthogonal categoryW in Mod(R), aW-totally acyclic complex is an acyclic complex of modules inWwhose acyclicity is preserved by Hom_R(−,W) and Hom_R(W,−); this includes the usual notions of total acyclicity for complexes of projective or injective modules, and is a special case of that in [7].

In this setting, that is, ifCis an additive self-orthogonal subcategory ofMod(S) andf isS-regular andM-regular for every M ∈C, then we prove in Theorem 3.5 that there is a full and faithful triangulated functor,

T:HF(C, f) //Ktac(R⊗SC),

whereKtac(R⊗SC) is the homotopy category ofR⊗SC-totally acyclic complexes.

This embedding extends work of Bergh and Jorgensen; indeed, its proof is closely modelled on that of [3, Theorem 3.5], which is recovered by settingC=prj(S).

The functorT sends aC-factorization off to a 2-periodic complex, see Propo- sition 2.5, and so we do not expect it to be an equivalence without additional assumptions onS andC. IfS is a regular local ring andC=Prj(S) is the category of projectiveS-modules, then we show in Theorem 4.2 that there is a triangulated equivalence:

HF(Prj(S), f) ^' //Ktac(Prj(R)).

Indeed, restricting to the subcategory of finitely generated projective modules, this is the equivalence due to Eisenbud [12] and Buchweitz [5] described above.

Parallel to this development, we consider a dual situation in terms of divisibility.

Iff isS-regular andM-divisible for everyM ∈C, we observe in Theorem 3.6 that there is an embeddingHF(C, f)→K_tac(Hom_S(R,C)). In particular, since injective S-modules are divisible, we obtain an equivalence for C=Inj(S), the category of injectiveS-modules, whenS is a regular local ring; see Theorem 4.5.

Another natural (torsion-free) self-orthogonal category to consider isFlatCot(S), the category of flat-cotorsionS-modules; see Section 5. We prove in Theorem 5.4 that ifS is a regular local ring, then there is a triangulated equivalence:

HF(FlatCot(S), f) ^' //Ktac(FlatCot(R)).

HereK_tac(FlatCot(R)) is the homotopy category of acyclic complexes of flat-cotorsion R-modules such that for every flat-cotorsionR-moduleF, application of Hom_R(F,−) and Hom_R(−, F) preserves acyclicity.

In addition to the classic equivalence described above, Buchweitz gave in [5] an equivalence, assuming S is a regular local ring, between the homotopy category of matrix factorizations of f and the singularity category of R; this was proven explicitly by Orlov [21]. Along these lines, and as a consequence of the previous equivalence, we observe in Corollary 5.6 a triangulated equivalence,

HF(FlatCot(S), f) ^' //DF-tac(Flat(R)),

where DF-tac(Flat(R)) is the subcategory of the pure derived category of flat R- modules consisting of F-totally acyclic complexes. This category plays the role of the singularity category in the context of the pure derived category, in that it vanishes if and only ifRis regular; see [18, Proposition 9.7] and [19].

1. Self-orthogonal categories of modules

Throughout this paper, letSbe a commutative ring. The category of allS-modules is denoted Mod(S). Tacitly, we assume all subcategories of Mod(S) are full and closed under isomorphisms. We use standard homological notation throughout, and anS-complex means a chain complex ofS-modules.

LetPrj(S),Inj(S),Flat(S) denote the categories of projective, injective, and flat S-modules, respectively;prj(S) denotes the category of finitely generated projective S-modules. LetCot(S) denote the category of cotorsionS-modules, that is, those S-modules C such that Ext¹_S(F, C) = 0 for every flat S-module F. For brevity, writeFlatCot(S) =Flat(S)∩Cot(S) for the category of flat-cotorsionS-modules.

Definition 1.1. Let C be a subcategory of Mod(S). The category C is called self-orthogonal¹if Extⁱ_S(C, C⁰) = 0 for allC, C⁰∈Cand alli≥1.

Example 1.2. Evidently bothPrj(S) andInj(S) are self-orthogonal.

The category FlatCot(S) is also self-orthogonal: LetF andF⁰ be flat-cotorsion S-modules. If P → F is a projective resolution over S, then coker(d^P_i ) is a flat S-module fori≥1, hence Extⁱ_S(F, F⁰)∼= Ext¹_S(coker(d^P_i ), F⁰) = 0 for alli≥1.

Definition 1.3. LetM be anS-module,f ∈S, andCbe a subcategory ofMod(S).

The element f is M-regular if f x = 0 implies x = 0 for each x ∈ M; f is C-regular iff isM-regular for everyM ∈C.

The elementf isM-divisibleif for everyx∈M, there existsy∈M withf y=x;

f isC-divisible iff is M-divisible for everyM ∈C.

Example 1.4. Letf ∈S be anS-regular element.

If C is a subcategory of Mod(S) contained in the category of torsion-free S- modules, thenf isC-regular. In particular,f isFlat(S)-regular,FlatCot(S)-regular, andPrj(S)-regular.

IfCis a subcategory ofMod(S) contained in the category of divisibleS-modules, thenf isC-divisible. In particular,f isInj(S)-divisible.

LetS→Rbe a ring homomorphism and letCbe a subcategory ofMod(S). The following subcategories ofMod(R) play a special role in this paper:

R⊗SC={W ∈Mod(R)|W ∼=R⊗SC, for someC∈C};

HomS(R,C) ={W ∈Mod(R)|W ∼= HomS(R, C), for someC∈C}.

Remark 1.5. For any ring homomorphismS→R, we haveR⊗SPrj(S)⊆Prj(R) and Hom_S(R,Inj(S))⊆Inj(R), see for example [9, Proposition 2.3]; the former is an equality if the homomorphism is local, the second is an equality if the homo- morphism is a surjection. The equality for projective modules uses that projective modules over a local ring are free. We justify the equality for injective modules here: Let I be an injective R-module and let I →ES(I) be its injective envelope as an S-module. Since the natural map HomS(R, I) → I is an isomorphism, it follows that the induced injection HomS(R, I)→HomS(R, ES(I)) ofR-modules is essential and splits, thus is an isomorphism. It follows thatI∼= HomS(R, ES(I)).

For anS-moduleM, denote by pd_SM, id_SM, and fd_SM the projective, injec- tive, and flat dimensions ofM overS.

1This differs from [7], where the term was used to refer to Ext¹-orthogonality and is implied by the definition given here; our usage here agrees with what would be written asC⊥Cin [23].

Remark 1.6. Let f ∈ S be an S-regular element, and set R = S/(f). If P is a projective R-module, then pd_SP = 1 (see [15, Part III, Theorem 3]); ifI is an injective R-module, then idSI= 1 (see [14, Theorem 202]). It thus follows that if F is a flatR-module, then fd_SF = 1; this uses the fact that anS-moduleM is flat if and only if its character dual Hom_Z(M,Q/Z) is injective.

Part (i) of the next change of rings result is due to Rees [22]; part (iii) is dual.

If M is an S-module, f ∈ S, and R = S/(f), it is often convenient to identify R⊗SM ∼=M/f M and HomS(R, M)∼= (0 :M f) ={x∈M |f x= 0} ⊆M. Lemma 1.7. Let f ∈S be anS-regular element and set R=S/(f).

If M is an S-module such thatf isM-regular andN is anR-module, then (i) Extⁱ⁺¹_S (N, M)∼= Extⁱ_R(N, R⊗SM)for alli≥0;

(ii) Extⁱ_S(M, N)∼= Extⁱ_R(R⊗SM, N)for alli≥0.

If M is an S-module such thatf isM-divisible andN is an R-module, then (iii) Extⁱ⁺¹_S (M, N)∼= Extⁱ_R(Hom_S(R, M), N) for alli≥0;

(iv) Extⁱ_S(N, M)∼= Extⁱ_R(N,Hom_S(R, M))for all i≥0.

Proof. (i) & (ii): See Matsumura [17, Lemma 2, p. 140] for a proof of these; (i) was originally shown by Rees [22, Theorem 2.1].

(iii): We give an argument dual to [22, Theorem 2.1], showing that the functor Eⁱ(−) = Extⁱ⁺¹_S (M,−) is theith right derived functor of HomR(HomS(R, M),−).

Apply Hom_S(−, N) to the short exact sequence

0 //HomS(R, M) //M ^f //M //0 to obtain the following exact sequence

Hom_S(M, N) //Hom_S(Hom_S(R, M), N) //Ext¹_S(M, N) ^f//Ext¹_S(M, N).

Sincef N = 0, we obtain HomS(M, N) = 0. Additionally, multiplication byfonM orN induce the same map on Ext¹_S(M, N): also multiplication byf. Asf N = 0, this map must be 0, thus yielding

Ext¹_S(M, N)∼= HomS(Hom_S(R, M), N)∼= HomR(Hom_S(R, M), N).

Hence E⁰(−) ∼= HomR(HomS(R, M),−). For any injectiveR-module I, we have idSI = 1 by Remark 1.6, hence Eⁱ(I) = 0 fori ≥ 1. Finally, for a short exact sequence 0→N⁰ →N →N⁰⁰→0 ofR-modules, Hom_S(M, N⁰⁰) = 0 and so there is a long exact sequence

0→E⁰(N⁰)→E⁰(N)→E⁰(N⁰⁰)→E¹(N⁰)→E¹(N)→E¹(N⁰⁰)→ · · ·, and it follows thatEⁱ(−) is theith right derived functor of HomR(HomS(R, M),−) and thus is isomorphic to Extⁱ_R(HomS(R, M),−).

(iv): Let P be a projective resolution of N over R; standard tensor–Hom ad- junction yields HomS(R⊗RP, M)∼= HomR(P,HomS(R, M)), and the desired iso-

morphism follows.

Proposition 1.8. Let Cbe a self-orthogonal subcategory of Mod(S), let f ∈S be S-regular, and setR=S/(f). The following hold:

(i) If f isC-regular, thenR⊗SCis self-orthogonal inMod(R).

(ii) If f isC-divisible, thenHom_S(R,C)is self-orthogonal in Mod(R).

Proof. (i): For S-modules C, C⁰ ∈ C and i ≥ 0, Lemma 1.7(ii) yields that Extⁱ_R(R⊗_SC, R⊗_SC⁰)∼= Extⁱ_S(C, R⊗_S C⁰). It will therefore be enough to show that Extⁱ_S(C, R⊗SC⁰) = 0 fori≥1. Asf isC-regular, there is an exact sequence

0 //C^{0 f} //C⁰ //R⊗SC⁰ //0.

Application of the functor HomS(C,−) yields a long exact sequence:

· · · //Extⁱ_S(C, C⁰) //Extⁱ_S(C, R⊗SC⁰) //Extⁱ⁺¹_S (C, C⁰) //· · · By assumption, Extⁱ_S(C, C⁰) = 0 = Extⁱ⁺¹_S (C, C⁰) for i ≥ 1, and it follows that Extⁱ_S(C, R⊗SC⁰) = 0 fori≥1.

(ii): This is proved dually to part (i), using instead Lemma 1.7(iv) and the existence of an exact sequence

0 //HomS(R, C) //C ^f //C //0

for eachC∈C.

2. C-factorizations and total acyclicity

Letf ∈S. Extending the classic notion of matrix factorizations [12], Dyckerhoff and Murfet define [11] alinear factorization of f to be aZ/2Z-graded S-moduleM = M0⊕M1together with anS-linear differentiald:M →M that is homogeneous of degree 1 and satisfiesd²=f1^M. We often write such a linear factorization as

(M, d) = (M1

d1 //M0 d₀

oo ),

whered₁d₀=f1^M0 and d₀d₁=f1^M1.

A morphism α : (M, d) → (M⁰, d⁰) of linear factorizations of f is a degree 0 map which commutes with the differentials on M and M⁰; it consists of maps α_i:M_i →M_i⁰, fori= 0,1, making the following diagram commute:

M1 d₁ //

α1

M0 d₀ //

α0

M1 α1

M₁^{0 d}

0

1 //M₀^{0 d}

0 0 //M₁⁰

Definition 2.1. Let C be a subcategory of Mod(S). A C-factorization of f is a linear factorization (M, d) off such thatM₀, M₁∈C.

Denote byF(C, f) the category whose objects areC-factorizations off and whose morphisms are those described above.

In particular, ifprj(S) is the category of finitely generated projectiveS-modules, then aprj(S)-factorization off is the same as the usual notion of a matrix factor- ization off, that is,F(prj(S), f) =MF(S, f).

We say two morphisms α, β : (M, d) → (M⁰, d⁰) of linear factorizations are homotopic, and write α∼β, if there exists homomorphisms h0 : M0 → M₁⁰ and h1:M1→M₀⁰ satisfying the usual homotopy conditions:

α₀−β₀=h₁d₀+d⁰₁h₀ and α₁−β₁=h₀d₁+d⁰₀h₁.

From this, we define the associatedhomotopy category ofC-factorizations off, de- notedHF(C, f), to be the homotopy category whose objects are the same asF(C, f) and whose morphisms are homotopy classes of morphisms ofC-factorizations.

Lemma 2.2. Let(M, d)∈F(C, f). Iff isM-regular, thend₁ andd₀ are injective.

If f isM-divisible, then d₁ andd₀ are surjective.

Proof. First assume f is M-regular. The equality d0d1 = f1^M1 implies that for x∈M1 withd1(x) = 0, we have 0 =d0d1(x) =f x. Sincef isM-regular, it follows thatx= 0, henced1is injective. Injectivity ofd0 is proved similarly.

Next assume f isM-divisible. Let x∈M0 be any element. Divisibility implies there existsy ∈M0 with f y=x. Sinced1d0=f1^M0, we have d1d0(y) =f y =x, henced1 is surjective. Surjectivity ofd0 is proved similarly.

Given a categoryCofS-modules, the notions of (left and right)C-totally acyclic complexes and (left and right)C-Gorenstein modules were defined in [7, Definition 1.1]; in the case whereCis self-orthogonal, these notions simplify to the following equivalent characterizations by [7, Propositions 1.3 and 1.5]. For anS-complexT, we set Zi(T) = ker(d^T_i ) for eachi∈Z.

Definition 2.3. LetCbe a self-orthogonal category ofS-modules.

(1) AnS-complexT isC-totally acyclicifT is acyclic,T_i ∈Cfori∈Z, and for everyC∈C, the complexes HomS(T, C) and HomS(C, T) are also acyclic.

(2) An S-moduleM is C-Gorenstein ifM = Z0(T) for some C-totally acyclic complexT.

The homotopy category ofC-totally acyclic complexes is denotedKtac(C). If Cis additive, thenKtac(C) is triangulated.

A Prj(S)-Gorenstein module is called a Gorenstein projective module and an Inj(S)-Gorenstein module is called a Gorenstein injective module; these are the standard notions appearing in the literature.

The next lemma is used below to relate cokernel modules ofC-factorizations to totally acyclic complexes.

Lemma 2.4. Let C be a self-orthogonal subcategory of Mod(S), let f ∈ S be S- regular andC-regular, and setR=S/(f). If(M, d)∈F(C, f), then coker(d1)and coker(d0) areR-modules, and for any C∈Candi≥1 the following hold:

(i) Extⁱ_R(R⊗SC,coker(d1)) = 0 = Extⁱ_R(R⊗SC,coker(d0)), (ii) Extⁱ_R(coker(d1), R⊗SC) = 0 = Extⁱ_R(coker(d0), R⊗SC).

Proof. We prove the statements for coker(d₁); proofs for coker(d₀) are similar.

Note first that coker(d₁) is an R-module, sincefcoker(d₁) = 0; indeed, we have f M₀⊆im(d₁) asf1^M0 =d₁d₀, and sof1^M0 induces the zero map on coker(d₁).

Asf isC-regular, Lemma 2.2 yields an exact sequence 0 //M1

d₁ //M0 //coker(d1) //0.

(1)

Let C ∈ C. Application of HomS(C,−) to the exact sequence (1) yields a long exact sequence:

· · · //Extⁱ_S(C, M0) //Extⁱ_S(C,coker(d1)) //Extⁱ⁺¹_S (C, M1) //· · ·

As M0 and M1 are in C, we obtain that Extⁱ_S(C, M0) = 0 = Extⁱ⁺¹_S (C, M1) for i≥1, and hence Extⁱ_S(C,coker(d1)) = 0 fori≥1. Since coker(d1) is anR-module, Lemma 1.7(ii) now yields Extⁱ_R(R⊗S C,coker(d1)) ∼= Extⁱ_S(C,coker(d1)) = 0 for i≥1. This gives (i).

For (ii), instead apply Hom_S(−, C) to the exact sequence (1) to obtain a long exact sequence fori≥1:

· · · //Extⁱ_S(M1, C) //Extⁱ⁺¹_S (coker(d1), C) //Extⁱ⁺¹_S (M0, C) //· · · As M0 and M1 are in C, we obtain that Extⁱ_S(M1, C) = 0 = Extⁱ⁺¹_S (M0, C) for i≥1. It follows that Extⁱ⁺¹_S (coker(d1), C) = 0 fori≥1. Employing Lemma 1.7(i), we obtain Extⁱ_R(coker(d₁), R⊗SC)∼= Extⁱ⁺¹_S (coker(d₁), C) = 0 for alli≥1.

If M is an S-module, α is an S-homomorphism, and R =S/(f), then we set M =R⊗SM andα=R⊗Sα; context should make this clear.

Proposition 2.5. Let C be a subcategory of Mod(S), let f ∈S be S-regular and C-regular, and setR=S/(f). Let(M, d)∈F(C, f). TheR-sequence

T^M := · · · ^d0 //M1

d1 //M0

d0 //M1

d1 //· · ·

is acyclic. IfCis self-orthogonal, then T^M isR⊗_SC-totally acyclic.

Proof. First, asd₁d₀ =f1^M0 and d₀d₁=f1^M1, we have d₁ d₀ = 0 =d₀d₁ and so the sequenceT^M is a complex ofR-modules.

We now show T^M is acyclic. Letx∈M1 such that x∈ker(d1). It follows that d1(x)∈f M0, whence there exists y∈M0such that d1(x) =f y. Asf y=d1d0(y), it follows that d1(x) = d1d0(y), hence d1(x−d0(y)) = 0. Injectivity of d1, see Lemma 2.2, implies that x=d0(y). Hence d0(y) =x, and soH2i+1(T^M) = 0 for every i ∈Z. A similar argument (using injectivity ofd0) yields H2i(T^M) = 0 for everyi∈Z, thus proving the complexT^M is acyclic.

Multiplication by f on the exact sequence 0 →M₁ −→^d1 M₀ → coker(d₁) →0, along with the snake lemma, yields an exact sequence

coker(d1) ^f //coker(d1) //coker(d1) //0.

Since coker(d₁) is anR-module (see Lemma 2.4), this implies coker(d₁)∼= coker(d₁);

similarly, coker(d0)∼= coker(d0). Acyclicity ofT^M gives Z2i(T^M)∼= coker(d0) and Z2i+1(T^M)∼= coker(d1) for everyi∈Z.

FixC∈C. To verify the complexes HomR(T^M, R⊗SC) and HomR(R⊗SC, T^M) are acyclic, it suffices to show that the exact sequences

0 //coker(d0) //M0 //coker(d1) //0 and

0 //coker(d1) //M1 //coker(d0) //0

remain exact upon application of HomR(R⊗SC,−) and HomR(−, R⊗SC). This follows from Lemma 2.4. Therefore, as R⊗S Cis self-orthogonal by Proposition

1.8, we obtain thatT^M isR⊗SC-totally acyclic.

We have the next dual results involving divisibility:

Lemma 2.6. Let C be a self-orthogonal subcategory of Mod(S), let f ∈ S be S- regular and C-divisible, and set R=S/(f). If (M, d)∈ F(C, f), thenker(d1)and ker(d0)are R-modules, and for any C∈Candi≥1the following hold:

(i) Extⁱ_R(HomS(R, C),ker(d1)) = 0 = Extⁱ_R(HomS(R, C),ker(d0)), (ii) Extⁱ_R(ker(d₁),Hom_S(R, C)) = 0 = Extⁱ_R(ker(d₀),Hom_S(R, C)).

Proof. Dual to the proof of Lemma 2.4; use instead Lemma 1.7(iii,iv).

Proposition 2.7. Let C be a subcategory of Mod(S), let f ∈S be S-regular and C-divisible, and setR=S/(f). Let(M, d)∈F(C, f). TheR-sequence

Te^M := · · · ^(d0)∗ //Hom_S(R, M₁) ^(d1)∗ //Hom_S(R, M₀) ^(d0)∗ //· · · is acyclic. IfCis self-orthogonal, then Te^M isHom_S(R,C)-totally acyclic.

Proof. Dual to the proof of Proposition 2.5; use instead Lemma 2.6.

3. A full and faithful functor

LetCbe a self-orthogonal subcategory ofMod(S). We denote byK(C) the homo- topy category ofC, whose objects are complexes of modules inCand morphisms are homotopy classes of degree zero chain maps. Further, we consider the full subcategoryK_tac(C) whose objects are theC-totally acyclic complexes inK(C).

Proposition 3.1. Let Cbe an additive self-orthogonal subcategory ofMod(S), let f ∈ S be S-regular and C-regular, and set R = S/(f). There is a triangulated functor

T:HF(C, f) //K_tac(R⊗_SC)

defined, in notation from Proposition 2.5, as T(M, d) =T^M andT([α]) = [α].

Proof. Let [α],[β] : (M, d) → (M⁰, d⁰) be morphisms in HF(C, f). Set T^M and T^M0 as the complexes constructed in Proposition 2.5 and associated toM andM⁰, respectively. Defineα, β:T^M →T^M0as the evident 2-periodic chain maps induced byα andβ. If [α] = [β], then there is a homotopyh from αto β; this induces a 2-periodic homotopyhfromαtoβ, implying that [α] = [β] inKtac(R⊗SC). Notice that as 1^M = 1^TM, if [α] = [1^M], then [α] = [1^TM].

Define a functorT:HF(C, f)→K_tac(R⊗_SC) as follows: For an object (M, d), set T(M, d) =T^M, and for a morphism [α] : (M, d)→(M⁰, d⁰), set T([α]) = [α].

The above remarks justify thatT is well-defined on both objects and morphisms, that T preserves identities, and that T preserves compositions by the following equalities:

T([α])T([β]) = [α][β] = [(α)(β)] = [αβ] =T([αβ]).

Moreover, the functorTrespects the triangulated structures, that is,Tis additive, T((M, d)[1]) =T^M[1]=T^M[1] =T((M, d))[1], andT preserves exact triangles.

Lemma 3.2. Let C be a self-orthogonal subcategory of Mod(S), let f ∈ S be C- regular, and setR=S/(f). IfM, M⁰ ∈Candϕ∈HomR(M , M⁰), then there exists ψ∈Hom_S(M, M⁰) such thatψ=ϕ.

Proof. Letϕ:M →M⁰ be anR-homomorphism. There is an exact sequence 0 //M^{0 f} //M^{0 π}

0 //M⁰ //0.

As Ext¹_S(M, M⁰) = 0, we obtain an exact sequence

0 //HomS(M, M⁰) //HomS(M, M⁰) //HomS(M, M⁰) //0.

Letπ:M →M be the canonical quotient map. The mapϕπ∈HomS(M, M⁰) lifts to a mapψ∈HomS(M, M⁰) such thatπ⁰ψ=ϕπ, that is,ψ=ϕ.

The following arguments to show that Tis full and faithful closely follow those given in [3], put into the more general setting of totally acyclic complexes from [7].

Proposition 3.3. The functor Tin Proposition 3.1 is faithful.

Proof. SetW=R⊗SC. Let [α] :M →M⁰ be a morphism inHF(C, f) such that T([α]) is the zero morphism inKtac(W). Our goal is to show [α] = [0], that is,αis null homotopic inF(C, f). Write α:M →M⁰ as:

M1 d₁ //

α1

M0 d₀ //

α0

M1 α1

M₁^{0 d}

0

1 //M₀^{0 d}

0 0 //M₁⁰

Letα:T(M, d)→T(M⁰, d⁰) denote the 2-periodic chain map induced byα. The assumptionT([α]) = [0] inKtac(W) implies that αis null homotopic (i.e.,α∼0).

Let σ be a null homotopy for α; notice, however, that σ need not be 2-periodic.

We have the following diagram:

· · · //M1 d₁ //

α₁

M0 d₀ //

α₀

σ2

~~

M1 d₁ //

α₁

σ1

~~

M0 d₀ //

α₀

σ0

~~

M1

α₁

σ−1

~~ //· · ·

· · · //M₁⁰

d⁰₁

//M⁰₀

d⁰₀

//M⁰₁

d⁰₁

//M⁰₀

d⁰₀

//M⁰₁ //· · ·

In particular, we have the following relations (coming from degrees 1 and 2):

α1=d⁰₀σ1+σ0d1, (2)

α0=d⁰₁σ2+σ1d0. (3)

Lemma 3.2 yieldsS-module homomorphism liftings h2i :M0 →M₁⁰ of σ2i and h2i+1:M1→M₀⁰ofσ2i+1fori∈Z. The exact sequence 0→M₁⁰ −→^f M₁⁰ −→^π M₁⁰ →0 induces an exact sequence:

0 //Hom_S(M₁, M₁⁰) ^f //Hom_S(M₁, M₁⁰) ^π∗ //Hom_S(M₁, M₁⁰) //0, where π∗ = HomS(M1, π). Sinceα1−d⁰₀h1−h0d1 ∈ ker(π∗) by (2), one obtains a map β₁ ∈ Hom_S(M₁, M₁⁰) such that f β₁ = α₁−d⁰₀h₁−h₀d₁. Similarly, using instead (3), one obtainsβ₂∈Hom_S(M₀, M₀⁰) such thatf β₂=α₀−d⁰₁h₂−h₁d₀.

Defines1=h1+d⁰₁β1. We claim that (h0, s1) is a null homotopy ofα:M →M⁰. First, we have:

d⁰₀s1+h0d1=d⁰₀(h1+d⁰₁β1) +h0d1

=d⁰₀h1+d⁰₀d⁰₁β1+h0d1

=d⁰₀h1+f β1+h0d1

=d⁰₀h1+α1−d⁰₀h1−h0d1+h0d1

=α₁.

Next, precomposing the equalityf β₁=α₁−d⁰₀h₁−h₀d₁ withd₀ gives:

f β₁d₀= (α₁−d⁰₀h₁−h₀d₁)d₀

=α₁d₀−d⁰₀h₁d₀−h₀f

=d⁰₀α₀−d⁰₀h₁d₀−h₀f

=d⁰₀(α₀−h₁d₀)−h₀f

=d⁰₀(f β2+d⁰₁h2)−h0f

=f d⁰₀β2+d⁰₀d⁰₁h2−f h0

=f(d⁰₀β2+h2−h0).

Asf isM₁⁰-regular, this yields

β1d0=d⁰₀β2+h2−h0. (4)

We therefore obtain:

d⁰₁h0+s1d0=d⁰₁h0+ (h1+d⁰₁β1)d0

=d⁰₁h0+h1d0+d⁰₁β1d0

=d⁰₁h0+h1d0+d⁰₁(d⁰₀β2+h2−h0), by (4),

=d⁰₁h0+h1d0+f β2+d⁰₁h2−d⁰₁h0

=h1d0+α0−d⁰₁h2−h1d0+d⁰₁h2

=α0.

Henceα:M →M⁰ is homotopic to 0, that is, [α] = [0] in HF(C, f).

Proposition 3.4. The functor Tin Proposition 3.1 is full.

Proof. Set W = R ⊗S C. Let (M, d) and (M⁰, d⁰) be objects in HF(C, f) and supposeα:T(M, d)→T(M⁰, d⁰) is a degree 0 chain map, not necessarily 2-periodic, that represents a morphism [α] in Ktac(W); in particular, we have a commutative diagram:

· · · //M1 d₁ //

α3

M0 d₀ //

α2

M1 d₁ //

α1

M0 //

α0

· · ·

· · · //M₁^{0 d}

0

1 //M₀^{0 d}

0

0 //M₁^{0 d}

0

1 //M₀⁰ //· · ·

By Lemma 3.2, for i ∈ Z we can lift α_2i to α_2i : M₀ → M₀⁰ and α_2i+1 to α_2i+1 :M₁ →M₁⁰. In particular, we obtain the following diagram that commutes

modulof:

M₀ ^d0 //

α2

M₁ ^d1 //

α1

M₀

α0

M₀^{0 d}

0

0 //M₁^{0 d}

0 1 //M₀⁰

The exact sequence 0→M₁⁰ −→^f M₁⁰ −→^π M₁⁰ →0 induces an exact sequence 0 //Hom_S(M₀, M₁⁰) ^f //Hom_S(M₀, M₁⁰) ^π∗ //Hom_S(M₀, M₁⁰) //0.

Sinceα1d0−d⁰₀α2∈ker(π∗), there exists a mapσ0∈HomS(M0, M₁⁰) such that α1d0−d⁰₀α2=f σ0.

(5)

Similarly, there existsσ1∈HomS(M1, M₀⁰) such that α0d1−d⁰₁α1=f σ1. (6)

We now define new maps in order to construct a morphism inF(C, f); define γ0=α0+d⁰₁σ0, and

γ1=α1+d⁰₀σ1+σ0d1.

We aim to verify that the following diagram is commutative:

M0 d₀ //

γ0

M1 d₁ //

γ1

M0 γ0

M₀^{0 d}

0

0 //M₁^{0 d}

0 1 //M₀⁰ (7)

The equality (6), along withd₁d₀=f1^M0 andd⁰₀d⁰₁=f1^M10, imply f d⁰₀σ1d0=d⁰₀(α0d1−d⁰₁α1)d0=f d⁰₀α0−f α1d0, and so asf isM₁⁰-regular, we have

d⁰₀σ1d0=d⁰₀α0−α1d0. (8)

First we verify the left square of (7) commutes:

γ1d0= (α1+d⁰₀σ1+σ0d1)d0

=α1d0+d⁰₀σ1d0+f σ0

=α1d0+ (d⁰₀α0−α1d0) +f σ0, by (8),

=d⁰₀α0+f σ0

=d⁰₀α0+d⁰₀d⁰₁σ0

=d⁰₀(α₀+d⁰₁σ₀)

=d⁰₀γ₀.

Next we verify the right square of (7) commutes:

d⁰₁γ1=d⁰₁(α1+d⁰₀σ1+σ0d1)

=d⁰₁α1+f σ1+d⁰₁σ0d1

=d⁰₁α1+α0d1−d⁰₁α1+d⁰₁σ0d1, by (6),

=α0d1+d⁰₁σ0d1

= (α0+d⁰₁σ0)d1

=γ0d1.

Thusγ= (γ0, γ1) is a morphism (M, d)→(M⁰, d⁰) inF(C, f).

We next claimα∼γ, i.e., thatT([γ]) = [α]. We start with the following diagram (displaying homological degrees 3 to−1):

· · · //M1 d₁ //

γ₁−α3

M0 d₀ //

γ₀−α2

M1 d₁ //

γ₁−α1

σ₁

~~

M0 d₀ //

γ₀−α0

σ₀

~~

M1 //

γ1−α−1

· · ·

· · · //M₁⁰

d⁰₁

//M⁰₀

d⁰₀

//M⁰₁

d⁰₁

//M⁰₀

d⁰₀

//M⁰₁ //· · ·

Evidently,σ₁ andσ₀give the start of a homotopy in degree 1:

γ1−α1=α1+d⁰₀σ1+σ0d1−α1=d⁰₀σ1+σ0d1.

Note that the subcategory W is self-orthogonal by Proposition 1.8. As T(M, d) and T(M⁰, d⁰) are W-totally acyclic complexes, the arguments in [10, Appendix]

show that we may extendσ1andσ0to a null homotopy of the displayed morphism, giving γ ∼ α. Indeed, extending the homotopy to the left is done by the proof of [10, Proposition A.3], with W-total acyclicity of T(M⁰, d⁰) standing in for the assumptions inloc. cit. (and [8, Lemma 2.4] in place of [8, Lemma 2.5]); extending the homotopy to the right uses the dual proof for [10, Proposition A.1]. It follows

thatT([γ]) = [α] henceT is full.

The following recovers [3, Theorem 3.5] when one takesC=prj(S).

Theorem 3.5. Let C be an additive self-orthogonal subcategory of Mod(S), let f ∈ S be S-regular and C-regular, and set R = S/(f). The triangulated functor T:HF(C, f)→Ktac(R⊗SC)is full and faithful.

Proof. Combine Propositions 3.1, 3.3, and 3.4.

In fact, the results of this section have dual statements involving divisibility. In summary, one can show the following:

Theorem 3.6. Let C be an additive self-orthogonal subcategory of Mod(S), let f ∈ S be S-regular and C-divisible, and set R = S/(f). There is a triangulated functorTe:HF(C, f)→Ktac(HomS(R,C))that is full and faithful.

Proof. One first notices that a version of Proposition 3.1 holds, by defining a functor Teusing Proposition 2.7. Then using a dual version of Lemma 3.2, one can establish

analogues of Propositions 3.3 and 3.4.

4. Equivalences for projective and injective factorizations In this section, we considerPrj(S)- andInj(S)-factorizations, referred to as projec- tive and injective factorizations, respectively. Our goal here is to show that ifSis a regular local ring,f ∈S is nonzero, andR=S/(f), then projective factorizations off correspond to Gorenstein projectiveR-modules; this can be considered as an extension of the classic bijection [12, Corollary 6.3] between matrix factorizations (having no trivial direct summand) and maximal Cohen-MacaulayR-modules (hav- ing no free direct summand). Dually, we observe a correspondence between injective factorizations off and Gorenstein injectiveR-modules.

If one considersprj(S) in place ofPrj(S) in the next result, then the classic proof, as in [12], uses the Auslander–Buchsbaum formula. However, we use an approach here that does not require the modules to be finitely generated.

Proposition 4.1. Assume S is a regular local ring, letf ∈S be nonzero, and set R=S/(f). IfM is a Gorenstein projectiveR-module, then there exists a projective factorization(P, d)∈F(Prj(S), f)withcoker(d₁) =M.

Proof. LetM be a Gorenstein projectiveR-module. Asf M = 0, a result of Bennis and Mahdou [2, Theorem 4.1] yields Gpd_SM = Gpd_RM + 1 = 1, where Gpd denotes Gorenstein projective dimension. AsS is regular, M has finite projective dimension overS, thus by [6, 4.4.7] we have pd_SM = Gpd_SM = 1.

Now a standard construction yields a projective factorization off which corre- sponds toM: First choose a projective resolution P ofM overS having the form 0 →P₁ −→^d1 P₀ →M →0. Application of Hom_S(P₀,−) to this sequence gives an exact sequence:

0 //Hom_S(P₀, P₁) //Hom_S(P₀, P₀) //Hom_S(P₀, M) //0.

As f M = 0, the mapf1^P0 is sent to 0, hence this sequence shows there exists a mapd0:P0→P1such that d1d0=f1^P0. Further, d1(d0d1) =f1^P0d1=d1(f1^P1), and since d1 is injective this implies that d0d1 =f1^P1. It follows that (P, d) is a projective factorization off such that coker(d₁) =M. Theorem 4.2. Assume S is a regular local ring, let f ∈ S be nonzero, and set R=S/(f). There is a triangulated equivalence

T:HF(Prj(S), f) ^' //Ktac(Prj(R)) given by the functor from Proposition 3.1.

Proof. The triangulated functorTgiven in Proposition 3.1, applied toC=Prj(S), is full and faithful by Theorem 3.5. Also note that R⊗S Prj(S) = Prj(R) (see Remark 1.5) and so the functor Thas the claimed codomain. It remains to show that Tis essentially surjective. LetT ∈Ktac(Prj(R)). Then Z0(T) is a Gorenstein projectiveR-module. By Proposition 4.1 there is aPrj(S)-factorization (P, d) such that coker(d₁) = Z₀(T).

We argue that T(P, d) is homotopic to T. Notice that Z₀(T(P, d)) = Z₀(T) by construction. There exists a degree 0 chain map φ : T(P, d) → T that lifts the identity map Z₀(T(P, d))−⁼→Z₀(T) by [7, Lemma 3.1]; the liftingφis a homotopy

equivalence by [7, Proposition 3.3(b)].

Corollary 4.3. Assume S is a regular local ring, let f ∈ S be nonzero, and set R=S/(f). There is a triangulated equivalence betweenHF(Prj(S), f)and the stable category of Gorenstein projectiveR-modules.

Proof. Combine Theorem 4.2 with the equivalence between K_tac(Prj(R)) and the stable category of Gorenstein projectiveR-modules; see e.g., [7, Example 3.10].

There are dual results for injective factorizations:

Proposition 4.4. Assume S is a regular local ring, letf ∈S be nonzero, and set R=S/(f). IfM is a Gorenstein injectiveR-module, then there exists an injective factorization(I, d)∈F(Inj(S), f)withker(d1) =M.

Proof. Dual to the proof of Proposition 4.1, where one instead uses [2, Theorem 4.2] in place of [2, Theorem 4.1] and [6, 6.2.6] in place of [6, 4.4.7].

Theorem 4.5. Assume S is a regular local ring, let f ∈ S be nonzero, and set R=S/(f). There is a triangulated equivalence

eT:HF(Inj(S), f) ^' //Ktac(Inj(R)) given by the functor from Theorem 3.6.

Proof. Similar to the proof of Theorem 4.2; appeal instead to Theorem 3.6 and

Proposition 4.4.

Corollary 4.6. Assume S is a regular local ring, let f ∈ S be nonzero, and set R=S/(f). There is a triangulated equivalence betweenHF(Inj(S), f)and the stable category of Gorenstein injectiveR-modules.

Proof. Combine Theorem 4.5 and the equivalence between Ktac(Inj(R)) and the stable category of Gorenstein injectiveR-modules; see [16, Proposition 7.2].

5. An equivalence for flat-cotorsion factorizations

In this section, assumeSis a commutative noetherian ring. We give an equivalence in the case of the self-orthogonal category FlatCot(S) of flat-cotorsion S-modules (that is, the category ofS-modules that are both flat and cotorsion). The approach is similar to the previous section, but requires some extra care; in particular, we must establish a fact corresponding to the one from [2] used above.

Denote byM_p^∧= lim

←−(S/pⁿ⊗SM) thep-adic completion of anS-moduleM. By [13], anS-moduleM is flat-cotorsion if and only if it is isomorphic to a product over p∈SpecS of completions of freeSp-modules, that is,M ∼=Q

p∈SpecS(L

B(p)Sp)^∧_p for some setsB(p).

Lemma 5.1. Let π :S →R be a surjective ring homomorphism. Then we have an equalityR⊗_SFlatCot(S) =FlatCot(R).

Proof. First notice that for a flat-cotorsionS-moduleQ

p∈SpecS(L

B(p)Sp)^∧_p, there is an isomorphism

R⊗S

Q

p∈SpecS(L

B(p)Sp)^∧_p

∼=Q

p∈SpecS(L

B(p)R_π(p))^∧_π(p),

sinceRis finitely presented as an S-module. It is now immediate that there is an inclusion R⊗SFlatCot(S)⊆FlatCot(R). The other inclusion follows by observing that every flat-cotorsion R-module can be expressed in a form given by the right

side of this isomorphism, since SpecR=π(SpecS).

The next lemma is needed in place of the change of rings facts for Gorenstein pro- jective and Gorenstein injective dimensions from [2]. As in [7, Definition 4.3], refer to aFlatCot(S)-totally acyclic complex as atotally acyclic complex of flat-cotorsion S-modules and a FlatCot(S)-Gorenstein module as aGorenstein flat-cotorsion S- module; see Definition 2.3. Gorenstein flat-cotorsionS-modules are by [7, Theorem 5.2] precisely the modules that are both Gorenstein flat—that is, isomorphic to Z₀(F) for someF-totally acyclic complexF of flatS-modules—and cotorsion.

Lemma 5.2. Let f ∈ S be a regular element and set R = S/(f). Let M be a Gorenstein flat-cotorsion R-module. There is an exact sequence of S-modules

0 //M⁰ //F //M //0,

withM⁰ a Gorenstein flat-cotorsion S-module andF a flat-cotorsion S-module.

Proof. As M is a Gorenstein flat-cotorsion R-module, there is a totally acyclic complex T of flat-cotorsion R-modules such that Z₀(T) = M. For each i ∈ Z, we may find—because flat covers exist for all modules [4]—a surjective flat cover F_i→Z_i(T) overS; the kernel K_i = ker(F_i →Z_i(T)) is cotorsion by Wakamatsu’s Lemma [24, Lemma 2.1.1]. In fact, since Zi(T) is a cotorsionR-module, it is also a cotorsion S-module for each i ∈ Z by [24, Proposition 3.3.3], hence Fi is flat- cotorsion for each i ∈ Z. Indeed, Zi(T) being a cotorsion S-module also yields Ext¹_S(F_i−1,Z_i(T)) = 0; from this and the snake lemma we obtain, for eachi∈Z, the following commutative diagram with exact rows and columns:

0

0

0

0 //Ki //

T_i⁰ //

Ki−1 //

0

0 //Fi //

Fi⊕F_i−1 //

F_i−1 //

0

0 //Zi(T) //

Ti //

Z_i−1(T) //

0

0 0 0

As Ki and Ki−1 are cotorsion S-modules, so is T_i⁰. Additionally, as Ti is a flat R-module, fdSTi = 1; see Remark 1.6. From [1, 2.4.F], we obtain thatT_i⁰ is a flat S-module.

Now glue together the short exact sequences from the top rows of these diagrams to obtain an acyclic complex T⁰ of flat-cotorsion S-modules with Z_i(T⁰) =K_i for eachi ∈Z. Fix a flat-cotorsion S-module N. Evidently, as each K_i is cotorsion, we obtain Hom_S(N, T⁰) is acyclic. Moreover, for eachi∈Z,

Ext¹_S(K_i, N)∼= Ext²_S(Z_i(T), N)∼= Ext¹_R(Z_i(T), R⊗_SN) = 0,

where the first isomorphism follows from the left vertical exact sequence in the diagram and the second follows from Lemma 1.7(i). For the last equality, note that as N is a flat-cotorsion S-module, we have R⊗S N is a flat-cotorsion R-module

by Lemma 5.1. It now follows that HomS(T⁰, N) is also acyclic. Thus T⁰ is a totally acyclic complex of flat-cotorsionS-modules. In particular, Z0(T⁰) =K0 is a Gorenstein flat-cotorsionS-module, and the claim follows.

Proposition 5.3. Assume S is a regular local ring, let f ∈ S be nonzero, and set R=S/(f). If M is a Gorenstein flat-cotorsion R-module, then there exists a flat-cotorsion factorization (F, d)∈F(FlatCot(S), f)withcoker(d₁) =M.

Proof. LetM be a Gorenstein flat-cotorsionR-module. Lemma 5.2 yields an exact sequence

0 //F1

d₁ //F0 //M //0,

with F₀ a flat-cotorsion S-module and F₁ a Gorenstein flat-cotorsion S-module.

By [7, Theorem 5.2],F₁ is cotorsion and Gorenstein flat. AsS is regular, we have fd_SM < ∞, hence fd_SM = Gfd_SM ≤ 1 by [6, 5.2.8]. Thus F₁ is also flat [1, 2.4.F], hence flat-cotorsion.

As in Proposition 4.1, a standard construction applied to the sequence above provides a flat-cotorsion factorization (F, d) off with coker(d1) =M. Theorem 5.4. Assume S is a regular local ring, let f ∈ S be nonzero, and set R=S/(f). There is a triangulated equivalence

T:HF(FlatCot(S), f) ^' //K_tac(FlatCot(R)) given by the functor in Proposition 3.1.

Proof. Similar to the proof of Theorem 4.2, using Proposition 5.3 in place of Propo-

sition 4.1, and Lemma 5.1 in place of Remark 1.5.

Corollary 5.5. Assume S is a regular local ring, let f ∈ S be nonzero, and set R=S/(f). There is a triangulated equivalence between HF(FlatCot(S), f)and the stable category of Gorenstein flat-cotorsionR-modules.

Proof. This equivalence follows from Theorem 5.4 and the triangulated equivalence between K_tac(FlatCot(R)) and the stable category of Gorenstein flat-cotorsion R-

modules given in [7, Summary 5.7].

One motivation for considering totally acyclic complexes of flat-cotorsion R- modules is their relation to the next analogue of the singularity category as de- scribed by Murfet and Salarian [19].

The pure derived category of flatS-modules is defined as the Verdier quotient D(Flat(S)) =K(Flat(S))/Kpac(Flat(S)) of the homotopy category of flatS-modules by its subcategory of pure acyclic complexes of flat S-modules. Neeman proves in [20, Theorem 1.2] that D(Flat(S)) is equivalent to K(Prj(S)), and moreover, Murfet and Salarian show [19, Lemma 4.22] thatDF-tac(Flat(S)), the subcategory ofD(Flat(S)) ofF-totally acyclic complexes, is equivalent toKtac(Prj(S)), assuming thatS is a commutative noetherian ring having finite Krull dimension.

Corollary 5.6. Assume S is a regular local ring, let f ∈ S be nonzero, and set R=S/(f). There is a triangulated equivalence

HF(FlatCot(S), f) ^' //DF-tac(Flat(R)).

Proof. Combine Theorem 5.4 and [7, Summary 5.7].