Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations

We show that equivariant tilting modules over equivariant algebras induce equivalences of derived factorization categories. As an application, we show that the derived category of a noncommutative resolution of a linear section of a Pfaffian variety is equivalent to the derived factorization category of a noncommutative gauged Landau-Ginzburg model $(\Lambda,\chi, w)^{\mathbb{G}_m}$, where $\Lambda$ is a noncommutative resolution of the quotient singularity $W/\operatorname{GSp}(Q)$ arising from a certain representation $W$ of the symplectic similitude group $\operatorname{GSp}(Q)$ of a symplectic vector space $Q$.


Backgrounds
Derived categories of coherent sheaves on varieties are one of the important invariants in algebraic geometry, and they have interesting links to other fields of mathematics. For example, if a smooth variety X admits a tilting bundle T , the derived category D b (coh X) of coherent sheaves on X is equivalent to the derived category D b (mod Λ) of finitely generated right modules over the endomorphism ring Λ · · = End X (T ) of T . Once we have such an equivalence, we can study the derived category of coherent sheaves by the representation theory of noncommutative algebras. However, if X is a smooth projective Calabi-Yau variety, X can never admit a tilting bundle.
Recently, Okonek-Teleman proved that the derived category of a regular zero section in a certain smooth variety is equivalent to the derived factorization category of a noncommutative gauged Landau-Ginzbubrg (LG) model [13]. In particular, they proved that the derived category D b (coh Z) of a Calabi-Yau complete intersection Z ⊂ P n is equivalent to the derived factorization category Dmod Gm (Λ, χ, w) of a noncommutative gauged LG model (Λ, χ, w) Gm , where Λ is a noncommutative crepant resolution of some quotient singularity. This result gives a new approach to the study of the derived categories of Calabi-Yau complete intersections. For example, it is interesting to interpret autoequivalences of D b (coh Z) with autoequivalences of Dmod Gm (Λ, χ, w) induced by G m -equivariant tilting modules/complexes over Λ, and it is expected that this interpretation enables us to study the fundamental group action on D b (coh Z), constructed in [5] using variations of GIT quotients, by equivariant tilting theory over Λ.
In this paper, for any reductive affine algebraic group G, we generalize Okonek-Teleman's result to a G-equivariant setting. More precisely, we prove that G-equivariant tilting modules over G-equivariant algebras induce equivalences of derived factorization categories of noncommutative gauged LG models. Moreover, combining Rennemo-Segal's results in [16] with our result, we prove that the derived category of a noncommutative resolution of a generic linear section of a Pfaffian variety is equivalent to the derived factorization category of a noncommutative gauged LG model (Λ, χ, w) Gm , where Λ is a non-commutative resolution of the quotient singularity W/ GSp(Q) arising from a certain representation W of the symplectic similitude group GSp(Q) of a symplectic vector space Q.

Equivariant algebras and noncommutative LG models
Let X be a scheme, and G an algebraic group acting on X. Denote by σ : G × X → X the morphism defining the G-action on X, and write π : G × X → X for the natural projection. Let ρ : O X → A be a (not necessarily commutative) O X -algebra that is a coherent O X -module. A G-equivariant structure on A is an isomorphism θ A : π * A ∼ −→ σ * A of sheaves of O G×X -algebras such that the pair A, θ A defines the G-equivariant coherent O X -module, and we call the pair A, θ A a G-equivariant coherent X-algebra. Similarly, we define a Gequivariant coherent A-module to be the pair M, θ M of a right A-module M and an isomorphism θ M : π * M ∼ −→ σ * M of π * A-modules such that the pair M, θ M defines a G-equivariant coherent O X -module. We write coh G A for the category of G-equivariant coherent A-modules. Then we have an equivalence

categories, where [A/G] is the associated coherent [X/G]-algebra (see Appendix A).
Let χ : G → G m be a character of G, and w : X → A 1 a χ-semi-invariant regular function on X, i.e., W (g · x) = χ(g)W (x) for any (g, x) ∈ G × X. We call the data (A, χ, w) G a gauged Landau-Ginzburg (LG) model, and it is said to be noncommutative if the algebra A is not commutative. We define a factorization of (A, χ, w) G to be a sequence consisting of G-equivariant coherent A-modules M i and G-equivariant A-linear maps ϕ i such that the compositions ϕ 0 • ϕ 1 and ϕ 1 (χ) • ϕ 0 are the multiplications by w. To a gauged LG model (A, χ, w) G we associate the derived factorization category Dcoh G (A, χ, w) whose objects are factorizations of (A, χ, w) G . If X = Spec R, the G-equivariant X-algebra A corresponds to the G-equivariant R-algebra A · · = Γ(X, A), and we write Dmod G (A, χ, w) for the corresponding derived factorization category.

Notation and convention
• Unless stated otherwise, categories and stacks we consider are over an algebraically closed field k of characteristic zero.
• For a character χ : G → G m of an algebraic group G, we denote by O X (χ), or simply O(χ), the G-equivariant invertible sheaf on a G-scheme X associated to χ.
• For an exact category E, we denote by Ch(E) (resp. Ch b (E)) the category of cochain complexes (resp. bounded cochain complexes) in E, and we write D(E) (resp. D b (E)) for the derived category (resp. the bounded derived category) of E.

Preliminaries
In this section, we recall definitions and basic properties about derived factorization categories, equivariant quasi-coherent sheaves and tilting objects.

Derived factorization categories
We recall the basics of derived factorization categories mainly to fix notation. See, for example, [2,8,15] for more details. Throughout this section, A is an abelian category with small coproducts such that the small coproducts of families of short exact sequences are exact. In what follows, we fix a triple (E, Φ, w) (2.1) consisting of an exact subcategory E ⊆ A of A, an exact autoequivalence Φ : A → A preserving E and a functor morphism w : id → Φ that is compatible with Φ, i.e., the equality w(Φ(E)) = Φ(w(E)) holds for every object E ∈ E. The functor morphism w : id → Φ is called a potential of E.
Definition 2.1. A factorization of (E, Φ, w) is a sequence . Objects E 1 and E 0 in the above sequence are called the components of E. Hom(E, F ) 2n · · = Hom E (E 1 , Φ n (F 1 )) ⊕ Hom E (E 0 , Φ n (F 0 )), where E i and F i are the components of E and F respectively, and

This defines the dg category
Fact(E, Φ, w) of factorizations of (E, Φ, w).
The category Fact(E, Φ, w) is an exact category, and the homotopy category K(E, Φ, w) is a triangulated category (see [8,Propositions 3.5 and 3.9]). To define the derived factorization categories, we define the totalizations of complexes of factorizations: For an object E ∈ Fact(E, Φ, w), let us set Then a periodic (up to twists by Φ) infinite sequence Taking totalizations defines an exact functor Definition 2.4. Let A(E, Φ, w) be the smallest thick subcategory of K(E, Φ, w) that contains totalizations of all short exact sequences in Fact(E, Φ, w). Then we define the derived factorization category D(E, Φ, w) of the triple (E, Φ, w) by the Verdier quotient If E is closed under coproducts in A, we denote by A co (E, Φ, w) the smallest thick subcategory of K(E, Φ, w) that contains totalizations of all short exact sequences in Fact(E, Φ, w) and closed under coproducts. Then we define the coderived factorization category D co (E, Φ, w) by Let F be an exact subcategory of another abelian category B satisfying the same properties of A, and let be a triple as in (2.1). Definition 2.5. An additive functor such that for every object E ∈ E, the following diagram commutes: If F : E → F is a factored functor with respect to (Φ, w) and (Ψ, v), then it induces a dg functor This dg functor defines the additive functors where the latter functor is an exact functor. To define the derived functors of exact functors between homotopy categories, we need the following results due to [2]. Proposition 2.6 ([2, Corollary 2.25]). Assume that A has enough injectives and that the coproducts of injectives are injective. Let I ⊂ A be the subcategory of injective objects. Then the natural functor is an equivalence. Proposition 2.7. Let P ⊂ A be the subcategory of projective objects in A, and assume that A has enough projectives. Let C ⊂ A be an abelian subcategory that is preserved by Φ, and let Q · · = C ∩ P.
1. Assume that all objects in C are compact in A. Then for any P ∈ K(Q, Φ, w) and A ∈ A co (A, Φ, w) we have In particular, the natural functor is fully faithful.
2. If every object in C has a finite projective resolution in C, the natural functor is essentially surjective.
Proof . (1) The latter statement follows from the former one by [12,Proposition B.2]. The vanishing Hom K(A,Φ,w) (P, A) = 0 reduces to the case when A ∈ A(A, Φ, w), since the components of P are compact by our assumption. If  1. Assume that A has enough injectives and that the coproducts of injectives are injective. If F is left exact, we define the right derived functor where the first functor is the equivalence in Proposition 2.6, and Q is the natural quotient functor.
2. Let C ⊂ A and D ⊂ B be abelian subcategories that are preserved by Φ and Ψ respectively, and assume that the factored functor F restricts to F : C → D. Assume that every object in C is compact in A and has a finite projective resolution in C, and that the natural where the first functor is the equivalence in Proposition 2.7.

Quasi-coherent modules over sheaves of algebras
In this subsection, we recall fundamental properties of quasi-coherent modules over sheaves of algebras over schemes. Let X be a scheme. An O X -module F is said to be quasi-coherent if for any point x ∈ X, there are an open neighborhood U of x and an exact sequence where I and J are sets. A quasi-coherent O X -module F is called a coherent O X -module, if for every affine open subset U = Spec R of X, Γ(U, F) is a finitely generated R-module. We denote by Qcoh X (resp. coh X) the category of quasi-coherent (resp. coherent) O X -modules.
Definition 2.9. An X-algebra is a sheaf A of not necessarily commutative algebras together with a morphism ρ : O X → A of sheaves of algebras such that the image of ρ lies in the center of A and that A is a quasi-coherent O X -module. An X-algebra ρ : For an X-algebra ρ : O X → A, a quasi-coherent (resp. coherent) A-module is a right A-module M that is a quasi-coherent (resp. coherent) O X -module. For quasi-coherent A-modules M and N , a morphism from M to N is a morphism of sheaves of right A-modules. We denote by Qcoh A the category of quasi-coherent A-modules, and we write coh A for the full subcategory of coherent A-modules.
Remark 2.10. Since the category of right A-modules and the category Qcoh X are both abelian, Qcoh A is also an abelian category. Furthermore, in Proposition 3.19 we will see that Qcoh A is a Grothendieck category.
Let ϕ : A → B be a morphism of X-algebras. For an object N ∈ Qcoh B, we define a quasicoherent A-module N ϕ by the composition where the second morphism is the right B-action of N . This defines the functor and we call the functor (−) ϕ the restriction by ϕ.
Conversely, for an object M ∈ Qcoh A, the tensor product M ⊗ A B has a natural right B-action, and this defines the functor and we call this functor the extension by ϕ. It is standard that we have an adjunction Although the following propositions might be well known, we give the proofs for reader's convenience.
Proposition 2.11. Let X be a scheme, and ρ : O X → A an X-algebra.
1. For every M ∈ Qcoh A and every point x ∈ X, there are an open neighborhood U of x and a surjective A| U -linear map where J is a set.

A right
in the category of right A-modules.
Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations 9 (2) (⇒) Assume that a right A-module M is quasi-coherent and let x ∈ X be a point. Then by (1) there are an open neighbborhood U of x and a surjective A| U -linear map p : A| ⊕J U M| U . Denote by K the Kernel of p and by i : K → A| ⊕J U the natural inclusion. Then K is also a quasicoherent A-module, and thus, by shrinking U if necessary, we have a surjective A| U -linear map q : A| ⊕I U K| U . Then the sequence is an exact.
(⇐) Assume that a right A-module M is locally isomorphic to the cokernel of a morphism between free modules. Then, since A is a quasi-coherent O X -module and M is locally isomorphic to the cokernel of a morphism of quasi-coherent modules, M is a locally quasi-coherent module. Thus M is a quasi-coherent O X -module.

Equivariant sheaves
We briefly recall the basics of equivariant quasi-coherent sheaves. For more details, see, for example, [3, Section 2].
Let G be an algebraic group, and denote by ε : Spec k → G, µ : G × G → G and ι : G → G the morphisms defining the identity, the multiplication and the inversion of G. Let X be a scheme with an algebraic left G-action σ : G × X → X. We denote by π : G × X → X the natural projection, and we write Definition 2.12. A G-equivariant structure on a quasi-coherent sheaf F ∈ Qcoh X is an isomorphism Remark 2.13. For any closed point g ∈ G, there is the corresponding morphism g : Spec k → G, and this induces the morphism g × id X : X → G × X. The pull-back of a G-equivariant structure θ : π * F ∼ −→ σ * F by g × id X defines the isomorphism where σ g : X → X is the isomorphism defined by σ g · · = σ • (g × id X ). The pull-back of the first equation in (2.4) by the morphism g × h × id X : X → G × G × X implies the equation and the pull-back of the second equation is nothing but the equation A G-equivariant quasi-coherent sheaf is a pair (F, θ) of a quasi-coherent sheaf F ∈ Qcoh X and a G-equivariant structure θ on F. We say that (F, θ) is coherent (resp. locally free) if F is coherent (resp. locally free). If there is no risk of confusion, a G-equivariant quasi-coherent sheaf (F, θ) is denoted simply by F.
Let F, θ F and G, θ G be G-equivariant quasi-coherent sheaves on X. A morphism ϕ: Remark 2.14. Let F, θ F and G, θ G be G-equivariant quasi-coherent sheaves on X. Assume that X is of finite type over k. Then the set of closed points of G × X is equal to the set of pairs (x, g) of closed points x ∈ X and g ∈ G. This implies that a morphism ϕ : F → G of the O X -modules is G-equivariant if and only if for any closed point g ∈ G the pull-back G-equivariant quasi-coherent sheaves on X and G-equivariant morphisms define the abelian category It is standard that the category Qcoh G X is equivalent to the category Qcoh[X/G] of quasicoherent sheaves on the quotient stack [X/G]; (2.6) and for F ∈ Qcoh G X we denote by [F/G] ∈ Qcoh[X/G] the image of F by the equivalence (2.6). We denote by coh G X the subcategory of G-equivariant coherent sheaves on X. This category is an exact subcategory, and if X is noetherian, it is an abelian subcategory. Furthermore, if X is of finite type over k, by Remark 2.14, we have where the right hand side Hom Qcoh X (F, G) G is the G-invariant subset with respect to the Gaction on Hom Qcoh X (F, G) defined by g · ϕ · · = (θ G g ) −1 • σ * g ϕ • θ F g . Let Y be another G-scheme, and f : X → Y a G-equivariant morphism that is quasi-compact and quasi-separated. Then f induces the direct image and the inverse image It is standard that f * is a left adjoint functor of f * .
Tensor products and sheaf Hom define bi-functors and it is also standard that for any F ∈ coh G X, the functor Hom(F, −) : Qcoh G X → Qcoh G X is right adjoint to the functor (−) ⊗ X F : Qcoh G X → Qcoh G X.

Tilting objects and derived equivalences
We recall that tilting objects induce derived equivalences. Throughout this subsection, A is an abelian category with small coproducts and enough injectives, and we assume that small coproducts of families of short exact sequences in A are exact. These properties of A are satisfied if A is a Grothendieck category.
Definition 2.15. Let T ∈ A be an object. The object T is called a tilting object if the following conditions are satisfied: is an isomorphism of abelian groups for any set I and any family {A • i } i∈I . 3. T is a generator of D(A), i.e., for an object A • ∈ D(A), We call T a partial tilting object if the conditions (1) and (2) are satisfied.
For an object T ∈ A, we denote by Add T the smallest additive subcategory containing all direct summands of small coproducts of T . We write add T ⊂ Add T for the subcategory consisting of direct summands of finite coproducts of T . If T is a partial tilting object, then for arbitrary sets I and J , we have isomorphisms This implies that for any X, Y ∈ Add T , we have Ext n A (X, Y ) = 0 for any n > 0. Thus any short exact sequence 0 → X → Z → Y → 0 in A with X, Y ∈ Add T splits, and in particular Add T is an exact subcategory of A.
In the remaining of this subsection, T ∈ A is a partial tilting object, and we assume that every right module over the endomorphism ring Λ · · = End A (T ) is of finite projective dimension. Consider the functor F · · = Hom A (T, −) : A → Mod Λ and assume that (i) F has a left adjoint functor G : Mod Λ → A such that the adjunction morphisms σ(Λ) : Λ → F G(Λ) and τ (T ) : G F(T ) → T are isomorphisms.
(ii) The right derived functor R F : D + (A) → D + (Mod Λ) restricts to the functor Note that both of the functors F and G commute with small coproducts since T is compact and G admits a right adjoint functor. In particular, F and G restrict to the following functors where Proj Λ is the category of projective right Λ-modules.
Proof . The adjunction morphisms σ(P ) : P → F G(P ) and τ (S) : G F(S) → S are bijective for any P ∈ Mod Λ and S ∈ Add T , since σ(Λ) and τ (T ) are bijective and Proj Λ = Add Λ. Hence, the functors F and G are equivalences, and G ∼ = F −1 .
By our assumptions, the functor F defines the right derived functor between bounded derived categories, and the functor G also defines the left derived functor between bounded derived categories. The following is well known to experts.
Proof . We have the following commutative diagram where the vertical arrows are the natural quotient functors. The left vertical functor is an equivalence, since every right Λ-module has a finite projective resolution. Moreover, by Lemma 2.16, the top horizontal functor is also an equivalence. Hence, for the former assertion, it is enough to show that the natural functor is fully faithful, and this follows from an identical argument as in [6, Chapter III, Lemma 2.1]. For the latter assertion, assume that T is a tilting object. It is enough to prove that R F is also fully faithful. For any A • ∈ D(A), consider the following triangle where τ is the adjunction morphism. Applying the functor R F to the above triangle and using the natural isomorphism R F • L G ∼ = id, we see that R F(C(τ )) ∼ = 0 in D b (Mod Λ). Since T is a generator of D(A), this implies that C(τ ) ∼ = 0 in D(A). Hence the adjunction morphism τ is an isomorphism. This completes the proof.
Remark 2.18. Our assumption that every Λ-module has a finite projective resolution might be weakened. However, the result under our assumption is enough for our purpose.

Equivariant algebras and equivariant tilting modules
In this section, we introduce equivariant algebras and equivariant modules, and define several functors between equivariant modules that are generalizations of functors in [3, Section 2]. We also show that equivariant tilting modules induce derived equivalences of equivariant modules, which are simultaneous generalizations of tilting equivalences induced by tilting bundles on schemes and tilting modules over noncommutative algebras.

Equivariant modules over equivariant algebras
In this subsection, we give the definitions of equivariant algebras and equivariant modules, and discuss basic properties. Notation is the same as in Section 2.3. Let ρ : O X → A be an X-algebra.
We write simply A for A, θ A if no confusion seems likely to occur.
where θ can is the canonical equivariant structure on the structure sheaf O X induced by the G-action on X.
Let X = Spec R be an affine scheme with an action from an affine algebraic group G, and set R G · · = k[G] ⊗ k R, where k[G] denotes the coordinate ring of G. We denote by If A, θ A is a G-equivariant X-algebra, taking global sections induces an R-algebra A · · = A(X) and an isomorphism This yields the following definition.
Remark 3.4. For any commutative ring S, via the natural equivalence Mod S ∼ −→ Qcoh Spec S, to give a G-equivariant algebra over Spec S is equivalent to give a G-equivariant S-algebra.
Example 3.5. Notation is the same as above.
1. Let F ∈ coh G X be a G-equivariant coherent sheaf on X. Then the algebra A · · = End X (F) over X has a natural G-equivariant structure. Indeed, the G-equivariant structure on the G-equivariant coherent sheaf End X (F) ∈ coh G X is a morphism of algebras, and so the pair (A, θ) is a G-equivariant algebra over X.

Let
We define equivariant modules over equivariant algebras.

We denote by
Qcoh G A the category of G-equivariant A-modules whose morphisms are G-equivariant, and write coh G A for the full subcategory of G-equivariant coherent A-modules.
Hence G-equivariant modules are generalizations of G-equivariant quasi-coherent sheaves.
Definition 3.8. Let X = Spec R be an affine scheme with an action from an affine algebraic group G, and (Λ, θ Λ ) a G-equivariant R-algebra. A G-equivariant structure on a right Λ-module M is an isomorphism We call such a pair M, θ M a G-equivariant Λ-module. We denote by Mod G Λ the category of G-equivariant Λ-modules whose morphisms are defined similarly to Definition 3.6, and we denote by the full subcategory consisting of equivariant modules that are finitely generated over Λ.
Remark 3.9. Let X = Spec R be an affine scheme with an action from an affine algebraic group G, and A, θ A a G-equivariant X-algebra. Then it induces a G-equivariant R-algebra A · · = A(X), and we have a natural equivalence Recall from Remark 2.13 that for each closed point g ∈ G, we have the induced isomorphism The following is a generalization of the equality in (2.7).
Proposition 3.10. Assume that X is of finite type over k. We have Moreover, if G is reductive, we have for any i ≥ 0.
Proof . The first equality follows from an identical argument as in Remark 2.14. Since G is reductive, the functor of taking G-invariant parts is exact, and thus the second equality follows from the first one.
Example 3.11. Notation is the same as in Example 3.5(1).
1. For any G ∈ Qcoh G X, the quasi-coherent A-module has a natural G-equivariant structure induced by the G-equivariant structure on 2. For M ∈ Mod G A and F ∈ Qcoh G X, the tensor product We define the sheaf of rings on the quotient stack [X/G] to be the image of A ∈ Qcoh G X by the equivalence of sheaf of rings and this makes the sheaf [A/G] an algebra over [X/G]. The following is a generalization of (2.6), and it follows from Proposition A.11.
Proposition 3.12. We have an equivalence of abelian categories.
Remark 3.13. If an affine algebraic group G is abelian, by a similar argument as in [4, Section 2.1] G-equivariant algebras correspond to G-graded algebras, where G is the character group of G. Since we do not need this correspondence in the present paper, we do not give a formulation of the correspondence and its proof.

Functors of equivariant modules
In this subsection, we define fundamental functors between equivariant modules. Let X be a quasi-compact and quasi-separated scheme, and G an algebraic group acting on X by σ : G×X → X. Denote by π : G × X → X the natural projection.

Restrictions and extensions
Let A and B be G-equivariant X-algebras, and let ϕ : A → B be a G-equivariant morphism of X-algebras. The functors in (2.2) and (2.3) define the restriction by ϕ and we have an adjunction

Direct image functors and pull-back functors
Let Y be another quasi-compact and quasi-separated G-scheme, and f : X → Y a G-equivariant morphism. Let A be a G-equivariant X-algebra, and B a G-equivariant Y -algebra. Recall from Example 3.5(2) that the push-forward f * A is a G-equivariant Y -algebra and that the pullback f * B is a G-equivariant X-algebra. The push-forwards and the pull-backs of G-equivariant modules define the following additive functors Using natural morphisms of equivariant algebras ϕ : By standard arguments, we see that the above functors induce adjunctions; Let G be another algebraic group acting on another scheme Z, and let α : G → G be a morphism of algebraic groups. Let h : X → Z be a morphism of schemes, and C an G - , and we have the associated G-equivariant X-algebra h * C. This functor extends to the functor

Tensor products and sheaf Homs
Let -bimodule M such that the left action of O X via ρ coincides with the right action of O X via ρ and that M is a quasicoherent (resp. coherent) sheaf on X and an isomorphism θ : π * M ∼ −→ σ * M of (π * A , π * A)bimodules such that θ M satisfies the condition (2.4). We denote by Qcoh G (A , A) Then we have the tensor product If F ∈ coh G (A , A), we also have the sheaf Hom and there is a functorial isomorphism Hom A (F, N ) .
In other words, if F ∈ coh G (A , A), the functor is left adjoint to the functor The following is a version of projection formula for equivariant A-modules.
Lemma 3.14 (Projection formula). Let Y be a quasi-compact and quasi-separated scheme with a G-action, and g : Y → X a G-equivariant morphism. If g is flat and affine, for any E ∈ Qcoh G Y and F ∈ Qcoh G A, we have an isomorphism of G-equivariant A-modules Proof . Since Rg * ∼ = g * and L g * ∼ = g * , by the projection formula [11, Proposition 3.9.4], we have a quasi-isomorphism where the third isomorphism follows since g * is an exact functor. This isomorphism of O X -modules is nothing but the composition where the first morphism is the morphism induced by the adjunction F → g * g * F and the second morphism is the canonical morphism defined by x ⊗ y → x ⊗ y. Since these morphisms are A-linear and G-equivariant, so is the composition ϕ. This finishes the proof.

Taking invariant sections.
Let H be a closed normal subgroup of G. Assume that the restriction σ H · · = σ| H×X : H ×X → X is the trivial action, so that σ H = π H , where π H · · = π| H×X : H×X → X is the natural projection. Then we have the induced G/H-action on X denoted by We write π : G/H × X → X for the natural projection. For F, θ F ∈ Qcoh G X, we define the subsheaf F H ⊆ F to be the kernel of the composition where µ is the adjunction morphism and θ F H · · = θ F | H×X : (π H ) * F ∼ −→ (σ H ) * F = (π H ) * F. Then the pair F H , θ F | π * F H is a G-equivariant quasi-coherent sheaf. Since the restriction of the isomorphism θ F | π * F H : where p : G → G/H is the natural projection. This defines the functor is left adjoint to the functor (−) H .
Proof . Let M ∈ Qcoh G/H A H and N ∈ Qcoh G A. The result follows from the following sequences of isomorphisms where N ι is the restriction of N by the natural inclusion ι : A H → A, the first isomorphism follows from (3.1), the second isomorphism follows from an identical argument as in the proof of [3, Lemma 2.22], and the last isomorphism follows from a natural isomorphism (N ι ) H ∼ = N H .

Restriction functors and induction functors
Let H be a closed subgroup of G, and α : H → G the natural inclusion morphism. Let A be a G-equivariant X-algebra. We define the restriction functor to be the pull-back by the identity morphism id X : X → X that is α-equivariant. If H is trivial, we write Res · · = Res G {1} , which is nothing but the forgetful functor M, θ M → M. Next, we will construct the adjoint functor of this restriction functor. We define an H-action on G × X by h · (g, x) · · = g h −1 , hx for any h ∈ H, g ∈ G and x ∈ X, and we write for the associated quotient stack. By [3, Lemma 2.16(a)], the quotient stack G × H X is representable by the scheme G/H × X. Since the morphism σ : G × X → X defining the G-action on X is H-invariant, we have the induced morphism We define the morphism induces the G-action on G × H X. With respect to this G-action on G × H X, the morphism ε H X is α-equivariant, and σ H is G-equivariant. Thus σ H * A is a G-equivariant G × H X-algebra, and we have the following functor Since the morphism σ H : G × H X → X is G-equivariant, we have the direct image functor σ H * : Qcoh G σ H * A → Qcoh G A, and we define the induction functor Lemma 3.17. Notation is the same as above. where p and q are the morphisms defining the base field k. If G/H is a reductive affine algebraic group, it is linearly reductive. Then the natural morphism k → p * p * k = k[G/H] is a split mono, and so is the adjunction morphism by the base change formula for the above cartesian square. For any M ∈ Qcoh G A, by Lemma 3.14 the adjunction µ(M) : M → σ H * σ H * M is isomorphic to the tensor product and therefore µ(M) is also a split mono.
We will apply the above result to the following: Proof . For a morphism f : A → B in C, assume that F (f ) = 0. It is enough to show that f = 0. We have a commutative diagram

Fundamental properties of equivariant modules
In this subsection, we prove that the category of equivariant modules over a certain equivariant algebra has enough injectives and enough locally free modules. Let X be a quasi-compact and quasi-separated scheme with an action from an affine algebraic group G, and ρ : O X → A a G-equivariant X-algebra.
Proposition 3.19. The category Qcoh G A is a Grothendieck category. In particular, it has enough injectives.
Proof . The category Qcoh G A is an abelian category with small direct sums, and so it suffices to prove that (1) filtered colimits are exact in Qcoh G A and that (2) Qcoh G A has a generator.
(1) For a point x ∈ X, denote by F where the latter is taking the stalk at x, and A x is the stalk of A at x. Then a sequence in Qcoh G A is exact if and only if for every x ∈ X the sequence in Mod A x induced by F x is exact. Therefore, since F x commutes with filtered colimits and Mod A x is a Grothendieck category, filtered colimits in Qcoh G A are also exact.
(2) It is well known that Qcoh G X is a Grothendieck category (more generally, the category of quasi-coherent sheaves on an algebraic stack is a Grothendieck category [19, Proposition 14.2; 0781], [1, Corollary 5.10]). In particular, Qcoh G X has a generator G ∈ Qcoh G X. We show that is a generator. Let M ∈ Qcoh G A be an object. Then, there is a set I and a surjective morphism M. This shows that G A is a generator in Qcoh G A.

Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations 23
Definition 3.20. Notation is the same as above.

A G-equivariant A-module M, θ M is said to be locally free, if there is an open covering
{U i → X} i∈I of X such that for every U i the restriction M| U i of the underlying Amodule M is a free A| U i -module.

We say that A, θ A satisfies the resolution property, if for any G-equivariant coherent
3. A G-scheme X satisfies the G-equivariant resolution property if the G-equivariant X-algebra (O X , θ can ) has the resolution property. (2) Assume that X is noetherian and that A is coherent. If A, θ A satisfies the resolution property, for every G-equivariant A-module M, there is a surjection E M from a Gequivariant locally free A-module E. This follows since there is a set {M i } i∈I of G-equivariant coherent submodules M i of M such that the natural map i∈I M i → M is surjective.
Proposition 3.22. Assume that A, θ A is coherent and that X has the G-equivariant resolution property. Then A, θ A satisfies the resolution property.
Proof . Let M ∈ coh G A be a G-equivariant coherent A-module. By the assumption, there is a surjective morphism p : E M ρ from a G-equivariant locally free coherent O X -module E. Then the extension Definition 3.23. A G-equivariant (resp. coherent) A-module M, θ M has a finite locally free resolution in Qcoh G A (resp. in coh G A) if there exists an exact sequence 0 → E −n → · · · → E 0 → M → 0 in Qcoh G A (resp. in coh G A) such that each E i is locally free.

Lemma 3.24.
Assume that X is a normal scheme of finite type over k and that G is affine. If X has an ample family of line bundles, then A, θ A satisfies the resolution property.

Proof . This follows from [3, Theorem 2.29] and Proposition 3.22.
Lemma 3.25. Let R be a normal ring of finite type over k, and G an affine reductive algebraic group acting on Spec R. Let A, θ A be a G-equivariant coherent R-algebra, and M, θ M ∈ Mod G A a G-equivariant A-module. If the underlying A-module M ∈ Mod A has a finite projective resolution, so does M, θ M . If M is finitely generated and has a finite projective resolution in mod A, then M, θ M has a finite projective resolution in mod G A.
Proof . By Lemma 3.24, the categories Mod G A and mod G A have enough projectives. Since A is coherent R-algebra, mod G A is an abelian subcategory of Mod G A. It is standard that an object x in an abelian category A has a finite projective resolution if and only if there exists n > 0 such that Ext i A (x, y) = 0 for any i > n and any y ∈ A. Thus the statements follow from Proposition 3.10.

Equivariant tilting modules and derived equivalences
In this subsection, we prove that an equivariant tilting module induces a derived equivalence of equivariant algebras, which is considered as an equivariant Morita theory.
Let X be a separated scheme of finite type over k, and G an affine algebraic group acting on X. Let H ⊆ G be a closed normal subgroup of G, and A a G-equivariant coherent X-algebra. T is called a (G, H)-tilting module (resp. partial (G, H)-tilting module), if the restriction Res G H (T ) is a tilting object (resp. partial tilting object) in Qcoh H A. 2. T is called a G-tilting module (resp. partial G-tilting module) if it is (G, {1})-tilting (resp. partial (G, {1})-tilting).
Let R be a normal ring of finite type over k, and suppose that there is a G-action σ : G × Spec R → Spec R on Spec R such that the H-action on Spec R, which is given by the restriction of σ, is trivial. Let f : X → Spec R be a G-equivariant morphism such that f * O X = R and the associated morphism f : of the underlying coherent A-module T is a G-equivariant R-algebra. We define the functor Then the composition is left adjoint to F, and it preserves equivariant coherent modules. Note that since the morphism f : [X/H] → Spec R is proper, the functor F also preserves equivariant coherent modules. The following is one of our motivations of considering equivariant algebras and equivariant modules.
Theorem 3.27. Assume that G/H is reductive and that every Λ H -module M ∈ Mod Λ H is of finite projective dimension.
1. If T , θ T is partial (G, H)-tilting, then the functor is fully faithful, and it restricts to the fully faithful functor 2. If T , θ T is (G, H)-tilting, then the functor is an equivalence, and it restricts to the equivalence Before we prove this theorem, we prepare the following lemmas.
Lemma 3.28. Notation is the same as above, and assume that G/H is reductive. Then the restrictions Proof . We only prove Res G H is faithful. Since G/H is affine, Ind G H : Qcoh H A → Qcoh G A is an exact functor, and so it extends to the functor the adjunction morphisms, and let P : C 1 → C 2 and Q : D 1 → D 2 be faithful exact functors. Assume that we have functor isomorphisms ϕ : and that the following diagrams are commutative where the top horizontal arrows are the isomorphisms induced by ϕ and ψ.
(2) follows from (1). We only prove (1) for F i , since the statement for G i follows by a similar argument. For an object C ∈ C 1 consider the following triangle It suffices to prove that Cone(τ 1 (C)) is the zero object. By the assumption, P (Cone(σ 1 (C))) is isomorphic to the cone of the morphism σ 2 (P (C)), which is the zero object since F 2 is fully faithful. Hence Cone(σ 1 (C)) is also the zero object since P is faithful.
Proof of Theorem 3.27. We prove (1) and (2) simultaneously. We define a functor to be the composition is fully faithful, and if T H is tilting, it is an equivalence. Now we have the following commutative diagram: (1) and (2) follows from Lemmas 3.28 and 3.29.
The following is a special case of Theorem 3.27, which is an equivariant version of derived equivalences induced by tilting bundles and tilting modules.
Corollary 3.30. Notation is the same as above. Assume that G is reductive.
1. Let T be a G-equivariant vector bundle on X and set Λ · · = End X (T ). If T is G-tilting and Λ is of finite global dimension, we have an equivalence If T is G-tilting and Λ is of finite global dimension, we have an equivalence

Equivariant tilting objects and factorizations
In this section, we prove that equivariant tilting modules induce equivalences of derived factorization categories.

Tilting objects and factorizations
In this subsection, we prove that a tilting object in a Gorthendieck category induces an equivalence of derived factorization categories. To prove this, we need the following lemmas: Let A be a Grothendieck category and (E, Φ, w) a triple as in (2.1).
Lemma 4.1. Assume that we have Ext i A (A, B) = 0 for arbitrary objects A, B ∈ E and all i > 0. Then for objects E, F ∈ Fact(E, Φ, w) and an injective resolution ε : F → J • in Ch (Fact(A, Φ, w)), the map ε * : Hom Fact(A,Φ,w) (E, F ) p → Hom Fact(A,Φ,w) (E, J • ) p of cochain complexes of abelian groups is a quasi-isomorphism for every p ∈ Z. Here injective resolution means that ε is a quasi-isomorphism given by an injection ε 0 : F → J 0 , and all components of the factorizations J n are injective objects in A.
Proof . Since Hom Fact(A,Φ,w) (E, F ) p ∼ = Hom Fact(A,Φ,w) (E[p], F ) 0 , we may assume that p = 0. If ε : F → J • is an injective resolution, then we have the induced injective resolutions ε i : of cochain complexes. The i-th cohomology of the cochain complex Hom Hence, by the assumption, the map (ε * ) i is a quasi-isomorphism, and so is ε * . Proof . Let E, F ∈ Fact(E, Φ, w) be objects. We need to show that the map On the other hand, we can consider the double complex Y •,• defined by is the differential of the complex Hom Fact(A,Φ,w) (E, I q ) • . Then the injective map ι : F → I 0 defining the resolution ι : F → I • gives the morphism of double complexes. By definition, we have isomorphisms Tot(X •,• ) = Hom Fact(A,Φ,w) (E, F ) and Tot(Y •,• ) ∼ = Hom Fact(A,Φ,w) (E, Tot(I • )), and the map Tot(ι) * in (4.1) is the totalization Tot(ι * ) of the map ι * in (4.2). By Lemma 4.1 the map ι * : X p,• → Y p,• of cochain complexes is a quasi-isomorphism for any p ∈ Z, and therefore the map in (4.1) is also a quasi-isomorphism by [9, Theorem 1.9.3] (see also [12,Lemma 2.46]). This completes the proof.
in the abelian category Ch(A) given by id ⊕(−ι i ) : in D + (A).
Let B be a Grothendieck category with enough projectives, and let F : A → B be an additive functor such that it commutes with small direct sums and has a left adjoint functor G : B → A. The adjunction G F implies that F is left exact and G is right exact. By Then, by the assumption, C k i = 0 for any k < m − d and i, and so i∈Z C • i lies in D + (A). Since LG admits a right adjoint functor, it commutes with small direct sums. Applying the functor LG to the triangle (4.3), we obtain a triangle i∈Z (2) Since R + F admits a left adjoint functor L + G by (1), it is enough to show that the adjunction morphism is an isomorphism of functors. Let A • ∈ D + (A) be an object. Then by Lemma 4.3 there are a family {A • i } i∈Z of objects in D b (A) and an exact triangle Consider the following commutative diagram where the horizontal sequences are triangles and the vertical arrows are induced by ε. Since Φ commutes with direct sums, the vertical arrows on the left and middle are the direct sums of morphisms . Hence the vertical arrow on the right hand side is also an isomorphism. Now we are ready to prove that a tilting object induces an equivalence of derived factorization categories. Let T ∈ A be a partial tilting object such that the exact subcategory Add T ⊂ A is preserved by Φ. Set Λ · · = End A (T ), and let (Ψ, v) be a potential on Mod Λ. We define D co Mod(Λ, Ψ, v) · · = D co (Mod Λ, Ψ, v), where the first equivalence follows from Proposition 2.7 and Q is the natural quotient functor.
Since we have the equivalence G : proj Λ ∼ −→ add T , the middle functor G is also an equivalence. Thus the statement follows from Lemma 4.2 and the assumption that the natural functor D(C, Φ, w) → D co (A, Φ, w) is fully faithful.

Derived factorization categories of noncommutative gauged LG models
In this short subsection, we define (noncommutative) gauged Landau-Ginzburg models, and its derived factorization categories.
Definition 4.6. We call the data (A, L, w) G a gauged Landau-Ginzburg model when G is an algebraic group acting on a scheme X, A is a G-equivariant coherent X-algebra, L is a G-equivariant line bundle on X, and w ∈ Γ(X, L) G is a G-invariant global section of L. A gauged Landau-Ginzburg model (A, L, w) G is said to be noncommutative, if the algebra A is not commutative. We write (X, L, w) G · · = (O X , L, w) G , and for a character χ : If (A, L, w) G is a gauged LG model, we have the triple is the tensor product with L, and w : id → L ⊗ O X (−) is the functor morphism defined by the multiplication by w. Then we define where lfr G A is the subcategory of coh G A consisting of locally free A-modules. We call Dcoh G (A, L, w) the derived factorization category of (A, L, w) G , and DMF G (A, L, w) the derived matrix factorization category of (A, L, w) G . If X = Spec R is an affine scheme and A · · = Γ(X, A) is the corresponding G-equivariant R-algebra, we define

Equivariant tilting modules and factorizations
In this subsection, we apply the general result in Section 4.1 to the following geometric setting.
Let the notation be the same as in Section 3.4. Let χ : G/H → G m be a character of G/H, and set χ · · = χ • p : G → G m , where p : G → G/H is the natural projection. Write for O R (χ) (resp. O X χ ) the associated G/H-equivariant invertible sheaf on Spec R (resp. G-equivariant invertible sheaf on X).
where add χ T is the subcategory of coh G A defined by the following additive closure Since the functors F : Qcoh G A → Mod G/H Λ H and G : Mod G/H Λ H → Qcoh G A are factored with respect to ( χ, w) and (χ, w R ), they induce the functors Since Qcoh G A has enough injectives, we have the right derived functor and it restricts to the functor If G/H is reductive and every Λ H -module has a finite projective resolution, then every finitely generated G/H-equivariant Λ H -module has a finite projective resolution in mod G/H Λ H by Lemma 3.25. In this case, we have the left derived functor 2. If T , θ T is (G, H)-tilting and every M ∈ Qcoh A has a finite injective resolution in Qcoh A, then we have the equivalence Proof . By an identical argument as in the proof of Lemma 3.28, we see that the restriction functors and if T , θ T is partial (G, H)-tilting (resp. (G, H)-tilting), L G H is fully faithful (resp. an equivalence) by Theorem 4.5. Hence the results follow from Lemma 3.29.

Linear sections of Pfaffian varieties and noncommutative resolutions
In this section, we prove that derived categories of noncommutative resolutions of linear sections of Pfaffian varieties are equivalent to the derived factorization categories of noncommutative gauged LG models.

Noncommutative resolutions of Pfaffian varieties and its linear sections
Following [16], we recall noncommutative resolutions of Pfaffian varieties and its linear sections. Let V be a vector space of dimension v. For an integer q with 0 ≤ 2q ≤ v, we have a Pfaffian variety and we denote its affine cone by Then we have dim Pf aff q = q(2v − 2q − 1). The Pfaffian variety Pf q is smooth if and only if q = 1, where it defines a Grassmannian Gr(2, V ), or q = v/2 , where it defines the whole space P 2 V * . In other cases, the singular locus is the subvairiety Pf (q−1) ⊂ Pf q . Let L ⊂ 2 V * be a subspace of codimension c such that Pf q | L · · = Pf q ∩ P(L) = ∅ and Pf q | L has the expected dimension dim Pf q −c. If c > dim Sing(Pf q ) = (q −1)(2v −2q +1)−1, we can take a generic L so that Pf q | L is smooth. If c is smaller than the bounds, then Pf q | L is never smooth, and in this case the usual bounded derived category D b coh Pf q | L does not behave well (for example, in the context of homological projective duality [16]).
Let (Q, ω) be a symplectic vector space of dimension 2q with a symplectic form ω ∈ 2 Q * . Let Sp(Q) · · = {f ∈ GL(Q) | f * ω = ω} be the symplectic group of Q, and GSp(Q) · · = {f ∈ GL(Q) | ∃ t ∈ k * such that f * ω = t ω} the symplectic similitude group of Q. Then we have a short exact sequence and consider the following quotient stacks Then the surjective morphism induces the morphisms π aff : Y aff → Pf aff q and π : Y → Pf aff q /G m , and if we set Y ss · · = Y \ p −1 (0) / GSp(Q) , the morphism π : Y → Pf aff q /G m restricts to a (stacky) resolution π : Y ss → Pf q of the Pfaffian variety Pf q .
Recall that irreducible representations of Sp(Q) are indexed by Young diagrams of height at most q. We denote by Y q,s the set of Young diagrams of height at most q and width at most s · · = v/2 − q. Since vector bundles on Y aff are nothing but Sp(Q)-equivariant vector bundles on Y , each representation of Sp(Q) defines a vector bundle on Y aff . For each Young diagram γ ∈ Y q,s , we can choose a vector bundle V γ ∈ vect Y whose pull-back to Y aff by the natural projection Y aff → Y is the vector bundle associated to γ (see [16,Section 2.4]). Then the vector bundle is a partial tilting bundle on Y. Restricting V to the open substack Y ss we obtain a partial tilting bundle V ss on Y ss , and the Pf q -algebra is a noncommutative resolution of Pf q by [18,Section 5]. By the noncommutative Bertini theorem [17], for a generic L the restriction B L := B| Pfq| L is a noncommutative resolution of the linear section Pf q | L . If Pf q | L is smooth, the category D b (coh B L ) is equivalent to D b (coh Pf q | L ).

Noncommutative resolutions of Pfaffian varieties and noncommutative gauged LG models
In this subsection, we prove that the derived category of a noncommutative resolution of a linear section of a Pfaffian variety is equivalent to the derived factorization category of a noncommutative gauged Landau-Ginzburg model. We use the same notation as in the previous section. We recall that the category D b (coh B L ) is equivalent to a full subcategory of the derived factorization category of a gauged LG model. We write L ⊥ ⊂ 2 V for the annihilator of L. We have a GSp(Q)-action on the product Y × L ⊥ defined by where σ : GSp(Q) → G m is the surjection in (5.1). Consider the following quotient stacks The function W : Y × L ⊥ → C defined by W (ϕ, x) · · = (ϕ * ω)(x) is GSp(Q)-invariant, and thus it defines a potential w : Y × Gm L ⊥ → C.
We define an additional G m -action, which is called an R-charge, on Y × L ⊥ by This G m -action commutes with the above GSp(Q)-action, and so we have an induced G m -action on Y × Gm L ⊥ . Then the potential w : Y × Gm L ⊥ → C is a semi-invariant regular function with respect to the character χ 1 · · = id Gm : G m → G m . Following [16], we define the B-brane category DB Y × Gm L ⊥ , w as follows: We write rep GSp(Q) for the category of finite dimensional algebraic representations of GSp(Q), and we define the full subcategory consisting of irreducible representations whose restriction to the subgroup Sp(Q) is an irreducible representation of Sp(Q) corresponding to some Young diagram in Y q,s . Since the origin 0 ∈ Y × L ⊥ is a fixed point of the above action from GSp(Q) × G m , the restriction to the origin defines a functor where the latter functor is a similarly equivalence as in [ For an interval I ⊂ Z, we define a subcategory of Y q,s consisting of representations whose restriction to the diagonal subgroup G m ⊂ GSp(Q) has weights in I. This subcategory defines a full subcategory consisting of objects F such that each cohomology H i (F | 0 ) of the restriction F | 0 lies in Y I q,s . By [16,Theorem 4.7] we have an equivalence In particular, if Pf q | L is smooth, we have an equivalence D b coh Pf q | L ∼ = Dmod Gm (Λ, χ, w).
Then the T -weights of Y are {χ i , χ 0 − χ i } 1≤i≤q with each weight occurring with multiplicity v, and T -weights of L ⊥ is −χ 0 with multiplicity dim L ⊥ = c. Thus, if c = qv, the sum of all T -weights of Y is trivial, and in this case the category D b (coh B L ) is Calabi-Yau of dimension 2qs − 1.

A Algebras over groupoids in algebraic spaces
In this appendix, we prove a generalization of Proposition 3.12. We freely use the terminology and notation from [19], and categories fibered in groupoids are defined over the big fppf site (Sch/S) fppf over a fixed base scheme S. First, we recall some definitions from [19], where the main references are Chapters Groupoids in Algebraic Spaces, Algebraic Stacks, and Sheaves on Algebraic Stacks.
Let p : X → (Sch/S) fppf be a category fibered in groupoids. Recall that we have the induced fppf topology on X , where a family of morphisms {x i → x} i in X is a covering of x ∈ X if the family {p(x i ) → p(x)} i is an fppf cover of the scheme p(x). We write X fppf for this fppf site. Then we have the structure sheaf O X : X op fppf → (Rings) of X fppf defined by and thus we have the associated ringed site X fppf , O X . An O X -module F : X op fppf → (Sets) is said to be quasi-coherent, if for any x ∈ X there is a covering {x i → x} of x such that for each x i there is an exact sequence of O X /x i -modules where X fppf /x i , O X /x i is the localization of the ringed site X fppf , O X at x i ∈ X and F| x i is the restriction of F to X fppf /x i , O X /x i . We denote by Qcoh X the category of quasi-coherent O X -modules. If X is an algebraic stack, Qcoh X is equivalent to the category of quasi-coherent sheaves on the lisse-étale site of X defined in [14, Definition 9.1.14]. In particular, if X is an algebraic space, Qcoh X is equivalent to the category of quasi-coherent sheaves on the smallétale site of X defined in [14,Definition 7.1.5], and if X is a scheme, it is equivalent to the category of usual quasi-coherent sheaves on the small Zariski site of X . If X is an algebraic space, we write Xé t (resp. X fppf ) for the smallétale site of X (resp. the big fppf site (Sch/X) fppf of X), and sheaves on X means sheaves on Xé t . Similarly, sheaves on a scheme X means sheaves on the small Zariski site of X.
Definition A.1. An X -algebra is a sheaf of (not necessarily commutative) rings A on X fppf together with a morphism of sheaves of rings ρ : O X → A such that the O X -module A is quasi-coherent and the image of ρ is in the center of A, i.e., for any x ∈ X the image of ρ(x) : O X (x) → A(x) is contained in the center of the ring A(x).

If (A, ρ)
is an X -algebra, we have the restriction by ρ where Mod A denotes the category of right A-modules.
Definition A.2. A right A-module M is said to be quasi-coherent if the O X -module M ρ is quasi-coherent. We denote by Qcoh A the full subcategory of Mod A consisting of quasi-coherent right A-modules.
Remark A.3. If X is an algebraic space, we tacitly consider X -algebras and quasi-coherent modules over X -algebras as sheaves on the smallétale site Xé t instead of big fppf sheaves as above.
Let A be an X -algebra, and denote by X the stackification of X . By [19,  where Sh(−) denotes the category of sheaves on a site (−). For an object F ∈ Sh(X fppf ), we denote by F ∈ Sh X fppf the image of F by the equivalence (A.1). By this equivalence the ring object A ∈ Sh(X fppf ) defines a ring object A ∈ Sh X fppf and a morphism ρ : O X → A.
By [19,Lemma 12.2; 06WR], the sheaf of rings A is quasi-coherent O X -module, and so A is an X -algebra. Proof . The first assertion is obvious, and for the second statement it is enough to show that a right A-module M is quasi-coherent if and only if the right A-module M is quasi-coherent. But this follows from the following commutative diagram and [19, Lemma 12.2; 06WR].
Let G = (U, R, s, t, c) be a groupoid in algebraic spaces over S (see [19,Definition 11.1; 043W] for the notation).
Definition A.5. An algebra over G, or G-algebra, is a U -algebra A : U oṕ et → (Rings) together with an isomorphism θ A : s * A ∼ −→ t * A of R-algebras such that the equations of morphisms of sheaves of rings hold, where p i : R × s,U,t R → R is the i-th projection and e : U → R is the identity.
Remark A.6. Note that the structure sheaf O U together with the canonical isomorphism is commutative, where the isomorphisms on the top level are natural isomorphisms.