ON TWO EXAMPLES BY IYAMA AND YOSHINO

BERNHARD KELLER AND MICHEL VAN DEN BERGH

Abstract. In a recent paper Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen-Macaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov’s result on the graded singularity category. We obtain some new results on the singularity category of isolated singularities which may be interesting in their own right.

Contents

1. Introduction 1 2. Acknowledgement 3 3. Notations and conventions 4 4. First approach to the second example 4 5. Observations on generation 9 6. Some consequences for the singularity category. 13 7. The singularity category of graded Gorenstein rings 14 8. The Iyama-Yoshino examples (again) 17 9. A remark on gradability of rigid modules 19 References 20

1. Introduction

Throughout k is a field. The explicit description of the stable category of maximal Cohen-Macaulay modules over a commutative Gorenstein ring (also known as the singularity category [7, 6, 17]) is a problem that has received much attention over the years. This seems to be in general a difficult problem and perhaps the best one can hope for is a reduction to linear algebra, or in other words: the representation theory of quivers.

In [10] Iyama and Yoshino consider the following two examples.

1991 Mathematics Subject Classification. Primary 13C14, 16E65. Key words and phrases. Singularity category, maximal Cohen-Macaulay modules. The second author is a director of research at the FWO. The results of this paper were partially obtained during visits of the second author to the

Université Paris Diderot-Paris 7 and the Centre de Recerca Matemàtica in Barcelona. He hereby thanks these institutions for their hospitality.

Example 1.1. Let S = k[[x_{1}, x_{2}, x_{3}]] and let C_{3} = ⟨σ⟩ be the cyclic group of three elements. Consider the action of C_{3} on S via σx_{i} = ωx_{i} where ω^{3} = 1, ω ≠ 1. Put R = S^{C_{3}}.

Example 1.2. Let S = k[[x_{1}, x_{2}, x_{3}, x_{4}]] and let C_{4} = ⟨σ⟩ be the cyclic group of four elements. Consider the action of C_{4} on S via σx_{i} = −x_{i}. Put R = S^{C_{4}}.

In both examples Iyama and Yoshino reduce the classification of maximal Cohen-Macaulay modules over R to the representation theory of certain generalized Kronecker quivers. They use this to classify the rigid Cohen-Macaulay modules over R. As predicted by deformation theory, the latter are described by discrete data. The proofs of Iyama and Yoshino are based on the machinery of mutation in triangulated categories, a general theory developed by them. In the current paper we present two alternative approaches to the examples. Hopefully the thus obtained additional insight may be useful elsewhere.

Our first approach applies to Example 1.2 and is inspired by the treatment in [14] of Example 1.1 where the authors used the fact that in this case the stable category \underline{MCM}(R) of maximal Cohen-Macaulay R-modules is a 2-Calabi-Yau category which has a cluster tilting object whose endomorphism ring is the path algebra kQ_{3} of the Kronecker quiver with 3 arrows. From their acyclicity result [14, §1, Thm] they obtain immediately that \underline{MCM}(R) is the corresponding cluster category D^{b}(mod(kQ_{3}))/(τ[−1]). This gives a very satisfactory description of \underline{MCM}(R) and implies in particular the results by Iyama and Yoshino.

In the first part of this paper we show that Example 1.2 is amenable to a similar approach. Iyama and Yoshino prove that \underline{MCM}(R) is a 3-Calabi-Yau category with a 3-cluster tilting object T such that End(T) = k [10, Theorem 9.3]. We show that under these circumstances there is an analogue of the acyclicity result of the first author and Reiten.

Theorem 1.3. (see §4.4) Assume that T is k-linear algebraic Krull-Schmidt 3-Calabi-Yau category with a 3-cluster tilting object T such that End(T) = k. Then there is an equivalence of T with the orbit category D^{b}(mod(kQ_{n}))/(τ^{1/2}[−1]), n = dim Ext_{T}^{−1}(T, T), where Q_{n} is the generalized Kronecker quiver with n arrows and τ^{1/2} is a natural square root of the Auslander-Reiten translate of D^{b}(mod(kQ_{n})), which on the pre-projective/pre-injective component corresponds to “moving one place to the left”.

In the second part of this paper, which is logically independent from the first we give yet another approach to the examples 1.1,1.2 based on the following observation which might have independent interest. It is obtained as a consequence of some results on the generation of the bounded derived category of coherent sheaves which are exhibited in §5.

Proposition 1.4. (see §6) Let A = k + A_{1} + A_{2} ⋯ be a finitely generated commutative graded Gorenstein k-algebra with an isolated singularity. Let Â be the completion of A at A_{≥1}. Let \underline{MCM}_{gr}(A) be the stable category of graded maximal Cohen-Macaulay A-modules. Then the obvious functor \underline{MCM}_{gr}(A) → \underline{MCM}(Â) induces an equivalence

(1.1)  \underline{MCM}_{gr}(A)/(1) ≅ \underline{MCM}(Â)

where M ↦ M(1) is the shift functor for the grading.

In this proposition the quotient \underline{MCM}_{gr}(A)/(1) has to be understood as the triangulated/Karoubian hull (as explained in [13]) of the naive quotient of \underline{MCM}_{gr}(A) by the shift functor ?(1). This result is similar in spirit to [3] which treats the finite representation type case. Note however that one of the main results in loc. cit. is that in case of finite representation type case every indecomposable maximal Cohen-Macaulay Â-module is gradable. This does not seem to be a formal consequence of Proposition 1.4. It would be interesting to investigate this further.

Hence in order to understand MCM(Â) it is sufficient to understand MCM_{gr}(A). The latter is the graded singularity category [16] of A and by [16, Thm 2.5] it is related to D^{b}(coh(X)) where X = Proj A.

In Examples 1.1,1.2 R is the completion of a graded ring A which is the Veronese of a polynomial ring. Hence Proj A is simply a projective space. Using Orlov’s results and the existence of exceptional collections on projective space we get very quickly in Example 1.1

\underline{MCM}_{gr}(A) ≅ D^{b}(mod(kQ_{3}))

and in Example 1.2

\underline{MCM}_{gr}(A) ≅ D^{b}(mod(kQ_{6}))

(where here and below ≅ actually stands for a quasi-equivalence between the underlying DG-categories). Finally it suffices to observe that in Example 1.1 we have ?(−1) = τ[−1] and in Example 1.2 we have ?(−1) = τ^{1/2}[−1] (see §8 below).

Our proof of Proposition 1.4 uses the deep general Neron desingularization theorem of Popescu (which implies the Artin approximation theorem). However in Remark 6.2 we point out that in the situation of Examples 1.1,1.2 a substantial simplification is possible.

Furthermore in §9 we show that rigid Cohen-Macaulay modules are in fact gradable so they are automatically in the image of MCM_{gr}(A). We expect this to be well known in some form but we have been unable to locate a reference.

Finally we mention the following side result which we think may also be of independent interest.

Proposition 1.5. (see §6) Let (R, m) be a local Gorenstein ring with residue field k which is essentially finite over k and has an isolated singularity. Then the natural functor

(1.2)  R̂⊗_{R}? : \underline{MCM}(R) → \underline{MCM}(R̂)

is an equivalence up to direct summands. In partular every maximal Cohen-Macaulay module over R̂ is a direct summand of the completion of a maximal Cohen-Macaulay module over R.

2. Acknowledgement

We thank Osamy Iyama and Idun Reiten for commenting on a preliminary version of this manuscript.

3. Notations and conventions

We hope most notations are self explanatory but nevertheless we list them here. If R is a ring then Mod(R) and mod(R) denote respectively the category of all left R-modules and the full subcategory of finitely generated R-modules. The derived category of all R-modules is denoted by D(R). If R is graded then we use Gr(R) and gr(R) for the category of graded left modules and its subcategory of finitely generated modules. The shift functor on Gr(R) is denoted by ?(1). Explicitly M(1)_{i} = M_{i+1}. If we want to refer to right modules then we use R^{∘} instead of R. If X is a scheme then Qch(X) is the category of quasi-coherent O_{X}-modules. If X is noetherian then coh(X) is the category of coherent O_{X}-modules. We are generally very explicit about which categories we use. E.g. we write D^{b}(mod(R)) rather than something like D_{f}^{b}(R). If R is graded and M, N are graded R-modules then Ext_{R}^{i}(M, N) is the ungraded Ext between M and N. If we need Ext in the category of graded R-modules then we write Ext_{Gr(R)}^{i}(M, N).

4. First approach to the second example

4.1. Some preliminaries on tilting complexes. Let C, E be rings. We denote the unbounded derived category of right C-modules by D(C^{∘}). We let Eq(D(C^{∘}), D(E^{∘})) be the set of triangle equivalences of D(C^{∘}) → D(E^{∘}) modulo natural isomorphisms. Define Tilt(C, E) as the set of pairs (ϕ, T) where T is a perfect complex generating D(E^{∘}) and ϕ is an isomorphism C → RHom_{E}(T). Associated to (ϕ, T) ∈ Tilt(C, E) there is a canonical equivalence θ : D(C^{∘}) → D(E^{∘}) such that θ(C) = T. It may be constructed either directly [19] or using DG-algebras [12]. The induced map

Tilt(C, E) → Eq(D(C^{∘}), D(E^{∘}))

is obviously injective (as it is canonically split), but unknown to be surjective. Below we will informally refer to the elements of Tilt(C, E) as tilting complexes.

4.2. A square root of τ for a generalized Kronecker quiver. Let W be a finite dimensional k-vector space and let C be the path algebra of the quiver^{1}

(4.1)  \xymatrix{1 & 2 \ar[l]_{W}}

Let E be the path algebra of the quiver

\xymatrix{1 & 2 \ar[l]_{W^{∗}}}

which we think of as being obtained from (4.1) by “inverting the arrows” and renumbering the vertices (1, 2) ↦ (2, 1).

Let P_{i},I_{i}, S_{i} be respectively the projective, injective and simple right C-module corresponding to vertex i. For E we use P′_{i}, I′_{i}, S′_{i}. Let r_{i} : mod(C^{∘}) → mod(E^{∘}) be the reflection functor at vertex i. Recall that if (U, V) is a representation of C then r_{1}(U, V) is given by (V, U′) where U′ = ker(V ⊗ W → U) (taking into account the renumbered vertices).

1We use the convention that multiplication in the path algebra is concatenation. So representations correspond to right modules.

The right derived functor Rr_{1} of r_{1} defines an equivalence D(C^{∘}) → D(E^{∘}). It is obtained from the tilting complex S′_{2}[−1] ⊕ P′_{1} [2]. One has (see [8])

(4.2)  Rr_{1} ∘ Rr_{1} = τ_{C}

where τ_{C} is the Auslander Reiten translate on D(C^{∘}). Assume now that W is equipped with an isomorphism π : W → W^{∗}. Then π yields an equivalence D(E^{∘}) ≅ D(C^{∘}), which we denote by the same symbol. We use the same convention for the transpose isomorphism π^{∗} : W → W^{∗}.

Lemma 4.2.1. We have r_{1} ∘ π^{−1} = π^{∗} ∘ r_{1} as functors D(C^{∘}) → D(C^{∘}).

Proof. Let (U, V) be a representation of C determined by a linear map c : V ⊗W → U and put (V, U′′) = (r_{1} ∘ π^{−1})(U, V). Then one checks that U′′ is given by the exact sequence

0 → U′′ → V ⊗ W^{∗} ⟶^{c∘(π^{−1}⊗id)} U → 0

where the first non-trivial map induces the action U′′ ⊗ W → V. Similarly if we put (V, U′) = (π^{∗} ∘ r_{1})(U, V) then one gets the same sequence

0 → U′ → V ⊗ W^{∗} ⟶^{c∘(π^{−1}⊗id)} U

where the first non-trivial map again yields the action U′ ⊗ W → V. Thus we have (V, U′) = (V, U′′). □

Below we put a = π ∘ Rr_{1}.

Lemma 4.2.2. One has (π^{∗} ∘ π^{−1}) ∘ a^{2} = τ. In particular τ ≅ a^{2} if and only if π is self-adjoint or anti self-adjoint.

Proof. This is a straightforward verification using Lemma 4.2.1 and (4.2). □

For use below we record

aP_{2} = P_{1}
aP_{1} = I_{2}[−1]
aI_{2} = I_{1}

4.3. A 3-Calabi-Yau category with a 3-cluster tilting object. We let the notations be as in the previous section,

Put H = D^{b}(mod(C^{∘})), D = H/a[−1]. As H is hereditary we have

Ind(D) = Ind(H)/a[−1]

Inspection reveals that

(4.3)  Ind(D) = Ind(H) ∪ {I_{2}[−1]}

Lemma 4.3.1. D is 3-Calabi-Yau if and only is π is self-adjoint or anti self-adjoint.

Proof. Let S be the Serre-functor for H. Being canonical S commutes with the auto-equivalence a[−1]. Hence S induces an autoequivalence on D which is easily seen to be the Serre functor of D.

In D we have S = τ[1] = (π^{∗} ∘ π^{−1}) ∘ a^{2}[1] = (π^{∗} ∘ π^{−1})[3]. Thus D is 3-Calabi-Yau if and only if π^{∗} ∘ π^{−1} is isomorphic to the identity functor. It is easy to see that this is the case if and only is π^{∗} ∘ π^{−1} = ±1 in End_{k}(W). □

6 BERNHARD KELLER AND MICHEL VAN DEN BERGH

Lemma 4.3.2. The object P_{1} in D satisfies

Ext_{D}^{i}(P_{1}, P_{1}) = 0 for i = 1, 2, Hom_{D}(P_{1}, P_{1}) = k and Ext_{D}^{−1}(P_{1}, P_{1}) = W. Proof. For N ∈ Ind(H) ∪ {I_{2}[−1]} one computes

(4.4)  Hom_{D}(P_{1}, N) = Hom_{H}(P_{1}, N)

Thus we find

Hom_{D}(P_{1}, P_{1}[−1]) = Hom_{D}(P_{1}, a^{−1}P_{1})

= Hom_{D}(P_{1}, P_{2}) = W

Hom_{D}(P_{1}, P_{1}) = k
Hom_{D}(P_{1}, P_{1}[1]) = Hom_{D}(P_{1}, aP_{1})

= Hom_{D}(P_{1}, I_{2}[−1]) = 0

and

Hom_{D}(P_{1}, P_{1}[2]) = Hom_{D}(P_{1}, aP_{1}[1])
= Hom_{D}(P_{1}, I_{2})
= 0 □

The following lemma is not used explicitly. Lemma 4.3.3. The object P_{1} in D has the properties of a 3-cluster tilting object, i.e. if Ext_{D}^{i}(P_{1}, N) = 0 for i = 1, 2 then N is a sum of copies of P_{1}. Proof. Assume that N ∈ Ind(H)∪{I_{2}[−1]} is such that Hom_{D}(P_{1}, N[1]) = Hom_{D}(P_{1}, N[2]) = 0. We have to prove N = P_{1}.

We may rewrite

Hom_{D}(P_{1}, N[2]) = Hom_{D}(P_{1}[−1], N[1])
= Hom_{D}(a^{−1}P_{1}, N[1])
= Hom_{D}(P_{2}, N[1])

Thus we find Hom_{D}(P_{1}, aN) = Hom_{D}(P_{2}, aN) = 0. Hence aN ∉ Ind(H). We deduce N ∈ {P_{1}, I_{2}[−1]}.

But if N = I_{2}[−1] then

Hom_{D}(P_{1}, N[2]) = Hom_{D}(P_{1}, I_{2}[1])

= Hom_{D}(P_{1}, aI_{2}) = Hom_{D}(P_{1}, I_{1}) ≠ 0

So we are left with the possibility N = P_{1} which finishes the proof. □

4.4. Proof of Theorem 1.3. Let T be an algebraic Ext-finite Krull-Schmidt 3-Calabi-Yau category containing a 3-cluster tilting object T such that End_{T} (T) = k.

Lemma 4.4.1. Let N ∈ T . Then there exists a distinguished triangle in T

(4.5)  T^{a} → T^{b} ⊕ T [−1]^{c} → N[1] →

Proof. Let Y be defined (up to isomorphism) by the following distinguished triangle^{2}

Y → T^{Ext_{T}^{1} (T,N)} ⊕ T [−1]^{Ext_{T}^{2} (T,N)} → N[1] →

A quick check reveals that Ext_{T}^{1} (T, Y) = Ext_{T}^{2} (T, Y) = 0. Hence Y = T^{a} for some a. □

We need to consider the special case N = T [1]. Then the distinguished triangle (4.5) (constructed as in the proof) has the form

(4.6)  T^{Ext_{T}^{−1}(T,T)} ⟶^{ϕ} T [−1] ⟶^{α} T [2] ⟶^{β}

where ϕ is the universal map (this follows from applying Hom_{T} (T, −)). Since End_{C}(T [2]) = k it follows that α,β are determined up to (the same) scalar.

This has a surprizing consequence. Applying Hom_{T} (−, T) to the triangle (4.6) we find that Hom_{T} (β[−1], T)^{−1} defines an isomorphism

π : Ext_{T}^{−1}(T, T) → Ext_{T}^{−1}(T, T)^{∗}

Thus W =^{def} Ext_{C}^{−1}(T, T) comes equipped with an isomorphism π : W → W^{∗} which is canonical up to a scalar. In other words we are in the setting of §4.2 and we now use the notations introduced in sections 4.2 and 4.3.

As a is obtained from the reflection in vertex 1, one verifies (see §4.2) that a is associated to the element of Tilt(C, C) given by (θ, I_{2}[−1] ⊕ P_{1}) where θ : C → End_{C}(I_{2}[−1] ⊕ P_{1}) is the composition

(4.7)  C = (\begin{matrix}k & 0 \\ W & k \end{matrix}) ⟶^{π} (\begin{matrix}k & 0 \\ W^{∗} & k \end{matrix}) = End_{C}(I_{2}[−1] ⊕ P_{1})

Since the autoequivalence a is a derived functor that commutes with coproducts

it is isomorphic to a derived tensor functor − ⊗^{L}_{C} X for some X ∈ D(C^{e}), by [11, 6.4]. As a right C-module we have X ≅ I_{2}[−1] ⊕ P_{1}.

Now we use the assumption that H is algebraic and we proceed more or less as in the appendix to [14]. By [11, Thm. 4.3] we may assume that T is a strict (= closed under isomorphism) triangulated subcategory of a derived category D(A) for some DG-category A. We denote by _{C}T the full subcategory of D(C ⊗ A) whose objects are differential graded C ⊗ A-modules which are in T when considered as A-modules. Clearly _{C}T is triangulated. By [14, Lemma A.2.1(a)] T may be lifted to an object in _{C}T , which we also denote by T. Put S = T ⊕ T [−1].

Lemma 4.4.2. One has an isomorphism in _{C}T

X ⊗^{L}_{B} S ≅ S[1]

2It would be more logical to write e.g. Ext_{T}^{1} (T, N) ⊗_{k} T for T^{Ext_{T}^{1} (T,N)} but this would take a lot more space.

Proof. As objects in T we have

X ⊗^{L}_{C} S = (I_{2}[−1] ⊕ P_{1}) ⊗^{L}_{C} S

= I_{2} ⊗^{L}_{C} S[−1] ⊕ P_{1} ⊗^{L}_{C} S

Clearly P_{1} ⊗^{L}_{C} S ≅ T. To compute I_{2} ⊗^{L}_{C} S we use the resolution

0 → P_{1}^{Ext_{T}^{−1}(T,T)} → P_{2} → I_{2} → 0

Tensoring with S we get a distinguished triangle

T^{Ext_{T}^{−1}(T,T)} → T [−1] → I_{2} ⊗^{L}_{C} S →

Comparing with (4.6) we find I_{2} ⊗^{L}_{C} S ≅ T [2]. Thus, we have indeed an isomorphism

φ : X ⊗^{L}_{B} S → S[1]

in T .

Now we check that φ is C-equivariant in T . The left C-module structure on

X ⊗^{L}_{B} S is obtained from the (homotopy) C-action on I_{2}[−1] ⊕ P_{1} as given in (4.7).

Let μ be an element of W = Hom_{C}(P_{1}, P_{2}) = Ext_{T}^{−1}(T, T). We need to prove that the following diagram is commutative in T .

I_{2}[−1] ⊗^{L}_{B} S ⟶^{≅} T [1]
π(μ)⊗^{L}_{B}id_{S} ↓  ↓^{μ}
P_{1} ⊗^{L}_{B} S ⟶_{≅} T

We write this out in triangles

T^{Ext^{−1}(T,T)} ⟶^{ϕ} T [−1] ⟶^{α} T [2] ⟶^{β}
π(μ)↓  ↓  ↓^{μ}
T  ⟶ 0 ⟶ T [1] ⟶_{id}

Rotating the triangles we need to prove that the following square is commutative

T [1] ⟶^{β[−1]} T^{Ext^{−1}(T,T)}
μ↓  ↓^{π(μ)}
T  T

This commutivity holds precisely because of the definition of π. So ϕ is indeed C-equivariant.

But according to [14, Lemma A.2.2], any C-equivariant morphism in T between objects in _{C}T may be lifted to a morphism in _{C}T . This finishes the proof. □

We now have a functor

? ⊗^{L}_{C} T : C → T

and by Lemma 4.4.2 one finds that a[−1](?) ⊗^{L}_{C} T is isomorphic to ? ⊗^{L}_{C} T. By the universal property of orbit categories [13] we obtain a triangulated functor

Q : D → T

which sends P_{1} to T.

Lemma 4.4.3. Q is an equivalence.

Proof. We observe that analogues of the distinguished triangles (4.5) exist in D (with P_{1} replacing T). Indeed, let N ∈ Ind(D). By (4.3) we have N ∈ Ind(H) ∪ {I_{2}[−1]}. If N ∈ Ind(H) then N[1] ≅ aN and the analog of (4.5) is simply the image in D of the projective resolution of aN in H (taking into account that P_{2} = a^{−1}P_{1} = P_{1}[−1]).

If N = I_{2}[−1] then N[1] = I_{2} and the analog of (4.5) is the image in D of the projective resolution of I_{2} in H.

To prove that Q is fully faithful we have to prove that Q induces an isomorphism Hom_{D}(M, N) → Hom_{T} (QM, QN). Using the analogues of (4.5) we reduce to M = P_{1}[i]. But since Hom_{D}(P_{1}[i], N) = Hom_{D}(P_{1}[−1], N[−i − 1]) we reduce in fact to M = P_{1}[−1]. It now suffices to apply Hom_{D}(P_{1}[−1], −) to

P_{1}^{a} → P_{1}^{b} ⊕ P_{1}[−1]^{c} → N[1] →

taking into account that Hom_{D}(M, N) → Hom_{T} (QM, QN) is an isomorphism for M = P_{1}, N = P_{1}, P_{1}[1], P_{2}[2] by Lemma 4.3.2.

As a last step we need to prove that Q is essentially surjective. But this follows from the distinguished triangles (4.5) together with the fact that QP_{1} = T. □

To finish the proof of Theorem 1.3 we observe that since T is 3-Calabi-Yau, so is D. Hence by Lemma 4.3.1 π is either self-adjoint or anti self-adjoint. By Lemma 4.2.2 we deduce a^{2} ≅ τ and hence we may write a = τ^{1/2}.

Remark 4.4.4. It would be interesting to deduce the fact that π is (anti) self-adjoint directly from the Calabi-Yau property of T , without going through the construction of D first. This would have made our arguments above more elegant.

Remark 4.4.5. Iyama and Yoshino also consider categories T equipped with a 2n+1cluster tilting object T such that End(T) = k and Ext^{−i}(T, T) = 0 for 0 < i < n. They relate such T to the representation theory of the generalized Kronecker quiver Q_{m} where m = dim Ext^{−n}(T, T).

One may show that our techniques are applicable to this case as well and yield T ≅ D(kQ_{m})/(τ^{1/2}[−n]). We thank Osama Iyama for bringing this point to our attention.

5. Observations on generation

We now start with the second approach to the examples 1.1,1.2. The results in this section are of the type where we have an exact functor A → B between triangulated categories and we would like to show that B is classically generated by objects in A (see below for terminology). We suspect most results are true in far greater generality then they are stated here.

5.1. Generalities. We use similar notations as in [5] and [20]. Thus if T is a triangulated category (not necessarily closed under direct sums) then T^{c} is the collection of objects X in T such that T (X, −) commutes with (existing) direct sums. If E ⊂ T then ⟨E⟩_{d} is the set of objects in T which can be obtained by taking finite sums, shifts, summands and at most d cones. \overline{⟨E⟩}_{d} is defined similarly except that we allow arbitrary direct sums. Finally we put^{3} ⟨E⟩ = ⋃_{d}⟨E⟩_{d}, \overline{⟨E⟩} = ⋃_{d} \overline{⟨E⟩}_{d}. We say that T is classically generated by E if T = ⟨E⟩. Below we use “generated by” as a synonym for “classically generated by”.

A crucial result, is the following

Proposition 5.1.1. [20, Corollary 3.13] Let E be contained in T^{c}. Then for any d we have

\overline{⟨E⟩}_{d} ∩ T^{c} = ⟨E⟩_{d}

This result is essentially due to Neeman (see e.g. [15, Lemma 1.5]). In [5, Proposition 2.2.4] the result was stated under somewhat stronger hypotheses on T . Note that by distributivity we get

\overline{⟨E⟩} ∩ T^{c} = ⟨E⟩

Let R be a left noetherian ring. Then by [20, Cor 6.16] we have

D^{b}(Mod(R))^{c} = D^{b}(mod(R))

Similarly if X is a separated noetherian scheme then

(5.1)  D^{b}(Qch(X))^{c} = D^{b}(coh(X))

The following is a slight strengthening of a beautiful result by Rouquier.

Proposition 5.1.2. Let X be a separated scheme of finite type over k. Then there is E ∈ D^{b}(coh(X)) and d ∈ ℕ such that

(5.2)  D^{b}(coh(X)) = ⟨E⟩_{d}
(5.3)  D^{b}(Qch(X)) = \overline{⟨E⟩}_{d}

Proof. By (5.1) and Proposition 5.1.1 it suffices to prove (5.3). If k is perfect then this is [20, Thm 7.39]. To do the general case we use base extension. If l/k is a field extension and F ∈ D^{b}(Qch(X) then we write F_{l} for pullback of F under π : X_{l} → X. Similarly if G ∈ D^{b}(Qch(X)) then we write _{l}G for the push forward of G under π.

Since k̄ is always perfect there is some E_{1} ∈ D^{b}(coh(Xk̄)) such that D^{b}(Qch(Xk̄)) = \overline{⟨E_{1}⟩}_{d}. Then there is a finite field extension l/k and an object E_{2} ∈ D^{b}(coh(X_{l})) such that E_{1} = (E_{2})k̄. We put E = _{l}(E_{2}) ∈ D^{b}(coh(X)). Hence _{k}(E_{1}) is a direct sum of copies of E.

Let F be an arbitrary object in D^{b}(Qch(X)). Then Fk̄ ∈ ⟨E_{1}⟩_{d}. Hence _{k}(Fk̄) ∈ ⟨_{k}(E_{1})⟩_{d} = ⟨E⟩_{d}. Since F is a summand of _{k}(Fk̄) we are done. □

5.2. Smooth descent. We say that a commutative k-algebra is essentially finitely generated if it is a localization of a finitely generated k-algebra. A k-algebra morphism R → S of essentially finitely generated k-algebras is essentially smooth if S/R is flat and S ⊗_{R} S has finite global dimension.

3Rouquier uses ⟨E⟩_{∞} instead of ⟨E⟩.

Proposition 5.2.1. Assume that R → S is an essentially smooth ring morphism of essentially finitely generated k-algebras. Then there exists E ∈ D^{b}(mod(R)) such that

(5.4)  D^{b}(mod(S)) = ⟨S ⊗_{R} E⟩_{d}
(5.5)  D^{b}(S) = \overline{⟨S ⊗_{R} E⟩}_{d}

for certain d.

Proof. As above it is sufficient to prove (5.5). Since S ⊗_{R} S is noetherian of finite global dimension we have S ∈ ⟨S ⊗_{R} S⟩_{e} for some e. Assume F ∈ D^{b}(S). We find immediately F ∈ ⟨S ⊗_{R} F ⟩_{e}.

Since R is a localization of a finitely generated ring over k we deduce from Proposition 5.1.2 the existence of E ∈ D^{b}(mod(R)) such that D^{b}(Mod(R)) = \overline{⟨E⟩}_{f} for certain f. In particular S ⊗_{R} F ∈ \overline{⟨S ⊗_{R} E⟩}_{f}. We conclude F ∈ \overline{⟨S ⊗_{R} E⟩}_{e+f}.

□

5.3. Graded descent. Here is our result.

Proposition 5.3.1. Let R be a graded ring, essentially finitely generated over k. Then there exists E ∈ D^{b}(gr(R)) such that

D^{b}(mod(R)) = ⟨E⟩_{d}

Proof. One reduces immediately to the case where R is finitely generated over k. One then checks easily that Rouquier’s proof of [20, Thm 7.39] furnishes a generator E which may be assumed to be graded (by Proposition 5.1.2 the hypothesis on perfectness of the ground field is unnecessary). □

5.4. Descent for complete local rings. Here is our result.

Proposition 5.4.1. Let (R, m) be a Cohen-Macaulay local ring essentially of finite type over a field k, with residue field k, and let R̂ be its completion. Then there exists E ∈ D^{b}(mod(R)) such that

D^{b}(mod(R̂)) = ⟨R̂ ⊗_{R} E⟩_{d}

Proof. Since R essentially of finite type over k it is excellent [9, §7.8] and hence the morphism R → R̂ is regular [9, (7.8.3)(v)]. By the general Neron desingularization theorem of Popescu (see e.g. [18, Thm 1.8]) R̂ is a filtered colimit of (R_{i})_{i∈I} where the R_{i} are smooth finitely generated R-algebras. It will be convenient to replace R_{i} by the localization at the kernel of R_{i} → R̂ → k. So now the R_{i} are local and essentially smooth, essentially of finite type over R.

Claim Assume that M is a finitely generated Cohen-Macaulay R̂-module. Then

there exist i ∈ I and M_{1} ∈ mod(R_{i}) such that M ≅ R̂ ⊗^{L}_{R_{i}} M_{1}.

This claim proves the proposition. Indeed as R̂ is Cohen-Macaulay D^{b}(mod(R̂)) is classically generated by R̂ together with the class of maximal Cohen-Macaulay R̂-modules. So by the claim we find that D^{b}(mod(R̂)) is classically generated by D^{b}(mod(R_{i}))_{i}. But by Proposition 5.2.1 D^{b}(mod(R̂)) is then generated by D^{b}(mod(R)) which in turn is generated by a single element using Proposition 5.1.2.

We now prove the claim. The idea of the proof is to show that the (infinite) projective resolution of M is determined by a finite amount of data. So it is defined over some R_{i}.

Let (x_{i})_{i=1,...,d} be a system of parameters of R̂. Then R̂ is a finite C =^{def} k[[x_{1}, …, x_{d}]]-module. Since R̂ is Cohen-Macaulay it follows that R̂ is a free Cmodule. Write R̂ = ⊕_{i=1}^{m}Cy_{i}. We have

(5.6)  y_{i}y_{j} = ∑_{k} c_{ijk}y_{k}

where the c_{ijk} ∈ C satisfy the associativity equation

(5.7)  ∑_{l} c_{ijl}c_{lkm} = ∑_{l} c_{jkl}c_{ilm}

Since we have assumed that M is maximal Cohen-Macaulay, M is free over C as well. Thus M = ⊕_{i=1}^{n}Cm_{i}. We again have

y_{i}m_{j} = ∑_{k} c′_{ijk}m_{k}

where the c′_{ijk} ∈ C satisfy the associativity equation

(5.8)  ∑_{l} c_{ijl}c′_{lkm} = ∑_{l} c′_{jkl}c′_{ilm}

We now consider the bar resolution of M (which is exact since it is contractible over C)

⋯ → R̂ ⊗_{C} R̂ ⊗_{C} M → R̂ ⊗_{C} M → M → 0

The differential

d : R̂^{⊗_{C}p} ⊗_{C} M → R̂^{⊗_{C}p−1} ⊗_{C} M

is given by (5.9) d(r_{1}⊗⋯⊗r_{p}⊗m) = r_{1}r_{2}⊗r_{3}⊗⋯⊗r_{p}⊗m−r_{1}⊗r_{2}r_{3}⊗⋯⊗r_{p}⊗m+⋯±r_{1}⊗⋯⊗r_{p}m

We now equip R̂^{⊗_{C}p} ⊗_{C} M with the R̂-basis m_{i_{1}···i_{p}} =^{def} 1 ⊗ y_{i_{1}} ⊗ ⋯ ⊗ y_{i_{p−1}} ⊗ m_{i_{p}}.

We have

d(m_{i_{1}···i_{p}}) = y_{i_{1}}m_{i_{2}···i_{p}}−∑_{k} c_{i_{1}i_{2}k}m_{ki_{3}···i_{p}}+∑_{k} c_{i_{2}i_{3}k}m_{i_{1}ki_{4}···i_{p}}−⋯±∑_{k} c′_{i_{p−1}i_{p}k}m_{i_{1}···i_{p−2}k}

and (5.6)(5.7)(5.8) insure that d^{2} = 0.

We may now find i ∈ I as well as ȳ_{i}, c̄_{ijk}, c̄′_{ijk} ∈ R_{i} such that (5.6)(5.7)(5.8) hold in R_{i} when replacing the symbols by their overlined versions.

Let P_{p}, p ≥ 0 be the free R_{i}-module with basis m̄_{i_{0}···i_{p}} where the latter are now just formal symbols. We make (P_{p})_{p} into a complex by defining dm̄_{i_{0}···i_{p}} using the formula (5.9), replacing again all symbols by their overlined versions. Then by construction R̂ ⊗_{R_{i}} P_{•} is the bar resolution of M. We claim that P_{•} is exact in homological degree ≥ 1. Since R̂_{i}/R_{i} is faithfully flat, it suffices to consider P̂_{•} = R̂_{i} ⊗_{R_{i}} P_{•}. Now since R_{i} is essentially smooth over R, the same is true for (R̂ ⊗_{R} R_{i})/R̂ and the map R̂ → R̂ ⊗_{R} R_{i} is split using the map R_{i} → R̂. Hence R̂_{i} = (R̂ ⊗_{R} R_{i}) = R̂[[z_{1}, …, z_{t}]] in such a way that the map R̂_{i} → R̂ (coming from the map R_{i} → R̂) is given by killing (z_{i})_{i}. Invoking lemma 5.4.3 below we deduce that P_{•} is indeed acyclic in degrees ≥ 1.

To finish put M_{1} = coker(P_{1} → P_{0}). Then P_{•} is a projective resolution of M_{1}

and we have R̂ ⊗^{L}_{R_{i}} M_{1} = R̂ ⊗_{R_{i}} P_{•} ≅ M. □

Remark 5.4.2. It is easy to see that the claim contained in the above proof is in fact true for M an arbitrary element of D^{b}(mod(R)). Indeed M is quasi-isomorphic to a complex of the form

0 → M′ → Q_{l} → ⋯ → Q_{0} → 0

with Q_{i} finitely generated projective and M′ maximal Cohen-Macaulay. It now suffices to choose R_{i} in such a way that the maps in this complex are also defined over R_{i}.

We have used the following lemma.

Lemma 5.4.3. Let R be a commutative local noetherian ring and let I be an ideal in the Jacobson radical of R such that for all n one has that I^{n}/I^{n+1} is projective over R/I. Let

P_{2} ⟶^{α} P_{1} ⟶^{β} P_{0}

be a complex of finitely generated projective R-modules such that R/I ⊗_{R} P_{•} is exact (in the middle). Then P_{•} was already exact.

Proof. By induction we show that the cohomology of P_{·} lies in the image of the cohomology of I^{n}P_{·} for arbitrary n. The lemma now follows by an application of Artin-Rees + Nakayama. □

Remark 5.4.4. Some condition on I is necessary for the result to be true. Consider the following example: R = k[[x, y]]/(y(x, y)) and I = (y). Consider the map R → R given by multiplication by x. This map is not injective (it kills y) but it becomes injective after tensoring by R/I.

6. Some consequences for the singularity category.

We can now prove two results mentioned in the introduction.

Proof of Proposition 1.5. We first claim that the functor (1.2) is fully faithful. Let M, N be two maximal Cohen-Macaulay R-modules. Then we have

Ext_{R}^{1}(M, N) = R̂ ⊗_{R} Ext_{R}^{1}(M, N) = Ext1 ; R̂(R̂ ⊗_{R} M, R̂ ⊗_{R} N)

The first equality follows from the fact that Ext_{R}^{1}(M, N) is finite dimensional. Since stable Ext^{1} and ordinary Ext^{1} coincide, fully faithfulness follows.

It remains to show that (1.2) is essentially surjective. Now \underline{MCM}(R̂) is a quotient of D^{b}(mod(R̂)) and by Proposition 5.4.1, D^{b}(mod(R̂)) is generated by objects in D^{b}(mod(R)). This finishes the proof. □

Remark 6.1. In the setting of the Proposition 1.5 it is of course not true that every maximal Cohen-Macaulay R̂-module is obtained from one over R. Consider the following example. Let R be the localization of k[x, y, z, t]/(xy − z^{2} − t^{2}(1 + t)) at (x, y, z, t). This is a three-dimensional terminal singularity of type A_{1}. It is easily seen to be factorial and hence there are no non projective maximal Cohen-Macaulay modules of rank one. On the other hand R̂ does have non projective maximal Cohen-Macaulay modules of rank one. For example I = (x, z − t\sqrt{1 + t}). In this case one may check directly that I is a direct summand of the completion of a Cohen-Macaulay R-module. Put S = R[\sqrt{1 + t}]. Then S/R is Galois. Let σ be the non-trivial automorphism. Put J = (x, z − t\sqrt{1 + t}) ⊂ S. We view J as a Cohen-Macaulay R module of rank two. Then we have

R̂ ⊗_{R} J = R̂ ⊗_{R} S ⊗_{S} J
= (R̂ ⊕ R̂) ⊗_{S} J

The morphism S → R̂⊕R̂ is given by \sqrt{1 + t} ↦ (\sqrt{1 + t}, −\sqrt{1 + t}). Hence R̂⊗_{R}J = Ĵ⊕_{σ}Ĵ = I ⊕ · · ·.

Proof of Proposition 1.4. We claim first that (1.1) is fully faithful. It suffices to check this on generators. Let M, N be two graded maximal Cohen-Macaulay Amodules. Then we have

Ext_{\underline{MCM}_{gr}(A)/(1)}^{1}(M, N) = ⨁_{n} Ext_{\underline{MCM}_{gr}(A)}^{1}(M, N(n))

= ⨁_{n} Ext_{Gr(A)}^{1}(M, N(n))

= Ext_{A}^{1}(M, N)

= Â ⊗_{A} Ext_{A}^{1}(M, N)

= Ext1 ; Â(Â ⊗_{A} M, Â ⊗_{A} N)

Essential surjectivity follows as in the proof of Proposition 1.5. Let Ã be the localization of A at A_{>0}. By Proposition 1.5 D^{b}(mod(Â)) is generated by objects in D^{b}(mod(Ã)). Obviously D^{b}(mod(Ã)) is generated by objects in D^{b}(mod(A)) and finally by Proposition 5.3.1, D^{b}(mod(A)) is generated by D^{b}(gr(A)). □

Remark 6.2. In the Iyama-Yoshino examples A is a Veronese of a polynomial ring. In that case there is a substantial shortcut for the essential surjectivity of (1.1) (which is the only subtle point). Put B = k[x_{1}, …, x_{m}], deg x_{i} = 1, and A = B^{(m)} = ⊕_{i}B_{mi}. For j̄ ∈ ℤ/mℤ put A⟨j̄⟩ = ⊕_{i}B_{mi+j}.

To prove essential surjectivity we have to prove that D^{b}(mod(Â)) is classically generated by gradable objects. Let M ∈ mod(Â). We consider B̂ as a ℤ/mℤgraded Â-algebra. Put M′ = B̂ ⊗Â M ∈ gr_{ℤ/mℤ} B. Then M′ has a finite resolution by finitely generated projective ℤ/mℤ-graded B̂-modules. It is easy to see that all such modules are free as ℤ/mℤ-graded B̂-modules. Restricting to degree ¯0 ∈ ℤ/mℤ we find that M has a resolution by completions of the A⟨j̄⟩. Since the A⟨j̄⟩ are obviously graded we are done.

7. The singularity category of graded Gorenstein rings

7.1. Orlov’s results. Let A = k+A_{1}+A_{2}+⋯ be a commutative finitely generated graded k-algebra. As in [1] we write qgr(A) for the quotient of gr(A) by the Serre subcategory of graded finite length modules. We write π : gr(A) → qgr(A) for the quotient functor. If A is generated in degree one and X = Proj A then by Serre’s theorem [21] we have coh(X) = qgr(A).

Now assume that A is Gorenstein. Then we have RHom_{A}(k, A) ≅ k(a)[−d] where d is the Krull dimension of R and a ∈ ℤ. The number a is called the Gorenstein parameter of A (see [16, Definition 2.1]).

Example 7.1.1. If A is a polynomial ring in n variables (of degree one) then d = n, a = n.

For use below we record another incarnation of the Gorenstein parameter. Let A′ be the graded k-dual of A. Then

(7.1)  RΓ_{A_{>0}}(A) ≅ A′(a)[−d]

where RΓ_{A_{>0}} denotes cohomology with support in the ideal A_{>0}.

The following is a particular case of [16, Thm 2.5].

Theorem 7.1.2. If a ≥ 0 then there are fully faithful functors

Φ_{i} : \underline{MCM}_{gr}(A) → D^{b}(qgr(A))

such that for T_{i} = Φ_{i}\underline{MCM}_{gr}(A) there is a semi-orthogonal decomposition

D^{b}(qgr(A)) = ⟨πA(−i + a + 1), …, πA(−i), T_{i}⟩

Hence under the hypotheses of the theorem we obtain in particular that

\underline{MCM}_{gr}(A) ≅ ^{⊥}⟨πA(−i + a + 1), …, πA(−i)⟩ ⊂ D^{b}(qgr(A))

for arbitrary i.

7.2. The action of the shift functor on the singularity category. Unfortunately the functors Φ_{i} introduced in the previous section are not compatible with ?(1). Our aim in this section is to understand how ?(1) acts on the image of Φ_{i}. This requires us to dig deeper into Orlov’s construction which has the unusual feature of depending on the category D^{b}(gr_{≥i} A) where gr_{≥i} A are the finitely generated graded A-modules with non zero components concentrated in degrees ≥ i. The quotient functor

D^{b}(gr_{≥i} A) ↪ D^{b}(gr A) ⟶^{π} D^{b}(qgr A)

has a right adjoint Rω_{i}A. Its image is denoted by D_{i}.

We let P_{i} be the graded projective A-module of rank one generated in degree i (i.e. P_{i} = A(−i)). Likewise S_{i} is the simple A-module concentrated in degree i. As in Orlov [16] we put P_{≥i} = ⟨(P_{j})_{j≥i}⟩, S_{≥i} = ⟨(S_{j})_{j≥i}⟩ and obvious variants with other types of inequality signs. In [16] it is proved that the image T_{i} of Φ_{i} is the left orthogonal to P_{≥i} inside D^{b}(gr_{≥i} A). The identification of T_{i} with the graded singularity category is through the composition

(7.2)  T_{i} ≅ D^{b}(gr_{≥i} A)/P_{≥i} ≅ D^{b}(gr A)/ perf(A) ≅ \underline{MCM}_{gr}(A)

Assume a ≥ 0. Then the relation between T_{i}, D_{i} is given by the following semiorthogonal decompositions

D^{b}(gr A) = ⟨S_{<i}, \overbrace{P_{≥i+a}, \underbrace{P_{i+a−1}, …, P_{i}, T_{i}}_{D_{i}≅D^{b}(qgr(A))}}^{D^{b}(gr_{≥i} A)}⟩

This is a refinement of Theorem 7.1.2.

The category \underline{MCM}_{gr}(A) comes equipped with the shift functor ?(1). We denote the induced endofunctor on T_{i} by σ_{i}. We will now compute it.

Lemma 7.2.1. For M ∈ T_{i} ⊂ D^{b}(qgr(A)) we have

(7.3)  σ_{i}M = cone(RHom_{qgr(A)}(πA(−i), M) ⊗_{k} πA(−i + 1) → M(1))

where the symbol “cone” is to be understood in a functorial sense, for example by computing it on the level of complexes after first replacing M by an injective resolution.

Proof. Let N ∈ T_{i} ⊂ D^{b}(gr(A)). To compute σ_{i}N we see by (7.2) that we have to find σ_{i}N ∈ T_{i} such that σ_{i}N ≅ N(1) up to projectives. It is clear we should take

σ_{i}N = cone(RHom_{gr(A)}(P_{i−1}, N(1)) ⊗_{k} P_{i−1} → N(1))
= cone(RHom_{gr(A)}(P_{i}, N) ⊗_{k} P_{i−1} → N(1))

Now we note that the RHom can be computed in D_{i} ≅ D^{b}(qgr(A). Furthermore since the result lies in T_{i} ⊂ D_{i} we can characterize it uniquely by applying π to it. Since π commutes with ?(1) we obtain (7.3) with M = πN. □

7.3. The Serre functor for a graded Gorenstein ring. Let A, a, d be as above but now assume that A has an isolated singularity and let M, N ∈ \underline{MCM}_{gr}(A). Then by a variant of [10, Thm 8.3] we have a canonical graded isomorphism

Ext_{A}^{d}(\underline{Hom}_{A}(M, N), A) ≅ \underline{Hom}_{A}(N, M[d − 1])

and furthermore an appropriate version of local duality yields

Ext_{A}^{d}(\underline{Hom}_{A}(M, N), A) = \underline{Hom}_{A}(M, N)^{∗}(a)

In other words we find

\underline{Hom}_{A}(M, N)^{∗} = \underline{Hom}_{A}(N, M[d − 1](−a))

and hence the Serre functor S on \underline{MCM}(A) is given by ?[d − 1](−a).

It is customary to write S = τ[1] so that we have the usual formula

\underline{Hom}_{A}(M, N)^{∗} = Ext^{1}(N, τM)

In this setting we find

(7.4)  τ =?[d − 2](−a)

7.4. The Gorenstein parameter of Veronese subring. We remind the reader of the following well-known result.

Proposition 7.4.1. Let B be a polynomial ring in n variables of degree one. Assume m | n and let B^{(m)} be the corresponding Veronese subring of B. I.e. B_{i}^{(m)} = B_{mi}. Then B^{(m)} is Gorenstein with Gorenstein parameter n/m.

Proof. The Gorenstein property is standard. To compute the Gorenstein invariant we first let A be the “blown up” Veronese. I.e.

A_{i} = \begin{cases}B_{i} & if m | i \\ 0 & otherwise \end{cases}

Let a, b = n be respectively the Gorenstein parameters of A and B. If M is a B-module write M^{+} for ⊕_{i}M_{mi}, considered as graded A-module. We have

A′(a)[−n] = RΓ_{A_{>0}}(A)  (see (7.1))
= RΓ_{A_{>0}}(B)^{+}
= RΓ_{B_{>0}}(B)^{+}
= (B′(b)[−n])^{+}
= A′(b)[−n]

In the 3rd equality we have used that local homology is insensitive to finite extensions. We deduce a = b = n. Since B^{(m)} is obtained from A by dividing the grading by m obtain n/m as Gorenstein parameter for B^{(m)}. □

Remark 7.4.2. In characteristic zero we could have formulated the result for invariant rings of finite subgroups of Sl_{n}(k) (with the same proof). However in finite characteristic Veronese subrings are not always invariant rings (consider the case where the characteristic divides m).

8. The Iyama-Yoshino examples (again)

8.1. Example 1.1. Let B = k[x_{1}, x_{2}, x_{3}] and A = B^{(3)}. We have X =^{def} Proj A = Proj B = ℙ^{2}. By Proposition 7.4.1 A has Gorenstein invariant 1.

Unfortunately we have to deal with the unpleasant notational problem that the shift functors on coh(ℙ^{2}) coming from A and B do not coincide. To be consistent with the sections 7.1,7.2 we will denote them respectively by ?(1) and ?{1}. Thus ?(1) =?{3}. Note that this choice is rather unconventional.

According to Theorem 7.1.2 we have a semi-orthogonal decomposition

D^{b}(coh(X)) = ⟨O_{ℙ^{2}}, T_{0}⟩

From the fact that D^{b}(coh(X)) has a strong exceptional collection O_{ℙ^{2}}, O_{ℙ^{2}}{1}. O_{ℙ^{2}}{2} we deduce that there is a semi-orthogonal decomposition

T_{0} = ⟨O_{ℙ^{2}}{1}, O_{ℙ^{2}}{2}⟩

In particular RHom_{ℙ^{2}}(O_{ℙ^{2}}{1} ⊕ O_{ℙ^{2}}{2}, −) defines an equivalence between T_{0} and the representations of the quiver Q_{3}

\xymatrix{1 & 2 \ar[l]_{V}}

where V = kx_{1} + kx_{2} + kx_{3} and where O_{ℙ^{2}}{i} corresponds to the vertex labeled by i. By (7.4) the Auslander-Reiten translate on \underline{MCM}_{gr}(A) is given by ?[1](−1). In other words: the shift functor on \underline{MCM}_{gr}(A) is given by (τ[−1])^{−1}. By Proposition 5.3.1 we find (using R = Â)

\underline{MCM}(R) ≅ \underline{MCM}_{gr}(A)/(1) ≅ D^{b}(mod(kQ_{3}))/(τ[−1])

which is what we wanted to show.

Remark 8.1.1. Note that this in this example we had no need for the somewhat subtle formula (7.3).

8.2. Example 1.2. We use similar conventions as in the previous section, Let B = k[x_{1}, x_{2}, x_{3}, x_{4}] and A = B^{(2)}. We have X = Proj A ≅ Proj B = ℙ^{3} and we denote the corresponding shift functors by ?(1), ?{1} so that ?(1) =?{2}. By Proposition 7.4.1 A has Gorenstein invariant 2. By Theorem 7.1.2 we have a semiorthogonal decomposition

D^{b}(coh(X)) = ⟨O_{ℙ^{3}}, O_{ℙ^{3}}{2}, T_{−1}⟩

Now D^{b}(coh(X)) has a strong exceptional collection O_{ℙ^{3}}, O_{ℙ^{3}}{1}. O_{ℙ^{3}}{2}, O_{ℙ^{3}}{3}. This sequence is geometric [4, Prop. 3.3] and hence by every mutation is strongly exceptional [4, Thm. 2.3]. We get in particular the following strongly exceptional collection

O_{ℙ^{3}}, O_{ℙ^{3}}{2}. Ω_{ℙ^{3}}^{∗}{1}, O_{ℙ^{3}}{3} where Ω_{ℙ^{3}} is defined by the exact sequence

(8.1)  0 → Ω_{ℙ^{3}} → V ⊗ O_{ℙ^{3}}{−1} → O_{ℙ^{3}} → 0

where V = kx_{1} + kx_{2} + kx_{3} + kx_{4}. Thus there is a semi-orthogonal decomposition

T_{−1} = ⟨Ω_{ℙ^{3}}^{∗}{1}, O_{ℙ^{3}}{3}⟩

An easy computation yields

RHom_{ℙ^{3}}(Ω_{ℙ^{3}}^{∗}{1}, O_{ℙ^{3}}{3}) = ∧^{2}V

RHom_{ℙ^{3}}(Ω_{ℙ^{3}}^{∗}{1} ⊕ O_{ℙ^{3}}{3}, −) defines an equivalence between T_{−1} and the representations of the quiver Q_{6}

\xymatrix{1 & 2 \ar[l]_{Λ^{2}}}

Put W = ∧^{2}V and choose an arbitrary trivialization ∧^{4}V ≅ k. Let π : W → W^{∗} be the resulting (self-adjoint) isomorphism. We are in the setting of §4.2 and hence can define τ^{1/2} as acting on the derived category of Q_{6}.

We will now compute σ_{−1}(Ω_{ℙ^{3}}^{∗}{1}), σ_{−1}(O_{ℙ^{3}}{3}). An easy computation yields

RHom_{ℙ^{3}}(O_{ℙ^{3}}{2}, Ω_{ℙ^{3}}^{∗}{1}) = V^{∗}
RHom_{ℙ^{3}}(O_{ℙ^{3}}{2}, O_{ℙ^{3}}{3}) = V

Using the formula (7.3) we find

(8.2)  σ_{−1}(O_{ℙ^{3}}{3}) = cone(V ⊗ O_{ℙ^{3}}{4} → O_{ℙ^{3}}{5}) = Ω_{ℙ^{3}}{5}[1]
(8.3)  σ_{−1}(Ω_{ℙ^{3}}^{∗}{1}) = cone(V^{∗} ⊗ O_{ℙ^{3}}{4} → Ω_{ℙ^{3}}^{∗}{3}) = O_{ℙ^{3}}{3}[1]

where in the second line we have used the dual version of (8.1).

Let P_{i} be the projective representation of Q_{6} generated in vertex i. The endofunctor on D^{b}(mod(kQ_{6})) induced by σ_{−1} will be denoted by the same letter. We will now compute it. From (8.3) we deduce immediately σ_{−1}(P_{1}) = P_{2}[1]. To analyze (8.2) we note that a suitably shifted slice of the Koszul sequence has the form

0 → ∧^{4}V ⊗ Ω_{ℙ^{3}}^{∗}{1} → ∧^{2}V ⊗ O_{ℙ^{3}}{3} → Ω_{ℙ^{3}}{5} → 0

Thus Ω_{ℙ^{3}}{5} corresponds to the cone of

∧^{4}V ⊗ P_{1} → ∧^{2}V ⊗ P_{2}

which is easily seen to be equal to ∧^{4}V ⊗ τ^{−1}P_{1}.

If we use our chosen trivialization ∧^{4}V ≅ k then we see that at least on objects σ_{−1} coincides with τ^{−1/2}[1]. It is routine to extend this to an isomorphism of functors by starting with a bounded complex of projectives in mod(kQ_{6}).

By Proposition 5.3.1 we find (using R = Â)

\underline{MCM}(R) ≅ \underline{MCM}_{gr}(A)/(1) ≅ D^{b}(mod(kQ_{6}))/(τ^{1/2}[−1])

which is what we wanted to show.

9. A remark on gradability of rigid modules

We keep notations as in the previous section. Since in the Iyama-Yoshino examples \underline{MCM}_{gr}(A) is the derived category of a hereditary category the functor

\underline{MCM}_{gr}(A) → \underline{MCM}_{gr}(A)/(1)

is essentially surjective [13] and hence

\underline{MCM}_{gr}(A) → \underline{MCM}_{gr}(Â)

is also essentially surjective. In more complicated examples there is no reason however why this should be the case. Nevertheless we have the following result which is probably well-known.

Proposition 9.1. Assume that k has characteristic zero. Let A = k+A_{1}+A_{2}+⋯ be a left noetherian graded k-algebra. Put R = Â. Let M ∈ mod(R) be such that Ext_{R}^{1}(M, M) = 0. Then M is the completion of a finitely generated graded Amodule N.

In the rest of this section we let the notations and hypotheses be as in the statement of the proposition (in particular k has characteristic zero). We denote the maximal ideal of R by m.

Let E be the Euler derivation on A and R. I.e. on A we have E(a) = (deg a)a and we extend E to R in the obvious way. If M ∈ mod(R) then we will define an Euler connection as a k-linear map ∇ : M → M such that ∇(am) = E(a)m + a∇(m). If M = N̂ for N a graded A-module then M has an associated Euler connection by extending ∇(n) = (deg n)n for n a homogeneous element of N.

Lemma 9.2. Let M be a finitely generated R module. Then M has an Euler connection if and only if M is the completion of a finitely generated graded Amodule.

Proof. We have already explained the easy direction. Conversely assume that M has an Euler connection. For each n we have that M/m^{n}M is finite dimensional and hence it decomposes into generalized eigenspaces for ∇.

M/m^{n}M = ∏_{α∈k}(M/m^{n}M)_{α} (finite product)

Considering right exact sequences

(m/m^{2})^{⊗n} ⊗ M/mM → M/m^{n+1}M → M/m^{n}M → 0

we easily deduce that the multiplicity of a fixed generalized eigenvalue in M/m^{n}M stabilizes a n → ∞. Thus M = ∏_{α∈k} M_{α} where M_{α} is a generalized eigenspace with eigenvalue α. We put N′ = ⊕_{α}M_{α}. Then N′ is noetherian since obviously any ascending chain of graded submodules of N′ can be transformed into an ascending chain of submodules in M. If particular N′ is finitely generated and we have M = N̂′.

Now N′ is k-graded and not ℤ-graded. But we can decompose N′ along ℤ-orbits and then by taking suitable shifts we obtain a ℤ-graded module with the same completion as N′. □

Proof or Proposition 9.1. Let ϵ^{2} = 0 and consider M[ϵ] where A acts via a · m = (a + E(a)ϵ)m. We have a short exact sequence of A-modules

0 → Mϵ → M[ϵ] → M → 0

which is split by hypotheses. Denote the splitting by m + ∇(m)ϵ. For a ∈ A we have

am + ∇(am)ϵ = (a + E(a)ϵ)(m + ∇(m)ϵ)

and hence

∇(am) = E(a)m + a∇(m)

Hence ∇ is an Euler connection and so we may invoke Lemma 9.2 to show that M = N̂. □

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UFR de Mathmatiques, Universit Denis Diderot - Paris 7, 2, place Jussieu, 75251 Paris Cedex 05, FRANCE

   E-mail address: keller@math.jussieu.fr

Departement WNI, Universiteit Hasselt, Universitaire Campus, Building D, 3590 Diepenbeek, Belgium

   E-mail address: michel.vandbenbergh@uhasselt.be