DUALIZABLE AND INVERTIBLE OBJECTS IN THE DERIVED CATEGORY OF O_{X}-MODULES

H. FAUSK

Abstract. We give a characterization of dualizable and invertible objects in the category of modules over a commutative unital ringed space. This extends to ringed spaced with an action of a discrete group. The original motivation of this work was to describe the group of isomorphism classes of invertible complexes of O_{X}-modules. This has since been done in a more efficient way elsewhere [5]. We give a description of the equivariant Picard group of a commutative unital ring with a continuous action by a profinite group.

1. Introduction

The goal of this paper is to calculate the group of isomorphism classes of invertible objects in D_{R}, the derived category of cochain complexes of R-modules for a commutative unital ring R. This group is called the Picard group of D_{R} and denoted Pic(D_{R}). We also describe the dualizable objects in the derived category D_{R}, and generalize this to the derived category of a ringed space with an action of a discrete group.

We first give some motivation before we state the main theorem. There is an inclusion of the ordinary Picard group Pic(R) of a ring R into Pic(D_{R}) given by sending an invertible R-module M to the complex which is M in degree zero, and trivial in all other degrees. By splitting the ring R into a product of rings R ≅ R_{1} ⊕ ⋯ ⊕ R_{N} we get for any invertible module M and a set of integers n_{1}, …, n_{N}, one for each R_{i}, an invertible complex with trivial differentials given as the sum of the complexes which are M ⊗ R_{i} shifted to degree n_{i} for i = 1, …, N. Since each M ⊗ R_{i} is a projective R-module these complexes are invertible both in the category of cochain complexes and in the derived category D_{R}. A splitting of R into a product of finitely many rings is in bijective correspondence with a partition of the space of prime ideals spec R into a disjoint union of open-closed sets. A convenient way of expressing the decomposition of the ring R and the degree shifting is to consider continuous maps from spec R into the integers ℤ. We denote the additive group of all continuous functions spec R → ℤ by C(spec R).

Date: 2001.

1991 Mathematics Subject Classification. Primary ; Secondary .

The following theorem says that the invertible complexes described above give all isomorphism classes of invertible objects in D_{R}.

Theorem 1.1. There is a natural split short exact sequence

0 → Pic(R) → Pic(D_{R}) → C(spec R) → 0.

When spec R has finitely many components (e.g. if R is Noetherian) then C(spec R) ≅ ⊕ℤ where the sum is over the components of spec R.

We generalize this result to ringed spaces (X, O) with a group action by a discrete group G. Let sh_{G}(X) be the (abelian) category of G-O-modules.

Theorem 1.2. Assume that for all points x ∈ X the stalk O_{x} at x has connected prime ideal spectrum and trivial Picard group. Then there is a natural split short exact sequence

0 → Pic_{G}(X) → Pic_{G}(D_{sh_{G}(X)}) → C^{G}(X) → 0.

Here Pic_{G}(X) is the group of isomorphism classes of invertible G-O-modules, and C^{G}(X) are the G-fixed points of C(X), the additive group of continuous functions X → ℤ.

We prove the following exact sequence relating Pic_{G}(R) to Pic(R) and group cohomology with coefficient in the units R^{×} of R.

Proposition 1.3. There is an exact sequence

0 → H^{1}(G; R^{×}) → Pic_{G}(R) → Pic(R)^{G} →^{d} H^{2}(G; R^{×})

There is a G-action on Pic(R) and Pic(R)^{G} denotes the G-fixed points. This generalizes to profinite groups if we use continuous group cohomology.

Outline

In section 1 we review the definition and some properties of dualizable objects in symmetric monoidal categories. All invertible objects are dualizable. In section 2 we prove that any dualizable object in D_{R} is equivalent to a perfect complex, i.e. a bounded complex of finitely generated projective modules. To prove our main theorem we use that any invertible complex is equivalent to a bounded complex with finitely presented top (nonzero) cohomology group. In section 3 we prove that there is a natural split short exact sequence

0 → Pic(R) → Pic(D_{R}) →^{Ψ} C(spec R) → 0.

We show in 4.2 that for a local ring R, any invertible complex in D_{R} is equivalent to a complex of the form R[n], which is R concentrated in some degree n. The second map Ψ is given by sending a complex M^{•} to the continuous function Ψ(M^{•}) : spec R → ℤ defined by mapping a prime ideal 𝔭 to the integer Ψ(M^{•})(𝔭), such that M_{𝔭}^{•} is equivalent to R_{𝔭}[Ψ(M^{•})(𝔭)], that is R_{𝔭} concentrated in degree Ψ(M^{•})(𝔭). We show that Ψ(M^{•}) is a continuous function from spec R into ℤ.

The rest of the paper is devoted to generalizations to ringed spaces. In section 4 we prove that a dualizable object in the derived category of complexes of O-modules is locally (restricted to an open cover) a retract of a bounded complex with finitely generated free O-modules in each degree. We also get that the top nonzero cohomology is finitely presented. In section 5 we prove that if the stalk O_{x} has connected prime ideal spectrum and Pic(O_{x}) is trivial for all points x in the space X, then there is a natural split short exact sequence

0 → Pic(X) → Pic(D_{sh(X)}) → C(X) → 0.

In particular, we have the split short exact sequence for sheaves of abelian groups on a topological space X, and for O_{X}-modules for a scheme (X, O_{X}).

In section 6 we observe that the main theorem generalizes to G-O-modules and prove 1.2 We also study the Picard group of G-R-modules for a profinite group G and prove 1.3. This is motivated by [10, 8.4] and a question by J. Morava.

2. Formal duality

Let (C, ⊗, I, F) be a symmetric monoidal (tensor) closed additive category [15]. Here ⊗ is the tensor product, F is the internal homfunctor right adjoint to ⊗, and I is the unit object for the smash product. Define a functor D : C^{op} → C by sending an object X to the dual DX = F (X, I) of X.

Definition 2.1. An object X in a symmetric monoidal (not necessarily closed) category is called invertible if there exists an object Y such that X ⊗ Y ≅ I. Assume there is a set of isomorphism classes of invertible objects. Then this set has a structure of an abelian group under tensor product with the class of the unit object I as the identity element. This group is called the Picard group of the category and denoted Pic(C).

Definition 2.2. An object X is said to be dualizable if there exists an object Y and maps η : I → X ⊗ Y and ϵ : Y ⊗ X → I such that the following two maps are the identity maps

X ≅ I ⊗ X ⟶^{η⊗1} X ⊗ Y ⊗ X ⟶^{1⊗ϵ} X ⊗ I ≅ X

Y ≅ Y ⊗ I ⟶^{1⊗η} Y ⊗ X ⊗ Y ⟶^{ϵ⊗1} I ⊗ Y ≅ Y.

The maps η and ϵ give rise to maps natural in A and B

ϵ_{♯} : C(A, B ⊗ Y) ⟶^{⊗X} C(A ⊗ X, B ⊗ Y ⊗ X) →^{ϵ_{∗}} C(A ⊗ X, B)

η_{♯} : C(A ⊗ X, B) ⟶^{⊗Y} C(A ⊗ X ⊗ Y, B ⊗ Y) →^{η^{∗}} C(A, B ⊗ Y).

The two maps are inverse equivalences. Conversely, given a map ϵ such that ϵ_{♯} is a natural isomorphism, then a map η satisfying the two identities above is given by the adjunction, hence X is dualizable, and similarly if η is given such that η_{♯} is a natural isomorphism. The adjunction also shows that ϵ and η uniquely determine each other.

The next Proposition state some properties of dualizable object. For a general discussion of the theory of dualizable objects and proofs, see [4, 13, 18].

Proposition 2.3. Let C be a symmetric monoidal closed category.

(1) An object X is dualizable if and only if we can choose the counit map ϵ in definition 2.2 to be the evaluation map ev : DX ⊗X → I (adjoint to the identity morphism on DX). The map ν : X ⊗DX → F (X, X) adjoint to 1_{X} ⊗ev is then an isomorphism, and the corresponding unit map η : S → X ⊗ DX is given as the composite (the first map is the adjoint of 1_{X})

I → F (X, X) ≅^{ν^{−1}} X ⊗ DX.

(2) If X is dualizable then DX is dualizable, and ρ : X → DDX is an isomorphism.

(3) The pairing

F (X_{1}, Y_{1}) ⊗ F (X_{2}, Y_{2}) → F (X_{1} ⊗ X_{2}, Y_{1} ⊗ Y_{2})

is an isomorphism if one of the following conditions is satisfied a) X_{1} and X_{2} are dualizable or b) both X_{1} and Y_{1} are dualizable or both X_{2} and Y_{2} are dualizable.

(4) The tensor product and finite sum of dualizable objects are again dualizable, and a retract of a dualizable object is dualizable.

The next result is from [18].

Proposition 2.4. An object X is invertible if and only if X ⊗ − : C → C is an equivalence of categories. If X is invertible, then X is dualizable, and I → F (X, X), η : I → X ⊗ DX, and ev : DX ⊗ X → I are isomorphisms.

Proof. If X ⊗ Y ≅ S then Y ⊗ − is an inverse of X ⊗ −. Conversely, if X ⊗ − is an equivalence of categories, then there exists an object Y in C such that X ⊗ Y is isomorphic to S. For the second part note that I → F (X, X) induces a map

C(−, I) → C(−, F (X, X)) ≅ C(− ⊗ X, X)

which is the map − ⊗ X, hence is an isomorphism. The Yoneda lemma now gives that I → F (X, X) is an isomorphism.

Let ϵ : Y ⊗ X ≅ I. Then

ϵ_{#} : C(W, Z ⊗ Y) ⟶^{⊗X} C(W ⊗ X, Z ⊗ Y ⊗ X) ≅ C(W ⊗ X, Z)

is a natural isomorphism for all Z and W in C, and X is dualizable. Both ev_{#} and η_{#} are isomorphisms. Since X, Y are invertible it follows from the first part that ev and η are isomorphisms.

□

Example 2.5. In the category of (left) R-modules M_{R} for a commutative unital ring R, an R-module M is dualizable if and only if there exist elements m_{1}, m_{2}, …, m_{n} and maps ϕ_{1}, ϕ_{2}, …, ϕ_{n} : M → R for some n, such that ∑_{i=1}^{n} m_{i}ϕ_{i} = 1_{M}. Hence the dualizable objects are precisely the finitely generated projective R-modules. By localizing the map η : R → P ⊗ hom(P, R) at the prime ideals of R, it follows that the invertible modules P are precisely the finitely generated rank one projective modules. The Picard group of M_{R} is the classical Picard group of the ring R.

Let (X, O) be a commutative ringed space. The category of (sheaves of) O-modules, denoted sh(X), is a symmetric monoidal closed category. Recall that F ⊗_{O_{X}} G is the sheaf associated to the presheaf U ↦ F(U) ⊗_{O(U)} G(U). The internal-hom Hom(F, G) is the sheaf U ↦ hom_{O_{X}|U}(F|U, G|U).

Proposition 2.6. An O-module F is dualizable if and only if F is locally a retract of a free O-module of finite rank, i.e. there exists an open covering {U_{α}} of X such that F|U_{α} is a retract of (O|U_{α})^{n_{α}} for some nonnegative integer n_{α}.

Proof. If F is dualizable then the map

ν : Hom(F, O) ⊗_{O} F → Hom(F, F)

is an isomorphism, so the identity map 1_{F} in Hom(F, F) comes from an element in (Hom(F, O))⊗_{O} F(X). From the construction of sheafification there is an open covering {U_{α}} of X such that 1_{F_{X}|U_{α}} is contained in the image of the map of presheaves

Hom_{O_{X}|U_{α}}(F|U_{α}, O_{X}|U_{α}) ⊗_{O_{X}(U_{α})} F(U_{α}) → Hom_{O_{X}|U_{α}}(F|U_{α}, F|U_{α}).

Hence, as in the case of modules, F|U_{α} is a retract of (O_{X}|U_{α})^{n_{α}} for some integer n_{α}.

It is clear that if F is locally a retract of a free module of finite rank then F is dualizable. □

Remark 2.7. The Proposition tells us that all dualizable O-modules are locally finitely presented, hence quasi-coherent.

Localization of an O-module at a point (taking the stalk) gives a functor from sh(X) to M_{O_{X,x}} which respects the tensor product. Restricted to the full sub category of dualizable O-modules, the localization functor also respects the inner hom. In particular if F is dualizable then F_{x} is a finitely generated projective O_{X,x}-module for all x ∈ X. If F is invertible then for all points x in X the stalk F_{x} is an invertible O_{X,x}-module. A dualizable O-module F is invertible if and only if F_{x} is an invertible O_{X,x}-module for all points x in X.

Remark 2.8. If M is a finitely generated projective R-module then there are finitely many elements {r_{1}, …, r_{n}} in R such that M_{r_{i}} is a free R_{r_{i}}-module and {spec R_{r_{i}}} covers spec R [1]. Hence when (X, O) is a scheme we observe that F is dualizable if and only if F is a locally free O-module of finite rank, and F is invertible if and only if F is a locally free O-module of rank 1. This shows that our terminology agrees with the classical one in the case of schemes.

We repeatedly use the following observation due to J.P.C. Greenlees (unpublished). He used it to characterize the dualizable G-spectra in equivariant stable homotopy theory [17, p.174]. We refer to the following result as Greenlees’ Lemma.

Lemma 2.9. Let X be a dualizable object, and let f : Z → X be a map such that η : I → X ⊗ DX factors through Z ⊗ DX via f ⊗ 1_{DX}. Then there is a map g : X → Z such that the composite f ∘ g = 1_{X}. Hence X is a retract of Z.

Proof. The map g is given by the following commutative diagram.

\xymatrix{ & Z ⊗ DX ⊗ X \ar[r]^{1⊗ev} \ar[d]^{f⊗1} & Z ⊗ I ≅ Z \ar[d]^{f} \\ X ≅ I ⊗ X \ar[ur] \ar[r]^{η⊗1} & X ⊗ DX ⊗ X \ar[r]^{1⊗ev} & X ⊗ I ≅ X}

From the definition (1 ⊗ ev) ∘ (η ⊗ 1) = 1_{X}. □

3. The dualizable objects in the derived category of modules over a ring

We first set the notation and recall some facts about derived categories. The complexes have differentials of degree +1. We say that two complexes are equivalent if they are isomorphic in the derived category D_{R} i.e. there is a sequence of quasi isomorphisms connecting one to the other.

Convention 3.1. For an R-module M let M[n] denote the cochain complex which is M in degree n and 0 in all other degrees. The nth shift M^{•}[n] of a cochain complex M^{•} is M^{i−n} in degree i, and the differential is (−1)^{n}d. For a cochain complex M^{•}, let τ_{≤n}M^{•} denote the sub-complex of M^{•} which is 0 in degrees above n, the kernel of d : M^{n} → M^{n+1} in degree n, and agrees with M^{•} in degrees below n. The map τ_{≤n}M^{•} → M^{•} induces an isomorphism on cohomology in degrees less than or equal to n.

Lemma 3.2. Let M^{•} be a cochain complex. If H^{∗}(M^{•}) = 0 for all

∗ ≠ n then M^{•} and H^{n}(M^{•})[n] are equivalent.

Proof. We have obvious cohomology isomorphisms

H^{n}(M^{•})[n] ← τ_{≤n}M^{•} → M^{•}

connecting H^{n}(M^{•})[n] and M^{•}. □

The derived category D_{R} is a symmetric monoidal closed category. We follow Spaltenstein [19] and Lipman’s notes [14] for this and other results. See also [2, 12] for alternative approaches.

Definitions 3.3. A complex P^{•} is called K-projective if for all acyclic complexes A^{•} the homomorphism complex Hom^{•}(P^{•}, A^{•}) is acyclic.

A K-projective resolution of a complex M^{•} is a quasi isomorphism P^{•} → M^{•} from a K-projective complex P^{•} to M^{•}.

If P^{•} is a K-projective complex then the derived tensor product P^{•} ⊗_{R}^{L} M^{•} is the tensor product of the chain complexes P^{•} ⊗_{R} M^{•} which in degree n is ⊕_{i+j=n}P^{i} ⊗_{R} M^{j}. The differential restricted to P^{i} ⊗_{R} M^{j} is given by d ⊗ 1 + 1 ⊗ (−1)^{i}d. In general derived tensor products are defined by replacing one of the complexes with an equivalent K-projective complex. The derived hom functor RHom^{•}(F^{•}, G^{•}) is defined by replacing F^{•} with a K-projective resolution, or by replacing G^{•} with a dually defined K-injective resolution.

We need a K-projective resolution of a particular form which we now describe. This is a special case of [19, 3.3].

Lemma 3.4. Let M^{•} be a cochain complex of R-modules. For any integer k there exists a sequence

P_{k}^{•} → P_{k+1}^{•} → P_{k+2}^{•} → ⋯

of complexes of projective R-modules mapping into M^{•} such that

1): P_{n}^{∗} = 0 for all ∗ > n and f_{n} : P_{n}^{•} → τ_{≤n}M^{•} is a quasi

isomorphism. 2): The map P^{•} = colim(P_{n}^{•}) → M^{•} is a K-projective resolution

of M^{•}.

Proof. For a given k let P_{k}^{•} → τ_{≤k}M^{•} be a projective resolution with P_{k}^{∗} = 0 whenever ∗ > k. We now construct P_{n}^{•} inductively. Assume that f_{n−1} : P_{n−1}^{•} → τ_{≤n−1}M^{•} is given. Let f′_{n−1} denote the composite of f_{n−1} with the inclusion τ_{≤n−1}M^{•} → τ_{≤n}M^{•}. Consider the following diagram:

[figure: P_{n−1}^{•}; −f′_{n−1}; τ_{≤n}M^{•}; C_{−f′_{n−1}}; P_{n−1}^{•}[−1]; f′_{n−1}[−1]; τ_{≤n}M^{•}[−1]; Q^{•}; ≃; g; P_{n−1}^{•}[1]; C_{g}] .

The cone C_{−f′_{n−1}} is a complex which is 0 in degrees above n, and we can choose a projective resolution Q^{•} → C_{−f′_{−1}} with Q^{i} = 0 for i > n.

Define P_{n}^{•} to be C_{g}[1] and f_{n} to be the map C_{g}[1] → τ_{≤n}M^{•}. By construction each P_{n}^{i} is a projective R-module which is equal to 0 for i > n. Note that f_{n−1} is the composite

P_{n−1}^{•} → P_{n}^{•} →^{f_{n}} τ_{≤n}M^{•}.

Since cohomology commutes with colimits P^{•} = colim(P_{n}^{•}) → M^{•} is a quasi isomorphism. It remains to prove that P^{•} = colim(P_{n}^{•}) is Kprojective. Let A^{•} be an acyclic complex. Since each P_{n}^{•} is K-projective the hom complex Hom^{•}(P_{n}^{•}, A^{•}) is acyclic for all n. There is a canonical isomorphism

H^{∗}(lim_{n}Hom^{•}((P_{n}^{•}), A^{•})) ≅ H^{∗}(Hom^{•}(colim_{n}P_{n}^{•}, A^{•})).

For each k ∈ ℤ the sequence

P_{n−1}^{k} → P_{n}^{k} → Q_{n}^{k}

is, by construction, a split short exact sequence. This gives that

hom(P_{n}^{k}, A^{j}) → hom(P_{n−1}^{k}, A^{j})

is surjective for all indices, hence

Hom^{•}(P_{n}^{•}, A^{•}) → Hom^{•}(P_{n−1}^{•}, A^{•})

is surjective. The inverse system now satisfies the Mittag-Leffler condition (so the lim^{1}-term vanishes) hence we have an isomorphism

H^{∗}(lim_{n}Hom^{•}(P_{n}^{•}, A^{•})) → lim_{n}H^{∗}(Hom^{•}(P_{n}^{•}, A^{•})).

This shows that Hom^{•}(P^{•}, A^{•}) is an acyclic complex. □

Lemma 3.5. Any finitely generated R-module N in a projective Rmodule P is contained in a finitely generated projective submodule Q of P.

Proof. Since P is projective there exists an R-module P′ such that P ⊕ P′ ≅ R^{⊕X} for some indexing set X. Since N is finitely generated the map

N ↪ P ⊕ P′ ≅ R^{⊕X}

factors through R^{⊕n}, a finitely generated submodule in R^{⊕X}. Let Q be R^{⊕n} ∩ (P ⊕ 0) ⊂ P. The module N is a submodule of P so Q contains N. Further Q ⊕ (R^{⊕n} ∩ (0 ⊕ P′) ≅ R^{⊕n}, so Q is a finitely generated projective R-module. □

Definition 3.6. A complex P^{•} is perfect if P^{•} is a bounded complex of finitely generated projective R-modules.

Proposition 3.7. The dualizable objects in D_{R} are precisely the retracts in D_{R} of perfect complexes. Moreover the top dimensional nonzero cohomology group (if it exists) is a finitely presented R-module.

Proof. Step 1: If P^{•} is a perfect complex then for all integers n

⨁_{i−j=n} P^{i} ⊗_{R} hom(P^{j}, R) → ∏_{i−j=n} hom(P^{j}, P^{i})

is an isomorphism. Retracts of dualizable objects are again dualizable, so retracts of perfect complexes are dualizable. Step 2: We show that if M^{•} is dualizable then M^{•} is a retract of a complex bounded from above. Assume that M^{•} is dualizable, and let {P_{n}^{•}} and colim_{n}P_{n}^{•} = P^{•} → M^{•} be as in Lemma 3.4. There exists a cochain map

η′ : R[0] → P^{•} ⊗ Hom^{•}(P^{•}, R[0])

representing the map η : R[0] → P^{•} ⊗^{L} RHom^{•}(P^{•}, R[0]) in D_{R}. Colimits commute with tensor products so P^{•} ⊗ Hom^{•}(P^{•}, R[0]) ≅ colim(P_{n}^{•}⊗Hom^{•}(P^{•}, R[0])). The element η′(1) is in P_{n}^{•}⊗hom(P^{•}, R[0]) for some n. Thus η′ factors through P_{n}^{•} ⊗hom(P^{•}, R[0]) and Greenlees’ Lemma gives that M^{•} is a retract in D_{R} of P_{n}^{•} hence the cohomology of M^{•} is bounded from above. If M^{•} is not equivalent to the zero complex, then let m be so that H^{m}(M^{•}) ≠ 0 and H^{i}(M^{•}) = 0 for i > m. Since τ_{≥m}M^{•} → M^{•} is a quasi isomorphism, the map P_{m}^{•} → M^{•} is a quasi-isomorphism. Step 3: We here show that P_{m}^{•} is a retract of a perfect complex Q^{•} with Q^{i} = 0 for i > m. Again let

η′ : R[0] → P_{m}^{•} ⊗ Hom^{•}(P_{m}^{•}, R[0])

represent the map η in D_{R}. Then η′(1) = ∑_{i,j} p_{i,j} ⊗ϕ_{i,j} where p_{i,j} ∈ P_{m}^{i} and ϕ_{i,j} : P_{m}^{i} → R, and almost all of the p_{i,j} are zero. In particular there exists an integer n such that p_{i,j} = 0 for all i < n.

We construct a perfect subcomplex Q^{•} of P_{m}^{•} containing each element p_{i,j}. Starting with Q^{∗} = 0 for all ∗ < n, assume inductively that Q^{i−1} is a finitely generated projective R-module. By Lemma 3.5 we can choose a finitely generated projective submodule Q^{i} of P_{m}^{i} which contains the finitely generated module (p_{i,1}, p_{i,2}, …) + d(Q^{i−1}). Repeat this process until i = m. For i > m, P_{m}^{i} = 0 and we choose Q^{i} = 0. This defines Q^{•} as a perfect sub-complex of P_{m}^{•}. By construction the map η′ factors through R[0] → Q^{•}⊗hom(P_{m}^{•}, R[0]) so by Greenlees’ Lemma P_{m}^{•}, hence M^{•}, is a retract of Q^{•}.

The top cohomology group H^{m}(M^{•}) is a retract of H^{m}(Q^{•}) which is the cokernel of d : Q^{m−1} → Q^{m}. A retract of a finitely presented module is again finitely presented. □

Remark 3.8. There are similar results in [12, III.5.7 p.72] and [4, 1.9].

We next prove that a retract in D_{R} of a perfect complex is again equivalent to a perfect complex. The proof we present is a variation of the proof of this result in [2, 3.4]. The result is also in [11, Expose I.4]. I am thankful to A. Neeman for explaining to me his and M. Bökstedt’s proof of [2, Prop. 3.4]. This result is included for completeness.

Lemma 3.9. Let M^{•} be a retract in D_{R} of a perfect complex. Then the sum M^{•} ⊕ M^{•}[1] is equivalent to a perfect complex.

Proof. We first recall that retracts split in derived categories. A retract 1_{M^{•}} : M^{•} →^{f} P^{•} →^{s} M^{•} gives rise to the following map of distinguished triangles

[figure: M^{•}; f; P^{•}; g; s⊕g; C_{f}; M^{•}[1]; M^{•}; M^{•} ⊕ C_{f}; C_{f}; M^{•}[1].]

Hence P^{•} ⟶^{s⊕g} M^{•} ⊕ C_{f} is a quasi isomorphism, so retracts split.

Define the (idempotent) map e as the composite map

P^{•} ≅ M^{•} ⊕ C_{f} ⟶^{pr} C_{f} ⟶^{i} M^{•} ⊕ C_{f} ≅ P^{•}.

In the following diagram the horizontal sequences are distinguished triangles.

\xymatrix{P^{•} \ar[r]^{e} \ar[d]^{≅} & P^{•} \ar[r] \ar[d]^{≅} & C_{e} \ar[r] & P^{•}[1] \ar[d]^{≅} \\ M^{•} ⊕ C_{f} \ar[r]^{i∘pr} & M^{•} ⊕ C_{f} \ar[r] & C_{i∘pr} \ar[r] & (M^{•} ⊕ C_{f})[1].}

So there exists a quasi isomorphism C_{e} → C_{i∘pr} making the diagram commute. The cone C_{e} is a perfect complex, and the cone C_{i∘pr} is equivalent to the sum M^{•} ⊕ M^{•}[1]. □

Proposition 3.10. A retract in D_{R} of a perfect complex is equivalent to a perfect complex.

Proof. Let M^{•} be a retract of a perfect complex. By the previous lemma there exists a perfect complex P^{•} and a quasi isomorphism ψ : P^{•} → M^{•} ⊕ M^{•}[1]. Let Q^{•} = ⊕_{i=0}^{∞}P^{•}[2i]. Then ψ induces a quasi isomorphism Ψ : Q^{•} → ⊕_{i=0}^{∞}M[i]. Let f : Q^{•}[1] → Q^{•} be a representative of the shift map Ψ^{−1} ∘ s ∘ Ψ, where s : ⊕_{i=1}^{∞}M[i] → ⊕_{i=0}^{∞}M[i] is the canonical inclusion map. Such a representative exists since Q^{•} is K-projective [19]. There is a quasi isomorphism from the mapping-cone C_{f} of f : Q^{•} → Q^{•}[1] to the mapping-cone C_{s} of s : ⊕_{i=1}^{∞}M[i] → ⊕_{i=0}^{∞}M[i]. The mapping-cone C_{s} is equivalent to M^{•}. We show that C_{f} is equivalent to a perfect complex. In degree n the complex C_{f} is ⊕_{i=0}^{∞}P^{n−(1+2i)} ⊕ P^{n−(1+2i)} Hence C_{f} is a complex of finitely generated projective R-modules which is bounded from below.

For brevity let us denote C_{f} by N^{•}. The complex N^{•} is K-projective since Q^{•} is a sum of K-projective complexes, and mapping-cones of K-projective complexes are again K-projective. Since N^{•} is equivalent to the dualizable complex M^{•}, there exists an integer n such that H^{∗}(N^{•}) = 0 for ∗ ≥ n.

The inclusion map τ_{≤n}N^{•} → N^{•} is a quasi isomorphism. To prove that τ_{≤n}N^{•}, hence that M^{•}, is a perfect complex, we need to show that the finitly generated module Z^{n}(N) = ker(d : N^{n} → N^{n+1}) = im(d : N^{n−1} → N^{n}) is a projective module.

By Lemma 3.9 we have that N^{•} ⊕ N^{•}[1] is equivalent to a perfect complex P^{•}. Since N^{•} is K-projective this implies that N^{•} ⊕ N^{•}[1] is homotopy equivalent to the perfect complex P^{•} [19, 1.4]. Let n be large enough so that τ_{≤n}P^{•} = P^{•} (P^{i} = 0 for i > n). Since truncation respects homotopy equivalences we get that τ_{≤n}N^{•} is a retract in the homotopy category of a K-projective complex. For all acyclic complexes K^{•} we then have that Hom^{•}(τ_{≤n}N^{•}, K^{•}) is a retract of an acyclic complex so is itself acyclic. Hence τ_{≤n}N^{•} is by definition a K-projective complex.

Given any surjection A^{−1} → A^{0} of R-modules we can extend it to an acyclic complex A^{•} of the form ⋯ → A^{−2} → A^{−1} → A^{0} → 0. Given a homomorphism ϕ : Z(N^{n}) → A^{0}, since each module N^{i} for i < n is projective, we can extend it to a cochain complex map Φ : τ_{≤n}N^{•} → A^{•}. Since the complex τ_{≤n}N^{•} is K-projective the complex Hom^{•}(τ_{≤n}N^{•}, A^{•}) is acyclic. The cochain map Φ : τ_{≤n}N^{•} → A^{•} is a cycle in Hom^{•}(τ_{≤n}N^{•}, A^{•}) hence a boundary. This gives a lifting of ϕ to A^{−1}. Hence Z^{n}(N) is a finitely generated projective R-module. □

The previous Proposition and Proposition 3.7 give the following.

Theorem 3.11. The dualizable objects in D_{R} are precisely the complexes equivalent to perfect complexes. □

Remark 3.12. Note that a complex which is a retract of a perfect complex is also a retract of a bounded complex of finitely generated free R-modules.

If R is a Noetherian ring then all the cohomology groups of a perfect complex are finitely generated. In general the cohomology groups of perfect complexes are finitely generated if and only if the kernels of all maps between finitely generated free R-modules are again finitely generated.

4. The Picard group of the derived category of modules over a ring

We describe the group of isomorphism classes of invertible objects in D_{R}. We use that all invertible objects are dualizable, and the characterization of dualizable complexes from Proposition 3.7.

Lemma 4.1. Let R be a local ring and let M and N be finitely generated R-modules. If M ⊗_{R} N = 0 then M = 0 or N = 0.

Proof. The Lemma follows by induction on the number of generators and by Nakayama’s lemma. □

The proof of the next Proposition follows [9, V.3.3]. It is also in [?]. Recall that M[n] is the complex which is M in degree n and zero in all other degrees.

Proposition 4.2. If R is a local ring then up to isomorphism the invertible objects in D_{R} are precisely the R[n] for any integer n.

Proof. The cocomplex R[n] is invertible for any n since R[n]⊗R[−n] ≅ R[0].

Let M^{•} ⊗_{R}^{L} N^{•} ≃ R[0] in D_{R}. Since the complexes M^{•} and N^{•} are bounded from above, there is a convergent spectral sequence

E_{2}^{p,q} = ⨁_{i+j=q} Tor_{R}^{p}(H^{i}(M^{•}), H^{j}(N^{•})) ⇒ H^{p+q}(M^{•} ⊗_{R}^{L} N^{•})

where the grading is so that Tor_{R}^{p} is zero for p > 0. Since M^{•} ⊗_{R}^{L} N^{•} ≃ R[0] the cohomology H^{r}(M^{•} ⊗_{R}^{L} N^{•}) ≅ R for r = 0 and is zero for r ≠ 0. If H^{i}(M^{•}) = 0 for i > m and H^{i}(N^{•}) = 0 for i > n then the spectral sequence gives us that H^{m}(M^{•}) ⊗_{R} H^{n}(N^{•}) ≅ H^{m+n}(M^{•} ⊗_{R} N^{•}). If furthermore H^{m}(M^{•}) ≠ 0 and H^{n}(N^{•}) ≠ 0, then by Proposition 3.7 and the Lemma above H^{m}(M^{•}) ⊗_{R} H^{n}(N^{•}) is nonzero, and hence isomorphic to R. The ring R is local so up to isomorphism R is the only invertible R-module, and we conclude that H^{m}(M^{•}) ≅ R and that H^{n}(N^{•}) ≅ R. Using the spectral sequence it is now easy to see that H^{i}(M^{•}) = 0 for i < m. The cohomology of M^{•} is concentrated in degree m where it is R, so by Lemma 3.2 M^{•} ≃ R[m]. □

The localization of a cochain complex M^{•} with respect to a multiplicative set S in R is M^{•} ⊗_{R} S^{−1}R. Since S^{−1}R is a flat R-module, localizations respect quasi-isomorphisms so we get an induced map D_{R} → D_{S^{−1}R} of tensor triangulated categories. If M^{•} is an invertible object in D_{R} then M^{•} ⊗_{R} R_{𝔭} is an invertible object in D_{R_{𝔭}} and by Proposition 4.2 M^{•} ⊗_{R} R_{𝔭} ≃ R_{𝔭}[ψ(𝔭)] for some integer ψ(𝔭). We now show that ψ(𝔭) depends continuously on 𝔭.

Lemma 4.3. Let M^{•} be a given invertible cochain complex in D_{R}. The function ψ : spec(R) → ℤ, defined by letting ψ(𝔭) be the unique integer such that M^{•} ⊗_{R} R_{𝔭} is equivalent to R_{𝔭}[ψ(𝔭)] is a continuous function where ℤ has the discrete topology.

Proof. By Proposition 3.7 M^{•} is a bounded complex so let n_{k} < ⋯ < n_{2} < n_{1} be the values of ψ(𝔭) for 𝔭 ∈ spec R. H^{n_{1}}(M^{•}) is a finitely presented R-module, hence the support {𝔭 ∈ spec R | H^{n_{1}}(M_{𝔭}^{•}) ≠ 0} of H^{n_{1}}(M^{•}) is a closed set in spec R, and the subset

{𝔭 ∈ spec R | H^{n_{1}}(M^{•})_{𝔭} is free of rank 1}

is an open set in spec R, see [16, 4.10]. Since H_{𝔭}^{n_{1}}(M^{•}) is either 0 or R_{𝔭} we conclude that ψ^{−1}(n_{1}) is both an open and a closed subset of spec R.

Now assume that ψ^{−1}(n_{1}), ψ^{−1}(n_{2}), …, ψ^{−1}(n_{i−1}) are all both open and closed. We show that ψ^{−1}(n_{i}) is also open and closed. Recall that the subsets spec R_{f} with f ∈ R form a basis for the topology on the (compact) space spec R. Let f_{1}, f_{2}, …, f_{l_{i}} be elements in the ring R such that

∪_{j=1}^{l_{i}}spec(R_{f_{j}}) = spec R − (ψ^{−1}(n_{1}) ∪ ψ^{−1}(n_{2}) ∪ ⋯ ∪ ψ^{−1}(n_{i−1}))

As noted before M^{•} ⊗_{R} R_{f_{j}} is invertible in D_{R_{f_{j}}}, and H^{∗}(M^{•} ⊗_{R} R_{f_{j}}) ≅ H^{∗}(M^{•}) ⊗_{R} R_{f_{j}} = 0 for ∗ > n_{i}. Applying the same argument to M^{•} ⊗_{R} R_{f_{j}} as we did to M^{•} gives that ψ^{−1}(n_{i})∩spec(R_{f_{j}}) is both open and closed in spec(R_{f_{j}}) for every j, hence ψ^{−1}(n_{i}) is both open and closed in spec R. We conclude that for all integers n the inverse image ψ^{−1}(n) is both an open and a closed subset of spec R. □

Let C(X) denote the additive group of continuous functions from a topological space X to the integers ℤ, where ℤ has the discrete topology. Localization induces a map

Ψ : Pic(D_{R}) → C(spec R)

defined by Ψ(M^{•})(𝔭) = ψ(𝔭) where, as above M_{𝔭}^{•} ≃ R_{𝔭}[ψ(𝔭)]. We also have an inclusion of Pic(R) into Pic(D_{R}) given by sending an invertible module M to the cochain complex M[0] with M concentrated in dimension zero. The map Ψ is a homomorphism and Pic(R) lies in the kernel of Ψ. The next Theorem describes Pic(D_{R}) as the direct sum of Pic(R) and C(spec R). First let us recall the connection between open closed subsets of spec R and idempotents of R [1, II.4.3. Prop. 15].

Lemma 4.4. There is a bijection between the open closed subsets of spec R and the idempotents in R. Let e_{U} denote the idempotent corresponding to the open closed set U. The bijection has the following properties.

(1) Any open closed subset U is the subset spec e_{U}R of spec R via the surjective ring map R → e_{U}R. The open closed set U = {𝔭 ∈ spec R | e_{U}R_{𝔭} = R_{𝔭}}. If 𝔮 ∉ U then e_{U}R_{𝔮} = 0.

(2) For any open closed sets U_{1} and U_{2} we have that e_{U_{1}∩U_{2}} = e_{U_{1}}e_{U_{2}}, e_{U_{1}∪U_{2}} = e_{U_{1}} + e_{U_{2}} − e_{U_{1}∩U_{2}}, e_{X} = 1 and e_{∅} = 0.

(3) If f : R → S is a map of rings and U is an open closed set in spec R then f(e_{U}^{R}) = e_{spec(f)^{−1}(U)}^{S} in S.

Proof. Let U_{1} be an open closed subset of spec R, and let U_{2} be the complement of U_{1} in spec R. Let I_{1} and I_{2} be two ideals in R such that \{𝔭 ∈ spec R | 𝔭 ⊃ I_{i}\} = U_{i}. Then I_{1} ∩ I_{2} is in the radical \sqrt{R} of R and I_{1} + I_{2} = R. Let i_{1} ∈ I_{1} and i_{2} ∈ I_{2} be elements such that i_{1} + i_{2} = 1. Then their product i_{1}i_{2} ∈ \sqrt{R} so there is an n such that i_{1}^{n}i_{2}^{n} = 0. Since (i_{1}^{n}, i_{2}^{n}) = R there exist elements a and b such that a(i_{1})^{n} + b(i_{2})^{n} = 1 in R. It now follows that e_{1} = a(i_{1})^{n} ∈ I_{2} and e_{2} = b(i_{2})^{n} ∈ I_{1} are orthogonal idempotents. The map of spectra induced from R → e_{i}R gives an inclusion of U_{i} into spec R.

We now show that the idempotent e associated to an open closed set U is unique. Let e and e′ be two idempotents corresponding to the same open closed set U i.e. spec (eR) = spec (e′R) = U. The product e(1 − e′) is in the radical of R hence (e and e′ are idempotent) e(1 − e′) = 0, also (1 − e)e′ = 0. We now get that

e = e(e′ + (1 − e′)) = ee′ = (e + (1 − e))e′ = e′.

The other statements in the Lemma follow easily using uniqueness of the idempotent. □

Remark 4.5. The lemma says that there is a natural isomorphism between the Boolean algebra of open-closed subsets of spec R and the Boolean algebra of idempotents of R.

Theorem 4.6. There is a natural split short exact sequence

0 → Pic(R) → Pic(D_{R}) →^{Ψ} C(spec R) → 0

Proof. We need to prove that ker(Ψ) ⊂ Pic(R) and that Ψ is split surjective.

If M^{•} is in ker(Ψ) then M_{𝔭}^{•} ≃ R_{𝔭}[0] for all prime ideals 𝔭. In particular the cohomology of M^{•} is concentrated in degree zero, so by Lemma 3.2 the complex M^{•} is equivalent to H^{0}(M^{•})[0]. By Proposition 3.7 the module H^{0}(M^{•}) is finitely presented, and since all the localizations H^{0}(M^{•})_{𝔭} are isomorphic to R_{𝔭}, we have that H^{0}(M^{•}) is an invertible R-module.

We construct a splitting of Ψ. The spectrum of a ring is a compact space so the image of any continuous function f : spec R → ℤ consists of a finite set of integers, say n_{1}, …, n_{m}. The disjoint subsets U_{i} = f^{−1}(n_{i}) of spec R are both open and closed, hence correspond to an orthogonal basis of idempotents e_{U_{1}}, …, e_{U_{m}} in R. Define the complex Φ(f) as ⊕_{i=1}^{m}e_{U_{i}}R[n_{i}]. The idempotents have the property that e_{U_{i}}R_{𝔭} = 0 if f(𝔭) ≠ i and e_{U_{i}}R_{𝔭} = R_{𝔭} if f(𝔭) = i. Hence Ψ(Φ(f)) is equal to f.

We now check that Φ is a homomorphism. Note that if f ≡ 0 then Φ(f) ≅ R[0]. For two finite open-closed partitions {U_{i}} and {V_{i}} of spec R we have that

⊕_{i=1}^{N}e_{U_{i}}R[n_{i}] ⊗_{R} ⊕_{j=1}^{M}e_{V_{j}}R[m_{j}] ≅ ⊕_{i,j=1}^{N,M}e_{U_{i}∩V_{j}}R[n_{i} + m_{j}].

Hence Φ is a homomorphism. It is easy to see that all three homomorphisms are natural. □

5. The dualizable objects in the derived category of modules over a ringed space

We characterize the dualizable complexes of O-modules for any commutative unital ringed space (X, O). Let D_{sh(X)} be the derived category of unbounded complexes of O-modules. D_{sh(X)} is obtained from the category of complexes by inverting all cochain maps which give an isomorphism on cohomology O-modules [14, 19]. The category of O-modules does not have enough projectives. We recall the following definition from [19].

Definition 5.1. A complex M^{•} is K-flat if for any acyclic complex A^{•} the tensor product M^{•} ⊗ A^{•} is again an acyclic complex.

The derived tensor product in D_{sh(X)} is calculated using K-flat resolutions. We now describe a set of flat O-modules.

Definition 5.2. Let PreO_{U⊂X} be the presheaf defined by letting the O(V)-module PreO_{U⊂X}(V) be O(V) if V ⊂ U, and 0 if V ⊄ U. Let O_{U⊂X} denote the associated sheaf.

If U is an open subset of X then (O_{U⊂X})_{x} ≅ O_{x} if x ∈ U and is zero if x ∉ U. Let j : U → X be the inclusion. Then O_{U⊂X} is the extension of U by zero, denoted j_{!}O. The functor j_{!} is left adjoint to the restriction to U functor j^{∗}.

Note that if U is an open closed set then O_{U⊂X} is equivalent to the sheaf V ↦ O_{X}(U ∩ V). It is easy to see (Prop. 7.3) that O-modules have enough flat objects of the form ⊕_{α}O_{U_{α}⊂X}, and that O-complexes have K-flat resolution complexes which in each degree are sheafification of modules of the form ⊕_{α}O_{U_{α}⊂X} [14].

The category D_{sh(X)} has enough K-flat resolutions. Let F^{•} be a complex of O-modules and k an integer. There exists as in Lemma 3.4 a sequence of presheaf O-modules PreP_{m}^{•}, and weak presheaf equivalences f_{m} : PreP_{m}^{•} → τ_{≤m}F^{•}, for m ≥ k, such that for all m the module P_{m}^{n} is zero when n > m, each complex PreP_{m}^{∗} is a sum of presheaves of the form O_{U⊂X} for U ⊂ X, and PreP_{m}^{•} is a sub complex of PreP_{m+1}^{•} such that f_{m+1}|PreP_{m}^{•} = f_{m}. Let P_{m}^{•} be the sheafification of PreP_{m}^{•}, and let P^{•} = colim_{m}P_{m}^{•}; P^{•} equals the sheafification of colim_{m}PreP_{m}^{•}. Then f_{m} : P_{m}^{•} → τ_{≤m}F^{•} and P^{•} → F^{•} are equivalences in D_{sh(X)}. The complex P^{•} is K-flat (see [19, 3.3]).

Next we characterizes the dualizable objects in D_{sh(X)} in analogy with the case of D_{R}.

Proposition 5.3. A complex F^{•} is dualizable in D_{sh(X)} if and only if F^{•} is locally a retract of a bounded complex of finitely generated free O-modules, i.e. there exists a covering V_{α} of X such that F^{•}|V_{α} is a retract of a bounded complex which in each degree is a finite sum of O|V_{α}. Moreover, if the cohomology sheaf of a dualizable complex is trivial in degrees above m then H^{m}(F^{•}) is locally (in the sense above) a finitely presented O-module.

Proof. Step 1: Assume that F^{•} is a dualizable complex of O-modules. Since the unit map η induces an isomorphism on cohomology sheaves there exists a covering {U_{α}} of X such that the unit map η restricted to U_{α} is given by a cochain map

η_{α} : O|U_{α} → P^{•}|U_{α} ⊗_{O|U_{α}} Hom^{•}(P^{•}, O)|U_{α}

which factors through PreP_{n_{α}}^{•}|U_{α} ⊗_{O|U_{α}} Hom^{•}(P^{•}, O)|U_{α} for some n_{α}. Hence by Greenlees’ lemma the cohomology of P^{•}|U_{α} is bounded above. Step 2: Let F be a dualizable complex with H^{∗}(F^{•}) = 0 for all ∗ > m. The map P_{m}^{•} → F^{•} is then an equivalence. By possibly restricting to a subcovering of X we can, as in step 1, assume that the unit map η is represented by a cochain map

η : O → P_{m}^{•} ⊗_{O} Hom^{•}(P_{m}^{•}, O)

which factors through the presheaf tensor product

PreP_{m}^{•} ⊗_{O} Hom^{•}(P_{m}^{•}, O).

The element η(X)(1) is Σp_{i,j} ⊗ ϕ_{i,j}, where p_{i,j} ∈ P_{m}^{i}(X), ϕ_{i,j} : P_{m}^{i} → O and almost all the p_{i,j} are zero.

Let l be the largest integer such that p_{i,j} = 0 for all i < l and j arbitrary. We now prove by induction on the length m − l that there exists a covering U_{α} of X such that η|U_{α} factors through Q_{α}^{•} ⊗_{O|U_{α}} Hom^{•}(P^{•}, O)|U_{α} where Q_{α}^{•} is a subcoplex of P_{m}^{•}|U_{α} with Q_{α}^{n} a finitely generated free O|U_{α}-module for each integer n between l and m and zero otherwise.

By definition of O_{V ⊂X}, if s ∈ O_{V ⊂X}(U) then there exists a subcovering {U_{α}} of U such that s|U_{α} = 0 if U_{α} ⊄ V. If U_{α} ⊂ V then O_{V ⊂X}(U_{α}) = O(U_{α}). Hence if l = m there exists a covering {V_{α}} of X such that p_{m,1}, p_{m,2}, … restricted to V_{α} lies in a finite sum of free submodules of P_{m}^{m}|V_{α}.

Assume by induction that the statement is true when the length is k, and let m − l = k + 1. There exists a covering {U_{α}} such that for each j, p_{l,j} ∈ Q_{α}^{l} is a finitely generated free O|U_{α}-module. Let g_{α}^{1}, g_{α}^{2}, …, g_{α}^{q_{α}} be a set of generators of Q_{α}^{l}. Then dg_{α}^{1}, dg_{α}^{2}, …, dg_{α}^{q_{α}} (d is the differential) and p_{i,j} whenever i > l are all in P^{m−k}|U_{α}, …, P^{m}|U_{α} hence by our inductive hypothesis there exist subcoverings {U_{α,β}} of each U_{α} and a subcomplex Q_{α,β}^{•} of P_{m}^{•}|U_{α,β} which is a finitely generated free O|U_{α,β}-module in degrees between l + 1 and m and zero otherwise. By construction Q_{α}^{l}|U_{α,β} and Q_{α,β}^{•} combine to give a subcomplex of P_{m}^{•} containing all the p_{i,j} which is a finitely generated free O|U_{α,β}-module in degrees inbetween l and m and zero otherwise.

We conclude that there exists a covering {V_{α}} of X such that each η|V_{α} factors through

Q_{α}^{•} ⊗_{O|V_{α}} Hom^{•}(P_{m}^{•}, O)|V_{α}.

By Greenlees’ lemma P_{m}^{•}|V_{α}, hence F|V_{α}, is a retract of Q_{α}^{•}, a bounded complex such that each Q_{α}^{n} is a finitely generated free O|V_{α}-module and Q_{α}^{n} is zero if n > m. In particular H^{m}(F)|V_{α} is a retract of a finitely presented O|V_{α}-module, hence is locally finitely presented. Step 3: We now prove the converse, that a complex which locally is a retract of a bounded complex Q^{•} such that for each n the O|V_{α}-module Q_{α}^{n} is a finitely generated free O|V_{α}-module, is a dualizable complex. Since a retract of an isomorphism is an isomorphism it is enough to show that a complex Q^{•} which in each degree is a finitely generated free O-module is dualizable.

Let G^{•} be an arbitrary K-flat complex of O-modules. Note that O is projective so Q^{•} is K-projective.

The map

Hom^{•}(Q^{•}, O) ⊗_{O} G^{•} → Hom^{•}(Q^{•}, G^{•})

is a weak equivalence. In fact it is an equivalence of complexes since

Hom^{•}(⊕_{i=1}^{k}O, O) ⊗_{O} G^{n} → Hom^{•}(⊕_{i=1}^{k}O, G^{n})

is equivalent to the isomorphism ⊕_{i=1}^{k}G^{n} → Π_{i=1}^{k}G^{n}. □

6. The Picard group of the derived category of modules over a commutative ringed space

Assume now that (X, O) is a commutative unital ringed space such that Pic(D_{O_{X,x}}) ≅ ℤ for all points x in X. Note that by Theorem 4.6 this condition on (X, O) is equivalent to requiring the Picard group Pic(O_{x}) to be zero and spec O_{x} to be connected for all points x ∈ X. A locally ringed space and the constant sheaf of the ring ℤ satisfy this.

The complex F_{x}^{•} is an invertible complex of O_{x}-modules for all x in X. Under our assumptions we can define a homomorphism

Ψ : Pic(D_{sh(X)}) → ∏_{x∈X} ℤ.

In the following Pic(X) is short for Pic(sh(X)), the group of isomorphism classes of invertible O-modules. Recall that C(X) is the additive group of continuous functions from the space X to the integers.

Theorem 6.1. If (X, O) is a commutative unital ringed space such that Pic(D_{O_{X,x}}) is isomorphic to ℤ for all points x ∈ X, then there is a natural split short exact sequence

0 → Pic(X) → Pic(D_{sh(X)}) ⟶^{Ψ} C(X) → 0.

Proof. We need to prove that ker Ψ ⊂ Pic(X), and that the homomorphism Ψ maps into C(X) and is split surjective. Let F^{•} be an invertible cochain complex of O-modules, and let G^{•} be a K-flat inverse complex of F^{•}. If Ψ(F^{•}) ≡ 0 then the cohomology of F^{•} is concentrated in degree zero, hence F^{•} ≃ H^{0}(F^{•})[0]. For brevity denote H^{0}(F^{•}) by F. Since each stalk F_{x} is isomorphic to O_{x} (and the stalk and the cohomology functors commute), the canonical map

F ⊗_{O} H^{∗}(G^{•}) → H^{∗}(F ⊗_{O} G^{•})

is an isomorphism. Since F ⊗ G^{•} ≅ O[0] in D_{sh(X)} we conclude that F ⊗_{O} H^{0}(G^{•}) ≅ O as O-modules. This shows that F is an invertible O-module, so F^{•} is equivalent to an invertible O-module.

We prove that the function ψ : X → ℤ defined by F_{x}^{•} ≃ O_{x}[ψ(x)] is continuous. It is enough to do this locally, hence from Proposition 5.3 we can assume that F^{•} is a retract of a bounded complex which in each degree is a finitely generated free O-module.

Let H^{m}(F^{•}) be the top nonzero cohomology group. It is locally finitely presented, so the set {x ∈ X | H^{m}(F^{•})_{x} ≠ 0} is a closed set. The kernel of a surjective map from a finitely generated to a finitely presented O-module is locally a finitely generated O-module. It then follows, as for rings, that the set of points x such that H^{n}(F^{•})_{x} is free of rank 1 is an open set. Hence {x ∈ X | H^{m}(F)_{x} ≅ O_{x}} is an open closed set. Applying the same argument to F|X − ψ^{−1}(⟨m − 1, ∞⟩) shows that ψ^{−1}(m − 1) is an open closed subset of X. It follows by induction that each ψ^{−1}(n) is open and closed.

We now define a splitting Φ of Ψ. For a continuous function f : X → ℤ let Φ(f) be a complex which in degree n is O_{f^{−1}(n)⊂X} whose differentials are all zero. For an open-closed partition {U_{i}}_{i∈ℤ} of X, The map

⊕_{i∈ℤ}O_{U_{i}⊂X} → O

is an isomorphism stalkwise hence ⊕_{i∈ℤ}O_{U_{i}⊂X} is equivalent to the unit O. For two open closed sets U and V the tensor product O_{U⊂X} ⊗_{O} O_{V ⊂X} is isomorphic to the module O_{U∩V ⊂X}, so Φ is a homomorphism. In particular Φ(f)⊗_{O} Φ(−f) is equivalent to O, so Φ(f) is an invertible complex. It is easy to show that the split short exact sequence is natural. □

Remark 6.2. For a scheme (X, O) we can find the Picard group of isomorphism classes of (quasi-coherent) O-modules more directly by reducing to affine subsets, hence to rings.

7. The Picard group of modules over equivariant ringed spaces

In this section we extend the results of previous sections to ringed spaces with an action of a discrete group G. For a G-ring R we use Gequivariant cohomology to discuss the group of isomorphism classes of invertible G-R-modules. We refer to [7] for results about G-equivariant ringed spaces and modules.

Definition 7.1. A G-action on a ringed space (X, O) is a continuous group action G × X → X on the topological space X, and for each element g in G a ring homomorphism

θ_{g} : O(−) → O(g−)

of ringed spaces such that for the unit element e we have θ(e) = 1, and for any two elements g, h ∈ G and an open set U in X the composite θ_{h}(gU) ∘ θ_{g}(U) = θ_{hg}(U).

We denote the action θ_{g} simply by g.

Definition 7.2. A G-O-module is an O-module F with a G-action such that the module map O × F → O respects the G-action, i.e. for all open sets U in X and elements g in G

\xymatrix{O_{X}(U) ⊗_{ℤ} F(U) \ar[r] \ar[d]^{g⊗g} & F(U) \ar[d]^{g} \\ O_{X}(gU) ⊗_{ℤ} F(gU) \ar[r] & F(gU)}

is a commutative diagram of O_{X}(U)-modules.

The category sh_{G}(X) of G-O-modules is a symmetric monoidal closed abelian category. The tensor product and the inner-hom functors are defined as in the nonequivariant case. The action on the tensor product is the diagonal action. The action

g : hom_{O_{X}|U}(F|U, G|U) → hom_{O_{X}|gU}(F|gU, G|gU)

on the inner-hom functor is given by sending a map μ to

g ∘ μ ∘ g^{−1} : F|gU ⟶^{g^{−1}} F|U ⟶^{μ} G|U ⟶^{g} G|gU.

The forgetful functor from G-(pre)sheaves to (pre)sheaves respects the sheafification functor. The forgetful functor from the category sh_{G}(X) of G-O-modules to the category sh(X) of O-modules respects the tensor and the inner-hom functors. The maps

ν : DF ⊗_{O} F → Hom(F, F) and ev : DF ⊗_{O} F → O

are G-maps. Hence a G-O-module F is dualizable (invertible) if and only if the underlying O-module is dualizable (invertible) in the category of nonequivariant O-modules. The dualizable modules are described in Proposition 5.3. Let Pic_{G}(X) denote the group of isomorphism classes of invertible G-O-modules. And let D_{sh_{G}(X)} denote the derived category of complexes in the category of G-O-modules.

Proposition 7.3. The derived category D_{sh_{G}(X)} has enough K-flat and K-injective resolutions. Moreover the underlying nonequivariant resolutions are respectively K-flat, K-injective in the nonequivariant derived category.

Proof. We show that the category sh_{G}(X) has enough flat objects. Let U be an open set of X, and let O_{U⊂X} be the sheaf defined in 5.2. Define a G-O-module structure on ⊕_{g∈G}O_{gU⊂X} by letting an element k ∈ G act by sending s ∈ O_{gU⊂X} to ks ∈ O_{kgU⊂X}. For each f ∈ F(U) there is a unique G-O-module map ⊕_{g∈G}O_{gU⊂X} → F sending 1 ∈ O_{eU⊂X} to f ∈ F(U), where e is the identity element in G. Forming the sum over all open sets U of X and all elements f ∈ F(U) gives an epimorphism

⊕_{U} ⊕_{f∈F(U)} ⊕_{g∈G}O_{gU⊂X} → F.

The domain is a flat G-O-module as is seen by localizing at the points of X. This extends to give a K-flat resolution of arbitrary complexes as in [14].

For the existence of injective resolutions of G-equivariant O-modules see [7]. This gives K-injective resolutions of arbitrary complexes as in [19, 4.6.]. □

A consequence is that D_{sh_{G}(X)} has derived tensor and inner hom functors. They make D_{sh_{G}(X)} into a symmetric monoidal closed additive category. The forgetful functor D_{sh_{G}(X)} → D_{sh(X)} respects the closed and the monoidal structure. With the same argument as for G-O-modules, a complex in D_{sh_{G}(X)} is dualizable (invertible) if and only if it is dualizable (invertible) as a complex in D_{sh(X)}.

Proposition 7.4. Let (X, O) be a G-ringed space such that Pic(O_{X,x}) ≅ 0 and spec O_{X,x} is connected for all x ∈ X. Then we have the split short exact sequence

0 → Pic_{G}(X) → Pic(D_{sh_{G}(X)}) → C^{G}(X, ℤ) → 0.

Proof. The proof is almost identical to the nonequivariant case. The proof of Theorem 6.1 extends to G-rings by replacing open-closed by G-invariant open closed, and idempotents by G-invariant idempotents. We construct the splitting as before. □

In the special case of a G-ring R, we have the following natural split short exact sequences

0 → Pic_{G}(R) → Pic_{G}(D_{R}) → C^{G}(spec R, ℤ) → 0.

We now introduce equivariant sheaf cohomology and use it to study Pic_{G}(X) for G-ringed spaces (X, O) whose invertible O-modules are locally free of rank 1. From Remark 2.7 we know that all schemes satisfies this property.

The composite of the global sections and the G-fixed point functors

Γ(F, X)^{G} = Γ(F^{G}, X)

is a left exact functor from G-sheaves to abelian groups.

Definition 7.5. The n-th G-equivariant sheaf cohomology H_{G}^{n}(X; F) is the n-th right derived functor of

F ↦ Γ(F, X)^{G} = Γ(F^{G}, X)

in the category of G-sheaves of abelian groups [7].

When X is a point this reduces to group cohomology, and when the group is trivial we recover sheaf cohomology. The group of global sections of an injective G-sheaf is an injective G-abelian group. We have a first quadrant spectral sequence associated to the composite Γ(F, X)^{G}:

E_{2}^{p,q} = H^{p}(G; H^{q}(X; F)) ⇒ H_{G}^{p+q}(X; F).

The spectral sequence gives rise to an exact sequence, which applied to the unit sheaf O_{X}^{×} is

0 → H^{1}(G, O_{X}^{×}(X)) → H_{G}^{1}(X, O_{X}^{×}) → H^{1}(X, O_{X}^{×})^{G} →^{d} H^{2}(G, O_{X}^{×}(X)).

The first G-equivariant sheaf cohomology group agrees with the first G-equivariant Čech cohomology group [7]. Since we assume all invertible O-modules to be locally free of rank 1 the first G-equivariant Čech cohomology group Ȟ_{G}^{1}(X; O_{X}^{×}) is isomorphic to Pic_{G}(X) [7]. Also Ȟ^{1}(X; O_{X}^{×}) ≅ H^{1}(X, O_{X}^{×}) as G-abelian groups. The isomorphism Pic(X) ≅ Ȟ^{1}(X; O_{X}^{×}) defines a G-action on Pic(X).

Under these identifications we get the natural exact sequence

0 → H^{1}(G, O_{X}^{×}(X)) → Pic_{G}(X) → Pic(X)^{G} →^{d} H^{2}(G, O_{X}^{×}(X)).

In particular for X = spec R we obtain Proposition 7.6. For any G-ring R there is a natural exact sequence

0 → H^{1}(G, R^{×}) → Pic_{G}(R) → Pic(R)^{G} →^{d} H^{2}(G, R^{×}).

We generalize the last exact sequence to profinite groups. Let G be a profinite group, R a discrete ring with a continuous G-action. Let Pic_{G}^{cont}(R) denote the Picard group of the category of discrete G-R-modules with a continuous G-action. For basic results about profinite groups and continuous group cohomology (Galois cohomology) see [20].

Proposition 7.7. Let G be a profinite group, and R a ring with a continuous G-action. There is an exact sequence

0 → H_{cont}^{1}(G, R^{×}) → Pic_{G}^{cont}(R) → Pic(R)^{G} →^{d} H_{cont}^{2}(G, R^{×}).

Proof. Each open normal subgroup H in G gives rise to an exact sequence

0 → H^{1}(G/H, (R^{×})^{H}) → Pic_{G/H}(R^{H}) → Pic(R^{H})^{G/H} →^{d} H^{2}(G/H, (R^{×})^{H}).

The exact sequence is natural so taking the colimit over all open normal subgroups H of G gives an exact sequence of the colimits.

A G-module M has a continuous G-action if and only if M = ∪_{H}M^{H}, where the sum is over all open normal subgroups H of G. The Galois cohomology H_{cont}^{∗}(G, R^{×}) ≅ colim_{H}H^{∗}(G/H, (R^{×})^{H}).

We need to prove that colim_{H}(Pic_{G/H}(R^{H})) → Pic_{G}^{cont}(R) is an isomorphism. To prove surjectivity, let M be an invertible G-R-module. Then M is dualizable hence finitely presented so we have an exact sequence

⊕_{j=1}^{n′}R ⟶^{ϕ} ⊕_{i=1}^{n}R → M → 0

of G-R-modules. Since R = colim_{H}R^{H} there exists an open normal subgroup K of G such that

R ⊗_{R^{K}} (⊕_{j=1}^{n′}R^{K}) ⟶^{1⊗ϕ^{K}} R ⊗_{R^{K}} (⊕_{i=1}^{n}R^{K}) → M → 0

is exact. Let N = coker(ϕ^{K}). The exact sequence gives that M ≅ R ⊗_{R^{K}} N. Since N is a finitely presented R^{K}-module the isomorphism ev : M ⊗ DM → R is realized as the colimit over all L < K of the evaluation maps

(R^{L} ⊗_{R^{K}} N) ⊗_{R^{L}} hom_{R^{L}}(R^{L} ⊗_{R^{K}} N, R^{L}) → R^{L}.

Hence there exists an open normal L such that

(R^{L} ⊗_{R^{K}} N) ⊗_{R^{L}} hom_{R^{L}}(R^{L} ⊗_{R^{K}} N, R^{L}) ≅ R^{L}.

We conclude that R^{L} ⊗_{R^{K}} N is an invertible G-R^{L}-module so that R ⊗_{R^{K}} N is G-R-isomorphic to M. This is true for any M hence the map is surjective.

To prove injectivity, let P, P′ ∈ Pic_{G/K}(R^{K}) satisfy R ⊗_{R^{K}} P ≅ R ⊗_{R^{K}} P′ as G-R-modules. The isomorphism is

colim_{L<K}R^{L} ⊗_{R^{K}} P ≅ colim_{L<K}R^{L} ⊗_{R^{K}} P′.

Since P and P′ are finitely generated R-modules there is an open normal subgroup L of G such that R^{L} ⊗_{R^{K}} P ≅ R^{L} ⊗_{R^{K}} P′. The map colim_{H}(Pic_{G/H}(R^{H})) → Pic_{G}^{cont}(R) is an isomorphism.

Finally, note that (Pic(R^{H}))^{G/H} = (Pic(R^{H}))^{G}, and that

colim_{H}(Pic(R^{H}))^{G} ≅ (colim_{H}Pic(R^{H}))^{G} ≅ (Pic(R))^{G}.

The isomorphism to the right follows by an argument similar to the one above. □

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Department of Mathematics, Northwestern University, Evanston IL 60208

  E-mail address: fausk@math.northwestern.edu