GEOMETRY OF CHAIN COMPLEXES AND OUTER

AUTOMORPHISMS UNDER DERIVED EQUIVALENCE

Birge Huisgen-Zimmermann and Manuel Saorín

The authors wish to dedicate this paper to Idun Reiten on the occasion of her sixtieth birthday.

Abstract. The two main theorems proved here are as follows: If A is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of A is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization Comp_{d}^{A} of the family of finite A-module complexes with fixed sequence d of dimensions and an “almost projective” complex X ∈ Comp_{d}^{A}, there exists a canonical vector space embedding

T_{X}(Comp_{d}^{A})/T_{X}(G.X) ⟶ Hom_{D^{b}(A-Mod)}(X, X[1]),

where G is the pertinent product of general linear groups acting on Comp_{d}^{A}, tangent spaces at X are denoted by T_{X}(−), and X is identified with its image in the derived category D^{b}(A-Mod).

1. Introduction

One of our primary goals is to prove that the identity component Out(A)^{0} of the outer automorphism group of a finite dimensional algebra A is invariant under derived equivalence; here we assume that the base field K of A is algebraically closed. This generalizes a result of Brauer (see [13]), guaranteeing that Out(A)^{0} is Morita invariant, as well as the next step beyond Brauer’s result which established tilting-cotilting invariance of Out(A)^{0} (the latter is due to Guil Asensio and the second-named author [7]). On the other hand, note that the full outer automorphism group Out(A) = Aut(A)/ Inn(A) is not even Morita invariant, in general, and a fortiori, invariance fails for the group Aut(A) of all algebra automorphisms of A. Our invariance theorem was independently and with different methods proved by Rouquier [16].

2000 Mathematics Subject Classification. 16E05, 16G10, 16P10, 18E30, 18G35. While carrying out this project, the first-named author was partially supported by an NSF grant, and

the second-named author by grants from the DGES of Spain and the Fundación ‘Séneca’ of Murcia. The contents of this article were presented at the Conference on Representations of Algebras at Saõ Paolo in July 1999 and at the Workshop on Interactions between Algebraic Geometry and Noncommutative Algebra at the Mathematical Sciences Research Institute (Berkeley) in February 2000, by the second- and first-named authors, respectively.

Typeset by AMS-TEX

In his seminal paper [14], Rickard extended results of Happel [8] and Cline-Parshall-Scott [4], to give an explicit description of the equivalences between derived categories of module categories (strictly speaking, between categories derived from the triangulated homotopy categories of chain complexes of modules). This characterization opened up the possibilty of exhibiting invariants under derived equivalence: Rickard himself showed that the center and the Grothendieck group of A are among them [14], as are the cyclic and Hochschild cohomologies of A (see [9] and [15]). The group Out(A)^{0}, which our final theorem below places on this list, can be viewed as a carrier of homological information as well; indeed, as was first observed by Fröhlich in [6], Out(A) naturally embeds into the Picard group of A, that is, into the group of (isomorphism types of) Morita self-equivalences of the category of left A-modules. The invariance status of the full Picard group relative to derived equivalence is negative, but self-injective algebras which are derived equivalent share at least the stable Picard group [12]. To round off the picture, we point out that the classical Picard group is an object of natural significance in the context of derived categories, in that it in turn embeds into the ‘derived Picard group’ DPic(A), which consists of the isomorphism types of derived self-equivalences of A induced by tilting complexes in D^{b}((A ⊗_{K} A^{op})-Mod) (see [17], [20], and [11]). The second of our main results entails that DPic(A) contains only finitely many A-A tilting complexes of fixed total dimension which are pairwise non-isomorphic when viewed as one-sided complexes over A.

This latter result is obtained (in Section 2) as an ingredient of our invariance proof for Out(A)^{0}. We describe it in some detail, since it holds substantial interest in its own right. Suppose X is a point in the classical variety Mod_{d}^{A} of all d-dimensional left A-modules. In [19], Voigt exhibited a natural vectorspace monomorphism

T_{X}(Mod_{d}^{A})/T_{X}(GL_{d} .X) ⟶ Ext_{A}^{1}(X, X),

where T_{X}(Mod_{d}^{A}) and T_{X}(GL_{d} .X) denote the tangent spaces at X to Mod_{d}^{A} and the GL_{d}-orbit of X in Mod_{d}^{A}, respectively. Here is a sketch of our generalization (Theorem 7): Roughly speaking, it relates tangent spaces of varieties of bounded finite dimensional complexes over A to Hom-groups in the derived category D^{b}(A-Mod), whenever A is finite dimensional over an algebraically closed field. More precisely, the geometrization of the d-dimensional modules in the framework of Mod_{d}^{A} can be carried over, in the same spirit, to the complexes of the form 0 → X_{m} → X_{m−1} → ⋯ → X_{0} → 0 for fixed m, where X_{0}, …, X_{m} ∈ A-Mod have prescribed dimensions d_{0}, …, d_{m}, respectively. In the predictable manner, this leads to a Zariski-closed subset Comp_{d}^{A} of

∏_{0≤i≤m} Hom_{K}(A, End_{K}(K^{d_{i}})) × ∏_{1≤i≤m} Hom_{K}(K^{d_{i}}, K^{d_{i−1}}),

where d = (d_{m}, …, d_{0}). Moreover, the orbits of the (only plausible) conjugation action of G = GL_{d_{0}} × ⋯×GL_{d_{m}} on Comp_{d}^{A} are in one-to-one correspondence with the isomorphism types of complexes of the described ilk. In this setting, the following is true for any complex X of finite dimensional left A-modules of the above format, with the additional property that all X_{i}, except for possibly X_{m}, are projective over A: If X is represented by a point X ∈ Comp_{d}^{A} say, there exists a canonical vector space embedding

T_{X}(Comp_{d}^{A})/T_{X}(G.X) ⟶ Hom_{D^{b}(A-Mod)}(X, X[1]).

In case all X_{i} are projective, this embedding is actually an isomorphism. Observe that, in D^{b}(A-Mod), every finite dimensional left A-module coincides with a complex of the addressed ilk – to wit, with the stalk complex of the module concentrated in degree m – and hence Voigt’s result is retrieved as a special case. Among the consequences, the one instrumental in establishing our result concerning Out(A)^{0} is this: For any finite sequence of non-negative integers, there are only finitely many complexes of projectives with this sequence of dimensions such that Hom_{D^{b}(A-Mod)}(X, X[1]) = 0. So, in particular, we see that A has only finitely many one-sided tilting complexes of prescribed total dimension.

Throughout, A will be a finite dimensional algebra over a field K. In the main results of Section 2 and throughout Section 3, K will be algebraically closed. Moreover, A-Mod, Mod-A, A-mod, and mod- A will denote the categories of all left/right A-modules, and their full subcategories of finite dimensional left/right A-modules, respectively.

2. The geometry of chain complexes

Suppose that A is generated by a_{1} = 1, a_{2}, …, a_{s} as a K-algebra, and recall that the objects in A-mod of vector space dimension d are parametrized by the points of the following subvariety of s · d^{2}-dimensional affine space:

Mod_{d}^{A} = {(A_{1}, …, A_{s}) | the A_{j} ∈ End_{K}(K^{d}) satisfy the same relations as a_{1}, …, a_{s}}.

This variety comes with a canonical morphic GL_{d}-action by conjugation, the orbits of which reflect basis change. So, if one assigns to each point in Mod_{d}^{A} the corresponding left Amodule, it is clear that the isomorphism types of d-dimensional modules are in one-to-one correspondence with the GL_{d}-orbits in Mod_{d}^{A}. Note that we can identify the points of Mod_{d}^{A} with points in Hom_{K}(A, End_{K}(K^{d})) by assigning to each sequence (A_{1}, …, A_{s}) ∈ Mod_{d}^{A} the K-algebra homomorphism A → End_{K}(K^{d}) which sends a_{j} to A_{j}.

We carry this idea over – in the same spirit – to finite chain complexes

X : 0 → X_{m} ⟶^{∂_{m}} X_{m−1} ⟶^{∂_{m−1}} ⋯ ⟶^{∂_{1}} X_{0} → 0

in A-mod with fixed sequence

d = (dim_{K} X_{m}, …, dim_{K} X_{0}) = (d_{m}, …, d_{0})

of dimensions. Accordingly, we define the variety Comp_{d}^{A} to be the following Zariski closed-subset of the affine space

𝔸^{N} = ∏_{0≤i≤m} Hom_{K}(A, End_{K}(K^{d_{i}})) × ∏_{1≤i≤m} Hom_{K}(K^{d_{i}}, K^{d_{i−1}}).

Namely,

Comp_{d}^{A} := {(A^{m}, …, A^{0}, ∂_{m}, …, ∂_{1}) ∈ 𝔸^{N} | A^{i} = (A_{ij})_{j≤s} ∈ Mod_{d_{i}}^{A}, ∂_{i}A_{ij} = A_{i−1,j}∂_{i}, and ∂_{i−1}∂_{i} = 0 for all i ≤ m and j ≤ s}.

We will also label the points of Comp_{d}^{A} in the form (A_{ij}, ∂_{m}, …, ∂_{1}), as convenience dictates. Clearly, the group GL_{d} = GL_{d_{m}} × ⋯ × GL_{d_{0}} acts morphically on Comp_{d}^{A} via

(g_{m}, …, g_{0})(A_{ij}, ∂_{m}, …, ∂_{1}) = (A_{ij}^{g_{i}}, g_{m−1}∂_{m}g_{m}^{−1}, …, g_{0}∂_{1}g_{1}^{−1}),

where A_{ij}^{g_{i}} stands for the conjugate g_{i}A_{ij}g_{i}^{−1}. Moreover, it is straightforward to check the following Observation 1. The natural map from Comp_{d}^{A} to the collection of complexes X as above, with fixed sequence d of dimensions, induces a one-to-one correspondence between the orbits of the GL_{d}-action on Comp_{d}^{A} on one hand, and the isomorphism classes of complexes of the described format on the other. □

An alternate way of geometrizing complexes is, of course, that of viewing complexes as graded modules over an enlarged algebra; but this approach would be ill-suited to our purpose of relating the geometry of Comp_{d}^{A} to the derived category of A.

In the following, we will denote points of Comp_{d}^{A} by capital letters and the corresponding complexes by bold versions of these letters. Furthermore, we will, from now on, assume that a_{1} = 1, …, a_{s} is a K-basis of A. This provides us with structure constants c_{jkl} ∈ K, arising in the equalities a_{j}a_{k} = ∑_{l=1}^{s} c_{jkl}a_{l}. In order to describe the coordinate ring of Comp_{d}^{A}, we let X_{ij} stand for a d_{i} × d_{i} matrix of variables representing the entries of the matrix A_{ij}, respectively, and Y_{i} for a d_{i−1} × d_{i} matrix of variables representing the entries of the map ∂_{i}. When we view a sequence (X_{ij}, Y_{m}, …, Y_{1}) as an element of the coordinate ring of Comp_{d}^{A}, our setup shows these coordinate matrices to be subject to the following equalities; in fact, Comp_{d}^{A} is determined by these requirements.

(α) X_{ij}X_{ik} = ∑_{l=1}^{s} c_{jkl}X_{il};

(β) Y_{i}X_{ij} = X_{i−1,j}Y_{i};

(γ) Y_{i−1}Y_{i} = 0. In the upcoming lemma, we will derive a convenient explicit description of the Zariski

tangent space T_{X}(Comp_{d}^{A}) of Comp_{d}^{A} at a point X. Recall that a derivation of a left A-module M is a K-linear map δ : A → End_{K}(M) such that δ(ab)x = a(δ(b)x) + δ(a)(bx) for all a, b ∈ A and x ∈ M.

Lemma 2. Given a point X = (A^{m}, …, A^{0}, ∂_{m}, …, ∂_{1}) ∈ Comp_{d}^{A}, the tangent space T_{X}(Comp_{d}^{A}) of Comp_{d}^{A} at X consists precisely of those sequences (δ_{m}, …, δ_{0}, σ_{m}, …, σ_{1}) in

𝔸^{N} = ∏_{0≤i≤m} Hom_{K}(A, End_{K}(K^{d_{i}})) × ∏_{1≤i≤m} Hom_{K}(K^{d_{i}}, K^{d_{i−1}})

which satisfy the following conditions:

(a) Each δ_{i} : A → End_{K}(K^{d_{i}}) is a derivation of the left A-module X_{i} determined by A^{i} = (A_{ij})_{j≤s}.

(b) σ_{i}A_{ij} + ∂_{i}δ_{i}(a_{j}) = A_{i−1,j}σ_{i} + δ_{i−1}(a_{j})∂_{i} for all 0 ≤ i ≤ m and 1 ≤ j ≤ s; here we identify A_{ij} with the left multiplication of X_{i} = K^{d_{i}} by the element a_{j} of A.

(c) σ_{i−1}∂_{i} + ∂_{i−1}σ_{i} = 0 for all 1 ≤ i ≤ m.

Proof. Note that each of the δ_{i} is determined by the values (δ_{i}(a_{j}))_{j≤s}. Hence the defining conditions (α) − (γ) of Comp_{d}^{A} yield the following equations pinning down T_{X}(Comp_{d}^{A}); they are given in the variables w_{ij} representing the maps δ_{i}(a_{j}), and v_{i} representing the K-endomorphisms σ_{i} : K^{d_{i}} → K^{d_{i−1}}, respectively:

(a) w_{ij}A_{ik} + A_{ij}w_{ik} = ∑_{l=0}^{s} c_{jkl}w_{il},

(b) v_{i}A_{ij} + ∂_{i}w_{ij} = w_{i−1,j}∂_{i} + A_{i−1,j}v_{i},
(c) v_{i−1}∂_{i} + ∂_{i−1}v_{i} = 0,

for all eligible indices i, j, k. But these conditions are clearly tantamount to the ones listed in our claim. □

Next we follow the traditional road of mapping the tangent space of a module variety at a point x to the group of self-extensions of the module represented by x. Namely, we fix X ∈ Comp_{d}^{A}, and assign to any point (δ_{m}, …, δ_{0}, σ_{m}, …, σ_{1}) in the tangent space T_{X}(Comp_{d}^{A}) a short exact sequence 0 → X → Z → X → 0 of complexes as follows.

Remark 3. Let X ∈ Comp_{d}^{A} as before.

(1) Condition (a) of Lemma 2 implies that the following is a left A-module structure on Z_{i} = K^{d_{i}} ⊕K^{d_{i}}: Namely, we define left multiplication of Z_{i} by a ∈ A as left multiplication by ( \begin{matrix}A^{i}(a) & δ_{i}(a) \\ 0 & A^{i}(a) \end{matrix} ), viewing the elements Z_{i} as column vectors.

(2) Condition (c) of Lemma 2 tells us that we can supplement the Z_{i} to a chain complex Z of K-vector spaces by introducing the differentials ∂_{i}^{Z} = ( \begin{matrix}∂_{i} & σ_{i} \\ 0 & ∂_{i} \end{matrix} ).

(3) Condition (b) of Lemma 2, finally, shows that the complex Z just defined is, in fact, a complex of A-modules.

(4) Let 0 → X_{i} → Z_{i} → X_{i} → 0 be the canonical sequences, given by injection of X_{i} into the first component of Z_{i}, followed by projection onto the second component. These short exact sequences in A-mod are compatible with the differentials, thus yielding a short exact sequence 0 → X → Z → X → 0 of chain complexes. □

Now suppose that X, Y are chain complexes which are bounded above, but not necessarily finite. We will consider the group Ext^{1}(X, Y) of (equivalence classes) of extensions 0 → Y → E → X → 0 in the category of complexes of A-modules. In analogy with Ext-groups of A-modules, Ext^{1}(X, Y) actually carries a K-vector space structure.

For any point X ∈ Comp_{d}^{A}, Remark 3 provides us with a K-linear map

χ : T_{X}(Comp_{d}^{A}) → Ext^{1}(X, X),

sending an element of T_{X}(Comp_{d}^{A}) to the class of the self-extension 0 → X → Z → X → 0 of X described above. The proof of the following proposition is quite similar to that of Voigt’s Lemma ([19, Chap.2, Section 3.4]; see also [5, Theorem 1.6]). Since Voigt’s original argument (couched in German) carries over smoothly to our setting, we include only a brief outline (for the sake of the reader not familiar with German). Recall that an extension 0 → Y → E → X → 0 is semisplit in the sense of Verdier [18, p. 272-273] if, in each degree i, the pertinent short exact sequence of A-modules splits.

Proposition 4. Suppose that K is an algebraically closed field. Then, given any X ∈ Comp_{d}^{A}, the kernel of the map χ equals the tangent space of G.X at X. In other words, χ induces a vectorspace embedding

T_{X}(Comp_{d}^{A})/T_{X}(G.X) → Ext^{1}(X, X).

Moreover, all semisplit self-extensions of X belong to the image of χ.

Proof. The canonical morphism G → G.X is separable (since the stabilizer subgroup of X in G arises as the solution set of a system of linear equations), i.e., this map has a surjective differential

T_{e}(G) → T_{X}(G.X) ⊆ T_{X}(Comp_{d}^{A}).

In the description of the tangent space T_{X}(Comp_{d}^{A}) given in Lemma 2, this identifies T_{X}(G.X) as being the following subset of T_{X}(Comp_{d}^{A}): Namely, the set of those elements (δ_{m}, …, δ_{0}, σ_{m}, …, σ_{1}) in T_{X}(Comp_{d}^{A}) for which there is a sequence t = (t_{m}, …, t_{0}) ∈ ∏_{i=0}^{m} End_{K}(K^{d_{i}}) such that the following equalities hold for all eligible i and j:

• δ_{i}(a_{j}) = t_{i}A_{ij} − A_{ij}t_{i}, and

• σ_{i} = t_{i−1}∂_{i} − ∂_{i}t_{i}. (These equalities can be gleaned directly from Voigt’s argument for modules by identifying the complex X with a graded module ⨁_{i=0}^{m} X_{i} over the finite dimensional algebra A[∂_{1}, …, ∂_{m}]/(∂_{1}^{2}, …, ∂_{m}^{2}); here the operation of A on the direct sum is extended via ∂_{i}.(x_{m}, …, x_{0}) = ∂_{i}(x_{i}) ∈ X_{i−1} for 1 ≤ i ≤ m.)

It is readily verified that the described subset of T_{X}(Comp_{d}^{A}) coincides with the kernel of χ. Indeed, splitness of the extension 0 → X → Z → X → 0 of complexes coming with the point (δ_{m}, …, δ_{0}, σ_{m}, …, σ_{1}) of T_{X}(Comp_{d}^{A}) is equivalent to the existence of a section for the epimorphism Z → X. Since the latter map just projects the Z_{i} = X_{i} ⊕ X_{i} onto the second components, such sections are precisely the chain maps X → Z of the form t′ = (t_{i}, 1_{X_{i}})_{0≤i≤m}, where the family (t_{i}) belongs to ∏_{i=0}^{m} End_{A}(X_{i}). But that t′_{i} : X_{i} → Z_{i} be an A-module homomorphism is clearly tantamount to the requirement that t_{i}A_{ij} = A_{ij}t_{i} + δ_{i}(a_{j}), while the stipulation that a family (t_{i}) ∈ ∏_{i=0}^{m} Hom_{K}(K^{d_{i}}) be compatible with the differentials of X and Z amounts to the equality t_{i−1}∂_{i} = ∂_{i}t_{i} + σ_{i}. This completes the proof of the fact that T_{X}(G.X) coincides with the kernel of χ.

For the final comment, note that the differential of any semisplit self-extension 0 → X → Z → X → 0 of X, with splitting Z_{i} = X_{i} ⊕X_{i} in degree i, has the form ∂_{i}^{Z} = ( \begin{matrix}∂_{i} & σ_{i} \\ 0 & ∂_{i} \end{matrix} ) relative to such a splitting, where ∂_{i} is the i-th differential of X and σ_{i} ∈ Hom_{A}(X_{i}, X_{i−1}) satisfies condition (c) of Lemma 2. Consequently, Remark 3 shows that the semisplit selfextensions of X are precisely the images under χ of the vectors (0, …, 0, σ_{m}, …, σ_{1}) in the tangent space T_{X}(Comp_{d}^{A}). □

Following Verdier’s conventions in [18], we denote the category of right bounded complexes of finite dimensional left A-modules by C^{−}(A-mod), its quotient category modulo homotopy by K^{−}(A-mod), and the corresponding derived category by D^{−}(A-mod). Analogously, C^{b}(A-mod), K^{b}(A-mod), and D^{b}(A-mod) stand for the category of bounded complexes, for the quotient category modulo homotopy, and the pertinent derived category, respectively. According to [18, pp. 294-295], every element 0 → Y → E → X → 0 of Ext^{1}(X, Y) gives rise to a distinguished triangle Y → E → X → Y[1] in D^{−}(A-mod). Here Y[1] denotes the shifted complex which carries Y_{n−1} in the slot labeled n. This provides us with a K-linear map ξ : Ext^{1}(X, Y) ⟶ Hom_{D^{−}(A-mod)}(X, Y[1]) which assigns to an extension as above the connecting morphism X → Y[1] of the corresponding triangle. Our next intermediate goal is to scrutinize this map for complexes X which are ‘close’ to being projective.

Definition. A complex X : ⋯ → X_{i+1} → X_{i} → ⋯ in C^{−}(A-mod) is called projective in case all of the terms X_{i} are projective A-modules.

Now suppose that X ∈ C^{b}(A-mod). If X is nonzero, then the largest integer m with X_{m} ≠ 0 will be referred to as the left degree of X. We will call X almost projective in case either X is projective or else X_{m} is the only non-projective term of the complex.

It is clear that, under the natural map C^{b}(A-mod) → D^{b}(A-mod), every object in the derived category is represented by an almost projective complex.

One of the assets of almost projective complexes is a convenient lifting property for maps in the derived category: If X and Y are both almost projective with coinciding left degree, then any map in Hom_{D^{b}(A-mod)}(X, Y) lifts to a chain map in Hom_{C^{b}(A-mod)}(X, Y): Indeed, we may restrict our attention to bounded almost projective complexes for that purpose. So suppose that X is of the form

0 → X_{m} ⟶^{∂_{m}} X_{m−1} ⟶^{∂_{m−1}} … ⟶^{∂_{1}} X_{0} → 0,

where all of the left A-modules X_{i}, except for X_{m} possibly, are projective; write Y in the same format, with terms Y_{i}. If we let ⋯ → P_{m+1} ⟶^{p_{m+1}} P_{m} ⟶^{p_{m}} X_{m} → 0 be a projective resolution of X_{m}, the image of X in D^{−}(A-mod) clearly coincides with that of the projective complex X′, given by

… P_{m+1} ⟶^{p_{m+1}} P_{m} ⟶^{∂_{m}p_{m}} X_{m−1} ⟶^{∂_{m−1}} … ⟶^{∂_{1}} X_{0} → 0.

Analogously, define a projective complex Y′ based on Y. Now any homomorhism in Hom_{D^{−}(A-mod)}(X′, Y′) is induced by a chain map η ∈ Hom_{C^{−}(A-mod)}(X′, Y′). But p_{m} is the cokernel of p_{m+1}, and therefore η induces a chain map in Hom_{C^{b}(A-mod)}(X, Y) having the same image as η in D^{b}(A-mod).

Again suppose that X ∈ C^{b}(A-mod) is almost projective. Along the line suggested by the previous paragraph, we associate another almost projective complex X^{(1)} to X, by potentially extending X to the left as follows: Set X^{(1)} = X if X is projective. Otherwise, we denote the left degree of X by m, let p : X_{m}^{(1)} → X_{m} be a projective cover of X_{m} with kernel ∂_{m+1}^{X^{(1)}} : X_{m+1}^{(1)} → X_{m}^{(1)}, and define X^{(1)} to be the complex

0 → X_{m+1}^{(1)} → X_{m}^{(1)} → X_{m−1} → ⋯ → X_{0} → 0,

where the m-th differential of the complex X^{(1)} is ∂_{m}^{X^{(1)}} = ∂_{m}^{X} p. Clearly, X^{(1)} comes with a natural map f : X^{(1)} → X of chain complexes, given by f_{m} = p, f_{i} = 0 for i ≥ m + 1, and f_{i} = id_{X_{i}} for i ≤ m − 1. By setting X^{(0)} = X and continuing inductively via X^{(r+1)} = (X^{(r)})^{(1)}, we thus obtain a sequence

⋯ → X^{(r)} → X^{(r−1)} → ⋯ → X^{(1)} → X^{(0)}

of chain maps in C^{−}(A-mod), which, in turn, induces a sequence

Ext^{1}(X, Y) → Ext^{1}(X^{(1)}, Y) → Ext^{1}(X^{(2)}, Y) → ⋯

of K-linear maps for each Y ∈ C^{−}(A-mod). Note that Verdier’s map ξ : Ext^{1}(X, Y) → Hom_{D^{−}(A-mod)}(X, Y[1]) factors through these vectorspace homomorphisms, since all of the canonical maps X^{(r)} → X^{(r−1)} become isomorphisms in the homotopy category K^{−}(A-mod).

We begin with an auxiliary point which sometimes allows us to replace X by X^{(1)} in the first argument of Ext^{1} without penalty.

Lemma 5. Let X, Y ∈ C^{b}(A-mod) be nonzero almost projective complexes whose left degrees coincide. Then the canonical K-linear map Ext^{1}(X, Y) → Ext^{1}(X^{(1)}, Y) is bijective.

Proof. The coinciding left degree of X and Y is denoted by m. Moreover, we write Z for X^{(1)}, and f for the canonical chain map Z → X; this means, in particular, that f_{m} : Z_{m} → X_{m} is a projective cover, and f_{i} = id_{X_{i}} for i ≤ m − 1. The K-linear map addressed in our claim, finally, is denoted by h : Ext^{1}(X, Y) → Ext^{1}(Z, Y).

We start by proving surjectivity of h. Suppose

ϕ : 0 → Y ⟶^{ρ} F ⟶^{σ} Z → 0

represents a class in Ext^{1}(Z, Y). To construct a preimage

ϵ : 0 → Y ⟶^{μ} E ⟶^{ν} X → 0

of (the class of) ϕ under h, we start by setting E_{i} = 0 for i ≥ m + 1 and E_{i} = F_{i} for i ≤ m−1; the pertinent components of the differential ∂^{E} and those of the chain maps μ, ν are as follows: ∂_{i}^{E} = ∂_{i}^{F}, μ_{i} = ρ_{i}, and ν_{i} = σ_{i} for i ≤ m − 1, the definitions to the left of the m-th position being obvious. The A-module E_{m} and the corresponding differential of E are defined by the requirements that g_{m} : F_{m} → E_{m} be the cokernel of ∂_{m+1}^{F}, and ∂_{m}^{E} be the unique map in Hom_{A}(E_{m}, F_{m−1}) such that ∂_{m}^{F} = ∂_{m}^{E} g_{m}. Moreover, set μ_{m} = g_{m}ρ_{m}, and let ν_{m} : E_{m} → X_{m} be the unique homomorphism with f_{m}σ_{m} = ν_{m}g_{m}; the map ν_{m} is, once more, furnished by the universal property of g_{m}. Clearly, μ : X → E is a chain map. The same is true for ν : E → X; indeed, a straightforward diagram chase yields ∂_{m}^{X} ν_{m} = ν_{m−1}∂_{m}^{E}. It is now routine to check that ϵ yields a preimage of ϕ under h.

To verify injectivity of h, suppose that the class of ϵ : 0 → Y ⟶^{μ} E ⟶^{ν} X → 0 is mapped to zero by h. This means that the upper row of the following pullback diagram of complexes splits:

[figure: 0; Y; ρ; F; σ; g; Z; f; 0; 0; Y; μ; E; ν; X; 0]

In order to show that ϵ is trivial, i.e., that the lower row splits as well, let τ = (τ_{n})_{n∈ℕ} be a section for σ. To construct a section π for ν, we start by setting π_{i} = 0 for i ≥ m + 1, and π_{i} = g_{i}τ_{i} for i ≤ m − 1. This setup is recorded in the following diagram:

[figure: F_{m+1}; σ_{m+1}; g_{m+1}; F_{m}; σ_{m}; g_{m}; F_{m−1}; σ_{m−1}; g_{m−1}; F_{m−2}; σ_{m−2}; g_{m−2}; Z_{m+1}; f_{m+1}; τ_{m+1}; Z_{m}; f_{m}; τ_{m}; Z_{m−1}; τ_{m−1}; Z_{m−2}; τ_{m−2}; 0; ν_{m+1}; E_{m}; ν_{m}; E_{m−1}; ν_{m−1}; E_{m−2}; ν_{m−2}; 0; π_{m+1}; X_{m}; X_{m−1}; π_{m−1}; X_{m−2}; π_{m−2}]

Note that all of the squares in this diagram commute, so that our task is reduced to finding a section π_{m} for ν_{m} such that the two bottom squares adjacent to π_{m} commute. This is automatic for the left-hand square, irrespective of our choice of π_{m}. To construct π_{m} with the property that ∂_{m}^{E}π_{m} = π_{m−1}∂_{m}^{X}, we recall that f_{m} is the cokernel of the differential ∂_{m+1}^{Z}. This provides us with a unique homomorphism π_{m} : X_{m} → E_{m} having the property that g_{m}τ_{m} = π_{m}f_{m}, for g_{m}τ_{m}∂_{m+1}^{Z}σ_{m+1} = ∂_{m+1}^{E}g_{m+1} = 0 yields g_{m}τ_{m}∂_{m+1}^{Z} = 0. It is now routine to check that π = (π_{i})_{i∈ℤ} is a section for ν, as required. This completes the argument. □

Modules M, N ∈ A-mod can of course be viewed as stalk complexes M, N concentrated in degree 0. In this situation, Verdier’s map provides us with isomorphisms

Ext^{1}(M, N) ≅ Hom_{D^{−}(A-mod)}(M, N[1]).

The following lemma shows that the good behavior of Verdier’s map extends, at least partially, to more general complexes, which provides the missing link towards the main result of this section.

Lemma 6. Let X, Y be nonzero complexes in C^{−}(A-mod). Then Verdier’s map

ξ : Ext^{1}(X, Y) ⟶ Hom_{D^{−}(A-mod)}(X, Y[1])

is injective if either (1) X is a projective complex, or else

(2) X and Y are bounded almost projective complexes such that the left degree of X is larger than or equal to the left degree of Y.

Moreover, if X is as under (1) or (2) and Y is projective, then ξ is bijective.

Proof. We show injectivity of ξ simultaneously for (1) and (2). In case (2), we denote by k the left degree of X, by m that of Y, and observe that Lemma 5 permits us to derive the claim for k = m from that for k = m + 1. In other words, we may, w.l.o.g., assume that either both X and Y are projective or both are bounded almost projective with k ≥ m + 1. Note that, in this situation, any extension ϵ : 0 → Y ⟶^{μ} E ⟶^{ν} X → 0 has the property that 0 → Y_{i} ⟶^{μ_{i}} E_{i} ⟶^{ν_{i}} X_{i} → 0 splits (i.e., the extension ϵ is semi-scindée in the sense of [18, pp. 272-273]). For each i ∈ ℤ, let π′_{i} be a section of ν_{i}. We will see that the assumption ‘ξ(ϵ) = 0’ allows us to adjust the family (π′_{i}) so as to furnish a section for ν. To that end, suppose that ξ(ϵ) is represented by a chain map ϕ : X → Y[1]. Letting P → X be the projective resolution of X in K^{−}(A-mod) and ϕ̃ : P → Y[1] its composition with ϕ, the morphism ϕ̃ in K^{−}(A-mod) also represents ξ(ϵ) in D^{−}(A-mod), in view of the canonical isomorphism Hom_{K^{−}(A-mod)}(X, Y[1]) ≅ Hom_{K^{−}(A-mod)}(P, Y[1]) (see [18, p. 299]). We further observe that ξ(ϵ) = 0 if and only if ϕ̃ is trivial in K^{−}(A-mod), for the quotient map Hom_{K^{−}(A-mod)}(P, Y[1]) → Hom_{D^{−}(A-mod)}(P, Y[1]) is injective. Our hypothesis on the left degrees of X and Y guarantees that ϕ̃ is nullhomotopic precisely when ϕ is nullhomotopic. Consequently, the assumption that ξ(ϵ) be zero supplies us with a homotopy ψ from ϕ to zero. We deduce that π = (π′_{n} − μ_{n}ψ_{n})_{n∈ℤ} is a chain map X → E such that νπ = 1_{X}, which shows that ϵ is indeed trivial.

Now suppose that Y is projective. In that case, we have isomorphisms

Hom_{D^{−}(A-mod)}(X, Y[1]) ≅ Hom_{K^{−}(A-mod)}(P, Y[1]) ≅ Hom_{K^{−}(A-mod)}(X, Y[1]),

where P is a projective cover of X. Hence all we need to check is that the homotopy class of any chain map ψ : X → Y[1] occurs in the image of the map Ext^{1}(X, Y) → Hom_{K^{−}(A-mod)}(X, Y[1]) induced by ξ. But if C is the mapping cone of ψ, then clearly the class of the extension 0 → Y → C[−1] → X → 0 in Ext^{1}(X, Y) is such a preimage of ψ. □

From the preceding lemmas it is now easy to glean

Theorem 7. Suppose that K is algebraically closed, and let X : 0 → X_{m} → ⋯ → X_{0} → 0 be an almost projective complex of finite-dimensional A-modules, represented by a point X in Comp_{d}^{A}, where d = (dim X_{m}, …, dim X_{0}). Moreover, let T_{X}(Comp_{d}^{A}) and T_{X}(G.X) be the tangent spaces at X of Comp_{d}^{A} and G.X, respectively. Then there exists a canonical K-linear embedding

T_{X}(Comp_{d}^{A})/T_{X}(G.X) ⟶ Hom_{D^{b}(A-mod)}(X, X[1]).

In case X is a projective complex, this embedding is an isomorphism.

Proof. We compose the vectorspace homomorphism χ : T_{X}(Comp_{d}^{A}) → Ext^{1}(X, X) introduced after Remark 3 with Verdier’s map ξ : Ext^{1}(X, X) → Hom_{D^{−}(A-mod)}(X, X[1]) to obtain a vectorspace homomorphism η : T_{X}(Comp_{d}^{A}) → Hom_{D^{−}(A-mod)}(X, X[1]). Observe that the kernel of η coincides with T_{X}(G.X), since Ker(χ) = T_{X}(G.X) by Proposition 4, while ξ is an injection by Lemma 6. Consequently, η induces an embedding of T_{X}(Comp_{d}^{A})/T_{X}(G.X) into Hom_{D^{−}(A-mod)}(X, X[1]) as required.

For the final claim, let X be projective. Then ξχ is surjective, since χ is surjective by the last part of Proposition 4, and ξ is surjective by the last statement of Lemma 6. □

As in the case of the variety Mod_{d}^{A} of d-dimensional A-modules, Theorem 7 entails a number of interesting consequences. We just point out two of them, the second of which we will need in the sequel. Since the proofs are analogous to those for the module-theoretic counterparts (see [5]), we leave them to the reader.

Corollary 8. Suppose that K is algebraically closed and X ∈ C^{b}(A-mod) a bounded almost projective complex such that Hom_{D^{b}(A-mod)}(X, X[1]) = 0. If X ∈ Comp_{d}^{A} is a point representing X, then the G-orbit of X is open in Comp_{d}^{A}.

Corollary 9. Again suppose that K is algebraically closed. Given any finite sequence d = (d_{m}, … d_{0}) of natural numbers, there are – up to isomorphism in C^{b}(A-mod) – only finitely many almost projective complexes X of A-modules satisfying the following two conditions:

(1) Hom_{D^{b}(A-mod)}(X, X[1]) = 0, and

(2) dim_{K} X_{n} = d_{n}, for all n ∈ ℤ.

3. Faithfully balanced two-sided complexes and invariance of Out(A)^{0}

Several of the ideas underlying this section have precursors which were developed in a paper by Guil-Asensio and the second-named author [7]. Major portions of our auxiliary results can be readily generalized to algebras over commutative rings; in particular, this is true for Proposition 10 below. To facilitate the reading with a unified blanket hypothesis, we will, however, continue to assume that A is a finite dimensional algebra over an algebraically closed field K.

Of course, we can view a chain complex X ∈ C(A-Mod) as a pair (X, λ), where X ∈ C(K-Mod) and λ : A → End_{C(K-Mod)}(X) is a K-algebra homomorphism. Accordingly, given another finite dimensional K-algebra B, an A-B-bimodule complex _{A}X_{B} amounts to a chain complex X of K-spaces, combined with an algebra homomorphism τ : A⊗_{K} B^{op} → End_{C(K-Mod)}(X). We will call a bimodule complex _{A}X_{B} almost projective if both _{A}X and X_{B} are almost projective in the sense of Section 2; recall that, in particular, this means that X is a right bounded complex of finite dimensional modules. By adapting standard terminology for modules to this context (see, e.g., [1]), we moreover define:

Definition. A bimodule complex _{A}X_{B} will be called faithfully balanced provided that both of the canonical K-algebra homomorphisms

λ = τ(−, 1) : A → End_{C(Mod-B)}(X_{B})  and  ρ = τ(1, −) : B → End_{C(A-Mod)}(_{A}X)^{op}

are isomorphisms.

Whenever we let endomorphisms of a complex Y of left A-modules act on the right of Y, we record this by referring to them as maps in End_{C(A-mod)}(Y)^{op}. The existence of a faithfully balanced bimodule complex _{A}X_{B} entails a fairly tight connection between the algebras A and B, as the following two results suggest.

Proposition 10. Whenever there exists a faithfully balanced bimodule complex _{A}X_{B}, the centers of A and B are isomorphic.

Proof. Set E = End_{C(K-Mod)}(X). The canonical embedding of algebras λ : A → E and ρ : B^{op} → E take the centers of A and B to the same subalgebra E′ of E, namely to E′ = End_{C(A-Mod-B)}(X). □

As in Section 1, we denote by Out(A) the group of outer automorphisms of A, and by Out(A)^{0} its identity component. It is well-known (cf. [6], Theorem 1) that Out(A) embeds naturally into the Picard group Pic(A), which can be thought of as the group of isomorphism types of Morita self-equivalences of A-Mod. Clearly, any such Morita self-equivalence induces category equivalences C(A-Mod) → C(A-Mod), C^{b}(A-mod) → C^{b}(A-mod), and D(A-Mod) → D(A-Mod), D^{b}(A-mod) → D^{b}(A-mod). This provides us with a natural action of Pic(A), and consequently also of Out(A), on sets of isomorphism classes of objects in C(A-Mod) (resp. D(A-Mod)). Given any complex X ∈ C(A-Mod), we will accordingly consider its Out-orbit \{^{σ}X ∈ C(A-Mod) : \overline{σ} ∈ Out(A)\} in C(A-Mod), and analogously that in D(A-Mod), as well as its Out-stabilizer Out^{X}(A) = \{\overline{σ} ∈ Out(A) : ^{σ}X ≅ X in D(A-mod)\} in the derived category. As we are dealing with bimodule complexes _{A}X_{B}, it will promote orientation to denote the actions of Out(A) and Out(B) by means of left and right superscripts, respectively.

Proposition 11. Suppose that _{A}X_{B} is a bounded, faithfully balanced bimodule complex of finite dimensional modules. If the Out-orbits of _{A}X and X_{B} in C^{b}(A-mod) and C^{b}(B-mod) are finite, then the groups Out(A)^{0} and Out(B)^{0} are isomorphic. In particular, this is the case when both_{A}X and X_{B} are almost projective with Hom_{D^{b}(A-mod)}(X, X[1]) = 0 and Hom_{D^{b}(mod-B)}(X, X[1]) = 0. If, in addition, char K = 0, the first Hochschild cohomology groups of A and B are isomorphic as well.

Proof. The first assertion is simply a translation of Corollary 2.6 in [7] into the present context; the proof given in [7] carries over. The second part is an immediate consequence of Corollary 9 of the previous section. Since the Lie algebra associated to Out(A)^{0} coincides with the first Hochschild cohomology group of A in characteristic 0, the last assertion follows as well. □

Example. The above propositions cover situations which may be far removed from the situation where A and B are derived equivalent. For instance, if M is any nonzero left Amodule with the properties that Hom_{A}(M, A) = 0 and the Out-orbit of M is finite, then, by taking X = A ⊕ M and B = End(_{A}X)^{op} ≅ [ \begin{matrix}A & M \\ 0 & End_{A}(M)^{op} \end{matrix} ] we obtain a faithfully balanced bimodule _{A}X_{B} satisfying the hypotheses of the preceding propositions. Consequently, the centers of A and B are isomorphic, as are Out(A)^{0} and Out(B)^{0}; the same is true for the first Hochschild cohomology groups of A and B, provided that char K = 0. But A and B fail to be derived equivalent, since they do not have the same number of simple modules. More specific examples of such modules M are the preprojective modules (in the sense of [2]) which are devoid of projective direct summands. In case Out(A) is finite, all modules M with vanishing A-dual Hom_{A}(M, A) qualify of course. Finiteness of Out(A), in turn, is guaranteed whenever A is a split algebra which is tilting-cotilting equivalent to a hereditary algebra of tree type (cf. [7], Theorem 2.10) □

Our principal aim is to obtain the conclusion of Proposition 11 in case A and B are derived equivalent algebras. For that purpose, we stretch the concept of a faithfully balanced complex of bimodules as follows.

Definition. A complex _{A}X_{B} of A-B-bimodules will be called derived faithfully balanced, provided that the canonical K-algebra homomorphisms λ : A ⟶ End_{D(Mod-B)}(X_{B}) and ρ : B ⟶ End_{D(A-Mod)}(_{A}X)^{op} are isomorphisms.

Remark 12. Suppose that _{A}X_{B} is a derived faithfully balanced bimodule complex. Clearly, the canonical K-algebra homomorphism ρ : B → End_{D(A-Mod)}(_{A}X)^{op} equals the composition

ρ : B → End_{C(A-Mod)}(_{A}X)^{op} → End_{D(A-Mod)}(_{A}X)^{op},

where the second map is the canonical one. If we set B̂ = End_{C(A-Mod)}(_{A}X)^{op}, our hypothesis on _{A}X_{B} thus yields a vectorspace decomposition B̂ = B ⊕ H, where H is the two-sided ideal of B̂ consisting of the chain maps h : _{A}X → _{A}X which become zero in D(A-Mod). In the special case where_{A}X is an almost projective complex of left A-modules, these chain maps h are precisely the ones which are homotopic to zero.

Of course, the preceding remark has a twin sibling, with the roles of A and B switched. The same is true for the following useful observation.

Lemma 13. Let _{A}X_{B} be a bounded almost projective complex of A-B-bimodules which is derived faithfully balanced. Then there is an idempotent e ∈ End_{C^{b}(A-mod)}(_{A}X)^{op} with the following properties:

(1) _{A}X(1 − e) is an acyclic complex of left A-modules; in particular,

B ≅ End_{D^{b}(A-mod)}(_{A}Xe)^{op}.

(2) The kernel of the canonical K-algebra homomorphism

End_{C^{b}(A-mod)}(_{A}Xe) → End_{D^{b}(A-mod)}(_{A}Xe)

is contained in the Jacobson radical of End_{C^{b}(A-mod)}(_{A}Xe).

(3) A chain map in End_{C^{b}(A-mod)}(_{A}Xe) is an isomorphism in C^{b}(A-mod) if and only if it turns into an isomorphism in D^{b}(A-mod).

Proof. As in Remark 12, we denote by B̂ the endomorphism ring End_{C^{b}(A-mod)}(_{A}X)^{op}, and by H the twosided ideal of those chain maps which become trivial in D^{b}(A-mod). Then B̂ is in turn a finite dimensional algebra, because the complex X is bounded by hypothesis. The twosided ideal (H + J(B̂))/J(B̂) of B̂/J(B̂) therefore gives rise to an idempotent f ∈ H +J(B̂) such that H +J(B̂) = fB̂+J(B̂) = B̂f +J(B̂). We verify that f is trivial on the homology groups of X. Indeed, if we write f in the form f = h+j, with h ∈ H and j ∈ J(B̂), then H^{n}(f) = H^{n}(j) for all n by Remark 12. But j being nilpotent, so is the induced map H^{n}(j), and hence H^{n}(f) = 0 due to idempotency. Thus the chain complex Xf is acyclic, and the choice e = 1−f satisfies condition (1). Now the kernel eHe of the canonical K-algebra homomorphism eB̂e ≅ End_{C^{b}(A-mod)}(_{A}Xe)^{op} → End_{D^{b}(A-mod)}(_{A}Xe)^{op} ≅ B is contained in eJ(B̂)e; for if it were not, H would fail to be contained in fB̂+J(B̂), contrary to our choice of f. To check the final point of our claim, we consider the canonical surjection End_{C^{b}(A-mod)}(_{A}Xe)^{op} ≅ eB̂e → eB̂e/eHe → eB̂e/eJ(B̂)e. If β ∈ End_{C^{b}(A-mod)}(_{A}Xe) is a quasi-isomorphism, i.e., if β becomes a unit in eB̂e/eHe, then, a fortiori, β + eJ(B̂)e is a unit in eB̂e/eJ(B̂)e. But J(eB̂e) = eJ(B̂)e being nilpotent, this implies that β is a unit in eB̂e = End_{C^{b}(A-mod)}(_{A}Xe)^{op}, which proves (3). □

Our strategy in showing that the isomorphism type of Out(A)^{0} is invariant under derived equivalence is to focus on a certain closed subgroup of finite index of Out(A); invariance of the identity component of this subgroup will then of course amount to invariance of the identity component of the full outer automorphism group. More precisely, given mutually inverse twosided tilting complexes _{A}X_{B} and _{B}X̃_{A} in the sense of [15], the subgroups of choice in Out(A) and Out(B) will be the stabilizer subgroups Out^{X}(A) and OutX̃(B) of X and X̃ in D^{b}(A-mod) and D^{b}(B-mod), respectively. The construction of an isomorphism between them hinges on the following concept of a ‘semilinear chain map’ between complexes of left A-modules: Fix σ ∈ Aut(A) and complexes Y, Z of left A-modules. Following the pattern established in [7], we call a chain map φ = (φ_{n})_{n∈ℕ} : Y → Z of K-complexes σ-semilinear, provided that φ_{n}(ay) = a^{σ}φ_{n}(y) for all n ∈ ℕ, y ∈ Y_{n}, and a ∈ A. We note that a σ-semilinear map Y → Z can alternately be viewed as a morphism ^{σ^{−1}}Y → Z in C(A-Mod). Moreover, we observe that, given any σ-semilinear automorphism φ of Y and any morphism β ∈ End_{C(A-Mod)}(Y)^{op}, the conjugate φβφ^{−1} is again a morphism in End_{C(A-Mod)}(Y)^{op}. To the endomorphisms (resp., automorphisms) of the K-complex Y which are σ-semilinear for some σ ∈ Aut(A) we collectively refer as the (A-)semilinear endomorphisms (resp., automorphisms) of Y. It is straightforward that the set S(Y) of all semilinear automorphisms of Y forms a subgroup of the group of automorphisms of the underlying K-complex.

In the following ⊗^{L} will denote the left derived functor of the tensor functor on chain complexes. Keep in mind that, under the hypothesis of the following lemma, the algebra eB̂e/eHe is canonically isomorphic to B.

Lemma 14. Again, let _{A}X_{B} be a bounded almost projective complex of A-B-bimodules, which is derived faithfully balanced. Moreover, let e ∈ End_{C^{b}(A-mod)}(_{A}X)^{op} be chosen as in Lemma 13, and let H be the twosided ideal of B̂ = End_{C^{b}(A-mod)}(_{A}X)^{op} as specified in Remark 12. Given automorphisms σ ∈ Aut(A) and τ ∈ Aut(B), conditions (1)−(3) below are related as follows: ‘(1) ⟹ (2)’ and ‘(2) ⟺ (3)’.

(1) The functors ^{σ}(X ⊗^{L}_{B} −), X ⊗^{L}_{B} ^{τ} (−) : D^{b}(B-mod) → D^{b}(A-mod) are isomorphic.

(2) There is a σ-semilinear chain map ξ : X → X, i.e., a map in Hom_{C^{b}(A-mod)}(^{σ^{−1}}X, X), which turns into an isomorphism ^{σ^{−1}}X → X in D^{b}(A-mod) and which satisfies the condition that ξb = b^{τ} ξ in D^{b}(A-mod), for all b ∈ B.

(3) There is a bijective σ-semilinear chain map φ : Xe → Xe such that the automorphism of eB̂e, given by β ↦ φβφ^{−1}, induces the same map as τ on B.

Proof. (1) ⟹ (2). One readily verifies that (1) is equivalent to the functors ^{σ^{−1}}(X⊗^{L}_{B} −)

and X ⊗^{L}_{B} ^{τ^{−1}}(−) being isomorphic. We assume that this latter condition is satisfied, and

recall that, when applied to stalk complexes of projectives, the functor X ⊗^{L}_{B} − is just the usual tensor product (cf [18], ch. 2). Now the tensor products X ⊗_{B} (^{τ^{−1}}B) and X^{τ} ⊗_{B} B are clearly canonically isomorphic; we will identify them in fact. Specifying an isomorphism χ : ^{σ^{−1}}(X ⊗^{L}_{B} −) → X ⊗^{L}_{B} ^{τ^{−1}}(−) of functors D^{b}(B-mod) → D^{b}(A-mod), we consider χ(B) : ^{σ^{−1}}(X ⊗_{B} B) → X ⊗_{B} ^{τ^{−1}}B and the resulting chain of isomorphisms in D^{b}(A-mod):

σ^{−1}X ≅ ^{σ^{−1}}(X ⊗_{B} B) ≅ X ⊗_{B} ^{τ^{−1}}B = X^{τ} ⊗_{B} B ≅ X^{τ} .

Since ^{σ^{−1}}X and X^{τ} are both almost projective complexes of left A-modules, the displayed isomorphism on the level of the derived categories is induced by a chain map ξ : ^{σ^{−1}}X → X^{τ} in C^{b}(A-mod) (see the remarks following the definition of an almost projective complex in Section 1). Alternately expressed, ξ is a σ-semilinear map X → X^{τ} which becomes bijective in D^{b}(A-mod). Finally, the automorphism τ of A makes a relevant appearance: Namely, the naturality of χ translates into the required additional property of ξ.

(2) ⟹ (3). Let ξ : X → X be a σ-semilinear map as specified in (2). The fact that the morphism X ⟶^{e} Xe turns into an isomorphism in D^{b}(A-mod) implies that, in the derived category, ξ can be viewed as an isomorphism ^{σ^{−1}}Xe → Xe, induced by the chain map φ = eξ(e|_{Xe}) : ^{σ^{−1}}Xe → Xe in C^{b}(A-mod). From Lemma 13, part (3), one easily derives that φ is actually an isomorphism in C^{b}(A-mod). Thus φ is a σ-semilinear automorphism Xe → Xe. Checking the remaining condition under (3) is routine.

(3) ⟹ (2). Using the equality X = Xe ⊕ X(1 − e), one extends φ : Xe → Xe to a σ-semilinear map ξ = φ ⊕ 0 : X → X and notes that, trivially, the extension has the required properties. □

The following proposition represents the second crucial step – next to Corollary 9 – on the road towards the main result of this section. We continue to study a complex X of A-B-bimodules as in Lemma 14, still denoting its endomorphism ring in C^{b}(A-mod) by B̂. Moreover, we introduce a set G which will turn out to be the graph of a group isomorphism ϕ : Out^{X}(A) → OutX̃(B), provided that _{A}X_{B} and _{B}X̃_{A} are mutually inverse twosided tilting complexes. Namely, we define G as the set of all pairs (\overline{σ}, \overline{τ}) ∈ Out(A) × Out(B) such that σ and τ satisfy the equivalent conditions (2), (3) of Lemma 14. Since, from the resulting definition of ϕ, it is not obvious that this isomorphism of abstract groups is actually an isomorphism of algebraic groups, we resort to viewing ϕ from an alternate angle. Namely, we will rewrite ϕ in the form p_{2}p_{1}^{−1}, with isomorphisms p_{1} : S → Out^{X}(A) and p_{2} : S → OutX̃(B) for a suitable choice of S, these latter maps having the benefit of being more readily recognizable as isomorphisms of algebraic groups. To that end, we consider the group S(Xe) of semilinear automorphisms of the complex Xe ∈ C^{b}(A-mod), together with the following two subgroups U and V: The subgroup U consists of the maps of the form x ↦ uxv, where u is a unit of A and v a unit of eB̂e (note that the latter assignment is semilinear with respect to the automorphism a ↦ uau^{−1} of A). The subgroup V of S(Xe) consists of those maps φ ∈ S(Xe) for which the algebra automorphism eB̂e → eB̂e, given by β → φβφ^{−1}, induces an inner automorphism on eB̂e/eHe ≅ B; here again, H ⊆ B̂ is the ideal of those chain maps which vanish in the derived category D^{b}(A-mod). One readily checks that U and V are normal subgroups of S(Xe) and that U is contained in V. This provides us with a canonical projection π : S(Xe)/U → S(Xe)/V.

Proposition 15. For any bounded almost projective complex _{A}X_{B} of A-B-bimodules, which is derived faithfully balanced, the following are true:

(1) The set G is the graph of a group homomorphism ϕ : Out^{X}(A) → Out(B), where Out^{X}(A) is the Out(A)-stabilizer of X in D^{b}(A-mod).

(2) In case X is a tilting complex over A and e ∈ End_{C^{b}(A-mod)}(X)^{op} an idempotent as before, the subgroups U and V of S(Xe) are closed, as is the subgroup Im(ϕ) of Out(B). Moreover, there are isomorphisms of algebraic groups,

p_{1} : S(Xe)/U → Out^{X}(A)  and  p_{2} : S(Xe)/V → Im(ϕ),

such that p_{2}π = ϕp_{1}, where π is the projection introduced above.

Proof. (1) The task of checking that an element \overline{σ} ∈ Out(A) cannot be the first component of two different pairs in G is easily reduced to the case where \overline{σ} = 1, i.e., to the case where σ(a) = uau^{−1} for some unit u of A. Suppose that τ ∈ Aut(B) is such that the pair (σ, τ) satisfies the equivalent conditions of Lemma 14, and let φ : Xe → Xe be a bijective σsemilinear endomorphism of Xe as specified in condition (3) of that lemma; this is to say that β ↦ φβφ^{−1} induces the same map as τ on eB̂e/eHe. The σ-semilinearity of φ yields A-linearity of the assignment v : Xe → Xe defined by xv = u^{−1}(xφ). In other words, v is a unit in End_{C^{b}(A-mod)}(Xe)^{op} = eB̂e. Keeping in mind that the algebra eB̂e/eHe is isomorphic to B, by our balancedness hypothesis and the choice of H, we let c be the canonical image of v in B. In view of the equality vβv^{−1} = φβφ^{−1} for β ∈ eB̂e, we finally observe that τ is just conjugation by c, for our construction entails τ(b) = φbφ^{−1} = cbc^{−1} for b ∈ B. Thus τ is in turn inner, as required.

Knowing that G is the graph of a function ϕ, we pin down the domain of ϕ. By construction, it consists of those classes \overline{σ} ∈ Out(A) for which there exists a bijective σ-semilinear map φ : Xe → Xe (note that conjugation by φ automatically induces an automorphism of eB̂e/eHe). The latter condition is tantamount to the requirement that Xe be isomorphic to ^{σ}(Xe) in C^{b}(A-mod), which is in turn equivalent to the existence of an isomorphism Xe ≅ ^{σ}(Xe) in D^{b}(A-mod); this last equivalence is readily deduced from Lemma 13(3) So the domain of ϕ is Out^{X}(A) as claimed. It is straightforward to check that ϕ is a group homomorphism.

(2) Now suppose that X is a tilting complex of left A-modules. In particular, this means that the category add(X), consisting of the finite direct sums of direct summands of copies of X, generates the homotopy category K^{b}(A-proj) of all bounded projective complexes of finitely generated left A-modules as a triangulated category. We infer that the left annihilator of X in A is zero. Hence so is the annihilator of Xe, for the projective complexes X and Xe coincide in K^{b}(A-proj). Following the model of [7, Lemma 1.2], we deduce that, for every semilinear bijective chain map φ : Xe → Xe, there is a unique automorphism σ ∈ Aut(A) with the property that φ is σ-semilinear: Namely, if ℓ_{a} denotes left multiplication of Xe by a, then σ(a) is determined by the requirement that ℓ_{σ(a)} = φℓ_{a}φ^{−1}. If we write S for the group of semilinear automorphisms of Xe, this provides us with a map S → Aut(A), which is actually a homomorphism of abstract groups. The proof of assertion (1) shows that, whenever this homomorphism takes φ to an inner automorphism ‘a ↦ uau^{−1}’ for a suitable unit u ∈ A, the map v : Xe → Xe sending x to u^{−1}(xφ) is a unit in eB̂e, and vice versa. But the latter condition just says that xφ = uxv, and thus shows that our group homomorphism S → Aut(A) induces a monomorphism S/U → Out(A) of abstract groups. In light of the previous paragraph, the image of this monomorphism is Out^{X}(A), and hence it gives rise to an isomorphism p_{1} : S/U → Out^{X}(A).

Set p′_{2} = ϕp_{1} : S/U → Out(B). Then, clearly Im(p′_{2}) = Im(ϕ). As for the kernel of p′_{2}, we have ϕp_{1}(φ + U) = 1 if and only if there is an inner automorphism ι of B such that b^{ι}φ equals φb in D^{b}(A-mod), for all b ∈ B. But this just means that the automorphism of eB̂e given by β → φβφ^{−1} induces an inner automorphism of the algebra eB̂e/eHe ≅ B. Hence Ker(p′_{2}) equals V/U, so that p′_{2} induces an isomorphism S/V → Im(ϕ), as required.

Finally, we need to ascertain that p_{1}, p_{2} are isomorphisms of algebraic groups. As for p_{1}, we start by identifying the algebra A with the subalgebra of Â = End_{C^{b}(mod- eB̂e)}(Xe) consisting of the left multiplications ℓ_{a} on Xe for a ∈ A; this identification is legitimate, since Xe is faithful over A. With A viewed from this angle, the group homomorphism S → Aut(A) underlying p_{1} maps φ ∈ S to the automorphism ℓ_{a} ↦ φℓ_{a}φ^{−1} of A, the latter conjugate being equal to ℓ_{σ(a)} if φ is σ-semilinear. This assignment is clearly a morphism of algebraic groups, and hence U is a closed (normal) subgroup of S. To see that the inverse of p_{1} is again a morphism, it suffices to show that the map p′_{1} : S → Out^{X}(A) which induces p_{1} modulo U has a surjective differential T_{e}(S) → T_{e}(Im(p′_{1})) (cf. [3, Proposition 6.13]).

In a first step, we focus on the following morphism of algebraic groups

γ : G → GL(V)  with  γ_{g}(v) = gvg^{−1},

where G = GL(Xe) and V = End_{K}(Xe). Our aim is to see that the restriction S → γ(S), again denoted by γ, has surjective differential d γ : T_{e}(S) → T_{e}(Im(γ)) ⊆ End_{K}(V). From [3, 3.10], we know that

(†)  (d γ)(X)(v) = (d 𝔬𝔯𝔟_{v})_{e}(X),

for X ∈ T_{e}(S) and v ∈ V, where 𝔬𝔯𝔟_{v} : S → γ(S)v is the orbit map g ↦ γ_{g}v. To check surjectivity, we let v_{1}, …, v_{n} be a basis for V, set \underline{v} = (v_{1}, …, v_{n}) ∈ V^{n}, and consider the ‘expanded orbit map’

\underline{𝔒𝔯𝔟} : G → γ(G).\underline{v}, g ↦ g.\underline{v} = (γ_{g}v_{1}, …, γ_{g}v_{n}),

and its restriction to S,

\underline{𝔬𝔯𝔟} : S → γ(S).\underline{v}.

The map \underline{𝔒𝔯𝔟} has reduced fibres, since the stabilizer subgroup Stab(\underline{v}) of \underline{v} in G clearly arises as the solution set of a system of linear equations (for background on reduced fibres, see [10, A.I.2.5, 2.6]). By [10, A.I.5.5, Satz 2], this implies that Ker(d \underline{𝔒𝔯𝔟})_{e} = T\underline{v}(Stab(\underline{v})). Cutting down Stab(\underline{v}) to the stabilizer subgroup stab(\underline{v}) of \underline{v} in S, we infer that Ker(d \underline{𝔬𝔯𝔟})_{e} equals T\underline{v}(stab(\underline{v})), and, in view of [3, Proposition 6.7], we deduce further that the differential (d \underline{𝔬𝔯𝔟})_{e} : T_{e}(S) → T\underline{v}(γ(S).\underline{v}) is surjective. In light of (†), this yields surjectivity of d γ.

If we enlarge the codomain of p′_{1} from Out^{X}(A) to Out(A), then p′_{1} can be factored in the form

S ⟶^{γ} γ(S) ⟶^{𝔯𝔢𝔰} Aut(A) ⟶^{𝔠𝔞𝔫} Out(A),

where 𝔯𝔢𝔰 : γ(S) → Aut(A) sends any K-automorphism of V in γ(S) to its restriction to the invariant subspace A = Â. Now 𝔯𝔢𝔰 : γ(S) → Im(𝔯𝔢𝔰) is clearly separable, and so is the canonical map 𝔠𝔞𝔫, the latter being actually a geometric quotient, whence both 𝔯𝔢𝔰 and 𝔠𝔞𝔫 have surjective differentials. This shows that our auxiliary map p′_{1} has surjective differential, and we conclude that p_{1} is indeed an isomorphism of algebraic groups.

To deal with p_{2}, we identify B with eB̂e/eHe via the canonical isomorphism. Clearly, the group homomorphism S → Aut(eB̂e) sending φ to the automorphism β ↦ φβφ^{−1} respects the variety structures, and since the normal subgroup of Aut(eB̂e) consisting of those automorphisms which induce inner automorphisms of eB̂e/eHe is closed, this subgroup gives rise to a geometric quotient of Aut(eB̂e). In particular, the auxiliary map p′_{2} : S/U → Out(B) as above is again a morphism of algebraic groups. This in turn entails that V ⊆ S and Im(ϕ) ⊆ Out(B) are closed subgroups. Now the geometric quotient property of the canonical map S → S/V guarantees that p_{2} is a morphism of algebraic groups as well. With an argument similar to that given for p_{1}, one finally shows that p_{2} also has a surjective differential, which guarantees that its inverse is in turn a morphism of algebraic groups. □

In the following, we adopt Rickard’s terminology in [15], but insignificantly deviate from his conventions by calling a category equivalence D^{−}(B-Mod) → D^{−}(A-Mod) (resp.,

D^{b}(B-mod) → D^{b}(A-mod)) standard if it is isomorphic to X ⊗^{L}_{B} −, where _{A}X_{B} is a twosided tilting complex; as is shown in [15, Theorem 3.3], this is harmless. Moreover, we note

that according to [15, loc. cit.], any equivalence X ⊗^{L}_{B} − between the derived categories D^{−}(B-Mod) and D^{−}(A-Mod) automatically restricts to an equivalence D^{b}(B-mod) → D^{b}(A-mod). To finally clear the road to the main theorem of this section, we record an easy subsidiary observation.

Lemma 16. Let B be a finite dimensional algebra and F : D^{−}(B-Mod) → D^{−}(B-Mod) a standard derived equivalence. Then the following assertions are equivalent:

(1) F is induced by an element \overline{τ} ∈ Out(B), i.e., there is a natural isomorphism F ≅ τ (−).

(2) F takes the stalk complex _{B}B to an isomorphic copy of itself.

Proof. Clearly, any τ ∈ Aut(B) gives rise to an isomorphism B ≅ ^{τ} B of left B-modules, which proves ‘(1) ⟹ (2)’. For the converse, let _{B}T_{B} be a two-sided tilting complex for

B such that F ≅ T⊗^{L}_{B} −. Then B ≅ F (B) ≅ T⊗^{L}_{B} B ≅ T in D^{−}(B-Mod). In other words, as a complex of left B-modules, T is isomorphic to the stalk complex B on the level of the derived category. By [15], the right derived functor of Hom_{B}(T, −) is a quasi-inverse of F – call it F^{−1}. We deduce that F^{−1} = R Hom_{B}(T, −) ≅ Hom_{D^{−}(B-Mod)}(B, −) is induced by a Morita equivalence Hom_{B}(C, −) : B-Mod → B-Mod, where C is some B-B-bimodule isomorphic to ^{1}B^{τ^{−1}} for some automorphism τ of B. This shows that F ≅ ^{τ} (−) is determined by \overline{τ} ∈ Out(B), and completes the proof of ‘(2) ⟹ (1)’. □

As announced, we will exploit the scenario we have rigged up in the special case where AX_{B} is a twosided tilting complex. In particular, we will see that, in this situation, the map ϕ constructed in Proposition 15 is a group isomorphism Out^{X}(A) → OutX̃(B), provided that _{B}X̃_{A} is a twosided tilting complex inverse to X. Moreover, we will obtain U = V, which will provide us with an alternate description of ϕ as p_{2}p_{1}^{−1}, in the terminology of the proposition.

Theorem 17. Suppose that A and B are finite dimensional algebras (over an algebraically closed field K). If A and B are derived equivalent, then Out(A)^{0} and Out(B)^{0} are isomorphic algebraic groups.

Proof. We begin by choosing two-sided tilting complexes _{A}X_{B} and _{B}X̃_{A} which are inverse to each other, that is, X ⊗^{L}_{B} X̃ ≅ _{A}A_{A} and X̃ ⊗^{L}_{A} X ≅ _{B}B_{B} in the derived categories D^{b}(A ⊗_{K} A^{op} -mod) and D^{b}(B ⊗_{K} B^{op} -mod), respectively. Recall that Out^{X}(A) and OutX̃(B) denote the subgroups consisting of those elements of Out(A) and Out(B) which stabilize X and X̃ in D^{b}(A-mod) and D^{b}(B-mod), respectively. Our first goal is to show that these closed subgroups of Out(A) and Out(B) have finite index in the corresponding full outer automorphism groups.

We show finiteness of the index [Out(A) : Out^{X}(A)], the result for B being symmetric. Note that the Out(A)-stabilizer of X in D^{b}(A-mod) equals that of Xe in D^{b}(A-mod), where e is an idempotent in the C^{b}(A-mod)-endomorphism ring of X as specified in Lemma 13, for, in the derived category, X and Xe become isomorphic. Hence the index under consideration coincides with [Out(A) : Out^{Xe}(A)] and thus equals the cardinality of the Out(A)-orbit of Xe in D^{b}(A-mod). So clearly it suffices to show that the Out(A)-orbit of Xe in C^{b}(A-mod) is finite. But this follows from Corollary 9 of Section 1, which shows that there are only finitely many complexes over A which share the K-dimension of Xe, as well as the property that Hom_{D^{b}(A-mod)}(Xe, Xe[1]) = 0. This takes us to our first goal.

Our argument is thus reduced to showing that Im(ϕ) = OutX̃(B) and U = V, in the terminology of Proposition 15. Once these equalities are established, Proposition 15 will provide us with isomorphisms of algebraic groups

Out^{X}(A) ⟵^{p_{1}} S(Xe)/U ⟶^{p_{2}} OutX̃(B).

We deduce that ϕ = p_{2}p_{1}^{−1} restricts to an isomorphism Out(A)^{0} → Out(B)^{0}, for the identity component of S(Xe)/U is mapped onto that of Out^{X}(A) – which equals Out(A)^{0} – by p_{1}, and onto that of OutX̃(B) – which equals Out(B)^{0} – by p_{2}.

We access the two remaining equalities by way of the group homomorphism ϕ from Out^{X}(A) to Out(B) constructed in Proposition 15. To that end, we give yet another description of ϕ in the present situation. For any element \overline{σ} ∈ Out^{X}(A), the equivalence

F = (X̃ ⊗^{L}_{A} −) ∘^{σ}(X ⊗^{L}_{B} −) from D^{b}(B-mod) to D^{b}(B-mod) preserves the isomorphism type of the stalk complex _{B}B, whence Lemma 16 shows F to be induced by some element \overline{τ} ∈ Out(B). In view of the choice of X and X̃ as inverse to each other, this provides us

with an isomorphism ^{σ}(X ⊗^{L}_{B} −) ≅ X ⊗^{L}_{B} ^{τ} (−) of functors D^{b}(B-mod) → D^{b}(A-mod). Consequently, Lemma 14 guarantees that the pair (\overline{σ}, \overline{τ}) satisfies the equivalent conditions (2), (3) of that lemma; in other words, the pair (\overline{σ}, \overline{τ}) belongs to the graph G of ϕ. Thus ϕ assigns to any \overline{σ} ∈ Out^{X}(A) the unique element \overline{τ} ∈ Out(B) with the property that

σ(X ⊗^{L}_{B} −) ≅ X ⊗^{L}_{B} ^{τ} (−). To obtain Im(ϕ) ⊆ OutX̃(B), we derive the following string of isomorphisms in D^{b}(B-mod):

τ X̃ ≅ (X̃ ⊗^{L}_{A} X) ⊗^{L}_{B} ^{τ} X̃ ≅ X̃ ⊗^{L}_{A} (X ⊗^{L}_{B} ^{τ} X̃) ≅ X̃ ⊗^{L}_{A} ^{σ}(X ⊗^{L}_{B} X̃) ≅ X̃ ⊗^{L}_{A} ^{σ}A.

Since the left A-modules^{σ}A and A are isomorphic, we conclude that^{τ} X̃ ≅ X̃ in D^{b}(B-mod). But this just means that \overline{τ} ∈ OutX̃(B).

The symmetric version of Proposition 15, obtained by switching the roles of A and B, permits us to ‘symmetrize’ the preceding paragraph. Indeed, it provides us with a group homomorphism ϕ̃ : OutX̃(B) → Out(A) which assigns to \overline{τ} ∈ OutX̃(B) the unique

element \overline{σ} ∈ Out(A) with ^{τ} (X̃ ⊗^{L}_{A} −) ≅ X̃ ⊗^{L}_{A} ^{σ}(−), and we obtain Im(ϕ̃) ⊆ Out^{X}(A) as above. From the fact that X and X̃ are inverse to each other, we moreover derive that ϕ and ϕ̃ are inverse group isomorphisms between Out^{X}(A) and OutX̃(B). In particular, Im(ϕ) = OutX̃(B). In view of part (2) of Proposition 15, finally, injectivity of ϕ guarantees that the canonical map π : S(Xe)/U → S(Xe)/V is the identity; in other words, we also have U = V. This completes the argument. □

Acknowledgement

The paper was prepared while the second-named author was visiting the University of California at Santa Barbara. He would like to thank the UCSB Mathematics Department for its hospitality. Moreover, he is grateful to the D.G.E.S. of Spain and the Fundación ‘Séneca’ of Murcia for their financial support.

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Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

E-mail address: birge@math.ucsb.edu

Departamento de Mátematicas, Universidad de Murcia, 30100 Espinardo-MU, Spain

E-mail address: msaorinc@fcu.um.es