RIGIDITY OF TILTING COMPLEXES AND DERIVED EQUIVALENCE FOR SELF-INJECTIVE ALGEBRAS

SALAH AL-NOFAYEE AND JEREMY RICKARD

Let Λ and Γ be derived equivalent rings. That is, the derived categories D(Mod(Λ)) and D(Mod(Γ)) are equivalent as triangulated categories. Then many properties of Λ and Γ are shared. One example of this, first proved in [2, Corollary 5.3], is that if Λ is a symmetric finite-dimensional algebra over a field k, then so is Γ.

In fact, the property of being symmetric can be easily characterized in terms of the derived category. Recall that an object of the derived category D(Mod(Λ)) is called perfect if it is isomorphic to a bounded complex of finitely-generated projective modules, and that the perfect objects P can be characterized as those that are compact, meaning that the functor Hom_{D(Mod(Λ))}(P, −) preserves arbitrary direct sums, and that if Λ is a finite-dimensional k-algebra, then the objects X of D(Mod(Λ)) isomorphic to objects of D(mod(Λ)) can be characterized as those objects such that, for every perfect object P, Hom_{D(Mod(Λ))}(P, X) is finite-dimensional. Thus such objects P and X are both intrinsically characterized in terms of the triangulated category D(Mod(Λ)).

Proposition 0.1. A finite-dimensional k-algebra Λ is symmetric if and only if, for every object X of D(mod(Λ)) and every perfect object P of D(Mod(Λ)), the k-vector spaces Hom_{D(Mod(Λ))}(P, M) and Hom_{D(Mod(Λ))}(M, P) are naturally dual.

Proof. It was shown in [3, Corollary 3.2] that if Λ is symmetric, then this duality holds. Conversely, if there is such a duality, then the case P = X = Λ shows that there is an isomorphism between Λ and its dual, which, by naturality, is an isomorphism of bimodules; thus Λ is symmetric. □

For the weaker property of being self-injective, there seems to be no such direct characterization in terms of the derived category. However, our main theorem in this paper is that, at least for finite-dimensional algebras over an algebraically closed field, the property of being self-injective is preserved by derived equivalence. The proof depends on a theorem of Zimmermann-Huisgen and Saorín [4] on rigidity of tilting complexes. In Section 1 we show how this theorem also follows from the results of [1].

1. Rigidity for tilting complexes

Let k be a field and let Λ be a finite-dimensional k-algebra.

In Corollary 9 of [4], Zimmermann and Saorín prove a result (at least if k is algebraically closed) that easily implies the following theorem.

Theorem 1.1. Let …, P^{−1}, P^{0}, P^{1}, … be a sequence of finitely-generated projective Λmodules, such that P^{i} = 0 for all but finitely many i. Then up to isomorphism there are only finitely many tilting complexes

P = ⋯ → P^{−1} → P^{0} → P^{1} → ….

Since, up to isomorphism there is only a countable set of possibilities for the sequence …, P^{−1}, P^{0}, P^{1}, …, and since every algebra derived equivalent to Λ is isomorphic to the endomorphism algebra of a tilting complex, the following corollary follows.

Corollary 1.2. Let Λ be a finite-dimensional algebra over an algebraically closed field k. Then up to isomorphism there are only countably many tilting complexes for Λ, and hence only countably many algebras derived equivalent to Λ.

In particular, this rules out (at least over an uncountable field) the possibility of a family of non-isomorphic but derived equivalent algebras, parametrized by an algebraic curve or variety of higher dimension.

A number of people have noticed that Theorem 1.1 also follows using the techniques of [1], but we do not know of any reference to this in the literature. So let us sketch an alternative proof along these lines.

The complexes of the form

P = ⋯ → P^{−1} →^{d^{−1}} P^{0} →^{d^{0}} P^{1} → …

are parametrized by the k-rational points of an affine variety V: namely, the closed subvariety of

∏_{i∈ℤ}Hom(P^{i}, P^{i+1})

determined by the equations d^{i+1}d^{i} = 0.

Let k[V] be the coordinate ring of this variety. Then there is a natural complex

P̃ = ⋯ → P^{−1} ⊗_{k} k[V] → P^{0} ⊗_{k} k[V] → P^{1} ⊗_{k} k[V] → …

of Λ ⊗_{k} k[V]-modules such that, if P_{v} is the complex corresponding to a k-rational point v ∈ V with associated maximal ideal 𝔪_{v}, then

P_{v} ≅ P̃ ⊗_{k[V]} (k[V]/𝔪_{v}).

For each k-rational point w ∈ V, there is also a complex

P̃_{w} = P_{w} ⊗_{k} k[V]

with the same terms as P̃, but with

P̃_{w} ⊗_{k[V]} (k[V]/𝔪_{v}) ≅ P_{w}

for every k-rational point v ∈ V.

Now suppose that w is a k-rational point for which P_{w} is a tilting complex, and let { \atop k[V]_{w}} be the completion of k[V] at the maximal ideal 𝔪_{w}. Then by [1, Theorem 3.3], the complexes P̃ ⊗_{k[V]} { \atop k[V]_{w}} and P̃_{w} ⊗_{k[V]} { \atop k[V]_{w}} are isomorphic. Using the Artin Approximation Theorem as in the proof of [1, Theorem 4.1], it follows that for some étale neighbourhood U of w, there is an isomorphism between P̃ ⊗_{k[V]} k[U] and P̃_{w} ⊗_{k[V]} k[U]. In particular, for any k-rational point v in the open set that is the image of U → V,

P_{v} = P̃ ⊗_{k[V]} (k[V]/𝔪_{v}) ≅ P̃_{w} ⊗_{k[V]} (k[V]/𝔪_{v}) ≅ P_{w}.

Thus, the complexes parametrized by the k-rational points of some open neighbourhood of w are all isomorphic to P_{w}, and it follows that there can only be finitely many isomorphism classes of complexes P_{v} that are tilting complexes.

In fact, as in Zimmermann-Huisgen and Saorín’s proof, it is only the property

Hom_{D(Mod(Λ))}(P, P [1]) = 0

that is needed, so in fact there are only finitely many isomorphism classes of complexes P_{v} satisfying this weaker condition.

2. The main theorem

For a finite-dimensional k-algebra Λ we define the Nakayama functor

ν_{Λ} : Mod(Λ) → Mod(Λ)

by

ν_{Λ}(M) = Λ^{∨} ⊗_{Λ} M,

where Λ^{∨} denotes the k-linear dual of Λ, regarded as a Λ-bimodule. Thus if Λ is symmetric the Nakayama functor is isomorphic to the identity functor.

In general, ν_{Λ} induces an equivalence between the categories of finitely generated projective and injective modules for Λ. It is a right exact functor, and has a total left derived functor

Lν_{Λ} : D(Mod(Λ)) → D(Mod(Λ)),

where Lν_{Λ}(X) is constructed in the usual way by applying the functor − ⊗_{Λ} X to a projective resolution of Λ^{∨}, or by applying the functor Λ^{∨} ⊗_{Λ} − to a projective resolution of X.

If Λ is self-injective, however, ν_{Λ} is exact, and so ν_{Λ} = Lν_{Λ}. In this case ν_{Λ} is a selfequivalence of the module category Mod(Λ) and therefore induces a self-equivalence of the derived category D(Mod(Λ)).

In [2, Proposition 5.2], it was shown that, if Λ and Γ are derived equivalent finitedimensional k-algebras then (at least for some equivalence of derived categories) the diagram

D^{−}(Mod(Λ)) ⟶^{Lν_{Λ}} D^{−}(Mod(Λ))
↓  ↓
D^{−}(Mod(Γ)) ⟶^{Lν_{Γ}} D^{−}(Mod(Γ))

commutes up to isomorphism of functors, and so if

F : D^{−}(Mod(Λ)) → D^{−}(Mod(Γ))

is a derived equivalence, then

F (Lν_{Λ}X) ≅ Lν_{Γ}(F X)

for any object X of D^{−}(Mod(Λ)).

Theorem 2.1. Let Λ and Γ be derived equivalent finite-dimensional algebras over a field k, with Λ self-injective. Then Γ is self-injective.

Proof. Since ν_{Γ} induces an equivalence between the categories of finitely generated projective and finitely generated injective Γ-modules, we just need to show that ν_{Γ}(Γ) is projective.

Since ν_{Λ} is a self-equivalence of the module category Mod(Λ), some power ν_{Λ}^{r} of the Nakayama functor has the property that ν_{Λ}^{r}(P) ≅ P for every projective Λ-module P.

Let F : D^{−}(Mod(Λ)) → D^{−}(Mod(Γ)) be a derived equivalence. Then F^{−1}(Γ) is isomorphic to some tilting complex T for Λ, and ν_{Λ}^{r}(T) is also a tilting complex which has components isomorphic to those of T. Thus, for some multiple s of r, ν_{Λ}^{s}(T) is isomorphic to T, by Theorem 1.1.

By the discussion preceding the statement of Theorem 2.1, it follows that Lν_{Γ} is a self-equivalence of D^{−}(Mod(Γ)) and that Lν_{Γ}^{s}(Γ) is isomorphic to Γ. We shall prove that Lν_{Γ}(Γ) is isomorphic to a projective module concentrated in degree zero for all 0 ≤ n ≤ s. For if not, then let n < s be maximal such that this is not true.

Since Lν_{Γ} preserves the property of having zero homology in positive degrees, Lν_{Γ}^{n}(Γ) has zero homology in positive degrees. Let

⋯ → 0 → Q^{−k} → Q^{−k+1} → ⋯ → Q^{0} → 0 → …

be a minimal projective resolution of Lν_{Γ}^{n}(Γ) with k > 0 and Q^{−k} ≅ 0̸. It is an indecomposable bounded complex of finitely generated projective modules, since Lν_{Γ} is a self-equivalence of D^{−}(Mod(Γ)) and therefore preserves the property of being an indecomposable perfect object.

Since ν_{Γ} is an equivalence between the categories of finitely generated projective and injective Γ-modules, when we apply ν_{Γ} we get an indecomposable complex

⋯ → 0 → ν_{Γ}Q^{−k} → ν_{Γ}Q^{−k+1} → ⋯ → ν_{Γ}Q^{0} → 0 → …

of injective Γ-modules with ν_{Γ}Q^{−k} ≅ 0̸. But this complex is isomorphic in D(Mod(Γ)) to Lν_{Γ}^{n+1}(Γ), which, by by the assumption on n, has homology concentrated in degree zero. Thus the differential

ν_{Γ}Q^{−k} → ν_{Γ}Q^{−k+1}

is injective, and therefore split, since ν_{Γ}Q^{−k} is an injective module. But this contradicts the indecomposability of the complex.

By contradiction it follows that ν_{Γ}(Γ) is a projective module, and hence Γ is a selfinjective algebra. □

It would be interesting to find a simple direct characterization of self-injective algebras in terms of properties of their derived categories as we did for symmetric algebras in Proposition 0.1, but it is not clear that this can be done.

References

[1] Jeremy Rickard. Lifting theorems for tilting complexes. J. Algebra, 142:383–393, 1991.

[2] Jeremy Rickard. Derived equivalences as derived functors. J. London Math. Soc. (2), 43(1):37–48, [3] Jeremy Rickard. Equivalences of derived categories for symmetric algebras. J. Algebra, 257:460–481,

2002.

[4] Birge Zimmermann-Huisgen and Manuel Saorín. Geometry of chain complexes and outer automorphisms under derived equivalence Trans. Amer. Math. Soc., 353:4757–4777, 2001.

University of Bristol, School of Mathematics, University Walk, Bristol BS8 1TW, ENG-LAND

E-mail address: S.Al-Nofayee@bristol.ac.uk

University of Bristol, School of Mathematics, University Walk, Bristol BS8 1TW, ENG-LAND

E-mail address: j.rickard@bristol.ac.uk