DEGENERATIONS FOR DERIVED CATEGORIES

BERNT TORE JENSEN, XIUPING SU, AND ALEXANDER ZIMMERMANN

To Claus Michael Ringel for his sixtieth birthday

Abstract. We propose a theory of degenerations for derived module categories, analogous to degenerations in module varieties for module categories. In particular we define two types of degenerations, one algebraic and the other geometric. We show that these are equivalent, analogously to the Riedtmann-Zwara theorem for module varieties. Applications to tilting complexes are given, in particular that any two term tilting complex is determined by its graded module structure.

Introduction

Geometrical methods were introduced in representation theory of finite dimensional algebras in order to parameterize possible module structures on a given vector space by algebraic varieties. These varieties carry an action of a reductive algebraic group G such that the orbits correspond to isomorphism classes of modules. One says that a module M degenerates to N if N is in the closure \overline{G · M} of the orbit of M under the G-action, and in this case one writes M ≤ N. Riedtmann defined in [10] a relation ≤_{alg} by setting M ≤_{alg} N if there is a module Z and a short exact sequence

0 ⟶ N ⟶ M ⊕ Z ⟶ Z ⟶ 0

of A-modules. She showed that M ≤_{alg} N implies M ≤ N and in [13] Zwara proved that M ≤ N implies M ≤_{alg} N.

Since the derived category became a powerful tool in representation theory it seems desirable to study derived categories from such a geometric point of view. De Concini and Strickland [3] studied geometric properties of varieties of bounded complexes of free modules. For a finite dimensional algebra A Huisgen-Zimmermann and Saorin [11] defined an affine variety which parameterizes bounded complexes of A-modules. For this variety no group action seems available so that the quasi-isomorphism classes correspond to orbits under the action. Bekkert and Drozd studied in [1] minimal right bounded complexes of projective modules, where quasi-isomorphism is the same as homotopy equivalence. There homotopy equivalence classes are obtained as orbits of an action of a group, however Bekkert and Drozd did not study the topology of their space and in particular they did not study degeneration.

The purpose of the present paper is to define and to study a geometric structure on a set of right bounded complexes of projective modules and to show a result analogous to the result of Zwara and Riedtmann. More precisely, we define a topological space comproj\underline{d} parameterizing right bounded complexes of projective modules depending on a dimension array \underline{d} replacing the dimension vector for module varieties. This topological space is a projective limit of affine varieties and a projective limit G of affine algebraic groups is acting on it. The G-orbits correspond to quasi-isomorphism classes of right bounded complexes of projective modules. For two right bounded complexes M and N we define M ≤_{Δ} N if there is a complex Z and a distinguished triangle

N ⟶ M ⊕ Z ⟶ Z ⟶ N[1] .

Date: December 4, 2003; revised September 24, 2004.

For two right bounded complexes M and N in comproj\underline{d} we say M ≤_{top} N if N ∈ \overline{G · M}. Our main result is the following.

Theorem. Let A be a finite dimensional k-algebra over an algebraically closed field k and let N and M be complexes in the bounded derived category of A-modules D^{b}(A). Then, there is a dimension array \underline{d} so that N and M belong to comproj\underline{d} and moreover M ≤_{Δ} N if and only if M ≤_{top} N.

Using ≤_{alg} and ≤_{Δ} we show that for two A-modules M and N one can choose a dimension array \underline{d} so that the module M degenerates to N in the module variety if and only if the projective resolution of M degenerates to the projective resolution of N in comproj\underline{d}. To illustrate how the topology of comproj\underline{d} can be used we show that a partial two-term tilting complex is determined, up to isomorphism, by its structure as a graded module. We give an example showing that this is not true for longer tilting complexes.

The paper is organized as follows. In Section 1 we define the variety comproj\underline{d}, define a group acting on it, and show some basic properties. In Section 2 we define ≤_{Δ} and show that ≤_{Δ} implies the topological degeneration for two complexes with bounded homology. In Section 3 we show the converse. Section 4 finally develops consequences for complexes without self-extensions.

1. General definitions and elementary properties

Let A be a finite dimensional algebra over an algebraically closed field k. Let mod(A, d) denote the affine variety of d-dimensional A-modules. The general linear group Gl_{d}(k) acts on mod(A, d) by change of basis and the orbits correspond to the isomorphism classes of d-dimensional modules.

Let P_{1}, …, P_{l} be a complete set of projective indecomposable A-modules one in each isomorphism class. For an element d = (d^{1}, …, d^{l}) ∈ ℕ^{l} let α(d) be defined by ⨁_{j=1}^{l} P_{j}^{d^{j}} ∈ mod(A, α(d)).

For every sequence \underline{d} : ℤ ⟶ ℕ^{l} for which there is an i_{0} ∈ ℤ with d_{i} = (0, …, 0) for i ≤ i_{0} we define comp(A, \underline{α}(\underline{d})) to be the subset of

(∏_{i∈ℤ}mod(A, α(d_{i}))) × (∏_{i∈ℤ}Hom_{k}(k^{α(d_{i})}, k^{α(d_{i−1})}))

consisting of elements ((M_{i})_{i∈ℤ}, (∂_{i})_{i∈ℤ}) with the properties that ∂_{i} is an A-homomorphism when viewed as a map from M_{i} to M_{i−1} and ∂_{i}∂_{i−1} = 0.

The group ∏_{i∈ℤ} Gl_{α(d_{i})} acts on comp(A, \underline{α}(\underline{d})) by change of basis and the orbits correspond to isomorphism classes of complexes.

We have a projection π_{M} : comp(A, \underline{α}(\underline{d})) ⟶ ∏_{i∈ℤ} mod(A, α(d_{i})) and we define

comproj\underline{d} := π_{M}^{−1}(∏_{i∈ℤ} ⨁_{j=1}^{l} P_{j}^{d_{i}^{j}}) .

We say that \underline{d} is bounded if there is an i_{1} ∈ ℤ with d_{i} = (0, …, 0) for i ≥ i_{1}. In this case we identify comp(A, \underline{α}(\underline{d})) with the affine variety of bounded complexes defined by Huisgen-Zimmermann and Saorin in [11], in particular it has the Zariski topology. Also comproj\underline{d} is then an affine variety, being a closed subset of comp(A, \underline{α}(\underline{d})).

Naive truncation on the left induces surjective morphisms of varieties

φ_{n} : comproj^{\underline{d}^{n}} ⟶ comproj^{\underline{d}^{n−1}}
(∏_{i≤n} M_{i}, ∏_{i≤n} ∂_{i}) ↦ ( ∏_{i≤n−1} M_{i}, ∏_{i≤n−1} ∂_{i})

and similarly surjective maps

π_{n} : comproj\underline{d} ⟶ comproj^{\underline{d}^{n}}
(∏_{i∈ℤ}M_{i}, ∏_{i∈ℤ}∂_{i}) ↦ (∏_{i≤n} M_{i}, ∏_{i≤n} ∂_{i}) .

We give comproj\underline{d} the weak topology with respect to the maps \{π_{n}\}. So, the open sets in comproj\underline{d} are of the form U = ⋃_{n≥n_{0}} π_{n}^{−1}(U_{n}) for open sets U_{n} in comproj^{\underline{d}^{n}} and n_{0} ∈ ℤ. Similarly, the closed sets in comproj\underline{d} are of the form C = ⋂_{n≥n_{0}} π_{n}^{−1}(C_{n}) for closed sets C_{n} in comproj^{\underline{d}^{n}} and an n_{0} ∈ ℤ. Note that comproj\underline{d} is the projective limit of the varieties comproj^{\underline{d}^{n}} in the category of topological spaces.

The group

G := ∏_{i∈ℤ}Stab_{Gl_{α(d_{i})}}(⨁_{j=1}^{l} P_{j}^{d_{i}^{j}}) ≅ ∏_{i∈ℤ}Aut_{A}(⨁_{j=1}^{l} P_{j}^{d_{i}^{j}})

acts on the space comproj\underline{d} by conjugation and the orbits correspond to isomorphism classes of complexes of projective A-modules. The action of G on comproj\underline{d} induces naturally an action of G on comproj^{\underline{d}^{n}} for all n such that π_{n} and φ_{n} are G-equivariant maps.

We see that G is a connected algebraic group if \underline{d} is bounded since the endomorphism ring is a linear space, hence irreducible, and the automorphism group is an open dense subvariety. Moreover, the action of G is the action of a connected algebraic group on an affine variety if \underline{d} is bounded.

The following lemma is well known to the experts, but we could not find a reference and include a proof below. We do not require the field to be algebraically closed for the remainder of this section.

Lemma 1. Let X = (⨁_{i∈ℤ} Q_{i}, ⨁_{i∈ℤ} ∂_{i}^{X}) and Y = (⨁_{i∈ℤ} Q_{i}, ⨁_{i∈ℤ} ∂_{i}^{Y}) be two right bounded complexes of projective A-modules with the same homogeneous components Q_{i} in each degree i ∈ ℤ. Then, X is isomorphic to Y if and only if X is homotopy equivalent to Y.

Proof. If X is isomorphic to Y in the category of complexes, then clearly X is homotopy equivalent to Y. So, suppose that X is homotopy equivalent to Y, that is there is a mapping of complexes φ : X ⟶ Y and a mapping ψ : Y ⟶ X of complexes so that there is a map h of degree 1 so that φψ − id_{X} = h∂^{X} + ∂^{X}h and likewise there is an h′ with ψφ − id_{Y} = h′∂^{Y} + ∂^{Y} h′. We shall show that X ≃ X′ ⊕ N_{X} where im(∂^{X}|_{X′}) ⊆ rad(X′) and N_{X} is contractible, and likewise for Y.

Suppose for the moment that this is shown. Then, since N_{X} and N_{Y} are contractible, we get X′ and Y′ are quasi-isomorphic, and therefore, since both are right bounded complexes of projective modules, homotopy equivalent. Once we can show that then X′ and Y′ are isomorphic as complexes, then also N_{X} and N_{Y} are isomorphic. Indeed, N_{X} and N_{Y} are isomorphic as graded modules. Now, since N_{X} and N_{Y} are contractible, they are both isomorphic to a direct sum of copies of shifted copies of complexes of the form

… ⟶ 0 ⟶ M ⟶^{≃} M ⟶ 0 ⟶ ….

Comparing the direct factors, and using that N_{X} and N_{Y} are isomorphic as graded modules, one sees that N_{X} ≃ N_{Y} as complexes. So, we suppose for the moment in the statement of the lemma that im(∂^{X}) ⊆ rad(X). But then, φψ − id_{X} = h∂^{X} + ∂^{X}h, and therefore, (φψ−id_{X})(X_{i}) ⊆ rad(X_{i}) for any degree i. Nakayama’s Lemma implies that φψ is invertible. Likewise ψφ is invertible. Hence, φ is an isomorphism.

We need to show that X ≃ X′ ⊕ N_{X} for a contractible N_{X} and a complex X′ with im(∂^{X}|_{X′}) ⊆ rad(X′). Since ∂^{X}(rad(X_{i})) ⊆ rad(X_{i−1}), the complex X induces a complex (X/rad(X), \overline{∂}^{X}) of semisimple modules. Let m be the smallest degree such that X_{m−1} is non-zero. Denote \overline{X} := X/rad(X). If

0 ≠ \overline{∂}_{m}^{X} : \overline{X_{m}} ⟶ \overline{X}_{m−1}, then \overline{X}_{m−1} ≃ \overline{X}′_{m−1} ⊕ \overline{X}′′_{m−1}

so that \overline{∂}_{m}^{X} : \overline{X}_{m} ⟶ \overline{X}′′_{m−1} is surjective. But then, ∂_{m}^{X} is also surjective onto the projective cover X′′_{m−1} of \overline{X}′′_{m−1}. Since X′′_{m−1} is projective, there is a splitting σ_{m−1} : X′′_{m−1} ⟶ X_{m} of ∂_{m}^{X} and therefore, X_{m} ≃ X′_{m} ⊕ X′′_{m−1} and X_{m−1} ≃ X′_{m−1} ⊕ X′′_{m−1} so that ∂_{m}^{X} is transformed

by these isomorphisms into ( \begin{matrix}∂_{m}^{X}′ & 0 \\ 0 & id_{X′′_{m−1}} \end{matrix} ). Now, by induction define N_{X} := X′′ and

one gets X ≃ X′ ⊕ N_{X} and ∂^{X}|_{X′} induces the 0-mapping modulo the radical. This is tantamount to saying that im(∂^{X}|_{X′}) ⊆ rad(X′).

As a consequence of the lemma we see that the orbits in comproj\underline{d} under the action of G correspond to homotopy equivalence classes, or equivalently quasi-isomorphism classes, of right bounded complexes of projective modules with fixed dimension array \underline{d}. Note however that \underline{d} is not preserved under quasi-isomorphism.

Lemma 2. Let M and N be right bounded complexes of finitely generated projective modules. Then, there is a dimension array \underline{d} and homotopy equivalent complexes M ≃ M′ and N ≃ N′ so that M′, N′ ∈ comproj\underline{d}.

Proof. Let n be the smallest degree such that the homogeneous component of M or N is nonzero.

M′_{m} := ⨁_{j=n}^{m} M_{j} ⊕ ⨁_{j=n}^{m} N_{j} and N′_{m} := ⨁_{j=n}^{m} N_{j} ⊕ ⨁_{j=n}^{m} M_{j}

where the differential d_{M′} is chosen to be d_{M} on M_{m}, and the differential d_{N′} is chosen to be d_{N} on N_{m}. Moreover,

d_{M′}|_{M_{k}} = \begin{cases}id & if m − k > 0 is even \\ 0 & else \end{cases} whereas d_{M′}|_{N_{k}} = \begin{cases}id & if m − k is odd \\ 0 & else \end{cases}

Define the differential on N′ likewise, and get this way that M′_{m} ≃ N′_{m} for all m, and also M ≃ M′ as well as N ≃ N′.

We define for a complex X the complex X[1] shifted by one degree to the left by (X[1])_{m} := X_{m−1} and ∂_{m}^{X[1]} := −∂_{m−1}^{X}.

2. Algebraic relation implies topological relation

Let A be an algebra over an algebraically closed field k, let D^{−}(A) be the derived category of right bounded complexes of finitely generated A-modules, and let D^{b}(A) be its full subcategory formed by bounded complexes of A-modules. Let K^{−}(A) be the homotopy category of right bounded complexes of finitely generated projective A-modules and K^{−,b}(A) the image of D^{b}(A) in K^{−}(A) under the equivalence K^{−}(A) ≃ D^{−}(A). Concerning conventions for derived categories we shall follow [7].

For any X and Y in D^{−}(A) let X ≤_{Δ} Y if there is a distinguished triangle

Y ⟶ X ⊕ Z ⟶ Z ⟶ Y [1]

for an object Z in D^{−}(A).

On the topological side we define a relation ≤_{top} by

X ≤_{top} Y :⇔ Y ∈ \overline{G · X}

for X, Y ∈ comproj\underline{d}.

We denote by \underline{dim}(X) the dimension array of a complex X ∈ K^{−}(A).

Observe that if X and Y are in D^{b}(A), then X ≤_{Δ} Y implies [X] = [Y] in K_{0}(D^{b}(A)).

Theorem 1. Let M and N be right bounded complexes of finitely generated projective modules with the same dimension array \underline{d}. Then, M ≤_{Δ} N implies M ≤_{top} N in comproj\underline{d}.

Proof. Let U be a subset of comproj\underline{d}. We show that \overline{U} = ⋂_{n} π_{n}^{−1}(\overline{π_{n}(U)}). The inclusion ⊆ is obvious. Let C be a closed subset of comproj\underline{d} with U ⊆ C. Then C = ⋂_{n} π_{n}^{−1}(C_{n}) for closed subsets C_{n} ⊆ comproj^{\underline{d}^{n}}. Hence U ⊆ π_{n}^{−1}(C_{n}) and so \overline{π_{n}(U)} ⊆ C_{n} for every n, which proves the other inclusion.

Now, if one can prove that whenever M ≤_{Δ} N, then π_{n}(M) ≤_{Δ} π_{n}(N), and moreover, if this implies that π_{n}(N) ∈ \overline{G · π_{n}(M)}, then by the above, N ∈ \overline{G · M}. This means, that M degenerates to N in the topological sense.

We still have to show that if M ≤_{Δ} N then π_{n}(N) ∈ \overline{G · π_{n}(M)}. We shall use the very same proof as in the module case by Riedtmann [10]. Let M ≤_{Δ} N. Then, there is a complex Z of projective modules so that

N ⟶ M ⊕ Z ⟶ Z ⟶ N[1]

is a distinguished triangle. This implies that

Z[−1] ⟶ N ⟶ M ⊕ Z ⟶ (Z[−1])[1]

is a distinguished triangle. Hence, M ⊕ Z ≃ cone(Z[−1] ⟶ N) in the homotopy category. Now, we use that the dimension array of N and of M coincide. Indeed,

\underline{dim}(cone(Z[−1] ⟶ N)) = \underline{dim}(Z) + \underline{dim}(N) = \underline{dim}(Z) + \underline{dim}(M) = \underline{dim}(M ⊕ Z) .

Hence, cone(Z[−1] ⟶ N) ≃ M ⊕Z in the category of complexes and so there is a sequence

0 ⟶ N ⟶^{(ϕ,α)} Z ⊕ M \underrightarrow{\binom{β}{ψ}} Z ⟶ 0

which is exact in the category of complexes. This shows at once that M ≤_{Δ} N implies π_{n}(M) ≤_{Δ} π_{n}(N) for any n.

The first assertion is that β is invertible if and only if α is invertible and in this case,

0 ⟶ N ⟶^{(ϕ,α)} Z ⊕ M \underrightarrow{\binom{β}{ψ}} Z ⟶ 0

is isomorphic to

0 ⟶ N ⟶^{(0,α)} Z ⊕ M \underrightarrow{\binom{β}{0}} Z ⟶ 0

and therefore N ≃ M. Indeed, we get an isomorphism of exact sequences

0 ⟶ N ⟶^{(ϕ,α)}  Z ⊕ M  \underrightarrow{\binom{β}{ψ}} Z ⟶ 0
∥  ↓ ( \begin{matrix}id_{Z} & 0 \\ ψβ^{−1} & id_{M} \end{matrix} )  ∥
0 ⟶ N ⟶^{(0,α)}  Z ⊕ M  \underrightarrow{\binom{β}{0}} Z ⟶ 0

and likewise for α invertible.

For any t ∈ k we have a homomorphism of complexes \binom{β+t·id_{Z}}{ψ}. Let

N_{t} := ker(\binom{β + t · id_{Z}}{ψ})

in the category of complexes. For any t with f_{t} := \binom{β+t·id_{Z}}{ψ} being surjective we have that f_{t} is locally split. Here we call a homomorphism of complexes g locally split if g is split in each degree, but not necessarily split as a homomorphism of complexes. For all such t we see that N_{t} is a complex of projective modules with the same dimension array as N. We now consider π_{n}(N_{t}), π_{n}(N), π_{n}(M), π_{n}(Z) and the induced mappings on the truncated complexes. Of course, we still have ker(π_{n}(f_{t})) = π_{n}(N_{t}).

We shall prove that

t ↦ π_{n}(N_{t}) ∈ comproj^{\underline{dim}π_{n}(N)}

is a rational morphism of varieties, imitating Christine Riedtmann’s proof in [10].

There is an open neighborhood U of 0 in k so that π_{n}(f_{t}) is surjective for all t ∈ U, using that being surjective is an open condition and that π_{n}(f_{0}) is surjective.

Let

(B, ∂^{B}) ⟶^{f} (A, ∂^{A}) ⟶ 0

be a surjective map of complexes of projective modules. We want to compute the kernel (C, ∂^{C}) of this map. Since the structure of C as graded module is clear, we may choose bases in B so that we can identify B with C ⊕ A as graded modules. Let g = (g_{C}, g_{A}) : C ⟶ B be the inclusion of the kernel. We have f = \binom{f_{C}}{f_{A}} where f_{A} is an isomorphism. Then g_{C} is an isomorphism as well and we may assume that g_{C} = id_{C}. But then,

g_{A} = −f_{C}f_{A}^{−1}

since f_{A} is invertible. The differential on ker(f) depending on f is

∂_{C} = (id_{C}, −f_{C}f_{A}^{−1}) · ∂_{B} · \binom{id_{C}}{0} .

Thus we get a rational morphism of varieties Hom(B, A) ⟶ comproj^{\underline{dim}(C)} defined on the open neighborhood of f ∈ Hom(B, A) for which f_{A} is an isomorphism.

We may now apply this construction to the map f_{0} and by composing with the map

t ↦ \binom{β + t · id_{Z}}{ψ}

we get the promised rational morphism of varieties.

Finally, for those t for which π_{n}(β +t·id_{Z}) is an isomorphism, that is for all but the finite number of eigenvalues of −β, we get π_{n}(N_{t}) ≃ π_{n}(M) and for t = 0 we get π_{n}(N_{0}) ≃ π_{n}(N). Therefore

π_{n}(N) ∈ \overline{G · π_{n}(M)}.

3. Geometric relation implies algebraic relation

We shall prove in this section that under some conditions the inverse implication of Theorem 1 is true as well.

Let \underline{d} = (d_{n}, …, d_{m}) be a bounded dimension array. We associate to the affine variety comproj\underline{d}(k) an affine k-scheme comproj\underline{d}(−). This k-scheme has the following functorial description. Let R be a commutative k-algebra. Let comproj\underline{d}(R) denote the subset of

∏_{i∈ℤ}Hom_{R}(R^{α(d_{i})}, R^{α(d_{i−1})})

consisting of elements (∂_{i})_{i∈ℤ} with the properties that ∂_{i} is an R ⊗_{k} A-homomorphism when viewed as a map from ⨁_{j=1}^{l}(R⊗_{k}P_{j}^{d_{i}^{j}}) to ⨁_{j=1}^{l}(R⊗_{k}P_{j}^{d_{i−1}^{j}}) and ∂_{i}∂_{i−1} = 0. For a k-algebra homomorphism f : S ⟶ R there is naturally a corresponding map f^{∗} : comproj\underline{d}(S) ⟶ comproj\underline{d}(R) sending a tuple of matrices (∂_{i}) to the tuple (f(∂_{i})). Similarly we may associate to the affine algebraic group G a smooth affine group scheme G(−) over k. The action of G on comproj\underline{d} extends to an action of G(−) on comproj\underline{d}(−).

We may verify Grunewald-O’Halloran’s conditions which are necessary to apply [5, Theorem 1.2].

Theorem 2. Let \underline{d} be a dimension array. Let M, N ∈ comproj\underline{d} be two complexes with bounded homology. If M ≤_{top} N then M ≤_{Δ} N.

Proof. Let n be an integer such that the homology of M and N vanishes in all degrees larger or equal to n. We will construct a short exact sequence of complexes

0 ⟶ π_{n}(N) ⟶ π_{n}(M) ⊕ Z_{(n)} ⟶ Z_{(n)} ⟶ 0

where Z_{(n)} is a complex of projective A-modules. We are going to follow the steps of Zwara’s proof for the module case.

By Grunewald-O’Halloran’s result [5, Theorem 1.2] there is a discrete valuation k-algebra R with maximal ideal 𝔪 and residue field k and with over k finitely generated quotient field K of transcendence degree 1 and a complex Y in comproj^{\underline{d}^{n}}(R) so that k ⊗_{R} Y = π_{n}(N) and as complexes of K ⊗_{R} A-modules, K ⊗_{R} Y = g · (K ⊗_{k} π_{n}(M)) for a g ∈ G(K). Since the valuation on R is discrete, 𝔪 is principal, generated by an element f.

Since \underline{d}_{n} is bounded there is a non-zero element z ∈ R so that zg is a tuple of matrices with entries in R. Using the explicit definition of the action we get,

K ⊗_{R} Y = g · (K ⊗_{k} π_{n}(M)) = zg · (K ⊗_{k} π_{n}(M)).

So, we may assume that g is a tuple of matrices with entries in R. Restricting the multiplication with g to R ⊗_{k} π_{n}(M) gives a morphism of complexes of R ⊗_{k} A-modules φ : R ⊗_{k} π_{n}(M) ⟶ Y. Let X denote the image of this morphism. Both X and Y are complexes of free R-modules, with equal rank in all degrees, therefore there exists some s such that 𝔪^{s}Y ⊆ X.

Now we take the point of view that the complexes X and Y are graded R ⊗_{k} A-modules with differentials. Fix a k-basis B of R. As complexes of A-modules we have

X = ⨁_{b∈B} X_{b}

where X_{b} = φ(< b > ⊗_{k}π_{n}(M)) ≅ π_{n}(M) and where < b > denotes the k-subspace of R generated by b.

For each h we have a short exact sequence of complexes

0 ⟶ Y/𝔪Y ⟶ Y/𝔪^{h+1}Y ⟶ Y/𝔪^{h}Y ⟶ 0

We will show that there exists an h such that Y/𝔪^{h+1}Y ≃ π_{n}(M) ⊕ (Y/𝔪^{h}Y ) as complexes of A-modules where the mapping (Y/𝔪Y) ≃ (𝔪^{h}Y/𝔪^{h+1}Y ) ⟶ (Y/𝔪^{h+1}Y ) is induced by multiplication by f^{h} and canonical inclusion. Let V = ⊕_{i}V_{i} be a graded vector space formed by taking vector space complements of X_{i} in Y_{i} in each degree i. Note that V is a finite dimensional vector space since Y is bounded and 𝔪^{s}Y ⊆ X. Let Z_{0} be the smallest A-subcomplex of Y containing V. Then Z_{0} is a finite dimensional complex of A-modules, since Y is bounded. Now Y = X + Z_{0} as complexes of A-modules. Let V be a finite subset of B such that Z_{0} ∩ ⨁_{b∈V} X_{b} = Z_{0} ∩ X. Such a subset exists since Z_{0} is finite dimensional over k. Let Z_{1} = Z_{0} + ⨁_{b∈V} X_{b}. Then Y = Z_{1} ⊕ ⨁_{b∉V} X_{b}. Since V is finite there exists an integer t such that 𝔪^{t+1}X ∩ ⨁_{b∈V} X_{b} = 0. Thus there is a finite subset W of B such that

𝔪^{t+1}X ⊕ ⨁_{b∈W} X_{b} ⊕ ⨁_{b∈V} X_{b} = X

Let Z_{2} = Z_{1} + ⨁_{b∈W} X_{b}. Then Y = 𝔪^{t+1}X ⊕ Z_{2}.

It follows that we have a chain of inclusions

𝔪^{s+t+2}Y ⊆ 𝔪^{t+2}X ⊆ 𝔪^{t+1}X ⊆ Y

where the last two inclusions have direct complements as complexes of A-modules. Thus

Y/𝔪^{s+t+2}Y ≅ (𝔪^{t+2}X/𝔪^{s+t+2}Y) ⊕ π_{n}(M) ⊕ (Y/𝔪^{t+1}X) ≃ π_{n}(M) ⊕ (Y/𝔪^{s+t+1}Y)

where the last isomorphism follows since

𝔪^{t+2}X/𝔪^{s+t+2}Y ≃ 𝔪^{t+1}X/𝔪^{s+t+1}Y and 𝔪^{t+1}X/𝔪^{s+t+1}Y ⊕ Y/𝔪^{t+1}X ≃ Y/𝔪^{s+t+1}Y.

Now since Y/𝔪Y ≅ π_{n}(N) we get the promised short exact sequence of complexes 0 ⟶ π_{n}(N) ⟶ π_{n}(M) ⊕ Z_{(n)} ⟶ Z_{(n)} ⟶ 0 by choosing Z_{(n)} = Y/𝔪^{s+t+1}Y.

Now construct a complex N′ by splicing π_{n}(N) with a projective resolution P_{N} of H_{n}(π_{n}(N)). Similarly we form a complex Z by splicing a projective resolutions P_{Z} of H_{n}(Z_{(n)}) with Z_{(n)}. By the horseshoe lemma there exists a short exact sequence 0 ⟶ P_{N} ⟶ P_{M⊕Z} ⟶ P_{Z} ⟶ 0 of projective resolutions where P_{M⊕Z} ≃ P_{N} ⊕ P_{Z} as graded modules and where P_{M⊕Z} is a projective resolution of H_{n}(π_{n}(M) ⊕ Z_{(n)}) ≅ H_{n}(π_{n}(M)) ⊕ H_{n}(Z_{(n)}). Moreover we have a short exact sequence of complexes

0 ⟶ N′ ⟶ M′ ⟶ Z ⟶ 0

where the complex M′ is formed by splicing P_{M⊕Z} with the complex π_{n}(M) ⊕ Z_{(n)}. Now N′, M′ are homotopy equivalent to N, M ⊕ Z, respectively. Thus we get a triangle N ⟶ M ⊕ Z ⟶ Z ⟶ N[1], which completes the proof of the theorem.

4. Consequences for the geometry of complexes

We continue with some consequences and observations on comproj\underline{d} and the orders ≤_{Δ} and ≤_{top}.

Example 4.1. We consider the quiver Q defined by •_{1} ⟶ •_{2}. Then, up to isomorphism, there are 3 indecomposable kQ-modules, the indecomposable projective module P_{1} corresponding to the vertex 1 and the two simple modules S_{1} and P_{2}. Moreover, in the representation variety mod(kQ, (1, 1)) of 2-dimensional kQ-modules with two different composition factors, one has that the projective module with top 1 degenerates to the direct sum of the two simple modules. The projective indecomposable module with top 1 can be considered as being in comproj^{(\binom{0}{0},\binom{1}{0})}, where \binom{a}{b} indicates that in a certain degree the module is P_{1}^{a} ⊕P_{2}^{b}. The semi-simple module S_{1} ⊕ S_{2} is in comproj^{(\binom{0}{1},\binom{1}{1})}. So, the modules are represented in different varieties comproj\underline{d} and here it is not possible to consider degenerations between them if one declares that a complex X degenerates to a complex Y if Y is in the closure of the orbit of X. Nevertheless, one may consider another non minimal projective resolution of P_{1} as

P_{2} ⟶^{(id_{P_{2}},0)} P_{2} ⊕ P_{1}.

This complex can be seen as being in comproj^{(\binom{0}{1},\binom{1}{1})}, and the minimal projective resolution of S_{1} ⊕ P_{2} is

P_{2} ⟶^{(0,ι)} P_{2} ⊕ P_{1}.

for ι being the embedding P_{2} ⟶ P_{1}. Therefore, P_{1} and S_{1} ⊕ P_{2} can be both visualized in comproj^{(\binom{0}{1},\binom{1}{1})}. Moreover it is easy to see that P_{1} ≤_{top} S_{1} ⊕ P_{2}. This observation is one of the motivations not to ask for the complexes to be minimal as is done in [1] and to allow zero homotopic direct summands.

Let M and N be d-dimensional A-modules. We write M ≤ N if M degenerates to N in mod(A, d).

Proposition 3. Let M, N ∈ mod(A, d) for some dimension d and let P_{M}, P_{N} ∈ comproj\underline{d} for some dimension array \underline{d} be a projective resolution of M and N, respectively. Then, M ≤ N in mod(A, d) if and only if P_{M} ≤_{top} P_{N} in comproj\underline{d}.

Proof. If M ≤ N, then by Zwara’s theorem [13] there is an exact sequence

0 ⟶ N ⟶ Z ⊕ M ⟶ Z ⟶ 0

for an A-module Z. This implies a distinguished triangle

P_{N} ⟶ P_{Z} ⊕ P_{M} ⟶ P_{Z} ⟶ P_{N}[1]

in K^{−}(A) where P_{Z} is a projective resolution of Z. Hence, P_{M} ≤_{Δ} P_{N} and so by Theorem 1 we have P_{M} ≤_{top} P_{N}.

Conversely suppose P_{M} ≤_{top} P_{N} and so by Theorem 2 we have P_{M} ≤_{Δ} P_{N}. Then, there is a complex Z and a distinguished triangle

P_{N} ⟶ Z ⊕ P_{M} ⟶ Z ⟶ P_{N}[1]

Taking homology of this triangle gives a long exact sequence

⟶ H_{i+1}(Z) ⟶ H_{i}(P_{N}) ⟶ H_{i}(Z) ⊕ H_{i}(P_{M}) ⟶ H_{i}(Z) ⟶ H_{i−1}(P_{N}) ⟶

where H_{i}(P_{N}) = H_{i}(P_{M}) = 0 for i > 0. For i = 0 one gets an exact sequence

0 ⟶ H_{1}(Z) ⟶ H_{1}(Z) ⟶ N ⟶ H_{0}(Z) ⊕ M ⟶ H_{0}(Z) ⟶ 0 .

This implies that

0 ⟶ N ⟶ H_{0}(Z) ⊕ M ⟶ H_{0}(Z) ⟶ 0

is a short exact sequence and hence M ≤ N in mod(A, d), again by Zwara’s theorem [13]. This proves the statement.

Lemma 4. Let \underline{d} be any dimension array and let T be an element in comproj\underline{d} so that G·T is open. Then, T is a minimal element for ≤_{Δ} and for ≤_{top}.

Proof. If G · T is open, then comproj\underline{d} \ \{G · T \} is closed and for any X ≄ T one has that

\overline{G · X} ⊆ comproj\underline{d} \ \{G · T \}.

Hence, T is minimal with respect to ≤_{top}, and since ≤_{Δ} implies ≤_{top}, the complex T is minimal also with respect to ≤_{Δ}.

Observe that we only used the topology of the space in the previous argument. We shall see that for bounded \underline{d} the orbits of T with Hom_{D^{b}(A)}(T, T [1]) = 0 are open.

Lemma 5. Let \underline{d} be a bounded dimension array and let X be a complex in comproj\underline{d}. If Hom_{D^{b}(A)}(X, X[1]) = 0 then G · X is open in comproj\underline{d}

Proof. First assume that \underline{d} is a bounded dimension array. From Theorem 7 in [11] we see that the orbit of X in comp(A, \underline{α}(\underline{d})) is open if Hom_{D^{b}(A)}(X, X[1]) = 0. The result now follows since comproj\underline{d} is a subvariety of comp(A, \underline{α}(\underline{d})) and since G · X = comproj\underline{d} ∩ (Gl_{\underline{α}(\underline{d})} · X).

Lemma 6. The relation ≤_{top} is a partial order on the set of isomorphism classes of complexes with bounded homology with fixed dimension array \underline{d}.

Proof. If N ∈ \overline{G · M} and M ∈ \overline{G · L}, then clearly N ∈ \overline{G · L}. If N ∈ \overline{G · M} and M ∈ \overline{G · N}, then by the proof of Theorem 1 we get π_{n}(N) ∈ \overline{G · π_{n}(M)} and π_{n}(M) ∈ \overline{G · π_{n}(N)} for all n ∈ ℤ. This implies π_{n}(N) ≃ π_{n}(M) for all n ∈ ℤ

We show that whenever X is a complex with bounded homology in comproj\underline{d}, then denoting by m an integer so that the homology of X is 0 in all degrees higher than m, then Y ∈ G · X if and only if Y ∈ π_{ℓ}^{−1}(G · π_{ℓ}(X)) for all ℓ ≥ m + 1. Indeed, assume that Y ∈ π_{ℓ}^{−1}(G · π_{ℓ}(X)) for all ℓ ≥ m + 1. Then we have an isomorphism of homology groups H_{n}(Y) ≃ H_{n}(X) for all n, which shows that H_{n}(Y) = 0 for all n ≥ m + 1. Then there is an isomorphism π_{m+1}(Y) ≃ π_{m+1}(X), which lifts to a homotopy equivalence Y ≃ X and so Y ∈ G · X. The reverse implication is trivial.

Hence, one has N ≃ M.

Remark 4.2. [11, Theorem 7] cited in the proof of Lemma 5 shows that the tangent space of the variety of complexes comp(A, \underline{α}) at some point X modulo the tangent space of the orbit of X under the group which is acting at X is isomorphic to Hom_{D^{b}(A)}(X, X[1]). A similar result can be proven for comproj\underline{d} and the action of our smaller group. We also mention that Lemma 5 has a converse in the case where \underline{d} is bounded. Namely, if G · X is open in comproj\underline{d} then Hom_{D^{b}(A)}(X, X[1]) = 0. This can again be seen from [11, Theorem 7].

Corollary 7. Complexes with Hom_{D^{b}(A)}(X, X[1]) = 0 are minimal with respect to both ≤_{Δ} and ≤_{top}. In particular partial tilting complexes are minimal with respect to both orders.

We also give a consequence which doesn’t require an algebraically closed field.

Corollary 8. Let A be an algebra over a field K. Then, up to homotopy equivalence there is at most one two-term partial tilting complex

T = … ⟶ 0 ⟶ P_{1} ⟶ P_{0} ⟶ 0 ⟶ …

with fixed homogeneous components P_{0} and P_{1}.

Proof. Since two-term complexes of projective modules are entirely determined by their homology, and since for any field extension L of K one has H(L ⊗_{K} X) ≃ L ⊗_{K} H(X) for any complex X, we may assume that K is algebraically closed. Let α_{i} = \underline{dim}(P_{i}) for i ∈ \{0, 1\} and \underline{α} := (α_{1}, α_{0}). The variety comproj^{\underline{d}(\underline{α})} is an affine space, and therefore irreducible as algebraic variety. Moreover, since T is a partial tilting complex, the orbit G · T is open in comproj^{\underline{d}(\underline{α})}. Therefore, G · T is dense. Let S be another partial tilting complex in comproj^{\underline{d}(\underline{α})}. Also G · S is open and dense, and therefore S ≤_{top} T as well as T ≤_{top} S. Hence, S ≃ T by Lemma 6.

Example 4.3. Corollary 8 does not hold for general dimension arrays. Let A be given by the quiver

[figure: •_{1}; •_{2}; α; β]

with relations αβα = βαβ = 0. For this algebra take the indecomposable complex (unique up to isomorphism so that P_{2} is in degree 0)

T_{1} := … ⟶ 0 ⟶ P_{1} ⟶ P_{1} ⟶ P_{2} ⟶ 0 ⟶ …

and T_{2} := P_{1} ⟶ P_{2}. Then, T := T_{1} ⊕ T_{2} is a tilting complex. Let \underline{d} be the dimension array of T. The complex S

P_{1} ⟶^{(0,id)} P_{1} ⊕ P_{1} \underrightarrow{( \begin{matrix}α & 0 \\ 0 & 0 \end{matrix} )} P_{2} ⊕ P_{2}

is homotopy equivalent to the tilting complex S′

P_{1} ⟶^{(α,0)} P_{2} ⊕ P_{2}.

Here both tilting complexes T and S have the same dimension array, but are not isomorphic, and therefore belonging to different irreducible components of comproj\underline{d}. Using [8] and a slightly more detailed examination of comproj\underline{d} one observes that comproj\underline{d} has exactly two irreducible components.

The complex T_{1} ⊕ P_{1}[2] is a tilting complex as well and denote by \underline{e} the dimension array of T_{1} ⊕ P_{1}[2]. A short examination yields that comproj\underline{e} has two irreducible components, one C_{3} of dimension 3 and another component C_{4} of dimension 4. The orbit of T_{1} ⊕ P_{1}[2] is open in C_{3}, whereas the complexes corresponding to the points in C_{4} are not partial tilting complexes. Observe however, in C_{4} there is an open orbit of a complex U ≃ P_{1}[2] ⊕ P_{2} with Hom_{D^{b}(A)}(U, U[1]) = 0 ≠ Hom_{D^{b}(A)}(U, U[2]).

Remark 4.4. Observe that a tilting complex T over A is the image F (B) of an equivalence F : D^{b}(B) ⟶ D^{b}(A) of triangulated categories. By the Rickard’s and Keller’s main theorem [9, 6] there is a so-called twosided tilting complex X of A⊗_{K} B^{op}-modules which are projective on the left and on the right, so that X ⊗_{B}^{𝕃} − is an equivalence. For any dimension array \underline{d} let X ⊗ \underline{d} be the dimension array which is obtained by tensoring a complex with dimension array \underline{d} by X, and taking the total complex of the resulting bi-complex. Then, by definition X ⊗_{B} − induces a morphism of varieties

comproj(X) : comproj\underline{d} ; B ⟶ comproj_{A}^{X⊗\underline{d}} .

It should be an interesting question to study the image of this morphism inside comproj_{A}^{X⊗\underline{d}}. Note that studying varieties using functors is already far from trivial in the module case (see Bongartz [2] and Zwara [14]).

There is another consequence of these statements. Indeed, define for any two complexes X and Y in K^{−,b}(A)

X ≤_{Hom} Y :⇔ ∀U ∈ D^{b}(A) : dim_{k}(Hom_{D^{b}(A)}(U, X)) ≤ dim_{k}(Hom_{D^{b}(A)}(U, Y))

Lemma 9. Let X and Y be two complexes in comproj\underline{d} for bounded dimension array \underline{d}. Then, X ≤_{top} Y ⇒ X ≤_{Hom} Y.

Proof. Define for any two complexes X and Y with appropriate bounded dimension array \underline{d} and \underline{e} the mapping

φ_{X,Y} : ∏_{i∈ℤ}Hom_{A}(X_{i}, Y_{i+1}) ⟶ Hom_{C^{b}(A)}(X, Y)

by φ_{X,Y} (f) := ∂_{X}f + f∂_{Y} . It is clear that this image is exactly the set of 0-homotopic homomorphisms. Hence, we have that

dim_{k}(Hom_{D^{b}(A)}(X, Y)) = dim_{k}(Hom_{C^{b}(A)}(X, Y)) − dim_{k}(im(φ_{X,Y})) .

We use the argument from [4, §3, Theorem 2, special case] to show that

\{U\} × comproj\underline{d} ⟶ ℕ

given by (U, X) ↦ dim_{k}(Hom_{D^{b}(A)}(U, X)) is upper semi-continuous. Then, setting n = dim_{k}(Hom_{D^{b}(A)}(U, X)), one gets {Z|dim_{k}(Hom_{D^{b}(A)}(U, Z)) ≥ n} is closed, and if Y ∈ \overline{G · X}, then Y ∈ \{Z|dim_{k}(Hom_{D^{b}(A)}(U, Z)) ≥ n\}. Hence,

dim_{k}(Hom_{D^{b}(A)}(U, Y)) ≥ dim_{k}(Hom_{D^{b}(A)}(U, X)).

This proves the statement.

Acknowledgment: We would like to thank the referee for helpful remarks.

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Bernt Tore Jensen, Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, UK

   E-mail address: bjensen@maths.leeds.ac.uk

Xiuping Su, Alexander Zimmermann Université de Picardie, Faculté de Mathématiques et LAMFA (UMR 6140 du CNRS), 33 rue St Leu, F-80039 Amiens Cedex 1, France

   E-mail address: xiuping.su@u-picardie.fr, alexander.zimmermann@u-picardie.fr