Derived tame and derived wild algebras

We prove that every finite dimensional algebra over an algebraically closed field is either derived tame or derived wild. We also prove that any deformation of a derived wild algebra is derived wild.

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Дата:	2004
Автор:	Drozd, Y.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2004
Назва видання:	Algebra and Discrete Mathematics
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Цитувати:	Derived tame and derived wild algebras / Y.A. Drozd // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 1. — С. 57–74. — Бібліогр.: 17 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1559502025-02-09T14:26:27Z Derived tame and derived wild algebras Drozd, Y.A. We prove that every finite dimensional algebra over an algebraically closed field is either derived tame or derived wild. We also prove that any deformation of a derived wild algebra is derived wild. 2004 Article Derived tame and derived wild algebras / Y.A. Drozd // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 1. — С. 57–74. — Бібліогр.: 17 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16G60; 15A21, 16D90, 16E05. https://nasplib.isofts.kiev.ua/handle/123456789/155950 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We prove that every finite dimensional algebra over an algebraically closed field is either derived tame or derived wild. We also prove that any deformation of a derived wild algebra is derived wild.
format	Article
author	Drozd, Y.A.
spellingShingle	Drozd, Y.A. Derived tame and derived wild algebras Algebra and Discrete Mathematics
author_facet	Drozd, Y.A.
author_sort	Drozd, Y.A.
title	Derived tame and derived wild algebras
title_short	Derived tame and derived wild algebras
title_full	Derived tame and derived wild algebras
title_fullStr	Derived tame and derived wild algebras
title_full_unstemmed	Derived tame and derived wild algebras
title_sort	derived tame and derived wild algebras
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2004
url	https://nasplib.isofts.kiev.ua/handle/123456789/155950
citation_txt	Derived tame and derived wild algebras / Y.A. Drozd // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 1. — С. 57–74. — Бібліогр.: 17 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT drozdya derivedtameandderivedwildalgebras
first_indexed	2025-11-26T20:28:37Z
last_indexed	2025-11-26T20:28:37Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 1. (2004). pp. 57 – 74 c© Journal “Algebra and Discrete Mathematics” Derived tame and derived wild algebras Yuriy A. Drozd Abstract. We prove that every finite dimensional algebra over an algebraically closed field is either derived tame or derived wild. We also prove that any deformation of a derived wild algebra is derived wild. Introduction This is a talk given by the author at the IV International Algebraic Conference in Ukraine (Lviv, August 2003). It is devoted to the notions of derived tameness and wildness of algebras and the recent progress in the topic. The main result, obtained by Viktor Bekkert and the author, is the tame-wild dichotomy for derived categories, which is an analogue of the author’s theorem for algebras [5]. The proof is also very much alike that of [5]; it relies on the technique of matrix problems (representations of boxes) and a reduction algorithm for matrices. The principal distinction is that here we have to consider non-free (even non-semi-free) boxes. Fortunately, the required class of boxes can be dealt with in a similar way; it is considered in Section 3 of this paper. Section 1 is devoted to the general notions related to derived categories, derived tameness, etc., while Section 2 presents a transition to matrix problems. Further, we consider the results related to deformations and degenerations of derived tame and derived wild algebras obtained by the author [7]. Namely, in Section 4 we construct some “almost versal” families of complexes with projective bases, analogous to the families of modules constructed in [8]. It makes possible to introduce the “number of parameters” defining complexes of given rank and to prove (in Section 5) that this number is upper semi- continuous in flat families of algebras. As a corollary, we get that a 2000 Mathematics Subject Classification: 16G60; 15A21, 16D90, 16E05. Key words and phrases: derived categories, derived tame and derived wild algebras, deformations of algebras, matrix problems, representations of boxes. Jo u rn al A lg eb ra D is cr et e M at h .58 Derived tame and derived wild algebras deformation of a derived tame algebra is derived tame; respectively, a degeneration of a derived wild algebra is derived wild. We also explain the situation that can arise if one considers families that are non-flat (especially the example of Brüstle [2]). Since it is a survey, we sometimes only sketch proofs referring to the papers [1, 7], though we try to give all necessary definitions, especially related to derived categories and boxes. I am grateful to Viktor Bekkert and Igor Burban for fruitful collabora- tion and useful discussions, which were of great influence in preparing this paper. I am also thankful to Birge Huisgen-Zimmermann who inspired me to start these investigations. 1. Derived categories In what follows we consider finite dimensional algebras over an alge- braically closed field k. Let A be such an algebra. As usually, we write ⊗ and Hom instead of ⊗k and Homk; we denote by V ∗ the dual vector space Hom(V,k). All considered categories A are also supposed to be linear categories over the field k, which means that all sets A(a, b) are vector spaces over k and the multiplication of morphisms is k-bilinear. Recall that the derived category D(A) of (finite dimensional) A-modules is defined as follows [13, 14]. First consider the category of complexes C(A). Its objects are sequences M• = (Mn, dn) of finite dimensional modules and their homomorphisms . . . dn+2 −−−−→ Mn+1 dn+1 −−−−→ Mn dn−−−−→ Mn−1 dn−1 −−−−→ . . . (1) such that dn+1dn = 0 for all n. A morphism φ• = (φn) of a complex (1) to another complex M ′ • = (M ′ n, d′n) is a commutative diagram . . . dn+2 −−−−→ Mn+1 dn+1 −−−−→ Mn dn−−−−→ Mn−1 dn−1 −−−−→ . . . φn+1 y φn y φn−1 y . . . d′ n+2 −−−−→ M ′ n+1 d′ n+1 −−−−→ M ′ n d′n−−−−→ M ′ n−1 d′ n−1 −−−−→ . . . (2) One says that such a morphism is homotopic to zero if there are homo- morphisms σn : Mn → M ′ n+1 (n ∈ N) such that φn = σn−1dn + d′n+1σn for all n. The factor category H(A) of C(A) modulo the ideal of mor- phisms homotopic to zero is called the homotopic category of A-modules. For each n the n-th homology of a complex (1) is defined as Hn(M•) = Ker dn/ Im dn+1. Obviously, a morphism φ• of complexes induces ho- momorphisms of homologies Hn(φ•) : Hn(M•) → Hn(M ′ •), and if φ• is Jo u rn al A lg eb ra D is cr et e M at h .Yu. A. Drozd 59 homotopic to zero, it induces zero homomorphisms of homologies. Hence, Hn can be considered as functors H(A) → vec, the category of (finite dimensional) vector spaces. One call a morphism φ• from C(A) or from H(A) quasi-isomorphism if all Hn(φ•) are isomorphisms. Now the de- rived category D(A) is defined as the category of fractions H(A)[Q−1], where Q is the set of all quasi-isomorphisms. One calls a complex (1) right bounded (left bounded, bounded) if there is n0 such that Mn = 0 for n < n0 (respectively, there is n1 such that Mn = 0 for n > n1, or there are both). The corresponding categories are denoted by C−,H−,D− (respectively, by C+,H+,D+, or by Cb,Hb,Db). In this paper we mainly deal with the bounded derived category Db(A). The category A-mod of (finite dimensional) A-modules naturally embeds into D(A) (even in Db(A)): a module M is identified with the complex M• such that M0 = M, Mn = 0 for n 6= 0. Since the category of modules has enough projectives, one can replace, when considering right bounded homotopic or derived category, arbitrary complexes by projective ones, i.e. consisting of projective modules only. We denote by P−(A) and by Pb(A) the categories of right bounded and bounded projective complexes. Actually, D−(A) ≃ H−(A) ≃ P−(A)/Ih, where Ih is the ideal of morphisms homotopic to zero [13, 14]. Moreover, in finite dimensional case every module M has a projective cover, i.e. there is an epimorphism p(M) : P (M) → M , where P (M) is projective, such that Ker p(M) ⊆ radP (M), the radical of P (M) [9]. Therefore, one can only consider minimal complexes, i.e. projective complexes such that Im dn ⊆ radPn−1 for all n. We denote by P− min(A) the category of minimal projective complexes. Thus D−(A) ≃ P− min(A)/Ih. One immediately checks that the image in D−(A) of a morphism φ• of minimal complexes is an isomorphism if and only if φ itself is an isomorphism. Hence, if we are interested in classification of objects from D−(A), we can replace it by P− min(A). Unfortunately, it is no more the case if we consider Db(A). Certainly, if gl.dimA < ∞, one has that Db(A) ≃ Pb min(A)/Ih, since any bounded complex has a bounded projective resolution. On the contrary, if gl.dimA = ∞, it is wrong even for modules. Nevertheless, we propose the following approximation of Db(A), which is enough for classification purpose. (Compare it with [16].) We denote by Pm(A) the category of bounded projective complexes M• such that Mn = 0 for n > m (note that the left bound is not pre- scribed). We say that a morphism φ• in Pm(A) is quasi-homotopic to zero if there are homomorphisms σn : Mn → M ′ n+1 such that φn = σn−1dn + d′n+1σn for all n < m. Let Hm(A) = Pm(A)/Iqh, where Iqh is the ideal of morphisms quasi-homotopic to zero. Especially, if P• is a minimal complex from Pm(A), it is isomorphic in Hm(A) to a minimal Jo u rn al A lg eb ra D is cr et e M at h .60 Derived tame and derived wild algebras complex P̃• such that P̃n = Pn, d̃n = dn for n 6= m, while Pm = P̃m⊕Q so that Q ⊆ Ker dm, d̃m = dm\|P̃m and Ker d̃m ⊆ rad P̃m. There is a natural functor Hm(A) → Hm+1(A), which maps a complex P• to the complex P+ • , where P+ n = Pn, d+ n = dn for n ≤ m, while P+ m+1 = P (Ker dm) and d+ m+1 is the epimorphism p(Ker dm) : P+ m+1 → Ker dm. Conversely, there is a functor Hm+1(A) → Hm(A), which just abandon Pm+1 in a complex P•. One easily verifies the following results. Proposition 1.1. 1. Db(A) ≃ lim −→m Hm(A); 2. D−(A) ≃ lim ←−m Hm(A). Moreover, if complexes P• and P ′ • ∈ Pm(A) are minimal, their images in Db(A) are isomorphic if and only if P̃• ≃ P̃ ′ • as complexes. Thus dealing with classification problems, one can deal with Hm or with Pm instead of Db. Note also that if we fix all Hn(P•) for a complex from Pm min(A), there are finitely many possibilities for the values of Pn. Taking into consideration these remarks, one can define derived wild and derived tame algebras as follows. Definition 1.2. 1. Let R be a k-algebra. A family of A-complexes based on R is a complex of finitely generated projective A ⊗ Rop- modules P•. We denote by Pm(A,R) the category of all bounded families with Pn = 0 for n > m (again we do not prescribe the right bound). For such a family P• and an R-module L we denote by P•(L) the complex (Pn ⊗R L, dn ⊗ 1). If L is finite dimensional, P•(L) ∈ Pm(A). 2. We call a family P• strict if for every finite dimensional R-modules L, L′ (a) P•(L) ≃ P•(L ′) if and only if L ≃ L′; (b) P•(L) is indecomposable if and only if so is L. 3. We call A derived wild if it has a strict family of complexes over every finitely generated k-algebra R. The following fact is well known. Proposition 1.3. An algebra A is derived wild if and only if it has a strict family over one of the following algebras: 1) free algebra k〈x, y〉 in two variables; 2) polynomial algebra k[x, y] in two variables; Jo u rn al A lg eb ra D is cr et e M at h .Yu. A. Drozd 61 3) power series algebra k[[x, y]] in two variables. Let A1, A2, . . . , As be a set of representatives of isomorphism classes of indecomposable projective A-modules. (For instance, one can choose a decomposition 1 = ∑m i=1 ei, where ei are primitive orthogonal idem- potents, take e1, e2, . . . , es all pairwise non-conjugate of them, and set Ai = Aei.) If P is an arbitrary projective A-module, it uniquely decom- poses as P ≃ ⊕s i=1 riAi. Set r(P ) = (r1, r2, . . . , rs) and call it the vector rank of P . For a projective complex P• set r•(P•) = (r(Pn) \|n ∈ Z). Given an arbitrary sequence v• = (vn \|n ∈ Z) denote by P(v•,A) the set of complexes P• such that r•(P•) = v•. Definition 1.4. 1. A rational algebra is a k-algebra k[t, f(t)−1] for a non-zero polynomial f(t). A rational family of A-complexes is a family over a rational algebra R. 2. For a rational family P• denote by r•(P•) = r•(P•(L)) , where L is a one-dimensional R-module. (This value does not depend on the choice of L.) 3. An algebra A is called derived tame if there is a set of rational families of bounded A-complexes S such that: (a) for every (bounded) v• the set S(v•) = {P• ∈ S \| r•(P•) = v• } is finite. (b) for every v• all indecomposable complexes from P(v•,A), ex- cept finitely many of them (up to isomorphism) are isomorphic to a complex P•(L) for some P• ∈ S and some finite dimen- sional L. We call S a parameterising set of A-complexes. These definitions do not formally coincide with other definitions of derived tame and derived wild algebras, for instance, those proposed in [11, 12], but all of them are evidently equivalent. It is obvious (and easy to prove) that neither algebra can be both derived tame and derived wild. The following result (“tame-wild dichotomy for derived categories”) has recently been proved by V. Bekkert and the author [1]. Theorem 1.5 (Main Theorem). Every finite dimensional algebra over an algebraically closed field is either derived tame or derived wild. 2. Reduction to matrix problems We recall now the main notions related to the matrix problems (represen- tations of boxes). More detailed exposition can be found in [6]. A box is a Jo u rn al A lg eb ra D is cr et e M at h .62 Derived tame and derived wild algebras pair A = (A,V), where A is a (k-linear) category and V is an A-coalgebra, i.e. an A-bimodule supplied with comultiplication µ : V → V ⊗A V and counit ι : V → A, which are homomorphisms of A-bimodules and satisfy the usual coalgebra conditions (µ ⊗ 1)µ = (1 ⊗ µ)µ, il(ι ⊗ 1)µ = ir(1 ⊗ ι)µ = id, where il : A⊗AV ≃ V and ir : V⊗AA ≃ V are the natural isomorphisms. The kernel V = Ker ι is called the kernel of the box. A representation of such a box in a category C is a functor M : A → C. Given another representation N : A → C, a morphism f : M → N is defined as a homomorphism of A-modules V ⊗A M → N , or, equivalently, as a homo- morphism of A-bimodules V → HomC(M, N), the latter supplied with the obvious A-bimodule structure. The composition gf of f : M → N and g : N → L is defined as the composition V ⊗A M µ⊗1 −−−−→ V ⊗A V ⊗A M 1⊗f −−−−→ V ⊗A N g −−−−→ L, while the identity morphism idM of M is the composition V ⊗A M ι⊗1 −−−−→ A⊗A M il−−−−→ M. Thus we obtain the category of representations Rep(A, C). If C = vec, we just write Rep(A). If f is a morphism and γ ∈ V(a, b), we denote by f(γ) the morphism f(b)(γ⊗_ ) : M(a) → N(a). A box A is called normal (or group-like) if there is a set of elements ω = {ωa ∈ V(a, a) \| a ∈ obA} such that ι(ωa) = 1a and µ(ωa) = ωa ⊗ ωa for every a ∈ obA. In this case, if f is an isomorphism, all morphisms f(ωa) are isomorphisms M(a) ≃ N(a). This set is called a section of A. For a normal box, one defines the differentials ∂0 : A → V and ∂1 : V → V ⊗A V setting ∂0(α) = αωa − ωbα for α ∈ A(a, b); ∂1(γ) = µ(γ) − γ ⊗ ωa − ωb ⊗ γ for γ ∈ V(a, b). Usually we omit indices, writing ∂α and ∂γ. Recall that a free category kΓ, where Γ is an oriented graph, has the vertices of Γ as its objects and the paths from a to b (a, b being two vertices) as a basis of the vector space kΓ(a, b). A semi-free category is a category of fractions kΓ[S−1], where S = { gα(α) \|α runs through a set of loops } (called the marked loops) of the graph Γ. The arrows of Γ are called the free (respectively, semi-free) generators of the free (semi-free) category. A normal box A = (A,V) is called free (semi-free) if such is the category A, moreover, the kernel V = Jo u rn al A lg eb ra D is cr et e M at h .Yu. A. Drozd 63 Ker ι of the box is a free A -bimodule and ∂α = 0 for each marked loop α. A set of free (respectively, semi-free) generators of such a box is a union S = S0∪S1, where S0 is a set of free (semi-free) generators of the category A and S1 is a set of free generators of the A-bimodule V. A set of free (or semi-free) generators S is called triangular if there is a function ̺ : S → N (we call it “weight”) such that, for every α ∈ S, the differential ∂α belongs to the sub-box generated by {β ∈ S \| ̺(β) < ̺(α) }; especially, if ̺(α) = 0, α is minimal, i.e. ∂α = 0. A free (semi-free) box having a triangular set of free (semi-free) generators is called triangular. For a triangular box a morphism of representations f : M → N is an isomorphism if and only if all f(ωa) are isomorphisms. In what follows we always suppose that our graphs are locally finite, i.e. only have finitely many arrows starting or ending at a given vertex. If such a graph Γ has no oriented cycles, then the category kΓ is locally finite dimensional, i.e. all its spaces of morphisms are finite dimensional. We call a category A trivial if it is a free category generated by a trivial graph (i.e. one with no arrows); thus A(a, b) = 0 if a 6= b and A(a, a) = k. We call A minimal, if it is a semi-free category with a set of semi-free generators consisting of loops only, at most one loop at each vertex. Thus A(a, b) = 0 again if a 6= b, while A(a, a) is either k or a rational algebra. We call a normal box A = (A,V) so-trivial if A is trivial, and so-minimal if A is minimal and all its loops α are minimal too (i.e. with ∂α = 0). In [5] (cf. also [6]) the classification of representations of an arbitrary finite dimensional algebra was reduced to representations of a free trian- gular box. To deal with derived categories we have to consider a wider class of boxes. First, a factor-box of a box A = (A,V) modulo an ideal I ⊆ A is defined as the box A/I = (A/I,V/(IV + VI) (with obvious comultiplication and counit). Note that if A is normal, so is A/I. Definition 2.1. A sliced box is a factor-box A/I, where A = (A,V) is a free box such that the set of its objects V = obA is a disjoint union V = ⋃ i∈Z Vi such that A(a, b) = 0 if a ∈ Vi, b ∈ Vj with j > i, A(a, a) = k, and V(a, b) = 0 if a ∈ Vi, b ∈ Vj with i 6= j. We call a sliced box A/I triangular if so is the free box A. The partition V = ⋃ i Vi is called a slicing. Certainly, in this definition we may assume that the elements of the ideal I are linear combinations of paths of length at least 2. Otherwise such an element is a linear combination of arrows, so we can just eliminate one of these arrows from the underlying graph. Note that for every representation M ∈ Rep(A), where A is a free (semi-free, sliced) box with the set of objects V, one can consider its di- Jo u rn al A lg eb ra D is cr et e M at h .64 Derived tame and derived wild algebras mension Dim(M), which is a function V → N, namely Dim(M)(a) = dimM(a). We call such a representation finite dimensional if its support supp M = { a ∈ V \|M(a) 6= 0 } is finite and denote by rep(A) the cate- gory of finite dimensional representations. Having these notions, one can easily reproduce the definitions of families of representations, especially strict families, wild and tame boxes; see [5, 6] for details. The following procedure, mostly modelling that of [5], let us replace derived categories by representations of sliced boxes. Let A be a finite dimensional algebra, J be its radical. As far as we are interested in A-modules and complexes, we can replace A by a Morita equivalent reduced algebra, thus suppose that A/J ≃ ks [9]. Let 1 = ∑s i=1, where ei are primitive orthogonal idempotents; set Aji = ejAei and Jji = ejJei; note that Jji = Aji if i 6= j. We denote by S the trivial category with the set of objects { (i, n) \|n ∈ N, i = 1, 2, . . . , s } and consider the S-bimodule J such that J ( (i, n), (j, m) ) = { 0 if m 6= n − 1, J∗ ji if m = n − 1. Let B = S[J ] be the tensor category of this bimodule; equivalently, it is the free category having the same set of objects as S and the union of bases of all J ( (i, n), (j, m) ) as a set of free generators. Denote by U the S-bimodule such that U ( (i, n), (j, m) ) = { 0 if n 6= m, A∗ ji if n = m and set W̃ = B⊗S U⊗SB. Dualizing the multiplication Akj⊗Aji → Aki, we get homomorphisms λr :B → B ⊗S W̃, λl :B → W̃ ⊗S B, µ̃ :W̃ → W̃ ⊗S W̃. In particular, µ̃ defines on W̃ a structure of B-coalgebra. Moreover, the sub-bimodule W0 generated by Im(λr − λl) is a coideal in W̃, i.e. µ̃(W0) ⊆ W0 ⊗B W̃ ⊕ W̃ ⊗B W0. Therefore, W = W̃/W0 is also a B- coalgebra, so we get a box B = (B,W). One easily checks that it is free and triangular. Dualizing multiplication also gives a mapping ν : J∗ ji → s⊕ k=1 J∗ jk ⊗ J∗ ki. (3) Jo u rn al A lg eb ra D is cr et e M at h .Yu. A. Drozd 65 Namely, if we choose bases {α } , {β } { γ } in the spaces, respectively, Jji, Jjk, Jki, and dual bases {α∗ } , {β∗ } , { γ∗ } in their duals, then β∗⊗γ∗ occurs in ν(α∗) with the same coefficient as α occurs in βγ. Note that the right-hand space in (3) coincide with each B ( (i, n), (j, n − 2) ) . Let I be the ideal in B generated by the images of ν in all these spaces and D = B/I = (A,V), where A = B/I, V = W/(IW + WI). If necessary, we write D(A) to emphasise that this box has been constructed from a given algebra A. Certainly, D is a sliced triangular box, and the following result holds. Theorem 2.2. The category of finite dimensional representations rep(D(A)) is equivalent to the category Pb min(A) of bounded minimal projective A-complexes. Proof. Let Ai = Aei; they form a complete list of non-isomorphic in- decomposable projective A-modules; set also Ji = radAi = Jei. Then HomA(Ai, Jj) ≃ Jji. A representation M ∈ rep(D) is given by vector spaces M(i, n) and linear mappings Mji(n) : J∗ ji = A ( (i, n), (j, n − 1) ) → Hom ( M(i, n), M(j, n − 1) ) subject to the relations s∑ k=1 m ( Mjk(n) ⊗ Mki(n + 1) ) ν(α) = 0 (4) for all i, j, k, n and all α ∈ Jji, where m denotes the multiplication of mappings Hom ( M(k, n), M(j, n − 1) ) ⊗ Hom ( M(i, n + 1), M(k, n) ) → → Hom ( M(i, n + 1), M(j, n − 1) ) . For such a representation, set Pn = ⊕s i=1 Ai ⊗ M(i, n). Then radPn =⊕n i=1 Ji ⊗ M(i, n) and HomA(Pn, radPn−1) ≃ ⊕ i,j HomA ( Ai ⊗ M(i, n), Jj ⊗ M(j, n − 1) ) ≃ ≃ ⊕ ij Hom ( M(i, n), HomA ( Ai, Jj ⊗ M(j, n − 1) )) ≃ ≃ ⊕ ij M(i, n)∗ ⊗ Jji ⊗ M(j, n − 1) ≃ ≃ ⊕ ij Hom ( J∗ ji, Hom ( M(i, n), M(j, n − 1) )) . Jo u rn al A lg eb ra D is cr et e M at h .66 Derived tame and derived wild algebras Thus the set {Mji(n) \| i, j = 1, 2, . . . , s } defines a homomorphism dn : Pn → Pn−1 and vice versa. Moreover, one easily verifies that the condi- tion (4) is equivalent to the relation dndn+1 = 0. Since every projective A-module can be given in the form ⊕s i=1 Ai ⊗ Vi for some uniquely de- fined vector spaces Vi, we get a one-to-one correspondence between finite dimensional representations of D and bounded minimal complexes of pro- jective A-modules. In the same way one also establishes one-to-one corre- spondence between morphisms of representations and of the correspond- ing complexes, compatible with their multiplication, which accomplishes the proof. Corollary 2.3. An algebra A is derived tame (derived wild) if so is the box D(A). 3. Proof of the main theorem Now we are able to prove the main theorem. Namely, according to Corol- lary 2.3, it follows from the analogous result for sliced boxes. Theorem 3.1. Every sliced triangular box is either tame or wild. Actually, just as in [5] (see also [6]), we shall prove this theorem in the following form. Theorem 3.1a. Suppose that a sliced triangular box A = (A,V) is not wild. For every dimension d of its representations there is a functor Fd : A → M, where M is a minimal category, such that every representation M : A → vec of A of dimension Dim(M) ≤ d is isomorphic to the inverse image F ∗N = N ◦ F for some functor N : M → vec. Moreover, F can be chosen strict, which means that F ∗N ≃ F ∗N ′ implies N ≃ N ′ and F ∗N is indecomposable if so is N . Remark. We can consider the induced box AF = (M,M⊗AV⊗AM). It is a so-minimal box, and F ∗ defines a full and faithful functor rep(AF ) → rep(A). Its image consists of all representations M : A → vec that factorise through F . Proof. As we only consider finite dimensional representations, we may assume that the set of objects is finite. Hence the slicing V = ⋃ i Vi (see Definition 2.1) is finite too: V = ⋃m i=1 Vi and we use induction by m. If m = 1, A is free, and our claim has been proved in [5]. So we may suppose that the theorem is true for smaller values of m, especially, it is true for the restriction A′ = (A′,V ′) of the box A onto the subset V′ = ⋃m i=2 Vi. Thus there is a strict functor F ′ : A′ → M, where M Jo u rn al A lg eb ra D is cr et e M at h .Yu. A. Drozd 67 is a minimal category, such that every representation of A′ of dimension smaller than d is of the form F ′∗N for N : M → vec. Consider now the amalgamation B = A ⊔A′ M and the box B = (B,W), where W = B ⊗A V ⊗A B. The functor F ′ extends to a functor F : A → B and induces a homomorphism of A-bimodules V → W; so it defines a functor F ∗ : rep(B) → rep(A), which is full and faithful. Moreover, every representation of A of dimension smaller than d is isomorphic to F ∗N for some N , and all possible dimensions of such N are restricted by some vector b. Therefore, it is enough to prove the claim of the theorem for the box B. Note that the category B is generated by the loops from M and the images of arrows from A(a, b) with b ∈ V1 (we call them new arrows). It implies that all possible relations between these morphisms are of the form ∑ β βgβ(α), where α ∈ B(a, a) is a loop (necessarily minimal, i.e. with ∂α = 0), gβ are some polynomials, and β runs through the set of new arrows from a to b for some b ∈ V1. Consider all of these relations for a fixed b; let them be ∑ β βgβ,k(α). Their coefficients form a matrix( gβ,k(α) ) . Making linear transformations of the set {β } and of the set of relations, we can make this matrix diagonal, i.e. make all relations being βfβ(α) = 0 for some polynomials fβ . If one of fβ is zero, the box B has a sub-box aα 99 β // b , with ∂α = ∂β = 0, which is wild; hence B and A are also wild. Otherwise, let f(α) 6= 0 be a common multiple of all fβ(α), Λ = {λ1, λ2, . . . , λr } be the set of roots of f(α). If N ∈ rep(B) is such that N(α) has no eigenvalues from Λ, then f(N(α)) is invertible; thus N(β) = 0 for all β : a → b. So we can apply the reduction of the loop α with respect to the set Λ and the dimension d = b(a), as in [5, Propositions 3,4] or [6, Theorem 6.4]. It gives a new box that has the same number of loops as B, but the loop corresponding to α is “isolated,” i.e. there are no more arrows starting or ending at the same vertex. In the same way we are able to isolate all loops, obtaining a semi-free triangular box C and a morphism G : B → C such that G∗ is full and faithful and all representations of B of dimensions smaller than b are of the form G∗L. As the theorem is true for semi-free boxes, it accomplishes the proof. Remark. Applying reduction functors, like in the proof above, we can also extend to sliced boxes (thus to derived categories) other results obtained before for free boxes. For instance, we mention the following theorem, quite analogous to that of Crawley-Boevey [3]. Jo u rn al A lg eb ra D is cr et e M at h .68 Derived tame and derived wild algebras Theorem 3.2. If an algebra A is derived tame, then, for any vector d = (dn \|n ∈ Z) such that almost all dn = 0, there is at most finite set of generic A-complexes of endolength d, i.e. such indecomposable minimal bounded complexes P• of projective A-modules, not all of which are finitely generated, that lengthE(Pn) = dn for all n, where E = EndA(P•). Its proof reproduces again that of [3], with obvious changes necessary to include sliced boxes into consideration. 4. Families of complexes We consider now algebraic families of A-complexes, i.e. flat families over an algebraic variety X. Such a family is a complex F• = (Fn, dn) of flat coherent A⊗OX -modules. We always assume this complex bounded and minimal ; the latter means that Im dn ⊆ JFn−1 for all n, where J = radA. We also assume that X is connected; it implies that the vector rank r•(F•(x)) is constant, so we can call it the vector rank of the family F and denote it by r•(F•) Here, as usually, F(x) = Fx/mxFx, where mx is the maximal ideal of the ring OX,x. We call a family F• non- degenerate if, for every x ∈ X, at least one of dn(x) : Fn(x) → Fn−1(x) is non-zero. Having a family F• over X and a regular mapping φ : Y → X, one gets the inverse image φ∗(F), which is a family of A-complexes over the variety Y such that φ∗(F)(y) ≃ F(φ(y)). If F• is non-degenerate, so is φ∗(F). Given an ideal I ⊆ J, we call a family F• an I-family if Im dn ⊆ IFn−1 for all n. Then any inverse image φ∗(F) is an I-family as well. Just as in [8], we construct some “almost versal” non-degenerate I-families. Consider again a complete set of non-isomorphic indecomposable pro- jective A-modules {A1, A2, . . . , As }. For each vector r = (r1, r2, . . . , rs) set rA = ⊕s i=1 riAi, and denote I(r, r′) = HomA(rA, I · r′A), where I is an ideal contained in J. Fix a vector rank of bounded complexes r• = (rk \|m ≤ k ≤ n) and set H = H(r•, I) = ⊕n k=m+1 I(rk, rk−1). Consider the projective space P = P(r•, I) = P(H) and its closed subset D = D(r•, I) ⊆ P consisting of all sequences (hk) such that hk+1hk = 0 for all k. Because of the universal property of projective spaces [15, Theorem II.7.1], the embedding D(r•, I) → P(r•, I) gives rise to a non-degenerate I-family V• = V•(r•, I): V• : Vn dn−−−−→ Vn−1 dn−1 −−−−→ . . . −→ Vm, (5) where Vk = OD(n − k) ⊗ rkA for all m ≤ k ≤ n. We call V•(r•, I) the canonical I-family of A-complexes over D(r•, I). Moreover, regular mappings φ : X → D(r•, I) correspond to non-degenerate I-families F• Jo u rn al A lg eb ra D is cr et e M at h .Yu. A. Drozd 69 with Fk = 0 for k > n or k < m and Fk = L⊗(n−k) ⊗ rkA for some invertible sheaf L over X. Namely, such a family can be obtained as φ∗(V•) for a uniquely defined regular mapping φ. Moreover, the following result holds, which shows the “almost versality” of the families V•(r•, I). Proposition 4.1. For every non-degenerate family of I-complexes F• of vector rank r• over an algebraic variety X, there is a finite open covering X = ⋃ j Uj such that the restriction of F• onto each Uj is isomorphic to φ∗ jV•(r•, I) for a regular mapping φj : Uj → D(r•, I). Proof. For each x ∈ X there is an open neighbourhood U ∋ x such that all restrictions Fk\|U are isomorphic to OU ⊗ rkA; so the restriction F•\|U is obtained from a regular mapping U → D(r•, I). Evidently it implies the assertion. Note that the mappings φj are not canonical, so we cannot glue them into a “global” mapping X → D(r•, I). Consider now the group G = G(r•) = ∏ k Aut(rkA), which acts on H(r•, I): (gk) · (hk) = (gk−1hkg −1 k ). It induces the action of G(r•) on P(R•, I) and on D(r•, I). The definitions immediately imply that V•(r•, I)(x) ≃ V•(r•, I)(x ′) (x, x′ ∈ D) if and only if x and x′ belong to the same orbit of G. Consider the sets Di = Di(r•, I) = {x ∈ D \| dimGx ≤ i } . It is known that they are closed (it follows from the theorem on dimen- sions of fibres, cf. [15, Exercise II.3.22] or [17, Ch. I, § 6, Theorem 7]). We set par(r•, I,A) = max i {dim Di(r•, I) − i } and call this integer the parameter number of I-complexes of vector rank r•. Obviously, if I ⊆ I′, then par(r•, I,A) ≤ par(r•, I ′,A). Especially, the number par(r•,A) = par(r•,J,A) is the biggest one. Proposition 4.1, together with the theorem on the dimensions of fi- bres and the Chevalley theorem on the image of a regular mapping (cf. [15, Exercise II.3.19] or [17, Ch. I, § 5, Theorem 6]), implies the following result. Corollary 4.2. Let F• be an I-family of vector rank r• over a variety X. For each x ∈ X set Xx = {x′ ∈ X \| F•(x ′) ≃ F•(x) } and denote Xi = {x ∈ X \| dimXx ≤ i } , par(F•) = max i {dimXi − i } . Then all subsets Xx and Xi are constructible (i.e. finite unions of locally closed sets) and par(F•) ≤ par(r•, I,A). Jo u rn al A lg eb ra D is cr et e M at h .70 Derived tame and derived wild algebras Note that the bases D(r•, I) of our almost versal families are projective, especially complete varieties. In the next section we shall exploit this property. Tame-wild dichotomy allows to establish the derived type of an al- gebra knowing the behaviour of the numbers par(r•,A). Namely, if rk = (rk1, rk2, . . . , rks), set \|r•\| = ∑ k,i rki. Since it is a maximal pos- sible number of indecomposable summands of complexes of vector rank r•, indecomposable complexes over derived tame algebra form at most one-parameter families, and the parameter number grows quadratically for derived wild algebras, the following corollary is evident. Corollary 4.3. An algebra A is derived tame if and only if par(r•,A) ≤ \|r•\| for all vector ranks r•. An important case is that of families of free modules, i.e. such that Fk(x) ≃ akA for some integer k. Namely, let r(A) = (a1, a2, . . . , as) (we do not suppose that A is Morita reduced). For every vector b = (bm, bm+1, . . . , bn) we set ba = (bma, bm+1a, . . . , bna) and write D(b, I), par(b, I,A), etc. instead of, respectively, D(ba, I), par(ba, I,A), etc. When we are interested in the asymptotical behaviour of parameter num- bers (it is enough, for instance, to establish the derived type), we can restrict our considerations by free complexes only. Indeed, for a vector r = (r1, r2, . . . , rs), denote [ r/a ] = max { b \| bai ≤ ri for all i } ,] r/a [ = min { b \| bai ≥ ri for all i } . If r• = (rm, rm+1, . . . , rn), set b = (bm, bm+1, . . . , bn), where bk = ] rk/a [ ; b′ = (b′m, b′m+1, . . . , b ′ n), where b′k = [ rk/a ] . Then, obviously, par(b′, I,A) ≤ par(r•, I,A) ≤ par(b, I,A). Especially, Corollary 4.3 can be reformulated as follows. Corollary 4.4. An algebra A is derived tame if and only if par(b,A) ≤ \|b\|dimA for every sequence b = (bm, bm+1, . . . , bn). 5. Families of algebras. Semi-continuity A (flat) family of algebras over an algebraic variety X is a sheaf A of OX - algebras, which is coherent and flat (thus locally free) as a sheaf of OX - modules. For such a family and every sequence b = (bm, bm+1, . . . , bn) Jo u rn al A lg eb ra D is cr et e M at h .Yu. A. Drozd 71 one can define the function par(b,A, x) = par(b,A(x)). (Recall that here bk denote the ranks of free modules in a free complex.) Our main result is the upper semi-continuity of these functions. Theorem 5.1. Let A be a flat family of finite dimensional algebras over an algebraic variety X. For every vector b = (bm, bm+1, . . . , bn) the func- tion par(b,A, x) is upper semi-continuous, i.e. all sets Xj = {x ∈ X \| par(b,A, x) ≥ j } are closed. Proof. We may assume that X is irreducible. Let K be the field of rational functions on X. We consider it as a constant sheaf on X. Set J = rad(A ⊗OX K) and J = R ∩ A. It is a sheaf of nilpotent ideals. Moreover, if ξ is the generic point of X, the factor algebra A(ξ)/J (ξ) is semisimple. Hence there is an open set U ⊆ X such that A(x)/J (x) is semisimple, thus J (x) = radA(x) for every x ∈ U . There- fore par(b,A, x) = par(b,J (x),A(x)) for x ∈ U ; so Xj = Xj(J ) ∪ X ′ j , where Xj(J ) = {x ∈ X \| par(b,J (x),A(x)) ≥ j } and X ′ = X \U is a closed subset in X. Using noetherian induction, we may suppose that X ′ j is closed, so we only have to prove that Xj(J ) is closed too. Consider the locally free sheaf H = ⊕n k=m+1 Hom(bkA, bk−1J ) and the projective space bundle P(H) [15, Section II.7]. Every point h ∈ P(H) defines a set of homomorphisms hk : bkA(x) → bk−1J (x) (up to a ho- mothety), where x is the image of h in X, and the points h such that hkhk+1 = 0 form a closed subset D ⊆ P(H). We denote by π the re- striction onto D of the projection P(H) → X; it is a projective, hence closed mapping. Moreover, for every point x ∈ X the fibre π−1(x) is iso- morphic to D(b,A(x),J (x)). Consider also the group variety G over X: G = ∏n k=m GLbk (A). There is a natural action of G on D over X, and the sets Di = { z ∈ D \| dimGz ≤ i } are closed in D. Therefore the sets Zi = π(Di) are closed in X, as well as Zij = { x ∈ Zi \| dimπ−1(x) ≥ i + j } . But Xj(J ) = ⋃ i Zij , thus it is also a closed set. Taking into consideration Corollary 4.4, we obtain Corollary 5.2. For a family of algebras A over X denote Xtame = {x ∈ X \| A(x) is derived tame } , Xwild = {x ∈ X \| A(x) is derived wild } . Jo u rn al A lg eb ra D is cr et e M at h .72 Derived tame and derived wild algebras Then Xtame is a countable intersection of open subsets and Xwild is a countable union of closed subsets. The following conjecture seems very plausible, though even its ana- logue for usual tame algebras has not yet been proved. Conjecture 5.3. For any (flat) family of algebras over an algebraic va- riety X the set Xtame is open. Recall that an algebra A is said to be a (flat) degeneration of an algebra B, and B is said to be a (flat) deformation of A, if there is a (flat) family of algebras A over an algebraic variety X and a point p ∈ X such that A(x) ≃ B for all x 6= p, while A(p) ≃ A. One easily verifies that we can always assume X to be a non-singular curve. Corollary 5.2 obviously implies Corollary 5.4. Suppose that an algebra A is a flat degeneration of an algebra B. If B is derived wild, so is A. If A is derived tame, so is B. If we consider non-flat families, the situation can completely change. The reason is that the dimension is no more constant in these families. That is why it can happen that such a “degeneration” of a derived wild al- gebra may become derived tame, as the following example due to Brüstle [2] shows. Example 5.5. There is a (non-flat) family of algebras A over an affine line A 1 such that all of them except A(0) are isomorphic to the derived wild algebra B given by the quiver with relations • • α // • β1 // • γ1 OO γ2 ²² • β2oo β1α = 0, • while A(0) is isomorphic to the derived tame algebra A given by the quiver with relations • • α // • ξ1 @@ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ β1 // • γ1 OO γ2 ²² • β2oo ξ2¡¡¢¢ ¢ ¢ ¢ ¢ ¢ ¢ β1α = γ1β1 = γ2β2 = 0. • (6) Jo u rn al A lg eb ra D is cr et e M at h .Yu. A. Drozd 73 Namely, one has to define A(λ) as the factor algebra of the path algebra of the quiver as in (6), but with the relations β1α = 0, γ1β1 = λξ1, γ2β2 = λξ2. Note that dimA = 16 and dimB = 15, which shows that this family is not flat. Actually, in such a situation the following result always holds. Proposition 5.6. Let A be a family (not necessarily flat) of algebras over a non-singular curve X such that A(x) ≃ B for all x 6= p, where p is a fixed point, while A(p) ≃ A. Then there is a flat family B over X such that B(x) ≃ B for all x 6= p and B(p) ≃ A/I for some ideal I. Proof. Note that the restriction of A onto U = X \ { p } is flat, since dimA(x) is constant there. Let n = dimB, Γ be the quiver of the algebra B and G = kΓ be the path algebra of Γ. Consider the Grassmannian Gr(n,G), i.e. the variety of subspaces of codimension n of G. The ideals form a closed subset Alg = Alg(n,G) ⊂ Gr(n,G). The restriction of the canonical vector bundle V over the Grassmannian onto Alg is a sheaf of ideals in G = G ⊗ OAlg, and the factor F = G/V is a universal family of factor algebras of G of dimension n. Therefore there is a morphism φ : U → Alg such that the restriction of A onto U is isomorphic to φ∗(F). Since Alg is projective and X is non-singular, φ can be continued to a morphism ψ : X → Alg. Let B = ψ∗(F); it is a flat family of algebras over X. Moreover, B coincides with A outside p. Since both of them are coherent sheaves on a non-singular curve and B is locally free, it means that B ≃ A/T , where T is the torsion part of A, and B(p) ≃ A(p)/T (p). Corollary 5.7. If a degeneration of a derived wild algebra is derived tame, the latter has a derived wild factor algebra. In the Brüstle’s example 5.5, to obtain a derived wild factor algebra of A, one has to add the relation ξ1α = 0, which obviously holds in B. By the way, as a factor algebra of a tame algebra is obviously tame (which is no more true for derived tame algebras!), we get the following corollary (cf. also [4, 8]). Corollary 5.8. Any deformation (not necessarily flat) of a tame algebra is tame. Any degeneration of a wild algebra is wild. References [1] Bekkert, V. I. and Drozd, Yu.A. Tame–wild dichotomy for derived categories. arXiv:math.RT/031035. To appear. Jo u rn al A lg eb ra D is cr et e M at h .74 Derived tame and derived wild algebras [2] Brüstle, Th. Tree Algebras and Quadratic Forms. Habilitation Thesis. Universität Bielefeld, 2002. [3] Crawley-Boevey, W. C. Tame algebras and generic modules. Proc. London Math. Soc. 63 (1991), 241–265. [4] Crawley-Boevey, W. C. Tameness of biserial algebras. Arch. Math. 65 (1995), 399– 405. [5] Drozd, Yu.A. Tame and wild matrix problems. Representations and quadratic forms. Institute of Mathematics, Kiev, 1977, 39–74. (English translation: Amer. Math. Soc. Translations 128 (1986), 31–55.) [6] Drozd, Yu.A. Reduction algorithm and representations of boxes and algebras. Comtes Rendue Math. Acad. Sci. Canada 23 (2001) 97-125. [7] Drozd, Yu.A. Semi-continuity for derived categories. arXiv:math.RT/0212015 (to appear in Algebras and Representation Theory). [8] Drozd, Yu. and Greuel, G.-M. Semi-continuity for Cohen–Macaulay modules, Math. Ann. 306 (1996), 371–389. [9] Drozd, Yu. and Kirichenko, V. V. Finite Dimensional Algebras. Vyscha Shkola, Kiev, 1980. (Enlarged English edition: Springer–Verlag, 1994.) [10] Geiß, Ch. On degenerations of tame and wild algebras. Arch. Math. 64 (1995), 11-16. [11] Geiß, Ch. Derived tame algebras and Euler-forms. Math. Z. 239 (2002), 829-862. [12] Geiß, Ch. and Krause, H. On the notion of derived tameness. Preprint, 2000. [13] Gelfand, S. I. and Manin, Yu. I. Methods of Homological Algebra. Nauka,1988. (English translation: Springer–Verlag, 1996.) [14] Happel, D. Triangulated Categories in the Representation Theory of Finite Di- mensional Algebras. London Mathematical Society Lecture Notes Series, 119, 1988. [15] Hartshorne, R. Algebraic Geometry. Springer–Verlag, 1997. [16] Huisgen-Zimmermann, B. and Saoŕin, M. Geometry of chain complexes and outer automorphisms under derived equivalence. Trans. Amer. Math. Soc. 353 (2001), 4757-4777. [17] Shafarevich, I. R. Basic Algebraic Geometry. Nauka, 1988. Contact information Yu. A. Drozd Department of Mechanics and Mathematics, Kyiv Taras Shevchenko University, 01033 Kyiv, Ukraine E-Mail: yuriy@drozd.org URL: http://drozd.org/˜yuriy Received by the editors: 16.10.2003 and final form in 31.01.2004.

Derived tame and derived wild algebras

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