Cyclic homology of truncated quiver algebras and notes on the no loops conjecture for Hochschild homology

Transcription

1 Cyclic homology of truncated quiver algebras and notes on the no loops conjecture for Hochschild homology Tomohiro Itagaki (joint work with Katsunori Sanada) November 13, 2013 Perspectives of Representation Theory of Algebras Conference honoring Kunio Yamagata on the occasion of his 65th birthday

2 1 Introduction Definitions and the notation Aim 2 Hochschild homology and cyclic homology of truncated quiver algebras Hochschild homology of truncated quiver algebras Cyclic homology of truncated quiver algebras over a field of characteristic zero 3 The outline of the proof 4 The cyclic homology of truncated cycle algebras 5 The 2-truncated cycles version of the no loops conjecture The m-truncated cycles version of the no loops conjecture

3 Section 1 Introduction

4 Definitions and the notation Notation K: a commutative ring N: the set of all natural numbers containing 0 : a finite quiver n : the set of all paths of length n(n 0) in R : the arrow ideal of K G n : a cyclic group of order n generated by t n A n := A K A K K A for a K-algebra A (the n-fold tensor product of A) A e := A K A op : the enveloping algebra of A

5 Definitions and the notation Hochschild homology groups Definition For K-algebra A, the Hochschild complex is the following complex: C(A) : b A n+1 b A n b b A 2 b A 0, where the K-homomorphisms b : A n+1 A n is given by n 1 b(x 0 x n ) = ( 1) i (x 0 x i x i+1 x n ) i=0 + ( 1) n (x n x 0 x 1 x n 1 ) HH n (A) := H n (C(A)): the n-th Hochschild homology group of A (n 0)

6 Definitions and the notation Hochschild homology groups If a unital K-algebra A is projective as a module over K, then there is an isomorphism HH n (A) = Tor Ae n (A, A) For a unital K-algebra A, the bar resolution C bar (A) of A is given by C bar (A) : b A n+1 b n b b A A 2 b A 0, where the differentials b are given by n 1 b (x 0 x n ) = ( 1) i (x 0 x i x i+1 x n ) i=0

7 Definitions and the notation Cyclic group action Let n be a positive integer The action of G n on the module A n is given by t n (x 1 x 2, x n ) := ( 1) n 1 (x n x 1 x 2 x n 1 ), where x i A Let D := 1 t n and N := 1 + t n + t 2 n + + tn 1 n of the group ring ZG n Then as elements and D : A n A n N : A n A n are K-homomorphisms

8 Definitions and the notation Cyclic homology groups Definition For a K-algebra A, the following is a first quadrant bicomplex CC(A) of A: b b CC(A),2 : A 3 D A 3 N A 3 D b b b CC(A),1 : A 2 D A 2 N A 2 D b b b b CC(A),0 : A D A N A D CC(A) 0, CC(A) 1, CC(A) 2, HC n (A) := H n (Tot(CC(A))): the n-th cyclic homology group of A(n 0)

9 Definitions and the notation Cyclic homology is developed as a non-commutative variant of the de Rham cohomology by Connes (Loday, 1992) Aside from this, cyclic homology is developed by Tsygan, Loday and Quillen Cyclic homology is closely related to Hochschild homology, because of the definition itself and also the periodicity exact sequence by Connes: HH n (A) I HC n (A) S HC n 2 (A) B HH n 1 (A) I

10 Definitions and the notation It is well known that cyclic homology is Morita invariant Moreover, the following theorem is shown by König, Liu, Zhou Theorem(König, Liu and Zhou, 2011) Let R and T be two finite dimensional algebras over an algebraically closed field which are stably equivalent of Morita type Then, (1) For n > 0 odd, dim HC n (R) = dim HC n (T ) (2) Suppose that R and T have no semisimple direct summands Then for any n 0 even, the following statements are equivalent: (i) the Auslander-Reiten conjecture holds for this stable equivalence of Morita type; (ii) dim HC n(r) = dim HC n(t )

11 Definitions and the notation The cyclic homology of algebras is investigated for classes of various algebras, for example, 2-nilpotent algebras (Cibils, 1990) truncated quiver algebras over a field of characteristic zero (Taillefer, 2001) quadratic monomial algebras (Sköldberg, 2001) monomial algebras over a field of characteristic zero (Han, 2006) etc We want to know the influence of a ground ring of an algebra on its cyclic homology In particular, we are going to focus on the module structure of cyclic homology

12 Aim Aim Aim (1) Find the dimension formula of the cyclic homology of truncated quiver algebras K /R m over a field K of positive characteristic (2) Show the m-truncated cycles version of the no loops conjecture for a class of algebras over an algebraically closed field

13 Section 2 Hochschild homology and cyclic homology of truncated quiver algebras

14 Hochschild homology of truncated quiver algebras Truncated quiver algebra K /R m Following the notation by Sköldberg(1999), we recall characteristics of truncated quiver algebras By adjoining an element, we will consider the following set: ˆ = { } i i=0 This set is a semigroup with the multiplication defined by { δγ if t(δ) = s(γ), δ γ = δ, γ i, otherwise, i=0 and γ = γ =, γ ˆ

15 Hochschild homology of truncated quiver algebras K ˆ is a semigroup algebra, and the path algebra K is isomorphic to K ˆ /( ) K is ˆ -graded, that is, K = γ ˆ (K ) γ, where (K ) γ = Kγ for γ i=0 i and (K ) = 0 K is N-graded, that is, K = i=0 K i R m is ˆ -graded and N-graded Thus the truncated quiver algebra K /R m is a ˆ -graded and an N-graded algebra

16 Hochschild homology of truncated quiver algebras Theorem(Sköldberg, 1999) The following is a projective ˆ -graded resolution of a truncated quiver algebra A = K /R m as a left Ae -module: P A : di+1 P i d i d 2 d 1 P1 P0 Here each terms are ˆ -graded modules defined by P i = A K 0 KΓ (i) K 0 A for i 0, ε A 0 P 0 = A K 0 K 0 K 0 A = A K 0 A, where Γ (i) is given by { Γ (i) cm if i = 2c (c 0), = cm+1 if i = 2c + 1 (c 0),

17 Hochschild homology of truncated quiver algebras Theorem(Sköldberg, 1999) and the differentials are defined by d 2c (α α 1 α cm β) = m 1 j=0 αα 1 α j α 1+j α (c 1)m+1+j d 2c+1 (α α 1 α cm+1 β) α (c 1)m+2+j α cm β, = αα 1 α 2 α cm+1 β α α 1 α cm α cm+1 β, where α i 1 for i 1 and α, β A and ε is the multiplication

18 Hochschild homology of truncated quiver algebras Cycle and Basic cycle A path is called a cycle if its source coincides with its target A cycle γ is called a basic cycle if there does not exist a cycle δ such that γ = δ j for some positive integer j( 2) c n : the set of all cycles of length n b n : the set of all basic cycles of length n

19 Hochschild homology of truncated quiver algebras Cycle and Basic cycle Let a path α 1 α n 1 α n be a cycle, where α i is an arrow in Then the action of G n on c n is given by Similarly, G n acts on b n t n (α 1 α n 1 α n ) := α n α 1 α n 1 a n : the cardinal number of the set of all G n -orbits on c n b n : the cardinal number of the set of all G n -orbits on b n

20 Hochschild homology of truncated quiver algebras Hcohschild homology of truncated quiver algebras L denotes the complex A K e 0 KΓ ( ), which is isomorphic to A A e P A by the following isomorphism φ: φ : A A e P A A A e A e K e 0 KΓ ( ) A K e 0 KΓ ( ) Since the complex L is decomposed into the subcomplexes L γ L = L γ, q=0 γ c q /G q the p-th Hochschild homology group HH p (A) is N-graded, so we have HH p (A) = HH p,q (A) for p 0 q=0

21 Hochschild homology of truncated quiver algebras Hochschild homology of truncated quiver algebras Theorem (Sköldberg, 1999) Let K be a commutative ring, A = K /R m and q = cm + e for 0 e m 1 Then the degree q part of the p-th Hochschild homology HH p,q (A) is the following as a K-module K aq if 1 e m 1 and 2c p 2c + 1, ( ( )) K gcd(m,r) 1 m b Ker gcd(m, r) : K K r r q if e = 0 and 0 < 2c 1 = p, ( ( )) K gcd(m,r) 1 m b Coker gcd(m, r) : K K r r q if e = 0 and 0 < 2c = p, K # 0 if p = q = 0, 0 otherwise

22 Cyclic homology of truncated quiver algebras over a field of characteristic zero Cyclic homology of truncated quiver algebras Theorem (Taillefer, 2001) Let K be a field of charactaristic zero The cyclic homology group of a truncated quiver algebra A is given by dim K HC 2c (A) = # 0 + dim K HC 2c+1 (A) = m 1 e=1 r > 0 st r (c + 1)m a cm+e, (gcd(m, r) 1)b r

23 Section 3

24 Theorem (Itagaki, Sanada) Let K be a field of characteristic ζ and A = K /R m Then the dimension formula of the cyclic homology of A is given by, for c 0, m 1 dim KHC 2c(A) = # c 1 c =0 e=1 + m 1 c c =1 e=1 a cm+e r > 0 st rζ c m + e r > 0 st rζ gcd(m, r)c b r + c c =1 r > 0 st r c m, gcd(m, r)ζ m (gcd(m, r) 1)b r, b r

25 Theorem(Itagaki, Sanada) dim K HC 2c+1 (A) = r > 0 st r (c + 1)m (gcd(m, r) 1)b r + c m 1 c =0 e=1 + c c =1 r > 0 st rζ c m + e r > 0 st rζ gcd(m, r)c b r + c+1 c =1 r > 0 st r c m, gcd(m, r)ζ m (gcd(m, r) 1)b r b r

26 The outline of the proof Cibils projective resolution Lemma (Cibils, 1989) Let I be an admissible ideal of K E denotes the subalgebra of A = K /I generated by 0 The following is a projective resolution of A as left A e -module: d n d 1 d 0 Q A : Q n Qn 1 Q 1 Q0 A 0 where Q n = A E (rad A) n E d n (x 0 x n+1 ) = E A and differntials are given by n ( 1) i (x 0 x i x i+1 x n+1 ) i=0 for x 0, x n+1 A, x 1,, x n rad A

27 The outline of the proof Cibils mixed complex Let K be a field For a K-algebra A with a decomposition A = E r, where E is a separable subalgebra and r is a two-sided ideal, Cibils(1990) gives the following mixed complex (A E e r n E, b, B): b : A E e r n+1 E B : A E e r n E A E e r n E A E e r n+1 E is the Hochschild boundary and is given by the formula B(x 0 E e x 1 x n ) n = ( 1) in (1 E e x i (x 0 ) r x i 1 ), i=0 where x 0 = (x 0 ) E + (x 0 ) r E r The cyclic homology groups of the mixed complex and the cyclic homology groups of A are isomorphic

28 The outline of the proof The cyclic homology groups of Cibils mixed complex is isomorphic to the total homology groups of the following first quadrant bicomplex: b 2 E A E e r b A E e r b B b A E e r b B B b A A 0 A 0 0 Denote the above bicomplex by C E (A) For a truncated quiver algebra A = K /R m, there exists the decomposition K /R m = K 0 rad A

29 The outline of the proof We consider the spectral sequence associated with the filtration F Tot(C E (A)) given by (F p Tot(C E (A))) n = i p C E (A) i,n i The cyclic homology groups HC n (A) is given by HC n (A) = H n (Tot(C E (A))) = p+q=n E 2 p,q E 1 -page of the spectral sequence is drawn by the following illustration: HH 2 (A) HH 1 (A) HH 0 (A) B B HH 1 (A) HH 0 (A) 0 0 B 0 HH 0 (A)

30 The outline of the proof B : HH n (A) HH n+1 (A) Ames, Cagliero and Tirao (2009) give chain maps ι : P A Q A, and π : Q A P A which induce isomorphism HH n (A) = HH n (A) for n 0 By means of these chain maps, we consider B : HH n (A) HH n+1 (A) as follows: Sköldberg: A K e 0 KΓ (n) A K e 0 KΓ (n+1) A A e P n A A e P n+1 id A ι id A π A A e Q n A A e Q n+1 Cibils: A K e 0 rad A n K 0 B A K e 0 rad A n+1 K 0

31 Section 4

32 The cyclic homology of truncated cycle algebras Let K be a field of characteristic ζ We consider the cyclic homoology of a truncated quiver algebra A = K /R m, where is the following quiver: α s 1 0 α 0 s 1 1 α s 2 α 1 α s 3 s 2 s 3 2 The truncated quiver algebra A is called a truncated cycle algebra Note that a r and b r is given by { { 1 if s r, 1 if s = r, a r = b r = 0 otherwise, 0 otherwise

33 The cyclic homology of truncated cycle algebras For x R, we denote the largest integer i satisfying i x by [x] The dimension formula of the cyclic homology of A is given by dim KHC 2c(A) [ ] (c + 1)m 1 [ cm = s + s s ([ + ] c 1 + ([ (c + 1)m 1 sζ c =0 ] [ ]) c m m 1 gcd(m, s)ζ gcd(m, s)ζ c =1 [ ] gcd(m, s)c + (gcd(m, s) 1) sζ ([ c m s ] ] [ ]) c m sζ [ ]) c m 1 s

34 The cyclic homology of truncated cycle algebras and dim K HC 2c+1 (A) ([ ] [ (c + 1)m (c + 1)m 1 = (gcd(m, s) 1) s s ([ ] [ ]) c+1 m m 1 ([ c m + gcd(m, s)ζ gcd(m, s)ζ s c =1 c ([ ] [ ]) (c + 1)m 1 c m + sζ sζ c =0 ] + [ ]) gcd(m, s)c sζ [ ]) c m 1 ] s

35 The cyclic homology of truncated cycle algebras When ζ = 0, the dimension of the cyclic homology of A is as follows: [ ] [ ] (c + 1)m 1 cm dim K HC 2c (A) = s +, s s dim K HC 2c+1 (A) ([ ] [ ]) (c + 1)m (c + 1)m 1 = (gcd(m, s) 1) s s

36 The cyclic homology of truncated cycle algebras A change of dim K HC 2c (A) by ζ We fix s = 7, m = 26 The graphs of dim K HC 2c (A) of the three cases that ζ = 0, ζ = 13 and ζ = 113 are as follows:

37 Section 5 of the chain map and truncated quiver algebras

38 The 2-truncated cycles version of the no loops conjecture The no loops conjecture is as follows: The no loops conjecture If A is a finite dimensional algebra of finite global dimension then Ext 1 A (S, S) = 0 for all simple A-modules S, or equivalently, A has no loops in its quiver It is shown that this conjecture holds for a large class of the finite dimensional algebra including all those which the ground field is an algebraically closed field by Igusa (1990), and this is derived from an earlier result of Lenzing (1969) Bergh, Han and Madsen show the 2-truncated cycles version of this conjecture The 2-truncated cycles version of the no loops conjecture If A is a finite dimensional algebra of finite global dimension then A has no 2-truncated cycles in its quiver

39 The 2-truncated cycles version of the no loops conjecture Definition (m-truncated cycle) Let be a quiver with a finite number of vertices and I( R 2 ) an ideal in the path algebra K A finite sequence α 1,, α u of arrows in K /I which satisfies the equations t(α i ) = s(α i+1 )(i = 1,, u 1) and t(α u ) = s(α 1 ) is an m-truncated cycle for an integer m 2 if α i α i+m 1 = 0 and α i α i+m 2 0 in K /I for all i, where the indices are modulo u Definition (Hochschild homology dimension) The Hochschild homology dimension of an algebra A, denoted by HHdim A, is defined by HHdim A = sup{n 0 HH n (A) 0}

40 The 2-truncated cycles version of the no loops conjecture Theorem (Bergh, Han, Madsen, 2012) Let K be a commutative ring, a quiver with a finite number of vertices and I an ideal contained in R 2 Suppose K /I contains a 2-truncated cycle x 1,, x u Then for every n 1 with un u (mod 2), the element (x 1 x u ) n represents a nonzero element in HH un 1 (K /I) In particular, HHdim K /I = By the above theorem, they show the 2-truncated cycles version of the no loops conjecture

41 The 2-truncated cycles version of the no loops conjecture Corollary (Bergh, Han, Madsen, 2012) Let K be a field, a finite quiver and I an admissible ideal in K If the algebra K /I has finite global dimension, then it contains no 2-truncated cycles Moreover, they propose the m-truncated cycles version of the no loops conjecture The m-truncated cycles version of the no loops conjecture If A is a finite dimensional algebra of finite global dimension then A has no m-truncated cycles in its quiver They show that the above conjecture holds for monoial algebaras We show that the conjecture holds for other algebras by the similar way as the proof of the 2-truncated cycles version

42 The m-truncated cycles version of the no loops conjecture Theorem (Itagaki, Sanada) Let K be a field, a finite quiver and I an ideal contained in R m Suppose that K /I contains an m-truncated cycle α 1,, α u Then for every n 1 with un 0 (mod m), the element α (c 1)m+2 α cm α 1 α 2 α m α m+1 α m+2 α 2m α 2m+1 α (c 2)m+2 α (c 1)m α (c 1)m+1 represents a nonzero element in HH 2c 1 (K /I), where c = un/m In particular, the Hochschild homology dimension HHdim (K /I) =

43 The m-truncated cycles version of the no loops conjecture For an algebra A = K /I with I R m, by the surjective map A A = K /R m and the chain map π which induce the map π : H n (A A e Q A ) H n (A A e P A ), we have the composition map as follows: H n (C(A)) H n (C(A )) H n (A K e 0 (rad A ) K 0 ) H n (A A e Q A ) π H n (A A e P A ) H n (L) = HH n (A ) We already find the basis of H n (L) in order to compute the dimension formula of the cyclic homology of truncated quiver algebras Therefore, it is enough to find an element of H n (C(A)) whose image of the above composition map coincides with a member of the basis of H n (L)

44 The m-truncated cycles version of the no loops conjecture Corollary (Itagaki, Sanada) Let K be an algebraically closed field, a finite quiver and I an admissible ideal in K with I R m If the algebra K /I has finite global dimension, then it contains no m-truncated cycles

45 The m-truncated cycles version of the no loops conjecture Let A be an algebra given by the quiver: α 1 β 1 with relations: α 2 β 2 (α i α i+1 ) 3 = (β i β i+1 ) 2 = 0, α 1 β 1 β 2 α 2 = (α 1 α 2 ) 2, where the indices are modulo 2 and i is a positive integer modulo 2 Then, A has the 4-truncated cycle β 1, β 2 Therefore the global dimension of A is infinite

46 The m-truncated cycles version of the no loops conjecture [1] G Ames, L Cagliero, P Tirao, Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras, J Algebra 322(5)(2009), [2] PA Bergh, Y Han, D Madsen, Hochschild homology and truncated cycles, Proc Amer Math Soc (2012), no 4, [3] C Cibils, Cohomology of incidence algebras and simplicial complexes, J Pure Appl Algebra 56(3) (1989), [4] C Cibils, Cyclic and Hochschild homology of 2-nilpotent algebras, K-theory 4 (1990), [5] Y Han, Hochschild (co)homology dimension, J Lond Math Soc (2) 73 (2006), [6] K Igusa, Notes on the no loops conjecture, J Pure Appl Algebra 69 (1990),

47 The m-truncated cycles version of the no loops conjecture [7] K Igusa, S Liu, C Paquette, A proof of the strong no loop conjecture, e-print arxiv [8] T Itagaki, K Sanada, The dimension formula of the cyclic homology of truncated quiver algebras over a field of positive characteristic, preprint [9] T Itagaki, K Sanada, Notes on the Hochschild homology dimension and truncated cycles, in preparation [10] S König, Y-M Liu, G Zhou, Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander-Reiten conjecture, Trans Amer Math Soc 364(2012) [11] H Lenzing, Nilpotence Elemente in Ringen von endlicher globaler Dimension, Math Z 108 (1969) [12] J-L Loday, Cyclic homology, Springer-Verlag, Berlin (1992)

48 The m-truncated cycles version of the no loops conjecture [13] E Sköldberg, Hochschild homology of truncated and quadratic monomial algebras, J Lond Math Soc (2) 59 (1999), [14] E Sköldberg, Cyclic homology of quadratic monomial algebras, J Pure Appl Algebra 156 (2001), [15] R Taillefer, Cyclic homology of Hopf algebras, K-Theory 24 (2001), 69 85