Nakayama algebras of Loewy length Claus Michael Ringel (Bielefeld) Paderborn, May 0, 00
We consider the algebra A(n) = A(n, ), with quiver n n (all arrows will be labelled α) Relations: α = 0 ndecomposable modules are serial and of length at most, those of length are projective-injective S(t) denotes the simple module corresponding to the vertex t P(t) or just t will denote the projective mdoule with top S(t) The Auslander-Reiten quiver for n = :
C(Λ) the category of bounded complexes M P(Λ) the category of perfect complexes (ie: all M i projective) P(Λ) the homotopy category of perfect complexes D b (mod Λ) C(Λ) P(Λ) P(Λ) Seidel, Happel Lenzing, Geigle, Kussin, Meltzer, de la Peña Typical maps between indecomposable projective modules: P(i + ) P(i), P(i + ) P(i) Note: Composing maps of the first type is non-zero, composing maps of the second type always yields zero M indecomposable with all M i indecomposable projective or zero: P(i 0 ) P(i ) P(i ) P(i t ) P(i t ) has jumps i s i s+ either 0 or We get a 0--sequence without two consecutive 0 s
Λ Type of P(Λ) A() A() A() A() A() A() A() A() A() A(0) A() A() A() A() A A A D D E E E Ẽ E wild not piecewise hereditary C(,,) C(,,) C(,,) C(,,) + C(p, q, r) ist the canonical algebra of type (p, q, r) C(p,q,r) + is its one-point extension using the simple projective module Seidel: A n piecewise hereditary n Happel-Zacharia: Piecewise hered bounded strong global dimension
Aims: For n = : The one-parameter family For n : ndecomposable perfect complexes of arbitrarily large width For any n: Search for complexes with τ s M = M [t] for some s,t The wings arising from jumping between the boundaries The quiver of C(A(n)): a horicontal stripe with arrows pointing downwards and from right to left δ α 0 here, δ is the differential (pointing to the left) The relations: α = 0, δ = 0, αδ = δα The vertices of the quiver are pairs (i, j) of integers, with i n P(A(n)): the objects with vertical restrictions being projective A(n)-modules The indecomposable projective A(n) module with top S(t) is denoted by t j
One-parameter families for n = A() A A() A A() A A() D A() D A() E A() E A() A() E Ẽ C(,,) A(0) E C(,,) A() wild C(,,) A() not piecewise hereditary C(,,) + A() A()
One-parameter families for n = We want to construct one-parameter families of indecomposable objects q q q u u epi q q u u mono u These are complexes, however not perfect ones: the problem arises at the top of the stairs: We have to prolong the stairs as follows:
Ker q q q q q u u q q u u u u
Let us repeat: There is a full exact embedding functor η W W Ker q W q V W W q q f V V u u U U η q V W V V q f V V V u U u V V V u U Better: For any column index t, there is a full exact embedding functor η t
For n = 0: a second one-parameter family, shifting the stair one step up (and modifying the lower part in order to obtain a perfect complex): Ker q q Ker q q q q q q u u u u u u Coku 0
For n =, we obtain a strictly wild category: Ker q q q q u u Coku u 0
A() A A() A A() A A() D A() D A() E A() E A() E A() Ẽ C(,,) A(0) A() E wild C(,,) C(,,) A() not piecewise hereditary C(,,) + A() A()
The wings Consider thin objects Two extreme cases: δ α jumps resolution encircled is the homology
Border sequences We consider the jumps in more detail Recall (Butler-Ringel, ): Any arrow γ : x y gives rise to an Auslander- Reiten sequence 0 K(γ) M(γ) Q(γ) 0 Q(γ) is the cokernel of P(γ): P(y) P(x) and K(γ) is the kernel of (γ): (y) (x) and M(γ) is indecomposable K(δ) δ M(δ) Q(δ) t t+ This is an Auslander-Reiten sequence in C(Λ), f δ does not belong to the upper two rows, then the sequence belongs to P(Λ)
Wings W W 0 0 W even 0 W odd
Wings for n even 0 0 W W 0 0 W even 0 0 W odd
The neighbors of a wing An example: The odd wing for n = W Here, = 0 is the projective resolution of the module in the column 0 ( indecomposable, with composition factors S(n) and S(n )), thus: 0 0
Recall: For any column index t, there is a full exact embedding functor η t W W Ker q W q V W W q q f V V u u U U η t q V W V V q f V V V u U u V V V u U t and we have η η 0 0 0 0 = W =
The odd-homology wing for n = W W W 0 0 homogeneous dim vectors 0 0 W 0
n = W 0 0 W
Next example: n = 0 This is a tubular catgeory There are (up to shift) two basic one-parameter families, the tubes of rank are obtained from our wings: W 0 W
n general: Objects which are neighbors of the wings = S(n) = S(n ) indecomposable of length, with composition factors and Lemma The left neighbor is either or, the right neighbor is or or n 0 mod (n,n ) (,) n n n,n nn n mod n (,) (n,n ) n n,n nn n mod n (,) n (n,n ) nn n,n
The main table () n = t + Even Homology Odd Homology [] [] [] [] [ ] [ ] [ ] [] [] [] [] [] [] [ ] n = 0
Further border sequences 0 (t +,t + ) 0 X (t +,t) 0 where X is as follows (for t = ): 0 0 Again, we obtain a wing!
The main table () n even Even Homology Odd Homology [] [] [] [] [ ] [ ] [ ] [] [] [] [] [] [] [ ] [] [ ] [ ] n = 0
The main table () n = t +, t + Even Homology Odd Homology [] [] [] [] [ ] [ ] [ ] [] [] [] [] [] [] [ ] [] [ ] [ ] n = 0
Consequences Theorem (a) f n = t +, then τ (n+)/ n (b) f n = t +, then τ n+ n (c) f n is even, τ (n+)/ n is the shift functor [ n 0 ] is the shift functor [ n 0 ] is the shift functor [ n 0 ]
The odd-homology wing for n = 0 W 0 0 W
0 W This yields indecomposable perfect complexes of arbitrary width Thus: The strong global dimension of A() is Even: The strong global dimension of A() is For A(), there are Auslander-Reiten sequences at the border which can be used in order to construct a module M with an extension of M and M[ ]
The main table () n even Even Homology Odd Homology [] [] [] [] [ ] [ ] [ ] [] [] [] [] [] [] [ ] [] [ ] [ ] n = 0
The case n = The case n =
The case n =
The case n =
The case n =