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D ERIVED T AME N AKAYAMA A LGEBRAS Jos A. Vlez-Marulanda V ALDOSTA S TATE U NIVERSITY J OINT - WORK WITH Viktor Bekkert Universidade Federal de Minas Gerais & Hernn Giraldo U NIVERSIDAD DE A NTIOQUIA M AURICE A USLANDER D ISTINGUISHED

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SLIDE 1

DERIVED TAME NAKAYAMA ALGEBRAS

José A. Vélez-Marulanda

VALDOSTA STATE UNIVERSITY JOINT-WORK WITH

Viktor Bekkert

Universidade Federal de Minas Gerais

& Hernán Giraldo

UNIVERSIDAD DE ANTIOQUIA

MAURICE AUSLANDER DISTINGUISHED LECTURES AND INTERNATIONAL CONFERENCE WOODS HOLE, MA, APRIL 25-30, 2018

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SLIDE 2

SET UP

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

In this talk:

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SLIDE 3

SET UP

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

In this talk:

  • k is an algebraically closed field.
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SLIDE 4

SET UP

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

In this talk:

  • k is an algebraically closed field.
  • Λ denotes a fixed basic connected finite-dimensional k-algebra.
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SLIDE 5

SET UP

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

In this talk:

  • k is an algebraically closed field.
  • Λ denotes a fixed basic connected finite-dimensional k-algebra.
  • Unless explicitly stated, all modules are finitely generated and from the

left side.

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SLIDE 6

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

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SLIDE 7

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 1. Let Λ = Λǫ1 ⊕ · · · ⊕ Λǫn, where each ǫi is a primitive orthogonal idem- potent in Λ.

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SLIDE 8

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 1. Let Λ = Λǫ1 ⊕ · · · ⊕ Λǫn, where each ǫi is a primitive orthogonal idem- potent in Λ.

  • For all projective Λ-modules P, let r(P) = (p1, p2, . . . , pn), where p1, p2, . . . , pn are

non-negative integers such that P =

n

  • i=1

piΛǫi.

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SLIDE 9

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 1. Let Λ = Λǫ1 ⊕ · · · ⊕ Λǫn, where each ǫi is a primitive orthogonal idem- potent in Λ.

  • For all projective Λ-modules P, let r(P) = (p1, p2, . . . , pn), where p1, p2, . . . , pn are

non-negative integers such that P =

n

  • i=1

piΛǫi.

  • For all complexes of projective Λ-modules (P•, δ•), the vector rank r•(P•) is de-

fined as r•(P•) = (. . . , r(Pn−1), r(Pn), r(Pn+1), . . .).

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SLIDE 10

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 1. Let Λ = Λǫ1 ⊕ · · · ⊕ Λǫn, where each ǫi is a primitive orthogonal idem- potent in Λ.

  • For all projective Λ-modules P, let r(P) = (p1, p2, . . . , pn), where p1, p2, . . . , pn are

non-negative integers such that P =

n

  • i=1

piΛǫi.

  • For all complexes of projective Λ-modules (P•, δ•), the vector rank r•(P•) is de-

fined as r•(P•) = (. . . , r(Pn−1), r(Pn), r(Pn+1), . . .).

  • A rational family of bounded minimal complexes over Λ is a bounded complex

(P•, δ•) of projective Λ-R-bimodules, where R = k[t, f (t)−1] with f a non-zero

polynomial, and Im δn ⊆ rad Pn+1.

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SLIDE 11

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

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SLIDE 12

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • For a rational family (P•, δ•), we define the complex P•(m, λ) = (P• ⊗k R/(t −

λ)m, δ• ⊗ 1) of projective Λ-modules, where m ∈ N, λ ∈ k, f (λ) = 0.

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SLIDE 13

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • For a rational family (P•, δ•), we define the complex P•(m, λ) = (P• ⊗k R/(t −

λ)m, δ• ⊗ 1) of projective Λ-modules, where m ∈ N, λ ∈ k, f (λ) = 0.

  • We set r•(P•) = r•(P•(1, λ)) (r• does not depend on λ).
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SLIDE 14

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • For a rational family (P•, δ•), we define the complex P•(m, λ) = (P• ⊗k R/(t −

λ)m, δ• ⊗ 1) of projective Λ-modules, where m ∈ N, λ ∈ k, f (λ) = 0.

  • We set r•(P•) = r•(P•(1, λ)) (r• does not depend on λ).

Definition 2 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). We say that Λ is de- rived tame if there exists a set P of rational families of bounded complexes over Λ such that: (i) For each bounded vector v• = (vi)i∈Z of non-negative integers, the set P(v•) = {P• ∈ P | r•(P•) = v•} is finite.

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SLIDE 15

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • For a rational family (P•, δ•), we define the complex P•(m, λ) = (P• ⊗k R/(t −

λ)m, δ• ⊗ 1) of projective Λ-modules, where m ∈ N, λ ∈ k, f (λ) = 0.

  • We set r•(P•) = r•(P•(1, λ)) (r• does not depend on λ).

Definition 2 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). We say that Λ is de- rived tame if there exists a set P of rational families of bounded complexes over Λ such that: (i) For each bounded vector v• = (vi)i∈Z of non-negative integers, the set P(v•) = {P• ∈ P | r•(P•) = v•} is finite. (ii) For each vector v•, all indecomposable complexes (P′•, δ′•) of projective Λ- modules with r•(P′•) = v•, except finitely many of them (up to isomorphism), are isomorphic to P•(m, λ) for some P• ∈ P.

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SLIDE 16

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • For a rational family (P•, δ•), we define the complex P•(m, λ) = (P• ⊗k R/(t −

λ)m, δ• ⊗ 1) of projective Λ-modules, where m ∈ N, λ ∈ k, f (λ) = 0.

  • We set r•(P•) = r•(P•(1, λ)) (r• does not depend on λ).

Definition 2 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). We say that Λ is de- rived tame if there exists a set P of rational families of bounded complexes over Λ such that: (i) For each bounded vector v• = (vi)i∈Z of non-negative integers, the set P(v•) = {P• ∈ P | r•(P•) = v•} is finite. (ii) For each vector v•, all indecomposable complexes (P′•, δ′•) of projective Λ- modules with r•(P′•) = v•, except finitely many of them (up to isomorphism), are isomorphic to P•(m, λ) for some P• ∈ P. Remark 3. The definition of derived tameness above is equivalent to the one given in (CH. GEISS, H. KRAUSE, 2002).

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SLIDE 17

DERIVED TAMENESS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • For a rational family (P•, δ•), we define the complex P•(m, λ) = (P• ⊗k R/(t −

λ)m, δ• ⊗ 1) of projective Λ-modules, where m ∈ N, λ ∈ k, f (λ) = 0.

  • We set r•(P•) = r•(P•(1, λ)) (r• does not depend on λ).

Definition 2 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). We say that Λ is de- rived tame if there exists a set P of rational families of bounded complexes over Λ such that: (i) For each bounded vector v• = (vi)i∈Z of non-negative integers, the set P(v•) = {P• ∈ P | r•(P•) = v•} is finite. (ii) For each vector v•, all indecomposable complexes (P′•, δ′•) of projective Λ- modules with r•(P′•) = v•, except finitely many of them (up to isomorphism), are isomorphic to P•(m, λ) for some P• ∈ P. Remark 3. The definition of derived tameness above is equivalent to the one given in (CH. GEISS, H. KRAUSE, 2002). Theorem 4 ((CH. GEISS, H. KRAUSE, 2002)). Derived tameness is preserved by de- rived equivalence.

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SLIDE 18

SOME EXAMPLES OF DERIVED TAME ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

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SLIDE 19

SOME EXAMPLES OF DERIVED TAME ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • Every derived discrete algebra (in the sense of (D. VOSSIECK, 2001)) is derived

tame.

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SLIDE 20

SOME EXAMPLES OF DERIVED TAME ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • Every derived discrete algebra (in the sense of (D. VOSSIECK, 2001)) is derived

tame.

  • If Λ has finite global dimension, then Λ is derived tame if and only if its repetitive

algebra ˆ Λ is tame ((DE LA PEÑA, 1998) & (CH. GEISS, H. KRAUSE, 2002)).

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SLIDE 21

SOME EXAMPLES OF DERIVED TAME ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • Every derived discrete algebra (in the sense of (D. VOSSIECK, 2001)) is derived

tame.

  • If Λ has finite global dimension, then Λ is derived tame if and only if its repetitive

algebra ˆ Λ is tame ((DE LA PEÑA, 1998) & (CH. GEISS, H. KRAUSE, 2002)).

  • If Λ is piecewise hereditary, then Λ is derived tame (CH. GEISS, 2002).
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SLIDE 22

SOME EXAMPLES OF DERIVED TAME ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • Every derived discrete algebra (in the sense of (D. VOSSIECK, 2001)) is derived

tame.

  • If Λ has finite global dimension, then Λ is derived tame if and only if its repetitive

algebra ˆ Λ is tame ((DE LA PEÑA, 1998) & (CH. GEISS, H. KRAUSE, 2002)).

  • If Λ is piecewise hereditary, then Λ is derived tame (CH. GEISS, 2002).

Definition 5. Assume that Λ has finite global dimension. The Euler form χΛ of Λ is defined on the Grothendieck group of Λ by χΛ(dim M) =

∞

∑

i=0

(−1)i dimk Exti

Λ(M, M)

for every Λ-module M.

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SLIDE 23

SOME EXAMPLES OF DERIVED TAME ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • Every derived discrete algebra (in the sense of (D. VOSSIECK, 2001)) is derived

tame.

  • If Λ has finite global dimension, then Λ is derived tame if and only if its repetitive

algebra ˆ Λ is tame ((DE LA PEÑA, 1998) & (CH. GEISS, H. KRAUSE, 2002)).

  • If Λ is piecewise hereditary, then Λ is derived tame (CH. GEISS, 2002).

Definition 5. Assume that Λ has finite global dimension. The Euler form χΛ of Λ is defined on the Grothendieck group of Λ by χΛ(dim M) =

∞

∑

i=0

(−1)i dimk Exti

Λ(M, M)

for every Λ-module M.

  • If Λ is a tree algebra, then Λ is derived tame if and only if χΛ is non-negative

((J.A. DE LA PEÑA, 1998) & (TH. BRÜSTLE, 2001)).

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SLIDE 24

SOME EXAMPLES OF DERIVED TAME ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

  • Every derived discrete algebra (in the sense of (D. VOSSIECK, 2001)) is derived

tame.

  • If Λ has finite global dimension, then Λ is derived tame if and only if its repetitive

algebra ˆ Λ is tame ((DE LA PEÑA, 1998) & (CH. GEISS, H. KRAUSE, 2002)).

  • If Λ is piecewise hereditary, then Λ is derived tame (CH. GEISS, 2002).

Definition 5. Assume that Λ has finite global dimension. The Euler form χΛ of Λ is defined on the Grothendieck group of Λ by χΛ(dim M) =

∞

∑

i=0

(−1)i dimk Exti

Λ(M, M)

for every Λ-module M.

  • If Λ is a tree algebra, then Λ is derived tame if and only if χΛ is non-negative

((J.A. DE LA PEÑA, 1998) & (TH. BRÜSTLE, 2001)).

  • If Λ is either gentle or skewed-gentle, then Λ is derived tame (V. BEKKERT, H.

MERKLEN, 2003) AND (V. BEKKERT, E. N. MARCOS, H. MERKLEN, 2003).

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SLIDE 25

DERIVED WILD ALGEBRAS AND THE DERIVED TAME-WILD DICHOTOMY THEOREM

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

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SLIDE 26

DERIVED WILD ALGEBRAS AND THE DERIVED TAME-WILD DICHOTOMY THEOREM

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 6 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). Let Σ = kx, y. We say that Λ is derived wild if there exists a bounded complex (P•, δ•) of projective Λ-Σ-bimodules such that Im δn ⊆ rad Pn+1, and for all Σ-modules L and L′ with finite dimension over k, we have: (i) P• ⊗Σ L ∼

= P• ⊗Σ L′ if and only if L ∼ = L′;

(ii) P• ⊗Σ L is indecomposable if and only if so is L.

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SLIDE 27

DERIVED WILD ALGEBRAS AND THE DERIVED TAME-WILD DICHOTOMY THEOREM

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 6 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). Let Σ = kx, y. We say that Λ is derived wild if there exists a bounded complex (P•, δ•) of projective Λ-Σ-bimodules such that Im δn ⊆ rad Pn+1, and for all Σ-modules L and L′ with finite dimension over k, we have: (i) P• ⊗Σ L ∼

= P• ⊗Σ L′ if and only if L ∼ = L′;

(ii) P• ⊗Σ L is indecomposable if and only if so is L. Theorem 7 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). A finite dimensional

k-algebra Λ is either derived tame or derived wild.

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SLIDE 28

DERIVED WILD ALGEBRAS AND THE DERIVED TAME-WILD DICHOTOMY THEOREM

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 6 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). Let Σ = kx, y. We say that Λ is derived wild if there exists a bounded complex (P•, δ•) of projective Λ-Σ-bimodules such that Im δn ⊆ rad Pn+1, and for all Σ-modules L and L′ with finite dimension over k, we have: (i) P• ⊗Σ L ∼

= P• ⊗Σ L′ if and only if L ∼ = L′;

(ii) P• ⊗Σ L is indecomposable if and only if so is L. Theorem 7 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). A finite dimensional

k-algebra Λ is either derived tame or derived wild.

  • If Λ is self-injective, then Λ is either derived discrete or derived wild (R. BAUTISTA,

2007).

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SLIDE 29

DERIVED WILD ALGEBRAS AND THE DERIVED TAME-WILD DICHOTOMY THEOREM

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 6 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). Let Σ = kx, y. We say that Λ is derived wild if there exists a bounded complex (P•, δ•) of projective Λ-Σ-bimodules such that Im δn ⊆ rad Pn+1, and for all Σ-modules L and L′ with finite dimension over k, we have: (i) P• ⊗Σ L ∼

= P• ⊗Σ L′ if and only if L ∼ = L′;

(ii) P• ⊗Σ L is indecomposable if and only if so is L. Theorem 7 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). A finite dimensional

k-algebra Λ is either derived tame or derived wild.

  • If Λ is self-injective, then Λ is either derived discrete or derived wild (R. BAUTISTA,

2007).

  • If Λ is a self-injective Nakayama algebra with Loewy length ll(Λ) ≥ 3, then Λ is

derived wild (C. ZHANG, 2018).

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SLIDE 30

DERIVED WILD ALGEBRAS AND THE DERIVED TAME-WILD DICHOTOMY THEOREM

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 6 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). Let Σ = kx, y. We say that Λ is derived wild if there exists a bounded complex (P•, δ•) of projective Λ-Σ-bimodules such that Im δn ⊆ rad Pn+1, and for all Σ-modules L and L′ with finite dimension over k, we have: (i) P• ⊗Σ L ∼

= P• ⊗Σ L′ if and only if L ∼ = L′;

(ii) P• ⊗Σ L is indecomposable if and only if so is L. Theorem 7 ((V. BEKKERT, YU. DROZD, ARXIV:MATH/0310352)). A finite dimensional

k-algebra Λ is either derived tame or derived wild.

  • If Λ is self-injective, then Λ is either derived discrete or derived wild (R. BAUTISTA,

2007).

  • If Λ is a self-injective Nakayama algebra with Loewy length ll(Λ) ≥ 3, then Λ is

derived wild (C. ZHANG, 2018). Question: Which Nakayama algebras Λ are derived tame?

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SLIDE 31

NAKAYAMA ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

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SLIDE 32

NAKAYAMA ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Recall that Λ is said to be a Nakayama algebra if every left or right indecompos- able projective Λ-module has a unique composition series.

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SLIDE 33

NAKAYAMA ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Recall that Λ is said to be a Nakayama algebra if every left or right indecompos- able projective Λ-module has a unique composition series. The next result is well-known.

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SLIDE 34

NAKAYAMA ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Recall that Λ is said to be a Nakayama algebra if every left or right indecompos- able projective Λ-module has a unique composition series. The next result is well-known. Theorem 8. Λ is a Nakayama algebra if and only if Λ = kQ/I, where Q is one of the following quivers: Ln : •

1 a1

2 a2 . . . an−1 • n

Cn : •

a0

1 a1 . . . an−2 • n−1 an−1

  • for some n ≥ 1, and I is an admissible ideal of kQ.
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SLIDE 35

NAKAYAMA ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Recall that Λ is said to be a Nakayama algebra if every left or right indecompos- able projective Λ-module has a unique composition series. The next result is well-known. Theorem 8. Λ is a Nakayama algebra if and only if Λ = kQ/I, where Q is one of the following quivers: Ln : •

1 a1

2 a2 . . . an−1 • n

Cn : •

a0

1 a1 . . . an−2 • n−1 an−1

  • for some n ≥ 1, and I is an admissible ideal of kQ.

Assume that Λ = kQ/I is a Nakayama algebra.

  • If Q = Ln, then we say that Λ is a line algebra.
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SLIDE 36

NAKAYAMA ALGEBRAS

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Recall that Λ is said to be a Nakayama algebra if every left or right indecompos- able projective Λ-module has a unique composition series. The next result is well-known. Theorem 8. Λ is a Nakayama algebra if and only if Λ = kQ/I, where Q is one of the following quivers: Ln : •

1 a1

2 a2 . . . an−1 • n

Cn : •

a0

1 a1 . . . an−2 • n−1 an−1

  • for some n ≥ 1, and I is an admissible ideal of kQ.

Assume that Λ = kQ/I is a Nakayama algebra.

  • If Q = Ln, then we say that Λ is a line algebra.
  • If Q = Cn, then we say that Λ is a cycle algebra.
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SLIDE 37

DERIVED TAME NAKAYAMA ALGEBRAS AND GORENSTEIN-PROJECTIVE MODULES

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

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SLIDE 38

DERIVED TAME NAKAYAMA ALGEBRAS AND GORENSTEIN-PROJECTIVE MODULES

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Theorem 9. (V. BEKKERT, H. GIRALDO, V-M,

IN PROGRESS) Assume that Λ is a

Nakayama algebra. Then Λ is derived tame if and only if one of the following conditions holds: (i) Λ is a line algebra whose Euler form is non-negative. (ii) Λ is either gentle or derived equivalent to some skewed-gentle algebra.

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SLIDE 39

DERIVED TAME NAKAYAMA ALGEBRAS AND GORENSTEIN-PROJECTIVE MODULES

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Theorem 9. (V. BEKKERT, H. GIRALDO, V-M,

IN PROGRESS) Assume that Λ is a

Nakayama algebra. Then Λ is derived tame if and only if one of the following conditions holds: (i) Λ is a line algebra whose Euler form is non-negative. (ii) Λ is either gentle or derived equivalent to some skewed-gentle algebra. Definition 10. (E. ENOCHS, O. JENDA, 1995) A Λ-module V is said to be Gorenstein- projective if there exists an acyclic complex of projective Λ-modules P• : · · · → P−2 δ−2

− − → P−1 δ−1 − − → P0 δ0 − → P1 δ1 − → P2 → · · ·

such that HomΛ(P•, Λ) is also acyclic and V = coker δ0. We denote by Λ-Gproj the category of Gorenstein-projective Λ-modules that are finitely generated, and by Λ-Gproj its stable category.

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SLIDE 40

THE SINGULARITY CATEGORY OF A DERIVED TAME NAKAYAMA ALGEBRA

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 11. The singularity category of Λ is defined to be the Verdier quotient

Dsg(Λ-mod) = Db(Λ-mod)/Kb(Λ-proj).

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SLIDE 41

THE SINGULARITY CATEGORY OF A DERIVED TAME NAKAYAMA ALGEBRA

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 11. The singularity category of Λ is defined to be the Verdier quotient

Dsg(Λ-mod) = Db(Λ-mod)/Kb(Λ-proj).

Recall that Λ is Gorenstein if the injective dimensions of Λ as a left Λ-module and as a right Λ-module are finite.

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SLIDE 42

THE SINGULARITY CATEGORY OF A DERIVED TAME NAKAYAMA ALGEBRA

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 11. The singularity category of Λ is defined to be the Verdier quotient

Dsg(Λ-mod) = Db(Λ-mod)/Kb(Λ-proj).

Recall that Λ is Gorenstein if the injective dimensions of Λ as a left Λ-module and as a right Λ-module are finite.

  • Gorensteinness is preserved by derived equivalence (A. BELIGIANNIS, 2005).
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SLIDE 43

THE SINGULARITY CATEGORY OF A DERIVED TAME NAKAYAMA ALGEBRA

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 11. The singularity category of Λ is defined to be the Verdier quotient

Dsg(Λ-mod) = Db(Λ-mod)/Kb(Λ-proj).

Recall that Λ is Gorenstein if the injective dimensions of Λ as a left Λ-module and as a right Λ-module are finite.

  • Gorensteinness is preserved by derived equivalence (A. BELIGIANNIS, 2005).
  • Gentle and skewed-gentle algebras are Gorenstein ((CH.

GEISS & I. REITEN, 2005) and (X. CHEN & M. LU, 2017)).

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SLIDE 44

THE SINGULARITY CATEGORY OF A DERIVED TAME NAKAYAMA ALGEBRA

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 11. The singularity category of Λ is defined to be the Verdier quotient

Dsg(Λ-mod) = Db(Λ-mod)/Kb(Λ-proj).

Recall that Λ is Gorenstein if the injective dimensions of Λ as a left Λ-module and as a right Λ-module are finite.

  • Gorensteinness is preserved by derived equivalence (A. BELIGIANNIS, 2005).
  • Gentle and skewed-gentle algebras are Gorenstein ((CH.

GEISS & I. REITEN, 2005) and (X. CHEN & M. LU, 2017)).

  • If Λ is Gorenstein, then Dsg(Λ-mod) = Λ-Gproj ((R.O. BUCHWEITZ, PREPRINT) and

(D. HAPPEL, 1992)).

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THE SINGULARITY CATEGORY OF A DERIVED TAME NAKAYAMA ALGEBRA

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

Definition 11. The singularity category of Λ is defined to be the Verdier quotient

Dsg(Λ-mod) = Db(Λ-mod)/Kb(Λ-proj).

Recall that Λ is Gorenstein if the injective dimensions of Λ as a left Λ-module and as a right Λ-module are finite.

  • Gorensteinness is preserved by derived equivalence (A. BELIGIANNIS, 2005).
  • Gentle and skewed-gentle algebras are Gorenstein ((CH.

GEISS & I. REITEN, 2005) and (X. CHEN & M. LU, 2017)).

  • If Λ is Gorenstein, then Dsg(Λ-mod) = Λ-Gproj ((R.O. BUCHWEITZ, PREPRINT) and

(D. HAPPEL, 1992)). Corollary 12. If Λ is a derived tame Nakayama algebra, then Λ is Gorenstein, and consequently, if Λ is further a cycle algebra, then Dsg(Λ-mod) = Λ-Gproj.

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THE SINGULARITY CATEGORY OF A DERIVED TAME NAKAYAMA ALGEBRA

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

By using the description of the singularity category of a gentle algebra in (M. KALCK, 2015), we obtain the following result.

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THE SINGULARITY CATEGORY OF A DERIVED TAME NAKAYAMA ALGEBRA

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

By using the description of the singularity category of a gentle algebra in (M. KALCK, 2015), we obtain the following result. Corollary 13. Let Λ = kQ/I is a derived tame cycle algebra, and let |RΛ| the min- imal number of relations defining I. If Λ has infinite global dimension, then there exists an equivalence of triangulated categories

Dsg(Λ-mod) ∼ = Db(k-mod)/[|RΛ|],

where Db(k-mod)/[|RΛ|] denotes the orbit category (in the sense of (B. KELLER, 2005)) .

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SLIDE 48

THE SINGULARITY CATEGORY OF A DERIVED TAME NAKAYAMA ALGEBRA

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

By using the description of the singularity category of a gentle algebra in (M. KALCK, 2015), we obtain the following result. Corollary 13. Let Λ = kQ/I is a derived tame cycle algebra, and let |RΛ| the min- imal number of relations defining I. If Λ has infinite global dimension, then there exists an equivalence of triangulated categories

Dsg(Λ-mod) ∼ = Db(k-mod)/[|RΛ|],

where Db(k-mod)/[|RΛ|] denotes the orbit category (in the sense of (B. KELLER, 2005)) . The following result classifies the isomorphism class of versal deformation rings of Gorenstein-projective modules (in the sense of (F. M. BLEHER, V-M, 2012)) over de- rived tame Nakayama algebras.

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THE SINGULARITY CATEGORY OF A DERIVED TAME NAKAYAMA ALGEBRA

Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

By using the description of the singularity category of a gentle algebra in (M. KALCK, 2015), we obtain the following result. Corollary 13. Let Λ = kQ/I is a derived tame cycle algebra, and let |RΛ| the min- imal number of relations defining I. If Λ has infinite global dimension, then there exists an equivalence of triangulated categories

Dsg(Λ-mod) ∼ = Db(k-mod)/[|RΛ|],

where Db(k-mod)/[|RΛ|] denotes the orbit category (in the sense of (B. KELLER, 2005)) . The following result classifies the isomorphism class of versal deformation rings of Gorenstein-projective modules (in the sense of (F. M. BLEHER, V-M, 2012)) over de- rived tame Nakayama algebras. Corollary 14. Let Λ be a derived tame Nakayama algebra, and let V be in Λ- Gproj. If V is indecomposable, then the versal deformation ring R(Λ, V) of V is universal and isomorphic either to k or to k[[t]]/(t2).

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Derived Tame Nakayama Algebras J.A. Vélez-Marulanda

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