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COMPLETE SPECIAL BISERIAL ALGEBRAS ARE g-TAME

TOSHITAKA AOKI AND TOSHIYA YURIKUSA

Abstract. The g-vectors of two-term presilting complexes are important invariant. We study a fanconsisting of all g-vector cones for a complete gentle algebra. We show that any complete gentlealgebra is g-tame, by definition, the closure of a geometric realization of its fan is the entire ambientvector space. Our main ingredients are their surface model and their asymptotic behavior underDehn twists. On the other hand, it is known that any complete special biserial algebra is a factoralgebra of a complete gentle algebra and the g-tameness is preserved under taking factor algebras.As a consequence, we get the g-tameness of complete special biserial algebras.

1. Introduction

Gentle algebras, introduced in 1980’s, form an important class of special biserial algebras and theirrepresentation theory has been studied by many authors (e.g. [AH81, AS87, BR87]). Moreover, thederived categories of gentle algebras are related to various subjects, such as discrete derived categories[Voß01], numerical derived invariants [AAG08, APS19], and Fukaya categories of surfaces [HKK17,LP20].

An aim of this paper is to study two-term silting theory for gentle algebras. In this paper, wedon’t assume that gentle algebras are finite dimensional. For our purpose, we consider the completegentle algebras (see Definition 7.10). They are module-finite over k[[t]] (i.e., finitely generated as ank[[t]]-module), where k[[t]] is the formal power series ring of one value over an algebraically closed fieldk. In particular, finite dimensional gentle algebras are complete gentle algebras.

We discuss two-term silting theory over module-finite k[[t]]-algebras, see Section 7 (cf. [Kim20]). Fora module-finite k[[t]]-algebra A, the homotopy category K

b(projA) of bounded complexes of finitelygenerated projective right A-modules is Krull-Schmidt. We denote by 2-presiltA (resp., 2-siltA)the set of isomorphism classes of basic two-term presilting (resp., silting) complexes for A. EachT ∈ 2-presiltA has a numerical invariant g(T ) ∈ Zn, called the g-vector of T , where n is the numberof non-isomorphic indecomposable direct summands of A. Then one can define a cone in Rn, calledthe g-vector cone of T , by

C(T ) :=

{∑

X

aXg(X) | aX ∈ R≥0

},

where X runs over all indecomposable direct summands of T . We denote by F(A) a collection ofg-vector cones of all basic two-term presilting complexes for A, by |F(A)| its geometric realization. Itfollows from [Kim20] that F(A) is a simplicial fan (i.e., every cone is a simplex) and its maximal facescorrespond to basic two-term silting complexes for A. Namely,

|F(A)| =⋃

C∈F(A)

C =⋃

T∈2-siltA

C(T ).

Such a fan plays an important role in the study of stability scattering diagrams and their wall-chamberstructures (see e.g. [Asa19, Bri17, BST19, Yur18]). The following result is well-known.

Theorem 1.1. [Asa19, DIJ19] Let A be a finite dimensional algebra. Then the following conditionsare equivalent:

(1) 2-siltA is finite;(2) |F(A)| = Rn.

This result naturally leads the following definition in a general setting.1

2 TOSHITAKA AOKI AND TOSHIYA YURIKUSA

Definition 1.2. Let A be a module-finite k[[t]]-algebra. We say that A is g-tame if it satisfies

|F(A)| = Rn,

where (−) is the closure with respect to the natural topology on Rn.

This means that g-vector cones are dense in the stability scattering diagram [Bri17] for a finitedimensional g-tame algebra. Note that a similar notion, called τ -tilting tame, was given in [BST19].

The g-tameness is known for path algebras of extended Dynkin quivers [Hil06], for complete pre-projective algebras of extended Dynkin graphs [KM19], and for Jacobian algebras associated withtriangulated surfaces [Yur20]. We prove the g-tameness to a new class.

Theorem 1.3. Any complete special biserial algebra is g-tame.

To prove Theorem 1.3, it suffices to prove that complete gentle algebras are g-tame. In fact, anycomplete special biserial algebra is a factor algebra of a complete gentle algebra (proposition 7.9), andg-tameness is preserved under taking factor algebras (Proposition 7.6(2)).

To prove the g-tameness of complete gentle algebras, their surface model plays a central role (see[APS19, OPS18, PPP19]). A similar construction has been developed in several area, such as [AAC18,KS02, OPS18]. For each dissection D of a ◦•-marked surface (S,M), one can define a complete gentlealgebra A(D). Conversely, any complete gentle algebra arises in this way (see Sections 2.1 and 7.2 forthe details). Note that the cardinally n of D is completely determined by (S,M) (Remark 2.3).

For a given dissection D of (S,M), we observe a certain class of non-self-intersecting curves of S,calledD-laminates, and finite multi-set of pairwise non-intersectingD-laminates, calledD-laminations.Notice that we take account of closed curves here. To eachD-laminate γ, we associate an integer vectorg(γ) ∈ Zn, called g-vector, whose entries are intersection numbers of γ and d ∈ D. The next resultis an analog of [FT18, Theorems 12.3, 13.6] and a generalization of [PPP19, Proposition 6.14] to anarbitrary dissection.

Theorem 1.4. (Theorem 4.1) The map X 7→∑

γ∈X g(γ) gives a bijection between the set of D-laminations and Zn.

We especially consider certain D-laminations. A D-lamination X is said to be reduced if it consistsof pairwise distinct non-closed D-laminates, and complete if it is reduced and maximal as a set. Wedenote by F(D) a collection of C(X ) of all reduced D-laminations X , where C(X ) is a cone in Rn

spanned by g(γ) for all γ ∈ X . In particular, F(D) is a simplicial fan whose maximal faces correspondto complete D-laminations (Proposition 2.8). We prove that the fan F(D) is dense in Rn. Namely,

Theorem 1.5. For a dissection D of a ◦•-marked surface (S,M), we have

|F(D)| = Rn.

On the other hand, we show in Section 7 that the surface model realizes a fan of g-vector cones fora complete gentle algebra A(D) of D. It completes a proof of Theorem 1.3.

Theorem 1.6 (Theorem 7.26). Let D be a dissection of a ◦•-marked surface (S,M) and A(D) thecomplete gentle algebra associated with D. Then there are bijections

T(−) : {reduced D-laminations} → 2-presiltA(D) and {complete D-laminations} → 2-siltA(D)

such that C(X ) = C(TX ). In particular, we have F(A(D)) = F(D).

A main ingredient of our proof of Theorem 1.5 is the asymptotic behavior of g-vectors under Dehntwists. This proof is inspired from the proof of [Yur20, Theorem 1.5]. In the forthcoming paper [Aok],this method plays a key role for analyzing the polytope associated with the fan F(A(D)).

This paper is organized as follows. Through to Section 6, we study the geometric and combinatorialaspects of our results. In Section 2, we recall the notions and results of [APS19, PPP19] in terms ofour notations. Before proving our results, we give some examples in Section 3. By using the examples,we prove Theorem 1.4 in Section 4. In Sections 5 and 6, we study g-vectors of D-laminates and theirasymptotic behavior under Dehn twists, and prove Theorem 1.5.

In Section 7, we study the algebraic aspects of our results. First, we recall two-term silting theoryover module-finite algebras, in particular, they include complete gentle algebras. Second, we give a

COMPLETE SPECIAL BISERIAL ALGEBRAS ARE g-TAME 3

geometric model of two-term silting theory over complete gentle algebras, and prove Theorem 1.6.Finally, we prove Theorem 1.3 and also give a relation with a special class of special biserial algebrascontaining Brauer graph algebras (see Section 7.5). These examples are given in Section 8.

2. Preliminary

In this section, we recall the notions and results of [APS19, PPP19] (see also [OPS18]). Ournotations are slightly different from theirs for the convenience of our purpose.

2.1. ◦•-marked surfaces.

Definition 2.1. A ◦•-marked surface is the pair (S,M) consisting of the following data:

(a) S is a connected compact oriented Riemann surface with (possibly empty) boundary ∂S.(b) M = M◦ ⊔M• is a non-empty finite set of marked points on S such that

– both M◦ and M• are not empty;– each component of ∂S has at least one marked point;– the points of M◦ and M• alternate on each boundary component.

Any marked point in the interior of S is called a puncture.

Let (S,M) be a ◦•-marked surface.

Definition 2.2. (1) A ◦-arc (resp., •-arc) γ of (S,M) is a curve in S with endpoints in M◦ (resp.,M•), considered up to isotopy, such that the following conditions are satisfied:

• γ does not intersect itself except at its endpoints;• γ is disjoint from M and ∂S except at its endpoints;• γ does not cut out a monogon without punctures.

(2) A ◦-dissection (resp., •-dissection) is a maximal set of pairwise non-intersecting ◦-arcs (resp.,•-arcs) on (S,M) which does not cut out a subsurface without marked points in M• (resp., M◦).

Remark 2.3. Let g be the genus of S, b be the number of boundary components and p◦ (resp., p•)be the number of punctures in M◦ (resp., M•). By [APS19, Proposition 1.11], a ◦-dissection (resp.,•-dissection) of (S,M) consists of |M◦|+ p• + b+2g− 2 = |M•|+ p◦ + b+2g− 2 ◦-arcs (resp., •-arcs).

By symmetry, the claims in this paper hold if we permute the symbols ◦ and •. Thus we state onlyone side of each claim. A dissection divides (S,M) into polygons with exactly one marked point.

Proposition 2.4. [APS19, Proposition 1.12] For a •-dissection D of (S,M), each connected compo-nent of S \D is homeomorphic to one of the following:

• an open disk with precisely one marked point in M◦ ∩ ∂S;• an open disk with precisely one marked point in M◦, but not in ∂S.

For a •-dissection D of (S,M), the closure of a connected component of S \D is called a polygonof D. Proposition 2.4 implies that any polygon of D has exactly one marked point in M◦. We denoteby △v the polygon with marked point v ∈M◦ (see Figure 1).

v

v

Figure 1. Polygon △v for a marked point v ∈M◦

Definition-Proposition 2.5. [PPP19, Proposition 3.6] For a •-dissection D of (S,M), there is aunique ◦-dissection D∗ whose each ◦-arc intersects exactly one •-arc of D. We have D∗∗ = D. Wecall D∗ the dual dissection of D. For d ∈ D, we write the corresponding ◦-arc by d∗ ∈ D∗.

4 TOSHITAKA AOKI AND TOSHIYA YURIKUSA

2.2. g-vectors of D-laminates and D-laminations. We fix a •-dissection D of (S,M).

Definition 2.6. (1) A ◦-laminate of (S,M) is a curve γ in S, considered up to isotopy relative to M ,that is either

• a closed curve, or• a curve whose ends are unmarked points on ∂S or spirals around punctures in M◦ eitherclockwise or counterclockwise (see Figure 2).

(2) A D-laminate is a non-self-intersecting ◦-laminate γ of (S,M) intersecting at least one •-arc ofD such that the following condition is satisfied:

(∗) Whenever γ intersects d ∈ D, the endpoints v and v′ of d∗ lie on opposite sides of γ in △v ∪△v′ .

Here, we consider that the point v lies on the right (resp., left) to γ if γ circles clockwise (resp.,counterclockwise) around v in △v.

Figure 2. Example of a ◦-laminate

A D-laminate is called a closed D-laminate if it is a closed curve. Remark that non-closed D-laminates coincide with D-slaloms in [PPP19]. Now, we treat a certain collection of D-laminates, thatis central in this paper.

Definition 2.7. We say that two D-laminates are compatible if they don’t intersect. A finite multi-setof pairwise compatible D-laminates is called a D-lamination. A D-lamination is said to be

• reduced if it consists of pairwise distinct non-closed D-laminates, and• complete if it is reduced and is the maximal as a set.

Let γ be a D-laminate. Using the notations in the condition (∗), let p be an intersection point of γand d such that γ leaves △v to enter △v′ via p. Then p is said to be positive (resp., negative) if v is toits right (resp., left), or equivalently, v′ is to its left (resp., right). See Figure 3. For d ∈ D, we definean integer

g(γ)d := #{positive intersection points of γ and d}(2.1)

−#{negative intersection points of γ and d}.

The g-vector g(γ) of γ is given by(g(γ)d

)d∈D

∈ Z|D|, where |D| is the number of •-arcs of D.

Remark that if γ and d intersect twice, then their intersection points are either positive or negativesimultaneously. Thus, the absolute value of g(γ)d just counts the number of intersection points of γand d. For a D-lamination X , we denote by C(X ) a cone in R|D| spanned by g(γ) for all γ ∈ Xand call it the g-vector cone of X . We denote by F(D) a collection of all g-vector cones of reducedD-laminations.

p q

Figure 3. Positive intersection point p and negative intersection point q

The invariants, g-vectors and g-vector cones, have good properties.

Theorem 2.8. [PPP19, Theorems 5.12 and 6.12]

(1) F(D) is a simplicial fan whose maximal faces correspond to complete D-laminations.(2) A reduced D-lamination is complete if and only if it has precisely |D| elements.

Theorem 2.9. [PPP19, Theorem 6.14] If F(D) is finite, then all D-laminates are non-closed. In thiscase, we have |F(D)| = R|D|.

COMPLETE SPECIAL BISERIAL ALGEBRAS ARE g-TAME 5

3. Examples

In this section, we examine our notions defined in the previous section.(1) Let (S,M) be a disk with |M | = 8 such that all marked points lie on ∂S. For a •-dissection of

(S,M)

D = 1

2

3 ,

all D-laminates and the corresponding g-vectors are given as follows:

(1, 0, 0) (−1, 0, 0) (0, 1, 0) (0,−1, 0) (0, 0, 1) (0, 0,−1) (1,−1, 0) (0, 1,−1) (1, 0,−1)

There are 14 complete D-laminations. The corresponding fan F(D) of g-vector cones for D is givenas in the left diagram of Figure 4.

(2) Let (S,M) be a disk with |M | = 7 such that one marked point in M◦ is a puncture and theothers lie on ∂S. For a •-dissection of (S,M)

D =3

2

1,

all D-laminates and the corresponding g-vectors are given as follows:

(1, 0, 0) (−1, 0, 0) (0, 1, 0) (0,−1, 0) (0, 0, 1) (0, 0,−1)

(1,−1, 0) (−1, 1, 0) (0, 1,−1) (0,−1, 1) (1, 0,−1) (−1, 0, 1)

There are 20 complete D-laminations. The fan F(D) is given as in the center diagram of Figure 4.(3) Consider a torus S = T 2 with ∂S = ∅ and |M | = 2 (i.e., both marked points are punctures).

Let D be a •-dissection of (S,M) given by

1 1

2

2

>>

>>

>

>

,

6 TOSHITAKA AOKI AND TOSHIYA YURIKUSA

where we identify the opposite sides of the square in the same direction. All D-laminates and thecorresponding g-vectors are given as follows:

ℓ γ−2 γ−1 γ0 γ1 γ2

· · · · · ·

(1,−1) (−2, 3) (−1, 2) (0, 1) (1, 0) (2,−1)

ℓ′ γ′−2 γ′

−1 γ′0 γ′

1 γ′2

· · · · · ·

(−1, 1) (−2, 1) (−1, 0) (0,−1) (1,−2) (2,−3)

where ℓ, ℓ′ are closed D-laminates and γm, γ′m are non-closed D-laminates for all m ∈ Z. We find that

the set {{γm, γm+1}, {γ′m, γ′

m+1} | m ∈ Z} provides all complete D-laminations. The fan F(D) is givenas in the right diagram of Figure 4.

For the closed D-laminate ℓ, its g-vector g(ℓ) = (1,−1) ∈ Z2 does not contained in |F(D)|. It willbe approximated by using the Dehn twist Tℓ along ℓ (we refer to Section 5 for the details). In fact, wehave Tℓ(γi) = γi+1 for any i ∈ Z>0 and hence

g(ℓ) = (1,−1) ∈⋃

m≥0

C(Tmℓ ({γ1})).

(1) (2) (3)

Figure 4. A fan F(D) of g-vector cones for Examples (1)-(3)

4. g-vectors and lattice points

The aim of this section is to prove the following result.

Theorem 4.1. Let D be a •-dissection of (S,M). Then there is a bijection

{D-laminations} → Z|D|

given by the map X 7→ g(X ) :=∑

γ∈X g(γ), where g(∅) := 0.

To prove Theorem 4.1, we first consider the following two cases:

COMPLETE SPECIAL BISERIAL ALGEBRAS ARE g-TAME 7

(a) Let (S1,M1) be a disk with |M1| = 2n+ 2 such that all marked points lie on ∂S1. Let D1 bea •-dissection of (S1,M1) as in the left diagram of Figure 5.

(b) Let (S2,M2) be a disk with |M2| = 2n+ 1 such that one marked point in (M2)◦ is a punctureand the others lie on ∂S2. Let D2 be a •-dissection of (S2,M2) as in the right diagram ofFigure 5.

In both cases, we have |Di| = n.

(a)

...

(b)

...

Figure 5. Special cases (a) and (b)

Proposition 4.2. For i ∈ {1, 2}, F(Di) is finite. In particular, we have |F(Di)| = R|Di|.

Proof. In the same way as (1) and (2) in Section 3, one can check that the number of D1-laminates isequal to 1

2n(n+3) and the number of D2-laminates is equal to n(n+1), in particular, they are finite.The latter assertion follows from Theorem 2.9. �

Corollary 4.3. Theorem 4.1 holds for D = D1 or D = D2.

Proof. For i ∈ {1, 2}, F(Di) is a simplicial fan satisfying |F(Di)| = R|Di| by Proposition 4.2. This im-plies that the map X 7→ g(X ) provides a one-to-one correspondence between the set of Di-laminationsconsisting only of non-closed D-laminates and Z|D|. More precisely, for any g ∈ Z|Di|, there is exactlyone reduced Di-lamination X ′ such that g is contained in the interior of C(X ′). Since C(X ′) is sim-plicial, g is uniquely written by g =

∑γ∈X ′ aγg(γ) for aγ ∈ Z>0. Then a Di-lamination X consisting

of aγ elements γ ∈ X ′ is a unique one such that g(X ) = g. �

Now, we are ready to prove Theorem 4.1.

Proof of Theorem 4.1. Let D be a •-dissection of (S,M) and g = (gd)d∈D an arbitrary integer vectorin Z|D|. In the following, we construct a D-lamination X such that g = g(X ).

Recall that (S,M) is divided into polygons △v for all v ∈M◦. For v ∈M◦, we can naturally embed△v into the above •-dissection Di of (Si,Mi) for i = 1 or 2. More precisely, (Si,Mi) is obtained from△v by gluing a digon with one ◦-marked point on each •-arc of D ∩ △v, where D ∩ △v form Di in(Si,Mi). By Corollary 4.3, there is a unique Di-lamination Xv such that g(Xv) = (gd)d∈D∩△v

. Weregard Xv∩△v as a set of pairwise non-intersecting curves in (S,M) with |gd| endpoints on d ∈ D∩△v.Then we can glue the curves of Xv ∩△v for all v ∈M◦ at their endpoints on D. As a result, we obtaina set X of pairwise non-intersecting ◦-laminates of (S,M). From our construction, every ◦-laminateof X is a D-laminate, and hence X forms a D-lamination such that g(X ) = g as desired.

On the other hand, the uniqueness of X follows from one of Xv for any v ∈M◦. �

5. Positive position and Dehn twists

In this section, we fix a •-dissection D of (S,M) and make preparations for proving Theorem 1.5.The proof of Theorem 1.5 appears in the next section.

5.1. Dehn twist along a closed D-laminate. We denote by Tℓ the Dehn twist along a closed curveℓ with the orientation defined as follows:

ℓTℓ−→

8 TOSHITAKA AOKI AND TOSHIYA YURIKUSA

In general, Tℓ(γ) is not a D-laminate for a given D-laminate γ. We will give a condition that Dehntwists work well.

Let γ and δ be D-laminates. For each intersection point p of γ and δ, we can assume that p lies inS \D, thus p ∈ △v for some v ∈M◦. We set orientations of the segments of γ and δ in △v such thatv lies on the right to them. We say that γ is in positive position for δ if γ and δ don’t intersect or γintersects δ from right to left at each intersection point (see Figure 6).

γ>

δ>

γ>

δ

>Figure 6. A D-laminate γ is in positive position for a D-laminate δ

Lemma 5.1. Let ℓ be a closed D-laminate and γ a non-closed D-laminate which is in positive positionfor ℓ. Then

(1) Tℓ(γ) is a non-closed D-laminate;(2) g(Tℓ(γ)) = g(γ) + #(γ ∩ ℓ)g(ℓ);(3) if a D-laminate γ′ does not intersect ℓ, then

#(γ′ ∩ γ) = #(γ′ ∩ Tℓ(γ)).

Proof. The assertions immediately follow from the assumption. �

In the situation of Lemma 5.1, we can repeat the Dehn twist Tℓ. Moreover, Lemma 5.1 is generalizedfor D-laminations. For closed curves ℓ1, . . . , ℓk and m1, . . . ,mk ∈ Z≥0, we write

T(m1,...,mk)(ℓ1,...,ℓk)

:= Tm1

ℓ1· · ·Tmk

ℓk.

Note that if ℓ1, . . . , ℓk are pairwise non-intersecting, then all Tℓi are commutative.

Proposition 5.2. Let ℓ1, . . . , ℓk be a D-lamination consisting only of closed D-laminates and γ1, . . . , γha D-lamination consisting only of non-closed D-laminates which are in positive position for any ℓi.

Then for any m1, . . . ,mk ∈ Z≥0 and T := T(m1,...,mk)(ℓ1,...,ℓk)

,

(1) {T(γ1), . . . ,T(γh)} is a D-lamination consisting only of non-closed D-laminates;(2) we have the equality

h∑

i=1

g(T(γi)) =

h∑

i=1

g(γi) +

h∑

i=1

k∑

j=1

mj#(γi ∩ ℓj)g(ℓj).

Proof. (1) Let X := {γ1, . . . , γh} and Y := {ℓ1, . . . , ℓk}. For any γ ∈ X and ℓ ∈ Y, by Lemma 5.1(1),Tℓ(γ) is a non-closed D-laminate. Lemma 5.1(3) says that Tℓ(γ)∩ ℓ′ is naturally identified with γ ∩ ℓ′

for any ℓ′ ∈ Y. In particular, Tℓ(γ) is also in positive position for ℓ′, thus Tℓ′Tℓ(γ) is a non-closedD-laminate. Repeating this process, T(γ) is a non-closed D-laminate. Since T(γ) and T(γ′) don’tintersect for any γ, γ′ ∈ X , the assertion holds.

(2) The equality is calculated from Lemma 5.1(2) since Lemma 5.1(3) says that the number of allintersection points of X and Y is not changed by the Dehn twists. �

5.2. Non-closed D-laminate ℓd for a closed D-laminate ℓ. Let X be a D-lamination consistingonly of non-closed D-laminates. We assume that there is a closed D-laminate ℓ such that X ⊔ {ℓ} isa D-lamination. By the definition of D-laminates, there exists d ∈ D such that g(ℓ)d > 0. From now,we construct a non-closed D-laminate ℓd such that

(a) ℓd is a non-closed D-laminate which is compatible with any D-laminate of X ;(b) ℓd intersects with ℓ so that ℓd is in positive position for ℓ.

It plays an important role to prove Theorem 1.5 in the next section.First, for d ∈ D, we define a D-laminate d∗+ (resp., d∗−) as follows (see Figure 7):

COMPLETE SPECIAL BISERIAL ALGEBRAS ARE g-TAME 9

• d∗+ (resp., d∗−) is a laminate running along d∗ in a small neighborhood of it;• If d∗ has an endpoint v ∈ M◦ on a component C of ∂S, then the corresponding endpoint ofd∗+ (resp., d∗−) is located near v on C in the counterclockwise (resp., clockwise) direction;• If d∗ has an endpoint at a puncture p ∈M◦, then the corresponding end of d∗+ (resp., d∗−) is aspiral around p counterclockwise (resp., clockwise).

d∗+

d∗

d∗−d∗

Figure 7. Two D-laminates d∗+ and d∗−

That is, g(d∗+)e = δed (resp., g(d∗−)e = −δed) for e ∈ D, where δ is the Kronecker delta.On this notation, g(ℓ)d > 0 implies ℓ ∩ d∗+ 6= ∅ and d∗+ is in positive position for ℓ. Without loss of

generality, we can assume that p ∈ ℓ ∩ d∗+ lie on d as in the left diagram of Figure 8 .Second, for each endpoint v of d∗, we define a curve ℓv of S as follows: Consider the segment

α := d∗+ ∩△v.

• If α intersects none of X , then let ℓv := α (see the center diagram of Figure 8);• Otherwise, let pv be the nearest intersection point of α and X from p, where pv ∈ α ∩ γ forγ ∈ X . We denote by q an endpoint of a connected segment in γ ∩ △v containing pv suchthat the intersection point q ∈ γ ∩ D is negative. Then ℓv is a curve obtained by gluing thefollowing two curves at pv (see the right diagram of Figure 8):(i) a segment of α between p and pv;(ii) a segment of γ obtained by cutting γ at pv, that contains q.

d

ℓ d∗+p

ℓv = α

vp

α

q

γ

ℓvv

ppv

Figure 8. A closed D-laminate ℓ and d ∈ D with g(ℓ)d > 0 (left), constructions of acurve ℓv (center, right)

Finally, we define ℓd as a curve obtained by gluing ℓv and ℓv′ at p for endpoints v and v′ of d∗. Asegment of ℓd between pv and pv′ is called its center segment, where pv is a point on ℓv sufficientlyclose to v if ℓv = α. It follows from the construction that ℓd satisfies (a) and (b) above. Moreover, (b)is generalized as follows.

Lemma 5.3. In the above situations, if a D-laminate γ is compatible with X ∪ {ℓ}, then ℓd is inpositive position for γ.

Proof. If γ intersects ℓd, then all the intersection points lie on the center segment of ℓd, thus theassertion holds. �

6. Proof of Theorem 1.5

In this section, we prove Theorem 1.5. The idea of its proof comes from [Yur20]. Fix a •-dissectionDof (S,M). Let g ∈ Z|D|. By Theorem 4.1, there is a D-lamination X such that g = g(X ) =

∑γ∈X g(γ).

10 TOSHITAKA AOKI AND TOSHIYA YURIKUSA

It is sufficient to construct D-laminations {Xm}m∈Z≥0consisting only of non-closed D-laminates such

that

X nc ⊆ Xm and g ∈⋃

m∈Z≥0

C(Xm).

where X = X nc ⊔ X cl is a decomposition such that X nc (resp., X cl) consists of all non-closed D-laminates (resp., closed D-laminates) in X .

If X cl = ∅, then a family of Xm := X for all m ∈ Z>0 is the desired one. Assume that X cl isnon-empty. For ℓ1 ∈ X cl and d1 ∈ D with g(ℓ1)d1

> 0, we obtain a non-closed D-laminate ℓd1

1 by the

construction of Section 5.2 for X nc. By Lemma 5.3, ℓd1

1 is in positive position for every ℓ ∈ X cl.

If the set {ℓ ∈ X cl | ℓ ∩ ℓd1

1 = ∅} is non-empty, then we take ℓ2 ∈ {ℓ ∈ X cl | ℓ ∩ ℓd1

1 = ∅} and

d2 ∈ D with g(ℓ2)d2> 0. By the construction of Section 5.2 for X nc ⊔ {ℓd1

1 }, we obtain a non-closed

D-laminate ℓd2

2 . Notice that ℓd2

2 consists of some of segments of D-laminates in X nc, the center segment

of ℓd2

2 and one of ℓd1

1 , where the third type may not appear. In the same way as Lemma 5.3, we can

show that ℓd2

2 is in positive position for every ℓ ∈ X cl. Repeating this process, we finally get an integer

h ∈ {1, . . . , k = |X cl|} and non-closed D-laminates ℓd1

1 , . . . , ℓdh

h such that

{ℓ ∈ X cl | ℓ ∩ ℓd1

1 = · · · = ℓ ∩ ℓdh

h = ∅} = ∅.

Moreover, our construction provides the following properties:

• X nc ∪ {ℓd1

1 , . . . , ℓdh

h } is a D-lamination consisting only of non-closed D-laminates;

• For i ∈ {1, . . . , h}, ℓdi

i is in positive position for every ℓ ∈ X cl.

We set X cl = {ℓ1, . . . , ℓh}⊔{ℓh+1, . . . , ℓk} and fix the notations n(i)j := #(ℓdi

i ∩ ℓj), nj :=∑h

i=1 n(i)j ,

and N := n1 · · ·nk. Set

T := T( Nn1

,..., Nnk

)

(ℓ1,...,ℓk).

By Proposition 5.2, Tm(ℓdi

i ) are non-closed D-laminates for all m ∈ Z≥0 and i ∈ {1, . . . , h}, and weget the equalities

h∑

i=1

g(Tm(ℓdi

i )) =

h∑

i=1

g(ℓdi

i ) +m

h∑

i=1

k∑

j=1

N

nj

n(i)j g(ℓj)

=

h∑

i=1

g(ℓdi

i ) +mN

k∑

j=1

g(ℓj)

=h∑

i=1

g(ℓdi

i ) +mNg(X cl).

It gives

limm→∞

∑hi=1 g(T

m(ℓdi

i ))

m= Ng(X cl).

Then the D-lamination

Xm := X nc ∪ {Tm(ℓdi

i )}hi=1

is the desired one because

g = g(X nc) + g(X cl) ∈ C(X nc) +⋃

m∈Z≥0

C({Tm(ℓdi

i )}hi=1) ⊆⋃

m∈Z≥0

C(Xm).

7. Representation theory

In this section, we study the algebraic aspects of our results. We can see their examples in the nextsection.

COMPLETE SPECIAL BISERIAL ALGEBRAS ARE g-TAME 11

7.1. Two-term silting complexes for module-finite algebras. Let R := k[[t]] be the formalpower series ring of one value over k. Let A be a basic R-algebra which is module-finite (i.e., A isfinitely generated as an R-module). We denote by projA the category of finitely generated projectiveright A-modules, by K

b(projA) the homotopy category of bounded complexes of projA. In particular,Kb(projA) is an R-linear category and HomKb(projA)(X,Y ) is a finitely generated R-module for any

X,Y ∈ Kb(projA).

We begin with the following observation.

Proposition 7.1. The category Kb(projA) is a Krull-Schmidt triangulated category.

Proof. For any X ∈ Kb(projA), E = EndKb(projA)(X) is a module-finite algebra over the complete

local noetherian ring R. Therefore, E is semiperfect by [CR62, p.132] and hence Kb(projA) is Krull-

Schmidt. �

Now, we study two-term silting theory for a module-finite R-algebra A. We refer to [AIR14, Aih13,DIJ19] for two-term silting theory of finite dimensional algebras, and to [ADI, Kim20] for ones ofmodule-finite algebras.

Definition 7.2. Let P = (P i, f i) be a complex in Kb(projA).

(1) We say that P is two-term if P i = 0 for any integer i 6= 0,−1.(2) We say that P is presilting if HomKb(projA)(P, P [m]) = 0 for any positive integer m.

(3) We say that P is silting if it is presilting and thickP = Kb(projA), where thickP is the

smallest triangulated subcategory of Kb(projA) which contains P and is closed under takingdirect summands.

We denote by 2-ipsA (resp., 2-presiltA, 2-siltA) the set of isomorphism classes of indecomposabletwo-term presilting (resp., basic two-term presilting, basic two-term silting) complexes for A. Here, wesay that a complex P is basic if all indecomposable direct summands of P are pairwise non-isomorphic.We denote by |P | the number of non-isomorphic indecomposable direct summands of P and by addPthe category of all direct summands of finite direct sums of copies of P .

Let A =⊕|A|

i=1 Pi be a decomposition of A, where Pi is an indecomposable projective A-module.

Definition 7.3. Let P = [P−1 f→ P 0] be a two-term complex in K

b(projA).

(1) The g-vector of P is defined by

g(P ) := (m1 − n1, . . . ,m|A| − n|A|) ∈ Z|A|

where mi (resp., ni) is the multiplicity of Pi as indecomposable direct summands of P 0 (resp.,P−1).

(2) The g-vector cone C(P ) is defined to be a cone in R|A| spanned by g-vectors of all indecom-posable direct summands of P .

We denote by F(A) a collection of g-vector cones of all basic two-term presilting complexes for A.

The following are basic properties of two-term presilting complexes.

Proposition 7.4. [Kim20] Let P = [P−1 f→ P 0] ∈ 2-presiltA. Then the following hold:

(1) P is a direct summand of some basic two-term silting complex for A.(2) P is silting if and only if |P | = |A|.(3) addP 0 ∩ addP−1 = 0.

Proposition 7.5. [Kim20] The collection F(A) is a simplicial fan whose maximal faces correspondto basic two-term silting complexes for A.

Let I be an ideal in A and B := A/I. In particular, B is also module-finite over R. The functor−⊗A B : projA→ projB induces a triangle functor

(−) := −⊗A B : Kb(projA)→ Kb(projB).

Proposition 7.6. [ADI] In the above, the following hold.

(1) If P is a two-term presilting complex for A, then P is a two-term presilting complex for B.

12 TOSHITAKA AOKI AND TOSHIYA YURIKUSA

(2) If A is g-tame, then so is B.

Proposition 7.7. [Kim20, Theorem 1.4] If I is generated by central elements and contained in the

radical, then the correspondence (−) induces bijections

2-presiltA→ 2-presiltB and 2-siltA→ 2-siltB.

In particular, we have F(A) = F(B).

7.2. Complete special biserial algebras. We define complete special biserial algebras and complete

gentle algebras. For a given finite connected quiver Q, let kQ be the complete path algebra, that is,the completion of a path algebra kQ of Q with respect to kQ+-adic topology, where kQ+ is the arrowideal. For arrows α and β, we denote by s(α) and t(α) the starting point and the terminal point of α,respectively. Also we write αβ for the path from s(α) to t(β).

Definition 7.8. Let Q be a finite connected quiver and I an ideal in the path algebra kQ of Q. We

say that kQ/I is a complete special biserial algebra, where I is the closure of I, if all the followingconditions are satisfied:

(SB1) For each vertex i of Q, there are at most two arrows starting at i and there are at most twoarrows ending at i.

(SB2) For every arrow α in Q there exists at most one arrow β such that t(α) = s(β) and αβ /∈ I.(SB3) For every arrow α in Q, there exists at most one arrow γ such that s(α) = t(γ) and γα /∈ I.

It is called complete gentle algebra if in addition:

(SB4) For every arrow α in Q, there exists at most one arrow β such that t(α) = s(β) and αβ ∈ I.(SB5) For every arrow α in Q, there exists at most one arrow γ such that s(α) = t(γ) and γα ∈ I.(SB6) The ideal I is generated by paths of length 2.

Here, we don’t assume that complete special biserial algebras are finite dimensional. Notice thatfinite dimensional special biserial (resp., gentle) algebras are complete special biserial (resp., gentle)algebras. The following observation is basic.

Proposition 7.9. Complete special biserial algebras are precisely factor algebras of complete gentlealgebras.

Proof. It is immediate from the definition. �

Therefore, to prove Theorem 1.3, it suffices to show the g-tameness of complete gentle algebras byProposition 7.6(2).

7.3. Gentle algebras from dissections. It is known in [PPP19, Theorem 4.10] that complete gentlealgebras are precisely obtained by the following construction.

Definition 7.10. For a •-dissection D of (S,M), we define a quiver Q(D) and an ideal I(D) in kQ(D)as follows:

• The set of vertices of Q(D) corresponds bijectively with D;• The set of arrows of Q(D) is a disjoint union of sets of arrows in Cv for all v ∈M◦ defined asfollows (see Figure 9):

– If v is a puncture and d1, . . . , dm ∈ D are sides of △v in counterclockwise order, then

there is a cycle Cv : d1a1→ d2

a2→ · · ·am−1

→ dmam→ d1 in Q(D), that is uniquely determined

up to cyclic permutation.– If v lies on a boundary segment, and d1, . . . , dm ∈ D are sides of △v in counterclockwise

order, then there is a path Cv : d1a1→ d2 · · ·

am−1

→ dm in Q(D).• I(D) is generated by all paths of length 2 which is not a sub-path of any Cv.

We denote A(D) := kQ(D)/I(D).

Proposition 7.11. [PPP19, Theorem 4.10] For a •-dissection D of (S,M), the algebra A(D) is acomplete gentle algebra, and any complete gentle algebra arises in this way.

We prepare a few terminology.

COMPLETE SPECIAL BISERIAL ALGEBRAS ARE g-TAME 13

d3

d2

d1

dm

dm−1

v

am−1

am a1

a2

am−2 a3

d2

d1dm

dm−1am−1 a1

am−2 a2

v

Figure 9. Sub-quiver of Q(D) in △v

Definition 7.12. Let Q(D) be the quiver in Definition 7.10.

• For a puncture v ∈ M◦, a cycle Cv is called a special cycle at v. If it is a representative of itscyclic permutation class starting and ending at d ∈ D, then we call it a special d-cycle at v.• For v ∈M◦, every non-constant sub-path of Cv is called a short path.

7.4. Two-term silting complexes for A(D) via D-laminates. Let D be a •-dissection of (S,M)

and A(D) := kQ(D)/I(D) the complete gentle algebra associated with D. In this subsection, weestablish a geometric model of two-term silting theory for A(D) and prove Theorem 1.3. To do it, weneed some preparation.

Let t be a sum of all special cycles of Q(D), and it is in the center of A(D).

Proposition 7.13. The complete gentle algebra A(D) is a module-finite k[[t]]-algebra.

Proof. It follows from the fact that A(D) is generated by all short paths and constant paths as ank[[t]]-module. �

We would discuss a class of complexes in Kb(projA(D)) obtained from D-laminates. Let Pd be an

indecomposable projective A(D)-module corresponding to d ∈ D. For d, e ∈ D, every short path inQ(D) from d to e provides a non-vanishing homomorphism Pe → Pd in projA(D), that we call shortmap.

Definition 7.14. An indecomposable two-term complex in Kb(projA(D)) is called a two-term string

complex if it can be written as one of the following forms:

(1) (2) (3)

Pd1

Pd3

Pdm−2

Pdm

Pd2

Pd4

Pdm−1

f21❱❱

++❱❱

f23❤❤

33❤❤

f43❱❱

++❱❱

❳❳❳❳

++❳❳❳

❳

fm−1 m❢❢❢

33❢❢

Pd1

Pd3

Pdm−1

Pd2

Pd4

Pdm−2

Pdm

f21❱❱

++❱❱

f23❤❤

33❤❤

f43❱❱

++❱❱

❢❢❢❢

33❢❢❢❢

fmm−1

❳❳

++❳❳❳

Pd1

Pd3

Pdm−2

Pdm

Pd2

Pd4

Pdm−1

33

f12❤❤

❤❤

++f32❱❱

❱❱

33

f34❤❤

❤❤

33❢❢❢❢

❢❢❢❢

++fm m−1❳❳

where each fij is of the form fij = thf ′ for a short map f ′ : Pj → Pi and a non-negative integer h. Itis called two-term short string complex if all fij are short maps.

We denote by 2-scxA(D) the set of isomorphism classes of two-term short string complexes P suchthat addP 0 ∩ addP−1 = 0. To prove the following proposition, we use τ -tilting theory (see [AIR14]for details).

Proposition 7.15. Any indecomposable two-term presilting complex in Kb(projA) is a two-term short

string complex, that is, 2-ipsA(D) ⊆ 2-scxA(D).

14 TOSHITAKA AOKI AND TOSHIYA YURIKUSA

Proof. Since B := A(D)/(t) is a finite dimensional special biserial algebra, every indecomposable non-projective B-module is either a string module or a band module (see [BR87, WW85]). A band moduleM satisfies M = τM , in particular, HomB(M, τM) 6= 0, where τ is the Auslander-Reiten translationfor B. By [AIR14, Lemma 3.4], if P ∈ 2-ipsB is a non-stalk complex, then it is a minimal projectivepresentation of a string module and hence a two-term string complex. In addition, P must be shortsince t = 0 on B. Therefore, Proposition 7.4(3) gives P ∈ 2-scxB. By Proposition 7.7, so is anycomplex in 2-ipsA(D) because the functor −⊗A(D) B preserves the property being short. �

From now on, we establish a geometric model of two-term silting theory for A(D). First, we give ageometric model of short maps in projA(D).

Definition 7.16. A D-segment is a non-self-intersecting curve, considered up to isotopy relative toM , in a polygon △v of D for some v ∈M◦ whose ends are unmarked points on sides of △v or spiralsaround v.

Let η be a D-segment in △v whose endpoints lie on d, e ∈ △v ∩ D. We orient it to satisfy that v

is to its right and its starting point lies on d. Then it corresponds to a short path d1a1→ · · ·

as−1

→ dsin Q(D), where d1, . . . , ds ∈ D are sides of △v in counterclockwise order with d1 = e and ds = d. Itinduces a short map σ(η) : Pd → Pe in projA(D).

Proposition 7.17. The map σ induces a bijection

σ : {D-segments whose endpoints lie on D} → {short maps in projA(D)}.

Proof. The assertion immediately follows from the definition of σ. �

Next, we give a geometric model of two-term short string complexes in Kb(projA(D)).

Definition 7.18. A generalized D-laminate is a ◦-laminate γ intersecting at least one •-arc of D suchthat the condition (∗) in Definition 2.6(2) and the following conditions are satisfied:

• Each connected component of γ in △v does not intersect itself for any v ∈M◦;• For any d ∈ D, all intersection points of γ and d are either positive or negative simultaneously.

Note that a D-laminate is precisely a non-self-intersecting generalized D-laminate.A non-closed generalized (NCG, for short) D-laminate γ is decomposed into D-segments γ0, . . . , γm

in polygons such that γi−1 and γi have a common endpoint pi on di ∈ D for every i ∈ {1, . . . ,m}.In particular, an end of γ0 (resp., γm) does not lie on D and both endpoints of γi lie on D for

i ∈ {1, . . . ,m− 1}. By Proposition 7.17, each γi provides a short map σ(γi) between T(i)γ and T

(i+1)γ

for i ∈ {1, . . . ,m− 1}, where T(i)γ := Pdi

. It yields a complex Tγ in Kb(projA(D)). More precisely, Tγ

is a two-term complex [T−1γ

f→ T 0

γ ] given by

T−1γ :=

⊕

pj :negative

T (j)γ , T 0

γ :=⊕

pi:positive

T (i)γ ,

f = (fij)i,j∈{1,...,m−1}, where fij :=

σ(γj−1) if i = j − 1,

σ(γj) if i = j + 1,

0 otherwise.

From our construction, we have the equality g(Tγ) = g(γ) under the identification of Z|D| and Z|A| via

the map d 7→ Pd, where the vector g(γ) ∈ Z|D| is defined by the equality (2.1).

Lemma 7.19. Suppose that two NCG D-laminates γ and γ′ are decomposed intoD-segments γ0, . . . , γmand γ′

0, . . . , γ′m′ as above, respectively. If m = m′ > 1 and γi = γ′

i for i ∈ {1, . . . ,m− 1}, then γ′ = γ.

Proof. The D-segment γ0 (resp., γm) only depends on the sign of p1 (resp., pm), that is uniquelydetermined by γ1 (resp., γm−1). Thus we have γ0 = γ′

0 and γm = γ′m. �

Proposition 7.20. The map T(−) : γ 7→ Tγ induces a bijection

T(−) : {NCG D-laminates} → 2-scxA(D)

such that g(γ) = g(Tγ) for any NCG D-laminate γ.

COMPLETE SPECIAL BISERIAL ALGEBRAS ARE g-TAME 15

Proof. It follows from our construction that Tγ is contained in 2-scxA(D). By Lemma 7.19, this mapis injective.

In order to prove surjectivity of the map, we give the inverse map. If P ∈ 2-scxA(D) is a stalkcomplex Pd with d ∈ D concentrated in degree 0 (resp., −1), then we just take γ = d∗+ (resp., γ = d∗−).On the other hand, let P ∈ 2-scxA(D) be a non-stalk complex which is one of (1)-(3) in Definition7.14. We only consider the form (1) since the others can be proved similarly. By Proposition 7.17,γ1 := σ−1(f21), γ2 := σ−1(f23), . . . , γm−1 := σ−1(fm−1m) are D-segments, and γi−1 and γi have acommon endpoint on di for i ∈ {2, . . . ,m − 1}. Then, by Lemma 7.19, there are two D-segments γ0and γm such that the curve γ obtained by gluing γ0, . . . , γm one by one is an NCG D-laminate. Fromour construction, we have P = Tγ . �

Finally, we give a geometric model of two-term presilting/silting complexes in Kb(projA(D)). We

use a description of morphisms between two-term short string complexes due to [ALP16].

Definition 7.21. For T, T ′ ∈ 2-scxA(D), a morphism f ∈ HomKb(projA(D))(T, T′[1]) is called a

singleton single map if it is induced by a short map p as one of the following forms:

(a)• •

• •

• •

p

q q′

......

T T ′

(b)

• •

•

• •

pq q′

p′p

......

...

T T ′

(c)

• •

•

• •

pq q′

pp′′

......

...

T T ′

(d)

• •

• •

• •

pq q′

p′p pp′′

......

......

T T ′

where p and q (resp., q′) have no common arrows as paths, and p′ and p′′ are not constant.

Definition 7.22. For T, T ′ ∈ 2-scxA(D), a morphism f ∈ HomKb(projA(D))(T, T′) is called a quasi-

graph map if it is induced by the following form:

• •

• •

• •

• •

...

...

T T ′

U

L

�

where U and Γ are

• •

(e)

P P ′

• •

q q′

......

or

• •

(f)

P P ′

• •

r r′

......

and there is no p ∈ HomA(D)(P, P′) such that pq = q′ or r = r′p. Note that it implies q′ 6= 0

and r 6= 0. A quasi-graph map in HomKb(projA(D))(T, T′) naturally induces a unique morphism in

HomKb(projA(D))(T, T′[1]), called a quasi-graph map representative.

Regarding two-term short string complexes as homotopy strings defined as in [ALP16] (see also[BM03]), we can obtain the following result from [ALP16].

16 TOSHITAKA AOKI AND TOSHIYA YURIKUSA

Proposition 7.23. [ALP16, Propositions 4.1 and 4.8] For T, T ′ ∈ 2-scxA(D), singleton single mapsand quasi-graph map representatives give a basis of HomKb(projA(D))(T, T

′[1]).

Let γ and δ be NCG D-laminates. By Propositions 7.20 and 7.23, HomKb(projA(D))(Tγ , Tδ[1]) hasa basis consisting of singleton single maps and quasi-graph map representatives. It follows from thedefinition that a singleton single map in HomKb(projA(D))(Tγ , Tδ[1]) is given by one of the followinglocal figures:

(a)

δγp (b)

δγp (c)

γ δ

p (d)

δγp

where p is the associated short map. For a quasi-graphmap representative in HomKb(projA(D))(Tγ , Tδ[1]),

each of U and L as in Definition 7.22 is given by one of the following local figures:

(e)δ

γ

δ

γ

(f)

γ

δ

γ

δ

where the left figure of (e) (resp., (f)) is in the case of q = 0 (resp., r′ = 0).

Proposition 7.24. The following conditions are equivalent for two NCG D-laminates γ and δ:

(1) HomKb(projA(D))(Tγ , Tδ[1]) = 0;(2) γ is in positive position for δ.

Proof. The assertion follows from Proposition 7.23 and the above observations. �

We are ready to state our results.

Proposition 7.25. The following hold.

(1) The map T(−) in Proposition 7.20 restricts to a bijection

{non-closed D-laminates} → 2-ipsA.

(2) Two non-closed D-laminates γ and η are compatible if and only if Tγ ⊕ Tη is presilting.

Proof. In general, two NCG D-laminates γ and η are compatible if and only if they are in positive po-sition each other. By Proposition 7.24, this is equivalent to the condition that Hom(Tγ , Tδ[1]) =Hom(Tδ, Tγ [1]) = 0. Since a non-closed D-laminate is precisely a non-self-intersecting NCG D-laminate, we get (1) and (2). �

Theorem 7.26. The map X 7→ TX :=⊕

γ∈X Tγ gives bijections

{reduced D-laminations} → 2-presiltA(D) and {complete D-laminations} → 2-siltA(D)

such that C(X ) = C(TX ) for all reduced D-laminations X . In particular, we have

F(D) = F(A(D)) and |F(D)| = |F(A(D))|.

Proof. It follows from Proposition 7.25. �

Now, we prove Theorem 1.3.

Proof of Theorem 1.3. By Proposition 7.11, any complete gentle algebra is given as A(D) for some•-dissection D of ◦•-marked surfaces, and it follows from Theorems 1.5 and 7.26 that complete gentlealgebras are g-tame. Thus the assertion follows from Proposition 7.6(2) since every complete specialbiserial algebras is a factor algebra of some complete gentle algebra. �

COMPLETE SPECIAL BISERIAL ALGEBRAS ARE g-TAME 17

7.5. Application to finite dimensional k-algebras. In this subsection, we introduce a class ofspecial biserial algebras which is a common generalization of finite dimensional gentle algebras andBrauer graph algebras. It has the same geometric model of two-term silting theory as complete gentlealgebras.

Let (S,M) be a ◦•-marked surface and D a •-dissection of (S,M). Remember that every d ∈ Ddetermines a special d-cycle Cv,d for each endpoint v of d∗ ∈ D∗ which is a puncture.

Definition 7.27. For a function m : M◦\∂S → Z>0, we define a finite dimensional special biserialalgebra B(D) := A(D)/J , where J is the closure of an ideal generated by

Cm(u)u,d − C

m(v)v,d

for all d ∈ D and endpoints u, v of d∗. Here, Cm(v)v,d is an m(v)-th of Cv,d if v is a puncture; otherwise,

it is zero.

Example 7.28. Let D be a •-dissection of a ◦•-marked surface (S,M).

(a) If all •-marked points lie on the boundary ∂S, then A(D) = B(D) and this is precisely a finitedimensional gentle algebra.

(b) If all marked points are punctures (i.e., ∂S = ∅), then B(D) is a Brauer graph algebra. Infact, the corresponding Brauer graph is given as follows:• The set of vertices corresponds bijectively with M◦;• The set of edges corresponds bijectively with the dual dissection D∗ of D;• The cyclic ordering around vertex is induced from the orientation of S;• A multiplicity of a vertex v is m(v).

Conversely, it is shown in [Lab13] that every Brauer graph algebra arises in this way (see also[Sch18]).

For these algebras, we have the following geometric model of two-term silting theory, which iscompatible with one of complete gentle algebras.

Proposition 7.29. Let (S,M) be a ◦•-marked surface and D a •-dissection of (S,M). Let B(D) bea special biserial algebra associated to D. Then there are bijections

{reduced D-laminations} → 2-presiltB(D) and {complete D-laminations} → 2-siltB(D)

that preserve their g-vectors. In particular, we have F(B(D)) = F(D).

Proof. Let A(D) be a complete gentle algebra associated to D. Let K be an ideal in kQ(D) generatedby all special cycles in Q(D) and t a sum of all special cycles in Q(D). We have diagram

(7.1) A(D)→A(D)

(t)→

A(D)

K∼=

B(D)

K←

B(D)

(t)← B(D)

of algebras, where each map in the diagram is a canonical surjection. It is easy to see that each factoralgebra is given by an ideal satisfying the assumption of Proposition 7.7. In particular, we have acanonical bijection between 2-presiltA(D) and 2-presiltB(D) by Proposition 7.7. Finally, Theorem7.26 yields the assertion. �

8. Examples for representation theory

(1) Let (S,M) be a disk with |M | = 10 such that one marked point in M◦ (resp., M•) is a punctureand the others lie on ∂S. For a •-dissection of (S,M)

D = ,

18 TOSHITAKA AOKI AND TOSHIYA YURIKUSA

the quiver Q(D) and the ideal I(D) are given by

Q(D) =

1

2 3

4 5

a

b

c

d e

f and I(D) = 〈cf, fc, ed〉.

(i) We consider an NCG D-laminate γ, but not a D-laminate, that is decomposed into D-segmentsγ0, . . . , γ5 as follows:

γ∗ ∗

∗

∗

∗

γ0

γ1

γ2γ3

γ4

γ5

Then the corresponding two-term string complex Tγ is not presilting. In fact, there is a nonzeroquasi-graph map representative in HomKb(projA(D))(Tγ , Tγ [1]) induced by the form

P2

P3

P4

P3

P5

σ(γ1)

σ(γ2)

σ(γ3)

σ(γ4)

P2

P3

P4

P3

P5

σ(γ1)

σ(γ2)

σ(γ3)

σ(γ4)

Tγ

Tγ ,

where σ(γ1) (resp., σ(γ2), σ(γ3), σ(γ4)) is the short map in projA(D) induced by a short path b(resp., f , dab, ef) in Q(D). There is no short map p from P4 to P2 (resp., from P5 to P4) such thatσ(γ1) = pσ(γ3) (resp., σ(γ2) = pσ(γ4)). Therefore, Tγ is not presilting.

(ii) We consider two D-laminates δ and δ′ as follows:

δ

δ′

COMPLETE SPECIAL BISERIAL ALGEBRAS ARE g-TAME 19

Then δ′ is a positive position for δ, but δ is not a positive position for δ′. We observe whetherTδ ⊕ Tδ′ is presilting. It is easy to see that Tδ and Tδ′ are presilting respectively. In addition, wehave HomKb(projA)(Tδ′ , Tδ[1]) = 0. However, there is a nonzero singleton single map from Tδ to Tδ′ [1]induced by a short path b, as (d) in Definition 7.21. Thus Tδ ⊕ Tδ′ is not presilting.

(2) We consider the ◦•-marked surface (S,M) and the •-dissection D = {1, 2} in Section 3(3). Thenthe associated quiver Q(D) and the ideal I(D) are given by

Q(D) =1 1

2

2

>>

>>

>

>

a1b1

a2 b2

and I(D) = 〈a1b1, b1a2, a2b2, b2a1〉.

Let m(◦) := 1, then we have

J = 〈a1b1a2b2 − a2b2a1b1, b1a2b2a1 − b2a1b1a2〉,

and the algebra B(D) defined in Definition 7.27 is a Brauer graph algebra whose Brauer graph is

with multiplicity 1 on the vertex ◦. In Section 3(3), we gave the complete lists of D-laminates andcomplete D-laminations. For i ∈ Z>0 and the non-closed D-laminate γi, the corresponding two-termstring complex Tγi

is given by

P1

P2

P1

P2

P1,

σ(a1)

σ(a2)

σ(a2)

...

where σ(ak) is a short map induced by ak and P1 only appears i times in degree 0 (resp., P2 onlyappears i − 1 times in degree −1). For i, j ∈ Z>0, it is easy to show that all nonzero maps betweenTγi

and Tγj[1] are quasi-graph map representatives induced by the form

P2

P1

P2

P2

P1

P2

σ(a2)

σ(a1)

σ(a2)

σ(a1)

P1

P2

P2

P1

σ(a1)

σ(a2)

TγiTγj

,...

...

...

...

that is, i > j + 1. Therefore, Tγi⊕ Tγj

is two-term silting for j = i, i± 1; otherwise it’s not.

20 TOSHITAKA AOKI AND TOSHIYA YURIKUSA

Acknowledgements. The authors would like to thank their supervisor Osamu Iyama for his guidanceand helpful comments. The authors would also like to thank Aaron Chan and Pierre-Guy Plamondonfor valuable discussion. They are Research Fellows of Society for the Promotion of Science (JSPS).This work was supported by JSPS KAKENHI Grant Numbers JP17J04270 and JP19J11408.

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COMPLETE SPECIAL BISERIAL ALGEBRAS ARE g-TAME 21

T. Aoki: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602 Japan

E-mail address: m15001d@math.nagoya-u.ac.jp

T. Yurikusa: Mathematical Institute, Tohoku University, Aoba-ku, Sendai, 980-8578, Japan

E-mail address: toshiya.yurikusa.d8@tohoku.ac.jp