Generating degrees for graded projective resolutions
aa r X i v : . [ m a t h . K T ] S e p Generating degrees for graded projective resolutions
E. Marcos, A. Solotar, Y. Volkov ∗ Abstract
We provide a framework connecting several well known theories related to the linearityof graded modules over graded algebras. In the first part, we pay a particular attention tothe tensor products of graded bimodules over graded algebras. Finally, we provide a tool toevaluate the possible degrees of a module appearing in a graded projective resolution oncethe generating degrees for the first term of some particular projective resolution are known.
Koszul algebras were introduced by S. Priddy in [13]. We will apply the notion of a Koszulalgebra for algebras presented by quivers with relations. It can be stated as follows. Supposethat k is a field, Q is a finite quiver, I is a homogeneous ideal of the path algebra k Q and A = k Q/I . The algebra A is called Koszul if the maximal semisimple graded quotient A of A has a graded A -projective resolution ( P • , d • ) such that for all i >
0, the A -module P i isgenerated in degree i . Such a resolution is called a linear resolution and it is minimal wheneverit exists, in the sense that d i ( P i +1 ) ⊂ P i A > for all i > A is Koszul if andonly if its Yoneda algebra, E ( A ) = ⊕ i > Ext iA ( A , A ) is generated in degrees 0 and 1, which inturn is equivalent to the Yoneda algebra being isomorphic to the quadratic dual A ! of A .Koszulness has been generalized to various settings. Next we describe some of these gener-alizations.R. Berger introduced in [3] the notion of “nonquadratic Koszul algebra” for algebras of theform A = T k V /I , where V is a finite dimensional k -vector space and I is a two-sided idealgenerated in degree s , for some s ≥
2. He required the trivial A -module k to have a minimalgraded projective resolution ( P • , d • ) such that each P i is generated in degree is for i even and ( i − s + 1 for i odd.The authors of [9] considered, under the name of s -Koszul algebras, non necessarily quadraticKoszul algebras of the form A = k Q/I , with Q a finite quiver and I an ideal generated byhomogeneous elements of degree s , connecting this notion with the Yoneda algebra: the algebra A is s -Koszul if and only if E ( A ) is generated in degrees 0, 1 and 2. Observe that 2-Koszulalgebras are just Koszul algebras. ∗ The first named author has been supported by the thematic project of Fapesp 2014/09310-5. The sec-ond named author has been partially supported by projects PIP-CONICET 11220150100483CO and UBA-CyT 20020130100533BA. The first and second authors have been partially supported by project MathAmSud-REPHOMOL. The third named author has been supported by a post-doc scholarship of Fapesp (Project num-ber: 2014/19521-3) and by Russian Federation Presedent grant (Project number: MK-1378.2017.1). The secondnamed author is a research member of CONICET (Argentina). δ -Koszul and δ -determinedalgebras. See [7] for details.Moreover, E. Green and E. Marcos also introduced in [8] a family of algebras that theycalled 2- s -Koszul. They proved that these algebras also have the property that their Yonedaalgebras are generated in degrees 0, 1, and 2.The main objective of the current work is to place all these definitions in a unique framework.We next sketch how we will do this.Let A = ⊕ i > A i be a graded k -algebra generated in degrees 0 and 1, such that A is a finitedirect product of fields and A is finite dimensional. Given a graded A -module X , we consider aminimal graded projective resolution ( P • ( X ) , d • ( X )) and we take into account in which degrees P i ( X ) is generated for each i >
0. We are specially interested in what we call S -determinedcase.In Section 2 In Section 2 we prove that, given graded k -algebras A, B and C , a graded A − B bimodule X and a graded B − C bimodule Y , if X has a linear minimal A − B -projective gradedresolution, Y has a linear minimal B − C -projective graded resolution, and Tor Bi ( X, Y ) vanishesfor i >
1, then X ⊗ B Y has a linear minimal A − C -projective graded resolution. This is aparticular case of Theorem 2 below. Note that this theorem shows that any graded bimoduleover a Koszul algebra which is linear as a right module and flat as a left module is also linearas a bimodule and the tensor product with such a module gives a functor from the categoryof linear graded modules to the category of linear graded modules. Moreover, it recovers andgeneralizes the fact that the tensor product of two Koszul algebras is Koszul. The same holdsfor the S -determined (see Definition 1).Section 3 is devoted to Gr¨obner bases. Loosely speaking, we show how one can use themto obtain generating degrees for the n -th term of the minimal graded projective resolution of amodule if one knows the generating degrees for terms of its projective presentation of a specialform.We fix a field k . All algebras will be k -algebras and all modules will be right A -modulesunless otherwise stated. We will simply write ⊗ for ⊗ k and N for the set of non negativeinteger numbers.We thank the referee for the suggestions and for a careful reading of a previous version ofthis paper. S -determined modules In this section we will prove Lemma 1, which is a graded version of the spectral sequences (2)and (3) from [4, page 345].Let
A, B and C be k -algebras. Let X be an A − B -bimodule, Y a B − C -bimodule and Z an A − C -bimodule. Given a ∈ A , we will denote left multiplication by a on X by L a ∈ End B ( X ).We recall that for each n ∈ N , Ext nC ( Y, Z ) is an A − B bimodule with the structure given by aT b := (cid:0) Ext nC ( L b , Z ) ◦ Ext nC ( Y, L a ) (cid:1) ( T ) , for T ∈ Ext nC ( Y, Z ) , a ∈ A, b ∈ B. Suppose now that A is a Z -graded algebra and M is a graded A -module. Given i ∈ Z , M [ i ] will denote the i -shifted graded A -module with underlying A -module structure as before,whose grading is such that M [ i ] r = M i + r . For any graded A -module N and any n ∈ N , wewill denote by Hom GrA ( M, N ) the set of degree preserving A -module maps from M to N andby Ext nGrA ( M, N ) the set of equivalence classes of exact sequences of graded A -modules withdegree zero morphisms 0 → N f n − −−−→ T n − f n − −−−→ . . . f −→ T f − −−→ M → . nA ( M, N ) := ⊕ i ∈ Z Ext nGrA ( M, N [ i ]), which is a subset of Ext nA ( M, N )in a natural way. Moreover, if M has an A -projective resolution with finitely generated modules,then both sets coincide.Suppose now that A , B and C are Z -graded algebras, and that the bimodules X, Y and Z aregraded. For each n >
0, the A − B -bimodule structure on Ext nC ( Y, Z ) induces a graded A − B -bimodule structure on ext nC ( Y, Z ) whose i -th component is Ext nGrC ( Y, Z [ i ]). Note also thatext nC ( Y, Z [ i ]) ∼ = ext nC ( Y, Z )[ i ] as graded A − B -bimodule, moreover for any n >
0, Tor Bn ( X, Y )is a graded A − C -bimodule in a natural way.The main tool of this section is the following lemma. Lemma 1.
Let A , B and C be Z -graded algebras, X a graded A − B -bimodule, Y a graded B − C -bimodule, and Z a graded A − C -bimodule. There are two first quadrant cohomologicalspectral sequences with second pages E i,j = Ext iGr ( A op ⊗ B ) (cid:0) X, ext jC ( Y, Z ) (cid:1) and ˜ E i,j = Ext iGr ( A op ⊗ C ) (cid:0) Tor Bj ( X, Y ) , Z (cid:1) that converge to the same graded space. Proof.
Let . . . d n ( X ) −−−−→ P n ( X ) d n − ( X ) −−−−−→ . . . d ( X ) −−−−→ P ( X )( µ X −−→ X )be a graded A − B -projective resolution of X and( Z ι Z −→ ) I ( Z ) d ( Z ) −−−→ . . . d n − ( Z ) −−−−−→ I n ( Z ) d n ( Z ) −−−→ . . . be a graded A − C -injective resolution of Z . Consider two bicomplexes whose ( i, j )-componentsare respectively Hom Gr ( A op ⊗ B ) (cid:16) P i ( X ) , hom C (cid:0) Y, I j ( Z ) (cid:1)(cid:17) and Hom Gr ( A op ⊗ C ) (cid:0) P j ( X ) ⊗ B Y, I i ( Z ) (cid:1) . Since there is an isomorphism of complexes F • = Hom Gr ( A op ⊗ C ) (cid:0) P • ( X ) ⊗ B Y, I • ( Z ) (cid:1) ∼ = Hom Gr ( A op ⊗ B ) (cid:16) P • ( X ) , hom C (cid:0) Y, I • ( Z ) (cid:1)(cid:17) , the respective total complexes are isomorphic. Here, as usually, for two complexes of gradedmodules (cid:0) U • , d U, • (cid:1) and (cid:0) V • , d V, • (cid:1) over the algebra D we denote by Hom GrD ( U • , V • ) the complexwith (cid:0) Hom
GrD ( U • , V • ) (cid:1) n = ⊕ i ∈ Z Hom
GrD ( U i − n , V − i ) and differential d • defined by the equality d n ( f ) = d V, − i f + ( − n f d U,i − n for f ∈ Hom
GrD ( U i − n , V − i ).The first two pages of the spectral sequence E corresponding to the first bicomplex are E i,j = Hom Gr ( A op ⊗ B ) (cid:0) P i ( X ) , ext jC ( Y, Z ) (cid:1) and E i,j = Ext iGr ( A op ⊗ B ) (cid:0) X, ext jC ( Y, Z ) (cid:1) , while the first two pages of the spectral sequence ˜ E corresponding to the second bicomplex are˜ E i,j = Hom Gr ( A op ⊗ C ) (cid:0) Tor Bj ( X, Y ) , I i ( Z ) (cid:1) and ˜ E i,j = Ext iGr ( A op ⊗ B ) (cid:0) Tor Bj ( X, Y ) , Z (cid:1) . Since both spectral sequences converge to the homology of F • , the lemma is proved. ✷ Z -graded algebra A is assumed to be non negatively graded, that is A = ⊕ i > A i , where A is isomorphic to a finite product of copies of k as an algebra, dim k A < ∞ ,and A is generated as an algebra by A ⊕ A . This is equivalent to say that A ∼ = ( k Q ) /I where Q is a finite quiver and I is an ideal generated by homogeneous elements of degree bigger orequal 2. Definition 1.
Let S = ( S i ) i > be a collection of subsets S i ⊂ Z . A graded A -module X iscalled S -determined if it has a graded projective resolution P • ( X ) such that P i ( X ) is generatedas A -module by elements of degrees belonging to S i , i.e. P i ( X ) = h⊕ j ∈S i P i ( X ) j i A for all i >
0. We say that X is S -determined up to degree r if the condition on P i ( X ) holds for 0 i r . If the set S i = { i } , i.e. each P i ( X ) is generated in degree i , then we say that the resolutionis linear. Equivalently, a graded A -module X is S -determined if and only if for any i > A -module Y with support not intersecting S i – that is, ⊕ j ∈S i Y j = 0 – the spaceExt iGrA ( X, Y ) is zero. Analogously, the graded A -module X is S -determined up to degree r if and only if the last mentioned condition holds for 0 i r .The notion of an S -determined module provides a general framework for some well-knownsituations. We will now exhibit some well known examples of S -determined modules. • Consider a function δ : Z > → Z , and define S i = { δ ( i ) } for all i >
0, the S -determinedmodules are called δ -determined modules . If A is a δ -determined module over A , thenthe graded algebra A is called δ -determined. • With the same notations, if moreover the Ext algebra, E ( A ), of A is finitely generated,then A is called δ -Koszul , see [7]. In particular, if δ is the identity, then δ -determinedmodules are called linear modules and δ -Koszul algebras are exactly Koszul algebras [13]. • Also, given s ∈ N , let us define χ s : N → Z by χ s ( i ) = (cid:26) is if i is even, ( i − s + 1 if i is odd.The χ s -linear modules are called s -linear modules and χ s -Koszul algebras are s -Koszulalgebras , see [3]. Denoting by S i the set { j | j χ s ( i ) } , S -linear modules correspond to2 - s -linear modules . If moreover A is a 2- s -determined module over A , then the gradedalgebra A is called 2 - s -determined , see [8].Using minimal graded projective resolutions, it is not difficult to see that the A -module A is S -determined if and only if A is an S -determined module over A op ⊗ A . This fact follows, forexample, from [14, Theorem 2].Given two collections S = ( S i ) i > and R = ( R i ) i > of subsets of Z we define the collection S ⊗ R = (cid:0) ( S ⊗ R ) i (cid:1) i > by( S ⊗ R ) i = [ j + k = ij, k > { n + m | n ∈ S j , m ∈ R k } . Lemma 1 allows us to prove the following theorem, which generalizes some well known resultsabout Koszul algebras and Koszul modules concerning tensor products.4 heorem 2.
Let S = ( S i ) i > and R = ( R i ) i > be two collections of subsets of Z . Let A , B and C be Z -graded algebras, and finally let X be a graded A − B -bimodule which is S -determined as bimodule and Y be a graded B − C -bimodule which is R -determined as C -module.If Tor Bi ( X, Y ) = 0 for i r − , then X ⊗ B Y is an S ⊗R -determined until r -th degree A − C -bimodule. In particular, if Tor Bi ( X, Y ) = 0 for all i > , then X ⊗ B Y is an S ⊗ R -determined A − C -bimodule. Proof.
Let us fix n and r such that 0 n r . For any graded A − C -bimodule Z such that ⊕ m ∈ ( S⊗R ) n Z m = 0, we will prove that Ext nGr ( A op ⊗ C ) ( X ⊗ B Y, Z ) = 0. By Lemma 1 there arespectral sequences E i,j = Ext iGr ( A op ⊗ B ) (cid:0) X, ext jC ( Y, Z ) (cid:1) and ˜ E i,j = Ext iGr ( A op ⊗ C ) (cid:0) Tor Bj ( X, Y ) , Z (cid:1) that converge to the same graded space T • . It follows easily from the condition on Tor B ∗ ( X, Y )that T n = Ext nGr ( A op ⊗ C ) ( X ⊗ B Y, Z ) if n < r and that T r = Ext rGr ( A op ⊗ C ) ( X ⊗ B Y, Z ) ⊕ V forsome V ⊂ Hom Gr ( A op ⊗ C ) (cid:0) Tor Br ( X, Y ) , Z (cid:1) .Thus, it is enough to prove that E i,j = 0 for all integers i, j > i + j = n .Let us fix such i and j . If k ∈ S i , then for any l ∈ R j it is clear that k + l ∈ ( S ⊗ R ) n and so Z [ k ] l = Z k + l = 0. Since Y is an R -linear C -module, we know that ext jC ( Y, Z ) k =Ext jGrC ( Y, Z [ k ]) = 0 for any k ∈ S i ; from this, since X is an S -linear A op ⊗ B -module, E i,j =Ext iGr ( A op ⊗ B ) (cid:0) X, ext jC ( Y, Z ) (cid:1) = 0. We have proven that for any 0 n r and any graded A − C -bimodule Z such that ⊕ m ∈ ( S⊗R ) n Z m = 0 one has Ext nGr ( A op ⊗ C ) ( X ⊗ B Y, Z ) = 0. Consequently, X ⊗ B Y is an S ⊗ R -determined until r -th degree A − C -bimodule. ✷ Example 1.
Let A be the k -algebra with generators x and y subject to the relations xy = yx = 0 and x = y . Let X = A/ h x i and Y = A/ h y i . Note that X is a graded A -module and Y is agraded A -bimodule in a natural way. We will show the the conclusion of the Theorem 2 to thetensor product X ⊗ A Y of k - A -bimodule X and A - A -bimodule Y , does not hold. One can easilysee that there are short exact sequences Y [1] ֒ → A ։ X and X [1] ֒ → A ։ Y of graded right A -modules. It follows, from these two short exact sequences that, X and Y arelinear as right A -modules. On the other hand, X ⊗ A Y ∼ = A/ h x, y i is the unique simple A -modulewhose minimal projective resolution P • is not linear at P . This example shows that Theorem2 is not valid without the vanishing condition on Tor B ∗ ( X, Y ) . Corollary 2.
Let X be a graded A − B -bimodule. If A is an S -determined right A -module and X is an R -determined B -module, then X is an S ⊗R -determined A − B -bimodule. In particular,if A is - s -determined and X is a - s -determined B -module, then X is a - s -determined A − B -bimodule. When s = 2 we obtain that if A is Koszul and X is a linear B -module, then X is alinear A − B -bimodule. Proof.
Since A is an S -determined right A -module, A is an S -determined module over A op ⊗ A .Since A is flat as a right A -module, the result follows from Theorem 2, since A ⊗ A X ∼ = X . ✷
5t is was proved in [2] and [10] that if A and B are Koszul algebras, then A ⊗ B is a Koszulalgebra too. In the next corollary we give a very short and easy proof of a generalization of thisfact. Note that this generalization follows from [12, Chapter 3, Proposition 1.1] for algebras A , B such that A = B = k . Corollary 3. If X is an S -linear A -module and Y is an R -linear B -module, then X ⊗ Y is an S ⊗ R -linear module over A ⊗ B . In particular, if A and B are - s -determined, then A ⊗ B is - s -determined. In particular, if A and B are Koszul, then A ⊗ B is Koszul. Proof.
It follows from Theorem 2 since any k -module is flat. The second part follows from thefact that ( A ⊗ B ) = A ⊗ B . ✷ It is interesting to mention also the following special case of Theorem 2.
Corollary 4.
Let X be a graded A − B -bimodule that is flat as a left A -module. If X is - s -determined as B -module, then the functor − ⊗ A X : Mod A → Mod B induces a functorfrom the category of - s -determined A -modules to the category of - s -determined B -modules.In particular, if X is a linear B -module, then − ⊗ A X induces a functor from the category oflinear A -modules to the category of linear B -modules In this section we will use Gr¨obner bases techniques to study graded projective resolutions of agraded module X over an algebra A = k Q/I , where Q is a finite quiver and I is a homogeneousideal contained in ( k Q > ) . Our aim is to estimate the degrees of the modules appearing in theminimal projective resolution of X using Gr¨obner basis of I and a particular graded projectivepresentation of X .We fix a set of paths S ⊂ Q > . Next we introduce some notation. Given two paths p and q in Q , we write p | q if there are paths u and v in Q – possibly of length 0 – such that q = upv .If q = pv (resp. q = up ) we say that p divides q on the left (resp. right) and we write p | l q (respectively p | r q ). We say that S is reduced if for any q ∈ S there is no p ∈ S , p = q suchthat p | q . Let us write len ( q ) = n if q ∈ Q n .We next define, for n ∈ N , the notion of n - overlap for S . These elements will provide aminimal set of generators for each projective module of the minimal projective resolution of A as A -module. Note that n -overlaps are called n -chains by D. Anick in [1] and n -ambiguities byS. Chouhy and A. Solotar in [5].Given a quiver Q we also denote by Q the set of paths in Q , the context makes it clear whatwe mean. Definition 3.
Let Q be a quiver and S ∈ Q > be a set of paths. We say that p ∈ Q is an S -path if there exists s ∈ S such that s | r p . We denote by Q S the set of S -paths in Q . Supposethat p = qu for some u, q ∈ Q . We say that q S -vanishes p if there is no s ∈ S dividing u . Wesay that q almost S -vanishes p if q does not S -vanish p and, for any presentation u = u u with u ∈ Q > , q S -vanishes qu . We write q | S p if q S -vanishes p and q | aS p if q almost S -vanishes p (note that the relations | S and | aS are not transitive). If q | aS p , then we automatically have p ∈ Q S .We next define the set of n -overlaps O n ( S ) ⊂ Q and the set of n -quasioverlaps QO n ( S ) ⊂ Q × Q > inductively on n . • For n = 0, we define O ( S ) = Q and QO ( S ) = { ( w, v ) | w ∈ Q , v ∈ Q > , vw = v } .6 For n = 1, we define O ( S ) = S and QO ( S ) = { ( w, v ) | w, v ∈ Q > , vw ∈ S } . • For n >
1, we define O n ( S ) = { w | ∃ w ∈ O n − ( S ) , w ∈ O n − ( S ) such that w | S w, w | aS w } and QO n ( S )= { ( w, v ) | ∃ ( w , v ) ∈ QO n − ( S ) , ( w , v ) ∈ QO n − ( S ) such that w | S w, w | aS w } . Lemma 5.
Suppose that S is reduced and n > . If w ∈ O n ( S ) , then there is unique w ′ suchthat w ′ | l w and w ′ ∈ O n − ( S ) . If ( w, v ) ∈ QO n ( S ) , then there is unique w ′ such that w ′ | l w and ( w ′ , v ) ∈ QO n − ( S ) . Proof.
We will only prove the assertion about n -overlaps since the proof for n -quasioverlapsis similar. The existence of w ′ is a direct consequence of the definition of an n -overlap. Thusthe only thing to prove is uniqueness. We proceed by induction on n . For n = 1, the assertionis obvious. For n = 2, the assertion follows directly from the fact that S is reduced. Supposenow that n > n − n −
2. Suppose thatthere are two different paths w ′ , w ′′ ∈ O n − ( S ) such that w ′ | l w and w ′′ | l w . Without lossof generality we may assume that w ′′ = w ′ u for some u ∈ Q > . By the definition of O n − ( S ),there exist w ′ ∈ O n − ( S ) and w ′ ∈ O n − ( S ) such that w ′ | S w ′ and w ′ | aS w ′ . The inductivehypothesis assures that w ′ and w ′ are unique elements of O n − ( S ) and O n − ( S ) respectivelysuch that w ′ | l w ′ | l w ′′ . Then w ′ | aS w ′′ . Since u ∈ Q > , we have w ′ | S w ′ . Since this is acontradiction, the proof is complete. ✷ Given w ∈ O n ( S ), and i such that 0 i n , ρ i ( w ) will denote the unique element of O i ( S )that divides w on the left. Analogously, for ( w, v ) ∈ QO n ( S ), 0 i n , ρ vi ( w ) will denotethe unique element such that ( ρ vi ( w ) , v ) ∈ QO i ( S ) and ρ vi ( w ) | l w . The next two lemmas givealternative definitions for n -overlaps and n -quasioverlaps. Lemma 6.
Consider a reduced set of paths S and an integer n such that n > . Given a path w ∈ Q , w ∈ O n ( S ) if and only if it can be represented in the form w = v u v u . . . u n − v n − u n v n ,where u , . . . , u n ∈ Q , v , . . . , v n − ∈ Q > , and v , v n ∈ Q are such that1. v i − u i v i ∈ S for i n ,2. for all i n − , there are no u, v ∈ Q > such that vv i u ∈ S , u | l u i +1 v i +1 and v | r u i ,3. len ( u ) > if n . Proof.
Suppose that w ∈ O n ( S ). Given i , 1 i n , there exists a unique w i ∈ S such that w i | r ρ i ( w ). It follows from the definition of O i ( S ) that, for 2 i n , there exist v i − , u ′ i ∈ Q > such that ρ i ( w ) = ρ i − ( w ) u ′ i and w i = v i − u ′ i . We also define u ′ = ρ ( w ). Using again thedefinition of i -overlap, we get v i | r u ′ i for 1 i n −
1. It remains to define u i from the equality u ′ i = u i v i for 1 i n − u n = u ′ n . It is clear that v and v n are simply the ending andthe starting vertices of w .Now, if u , . . . , u n ∈ Q , v , . . . , v n − ∈ Q > , and v , v n ∈ Q satisfy all the required condi-tions, then the induction on 1 i n shows that u v . . . u i v i ∈ O i ( S ).7 Lemma 7.
Let S be a reduced set of paths and let n be an integer, n > . Given paths w ∈ Q and v ∈ Q > , the element ( w, v ) ∈ QO n ( S ) if and only if w can be represented in the form w = u v u . . . u n − v n − u n v n , where u , . . . , u n ∈ Q , v , . . . , v n − ∈ Q > , and v n ∈ Q are suchthat1. v i − u i v i ∈ S for i n ,2. for all i n − , there are no u, v ∈ Q > such that vv i u ∈ S , u | l u i +1 v i +1 and v | r u i ,3. len ( u ) > if n = 1 . Proof.
The proof is analogous to the proof of Lemma 6 and so it is left to a reader. ✷ Example 2.
Let Q be the quiver with Q = { e } and Q = { x, y } . Fix S = { x y , x } . Theelement ( w , v ) = ( xxxyyy, xx ) is a -quasioverlap with ρ v ( w ) = xxx and ρ v ( w ) = x .The elements u = e , v = u = v = x , and u = yyy provide the partition of Lemma 7.At the same time, the element w = v w = xxxxxyyy is a -overlap with ρ ( w ) = xxxx and ρ ( w ) = xxx ∈ S . In this case the paths u = x , v = xx , u = e , v = x , and u = xyyy providethe partition of Lemma 6. Note that, though v w is a -overlap, we have ρ ( v w ) = v ρ v ( w ) and this fact causes differences in the partitions of v w and ( w , v ) . Example 3.
Let us take Q as in the previous example and S = { x , xy } . Consider ( w , v ) =( xxxyy, xx ) . It is a -quasioverlap with ρ v ( w ) = xxx and ρ v ( w ) = x . The elements u = e , v = u = v = x , and u = yy provide the partition of Lemma 7. At the same time, v w = xxxxxyy O ( S ) while w = xxxxyy is a -overlap with ρ ( w ) = xxxx and ρ ( w ) = xxx ∈ S .The paths u = x , v = xx , u = e , v = x , and u = yy provide the partition of Lemma 6.This example shows that it is possible that ( w , v ) ∈ QO n ( S ) while v w O n ( S ) . Let us introduce the following notation. Given n ∈ N , maxo n ( S ) = sup { len ( w ) | w ∈ O n ( S ) } ,mino n ( S ) = inf { len ( w ) | w ∈ O n ( S ) } ,maxqo n ( S ) = sup { len ( w ) | ∃ v such that ( w, v ) ∈ QO n ( S ) } ,minqo n ( S ) = inf { len ( w ) | ∃ v such that ( w, v ) ∈ QO n ( S ) } . By definition, we set mino n ( S ) = + ∞ and maxo n ( S ) = −∞ if O n ( S ) is empty and minqo n ( S ) = + ∞ and maxqo n ( S ) = −∞ if QO n ( S ) is empty. Note that under this con-vention we have mino n ( S ) > n + 1 and maxo n ( S ) len ( S ) n − n + 1, where len ( S ) denotes themaximal length of the paths in S .Now we are going to prove a Theorem that allows to estimate the values of maxo n ( S ) and mino n ( S ) using maxqo n ( S ) and minqo n ( S ). Theorem 8.
Given n > , for any reduced set S we have • maxqo n ( S ) maxo n ( S ) − , • minqo n ( S ) > mino n ( S ) − len ( S ) + 1 . roof. The result is obvious for n = 0. For n >
1, we are going to prove the following assertion.If ( w, v ) is an n -quasioverlap, then there is v ′ ∈ Q > such that v ′ | r v and v ′ w is an n -overlap.Since it follows easily from the definition of 1-overlap that len ( v ) < len ( S ), we have len ( v ′ w ) − len ( S ) + 1 len ( w ) + len ( v ) − len ( S ) + 1 len ( w ) len ( v ′ w ) − v ′ . Thus, after proving the existence of v ′ we will be done.More precisely, we will prove the following statements by induction on n . If ( w, v ) ∈ QO n ( S ),then there exists v ′ ∈ Q > such that v ′ | r v , v ′ w ∈ O n ( S ), v ′ ρ vi ( w ) | l ρ i ( v ′ w ) for odd i , and ρ i ( v ′ w ) | l v ′ ρ vi ( w ) for even i .For n = 1, we define v ′ = v .Let us now consider the case n = 2. If vw cannot be presented in the form vw = u ′ su with u, u ′ ∈ Q > and s ∈ S , then we can take v ′ = v and obtain the 2-overlap wv satisfying all therequired conditions. Suppose that it is possible to write vw = u ′ su with u, u ′ ∈ Q > and s ∈ S .Choose such a presentation with minimal len ( u ). It follows from the definition of 2-quasioverlapthat len ( u ′ ) < len ( v ), i.e. there is v ′ ∈ Q > such that v = u ′ v ′ . It is easy to see that v ′ w is a2-overlap with ρ ( v ′ w ) = s satisfying all the required conditions.Let us now prove the inductive step. Suppose that the assertion above is true for all ( n − n by induction on len ( v ). Since the assertion is obvious forany ( w, v ) ∈ QO n ( S ) with len ( v ) = 1, we may assume that, when we try to prove the assertionfor some n -quasioverlap, we have already proved it for all n -quasioverlaps with length smallerthan len ( v ).Let us consider ( w, v ) ∈ QO n ( S ) and denote by w ′ the path ρ vn − ( w ). Since ( w ′ , v ) ∈ QO n − ( S ), we can apply the induction hypothesis. Thus, there exists ˜ v ∈ Q > such that ˜ v | r v ,˜ vw ′ ∈ O n − ( S ), ˜ vρ vi ( w ′ ) | l ρ i (˜ vw ′ ) for odd i , and ρ i (˜ vw ′ ) | l ˜ vρ vi ( w ′ ) for even i . Note also that ρ vi ( w ′ ) = ρ vi ( w ) for 0 i n −
1. Let us consider three cases:1. ρ n − (˜ vw ′ ) | aS ˜ vw . In this case we can simply define v ′ = ˜ v . It is easy to see that v ′ w is an n -overlap with ρ n − ( v ′ w ) = v ′ w ′ satisfying all the required conditions.2. ρ n − (˜ vw ′ ) | S ˜ vw . In this case ˜ vρ vn − ( w ) | l ρ n − (˜ vw ′ ), ˜ vρ vn − ( w ) = ρ n − (˜ vw ′ ), and, hence,2 ∤ n .Suppose that i is odd and ρ i (˜ vw ′ ) | S ˜ vρ vi +2 ( w ). Since ρ i (˜ vw ′ ) | l ρ i +1 (˜ vw ′ ) , ρ i +1 (˜ vw ′ ) | l ˜ vρ vi +1 ( w ′ ) , ˜ vρ vi +1 ( w ′ ) | l ˜ vρ vi +2 ( w ) , and ρ i (˜ vw ′ ) = ρ i +1 (˜ vw ′ ) , we know that ρ i (˜ vw ′ ) = ˜ vρ vi +2 ( w ).Also, ˜ vρ vi ( w ) | aS ˜ vρ vi +2 ( w ) implies ˜ vρ vi ( w ) | S ρ i (˜ vw ′ ) and ˜ vρ vi ( w ) = ρ i (˜ vw ′ ) . Thus, it followsfrom ρ i − (˜ vw ′ ) | aS ρ i (˜ vw ′ ) that ρ i − (˜ vw ′ ) | S ˜ vρ vi ( w ).Now, the descending induction on i gives us ρ i (˜ vw ′ ) | S ˜ vρ vi +2 ( w ) and ˜ vρ vi ( w ) | S ρ i (˜ vw ′ ) forall odd i such that 1 i n −
2. As before, we have ˜ vρ v ( w ) = ρ (˜ vw ′ ). Consequently, ˜ v = v and there is v ∈ Q > such that v = v ˜ v . Then (˜ vw, v ) ∈ QO n − ( S ) and len ( v ) < len ( v ).Thus we have v ′ such that v ′ ˜ vw ∈ O n ( S ) satisfies all required conditions. It is easy to checkthat we can take v ′ = v ′ ˜ v in this case.3. There is a presentation ˜ vw = ρ n − (˜ vw ′ ) u ′ su with u ∈ Q > , u ′ ∈ Q and s ∈ S . Let uschoose such a presentation with minimal len ( u ′ ). Since ρ n − (˜ vw ′ ) | l ˜ vρ vn − ( w ) and ρ n − (˜ vw ′ ) =˜ vρ vn − ( w ), n must be even.It follows from the minimality of u ′ that ρ n − (˜ vw ′ ) | aS ρ n − (˜ vw ′ ) u ′ s . Since ρ n − (˜ vw ′ ) | S ˜ vw ′ ,we know that ˜ vw ′ | l ρ n − (˜ vw ′ ) u ′ s and, hence, ρ n − (˜ vw ′ ) u ′ s | S ˜ vw . Since ˜ vρ vn − ( w ) | aS ˜ vw , wehave also ˜ vρ vn − ( w ) | S ρ n − (˜ vw ′ ) u ′ s. For even 2 i n −
2, we have ˜ vρ vi − ( w ) | l ρ i − (˜ vw ′ ) | l ρ i (˜ vw ′ ) | l ˜ vρ vi ( w ) and ˜ vρ vn − ( w ) | S ˜ vρ vi ( w ). Thus, ρ i (˜ vw ′ ) | S ˜ vρ vi ( w ) for such i . Suppose that ρ i (˜ vw ′ ) | S ˜ vρ vi ( w ) and ρ i (˜ vw ′ ) =9 vρ vi ( w ) for some 2 i n −
2. Since ˜ vρ vi − ( w ) | aS ˜ vρ vi ( w ), we have ˜ vρ vi − ( w ) | S ρ i (˜ vw ′ ).Moreover, ρ i − (˜ vw ′ ) | aS ρ i (˜ vw ′ ) implies ρ i − (˜ vw ′ ) = ˜ vρ vi − ( w ).Thus, the descending induction on i gives us ρ i (˜ vw ′ ) | S ˜ vρ vi ( w ) and ˜ vρ vi − ( w ) | S ρ i (˜ vw ′ )for even i , 2 i n −
2. In particular, we get ˜ v | S ρ (˜ vw ′ ). On the other hand, there is r ∈ S such that r | r ρ (˜ vw ′ ) and r = ρ (˜ vw ′ ). As a consequence, ρ (˜ vw ′ ) = v ′ r and ˜ v = v ′ v for some v , v ′ ∈ Q > . In particular, len ( v ) < len ( v ). It follows from our arguments that( w, v ) ∈ QO n ( S ) with v ′ r v n − ( w ) = ρ n − (˜ vw ′ ) u ′ s . We thus have got v ′ such that v ′ w ∈ O n ( S )satisfies all required conditions. ✷ Corollary 9.
For any reduced set of paths S , maxo n + m ( S ) maxo n ( S ) + maxo m ( S ) − and mino n + m ( S ) > mino n ( S ) + mino m ( S ) − len ( S ) + 1 . Proof.
The proof follows from Theorem 8 and the fact that any w ∈ O n + m ( S ) can be repre-sented in the form w = w ′ w ′′ with w ′ ∈ O n ( S ) and w ′′ ∈ QO m ( S ). ✷ From now on we fix an admissible order > on the set of paths in Q , [6]. More precisely, thismeans that there is a well order > such that for any paths p, q, u, v ∈ Q , • if p > q , then upv > uqv if the products are paths. • p > q if q | p .Given a linear space V , its basis B , and x ∈ V , we call the sum m P i =1 α i b i a reduced expression of x as a linear combination of the elements of B if α i ∈ k ∗ , b i ∈ B for 1 i m , b i = b j for1 i < j m , and x = m P i =1 α i b i .For x ∈ k Q , tip ( x ) is maximal path of Q , with respect to the order > , appearing in thereduced expression of x as a linear combination of paths. A subset G ⊂ I is called a Gr¨obnerbasis of I if for any x ∈ I , there exists g ∈ G such that tip ( g ) | tip ( x ).From now on we fix A = k Q/I , where I is an ideal of k Q that has a finite Gr¨obner basis G .We will use the notation of Green and Solberg in [11]. Let X be a graded A -module and let P ( X ) µ X −−→ X be its minimal graded projective cover. Suppose also that X is finitely presented.The projective module P ( X ) can be presented in the form P ( X ) = ⊕ i ∈ T f i A , where T isa finite set and, for any i ∈ T there exist e i ∈ Q , m i ∈ Z , and an isomorphism of gradedmodules f i A ∼ = e i A [ m i ] that sends f i to e i .Consider the graded space V X = ⊕ i ∈ T f i k Q . The set whose elements are of the form f i p ,where i ∈ T and p is a path ending in e i , is a basis of V X , we denote this set by B X andintroduce the following well order on it. We set f i p > f j q if either p > q or p = q and i > j .We say that f i p divides f j q on the right if i = j and p | r q . For x ∈ V X , we write tip ( x ) forthe maximal element of B X , with respect to the order > , appearing in the reduced expressionof x as a linear combination of elements of B X . The set x , . . . , x l of nonzero elements of V X iscalled right tip reduced if tip ( x i ) does not divide tip ( x j ) on the right for any 1 i, j l , i = j .By [11, Proposition 5.1], there are finite sets T and T ′ , elements h i ∈ V X ( i ∈ T ), andelements h i ′ ∈ V X ( i ∈ T ′ ) such that 10. any element h in the kernel of the composition V X ։ P ( X ) µ X −−→ X can be uniquelyrepresented in the form h = P i ∈ T f i + P i ∈ T ′ f ′ i , where f i ∈ h i k Q and f ′ i ∈ h i ′ k Q ,2. for any element h ∈ { h i } i ∈ T ∪ { h i ′ } i ∈ T ′ there exists e h ∈ Q such that he h = h ,3. h i ′ ∈ ⊕ j ∈ T h j I for any i ∈ T ′ ,4. the set { h i } i ∈ T ∪ { h i ′ } i ∈ T ′ is right tip reduced.Moreover, it is clear that all the elements in the set { h i } i ∈ T ∪ { h i ′ } i ∈ T ′ can be chosen homoge-neous. Let us define ¯ P ( X ) = ⊕ i ∈ T f i A , where, for any i ∈ T there exists an isomorphism ofgraded modules f i A ∼ = e h i A [ − deg ( h i )] that sends f i to e h i . Here deg ( h i ) denotes the degreeof h i . Note that the order in the basis of V X induces an order on the set { f i } i ∈ T . In this waywe obtain a graded projective presentation¯ P ( X ) ¯ d ( X ) −−−−→ P ( X ) µ X −−→ X of X , where ¯ d ( X ) sends f i to the class of h i in P ( X ) for any i ∈ T . Let also . . . d n ( X ) −−−−→ P n ( X ) d n − ( X ) −−−−−→ . . . d ( X ) −−−−→ P ( X )( µ X −−→ X )be the minimal graded A -projective resolution of X .As before, given a graded A -module M , we will denote by M j its j -th homogeneous com-ponent. The next Theorem will be crucial for us. It allows to estimate the degrees of thegenerators of the terms of the resolution constructed using the algorithm from [11]. Theorem 10.
Let us fix the notation as above. If k and l are the minimal and the maximaldegrees of f i , i ∈ T , then, for any n > , P n ( X ) = l + maxqo n ( tip ( G )) L j = k + minqo n ( tip ( G )) P n ( X ) j ! A . Proof.
The algorithm described in [11, Section 3] gives a graded projective resolution . . . ¯ d n ( X ) −−−−→ ¯ P n ( X ) ¯ d n − ( X ) −−−−−→ . . . ¯ d ( X ) −−−−→ P ( X )( µ X −−→ X ) , of X . Due to this algorithm ¯ P n ( X ) can be presented in the form ¯ P n ( X ) = ⊕ i ∈ T n f ni A and¯ d n − ( X ) sends f ni to h ni ∈ ⊕ i ∈ T n − f n − i A , where all these elements satisfy the following property.If tip ( h ni ) = f n − j p , then tip ( h n − j ) p ∈ Q S and if q | l tip ( h n − j ) p , q = tip ( h n − j ) p , then q is not S -path.For i ∈ T n , let us introduce p i ∈ Q and t ( i ) ∈ T n − by the equality tip ( h ni ) = f n − t ( i ) p i . Itis possible to prove inductively on n that ( w, v ) ∈ QO n ( S ) for w = p t n − ( i ) . . . p t ( i ) p i and some v ∈ Q > such that v | r tip (cid:16) h t n − ( i ) (cid:17) . Consequently, the required statement follows from theequality deg ( f ni ) = deg (cid:16) f t n − ( i ) (cid:17) + n − X m =0 len (cid:0) p t m ( i ) (cid:1) = deg (cid:16) f t n − ( i ) (cid:17) + len ( w ) . ✷ In other words, for any n >
1, we are able to deduce the possible degrees of the generatorsof P n ( X ) from those of the generators of ¯ P ( X ) and the lengths of n -quasioverlaps for tip ( G ).We obtain two corollaries. 11 orollary 11. Let A = k Q/I , X be a graded finitely presented A -module and let G bea homogeneous Gr¨obner basis for I such that tip ( G ) is reduced. Let P • ( X ) be a minimalgraded A -projective resolution of the graded A -module X and ¯ P ( X ) be as above. If k and l are the minimal and the maximal degrees of f i , i ∈ T , then, for any n > , P n ( X ) = l + maxo n ( tip ( G )) − L j = k + mino n ( tip ( G )) − len ( tip ( G ))+1 P n ( X ) j ! A . Proof.
It follows directly from Theorems 8 and 10. ✷ Corollary 12.
Let A = k Q/I where I has a homogeneous Gr¨obner basis G such that len ( tip ( G )) s and maxo ( tip ( G )) s + 1 . If additionally mino ( tip ( G )) = s , then thealgebra A is s -Koszul. Proof.
It follows from Corollaries 9 and 11 since ¯ P ( A ) = ⊕ α ∈ Q e α A [ − e α is thestarting vertex of the arrow α . More precisely, we get by induction on i > maxo i ( tip ( G )) maxo i − ( tip ( G )) + maxo ( tip ( G )) − χ s ( i −
2) + s = χ s ( i ) . If additionally mino ( tip ( G )) = s , then all the elements of tip ( G ) have length s . Then we getthat P ( A ) is generated in degree s + 1 and that if the minimal generating degree for P i ( A )is m , then the minimal generating degree for P i +2 ( A ) is not less than m + s . Then we get byinduction that P i ( A ) is generated in degree χ s ( i ). ✷ References [1] D. Anick, On the homology of associative algebras, Trans. Amer. Math. Soc. (1986),641–659.[2] J. Backelin, R. Froberg, Koszul Algebras,Veronese subrings, and rings with linear resolu-tion, Review of Romaine Math Pures App 30, 1980(85-97).[3] R. Berger, Koszulity of nonquadratic algebras, J. Algebra (2001), 705–734.[4] H. Cartan, S. Eilenberg, Homological algebra, Princeton Landmarks in Mathematics,Princeton University Press, Princeton, NJ (1999).[5] S. Chouhy, A. Solotar, Koszulity of nonquadratic algebra, J. Algebra (2015), 22–61.[6] E. Green, D. Farkas, C. Feustel, Synergy of Gr¨obner basis and Path Algebras, Can. J.Math, 45, (1993), 727-739.[7] E. Green, E. N. Marcos, δ -Koszul algebras, Comm. Algebra (6) (2005), 1753–1764.[8] E. Green, E. N. Marcos, d -Koszul, 2- d determined algebras and 2- d -Koszul algebras, J.Pure Appl. Algebra (4) (2011), 439–449.[9] E. Green, E. N. Marcos, R. Martinez-Villa, P. Zhang, d -Koszul algebras, J. Pure Appl.Algebra (2004), 141–162. 1210] E. Green, R. Mart´ınez-Villa, Koszul and Yoneda Algebras , Proceedings of the ICRA, CMSProceedings, Canadian Math. Soc., (1994), 227–244.[11] E. Green, Ø. Solberg, An Algorithmic Approach to Resolutions, J. Symbolic Comput. (2007), 1012–1033.[12] A. Polishchuk, L. Positselski, Quadratic algebras, University Lecture Series, , Amer.Math. Soc., Providence, RI (2005).[13] S. Priddy, Koszul resolutions, Trans. Amer. Math. Soc., (1970), 39–60.[14] E. Sk¨olberg, Going from cohomology to Hochschild cohomology, J. Algebra, (2005),263–278. Eduardo N. Marcos: IME-USP (Departamento de Matem´atica), Rua Mat˜ao 1010 Cid. Univ., S˜aoPaulo, 055080-090, Brasil. [email protected]
Andrea Solotar: IMAS and Dto de Matem´atica, Facultad de Ciencias Exactas y Naturales, Univer-sidad de Buenos Aires, Ciudad Universitaria, Pabell´on 1, (1428) Buenos Aires, Argentina. [email protected]
Yury Volkov: Saint-Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, Russia.Dto de Matem´atica, Instituto de Matem´atica e Estat´ıstica, Universidade S˜ao Paulo, Rua de Mat˜ao1010, Cidade Universit´aria, S˜ao Paulo-SP, 055080-090, Brasil. wolf86 [email protected] [email protected]