Finite generation for Hochschild cohomology of Gorenstein monomial algebras
aa r X i v : . [ m a t h . K T ] S e p FINITE GENERATION FOR HOCHSCHILD COHOMOLOGYOF GORENSTEIN MONOMIAL ALGEBRAS
VLADIMIR DOTSENKO, VINCENT G´ELINAS, AND PEDRO TAMAROFF
To Ed Green with deep admiration of his work on the homology theory of monomial algebras
Abstract.
We show that a finite dimensional monomial algebra satisfies thefinite generation conditions of Snashall–Solberg for Hochschild cohomology ifand only if it is Gorenstein. This gives, in the case of monomial algebras, theconverse to a theorem of Erdmann–Holloway–Snashall–Solberg–Taillefer. Wealso give a necessary and sufficient combinatorial criterion for finite generation.
Contents
Introduction 11. Tor and Ext of monomial algebras, and their higher structures 42. Perfect paths and Gorenstein algebras 103. The Fg conditions and higher structures 184. Periodicity operators for Gorenstein algebras 215. Main theorem 306. Further results 33References 36 Introduction
The present paper is intended as a contribution to the study of support varieties forfinite dimensional algebras. The introduction of support varieties in modular repre-sentation theory [Car83], relying on the finite generation theorems of Evens [Eve61],Golod [Gol59], and Venkov [Ven59] for group cohomology, has revolutionised thesubject since their appearance and has led to deep structural insight into the stablemodule category.A good theory of support varieties for finite dimensional algebras Λ was developedby Snashall and Solberg [SS04] under the hypothesis that certain finite generation( Fg ) conditions hold for Hochschild cohomology. Determining which finite dimen-sional algebras Λ satisfy Fg is an important open problem of representation theory.Furuya and Snashall [FS11] gave some explicit examples of Gorenstein, non self-injective monomial algebras satisfying Fg . Furthermore, a theorem of Erdmann,Holloway, Snashall, Solberg, and Taillefer [EHT +
04, Thm. 1.5] states that anyalgebra satisfying Fg is necessarily Gorenstein. We establish the converse in themonomial case: Theorem (Th. 5.4) . A monomial algebra satisfies the conditions Fg if and onlyif it is Gorenstein. In [Nag11], Nagase proved that a Nakayama algebra satisfies the conditions Fg ifand only if it is Gorenstein. This is a particular case of our result, since Nakayamaalgebras are monomial algebras for special type of quivers (cycles or type A quiv-ers). Our methods are however very different, and rely on the recent work [BG17]of Briggs and the second named author who proposed, for a finite dimensional al-gebra Λ, an approach towards studying the conditions Fg in terms of the canonical A ∞ -structure on the Yoneda algebra Ext ∗ Λ ( k , k ) of the module k = Λ / rad Λ. Theirwork relies on earlier ideas of Green, Snashall, and Solberg [GSS06] to study the ringstructure on Hochschild cohomology in terms of its image under the characteristichomomorphism ϕ k : HH ∗ (Λ , Λ) → Ext ∗ Λ ( k , k ) . We regard this as a useful concrete illustration of how one can apply the canonical A ∞ -algebra structure on the Yoneda algebra towards determining more classicalhomological invariants, such as Hochschild cohomology. This philosophy was advo-cated in [LPWZ04, LPWZ09], but since higher structures of the Yoneda algebrasare generally quite hard to compute, not many applications emerged so far.Knowledge of the canonical A ∞ -algebra structure of Ext ∗ Λ ( k , k ) is equivalent, byProut´e’s theorem [Pro11], to knowing the minimal quasi-free dg algebra resolution e Λ ∼ −→ Λ, what is known as the minimal model of Λ. In the case of monomial algebras,this was done successfully by the third named author, who determined in [Tam18]the minimal model of each monomial algebra Λ and used it to compute the canonical A ∞ -algebra structure on Ext ∗ Λ ( k , k ). This class of algebras is therefore a naturalstarting point for exploring the philosophy of [LPWZ04, LPWZ09]. Moreover, asone can approximate general algebras by monomial ones by means of Gr¨obner bases,a thorough understanding of the monomial case should serve as the basis for a moregeneral line of investigation.One of the main reasons that various invariants of monomial algebras can be com-puted explicitly is the extra grading that can be utilised. All important vector spaceand homology groups associated to a monomial algebra Λ = kQ/I have a gradingby the category C ( Q ) freely generated by the quiver Q . This means that somegenerally structureless vector space acquire distinguished bases, and thus becomenaturally identified with their linear duals, leading to elegant formulas that are notavailable otherwise; in particular, the Tor groups Tor Λ ∗ ( k , k ) have combinatorialbases of the so called Anick chains. This was already noted and used in a recentpreprint [Her18a]. For us, this observation leads to intricate vanishing patterns forhigher structures of Yoneda algebras that can be regarded as analogues of the for-mulas of He and Lu for higher structures associated to N -Koszul algebras [HL05].Those vanishing patterns are at heart of some of our arguments, as they allow us toproduce some explicit A ∞ -central elements of Yoneda algebras, arising from whatwe call “stable relation cycles”. A slightly weaker notion of stability for relationcycles was introduced by Green, Snashall and Solberg in [GSS06]; our approach tostability is directly motivated by applications to higher structures of the Yonedaalgebra.Our results also allow us to give a combinatorial characterisation of the Gorensteinproperty for monomial algebras in terms of Anick chains of sufficiently large length.This builds on recent work of Chen, Shen, and Zhou [CSZ18] who introduced thenotion of perfect paths for monomial algebras in their classification of indecom-posable Gorenstein-projective modules. Their work also shows that the minimal INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 3 resolution of k over a Gorenstein monomial algebra is eventually periodic. A con-sequence of our work is the following reflection of that periodicity property on thelevel of Hochschild cohomology as follows. Theorem (Th. 6.3) . Let Λ be a Gorenstein monomial algebra, of Gorensteindimension d . Then Hochschild cohomology is eventually periodic; more precisely,there exists an element χ ∈ HH ∗ (Λ , Λ) of even degree p such that taking cup productwith χ gives an isomorphism χ ⌣ − : HH n (Λ , Λ) ∼ = −→ HH n + p (Λ , Λ) for all n ≥ d + 1 . For a Gorenstein algebra Λ, we let d HH ∗ (Λ , Λ) denote its Tate–Hochschild cohomol-ogy. Our results imply the following elegant statement.
Theorem (Cor. 6.4) . Let Λ be a Gorenstein monomial algebra. Then Tate-Hochschild cohomology is given by periodic Hochschild cohomology: d HH ∗ (Λ , Λ) ∼ = HH ∗ (Λ , Λ)[ χ − ] . Structure of the paper.
In Section 1, we discuss various special features of(co)homology monomial algebras and the related higher structures. In particular,we establish some vanishing patterns for higher structures of Yoneda algebras thatare at heart of some of the main results of this paper. We also present a newperspective of some classical results, including the bimodule resolution of Bardzell,and a previously unpublished example of a monomial algebra with highly nontrivialhigher structure in its Yoneda algebra. In Section 2, we discuss the combinatoricsof perfect cycles for monomial algebras due to Chen, Shen, and Zhou, and presentan elegant combinatorial description of Anick chains for Gorenstein monomial alge-bras. In Section 3, we recall the Fg conditions of Snashall–Solberg, and reinterpretthem in terms of higher commutators and A ∞ -centres of Ext ∗ Λ ( k , k ). We use thepreviously established vanishing patterns of higher structures to obtain a combi-natorial criterion for A ∞ -centrality in the monomial case. In Section 4, we attach“periodicity operators” in Ext ∗ Λ ( k , k ) to any stable relation cycle, prove that pe-riodicity operators are A ∞ -central, and present the ring of periodicity operatorsexplicitly. In Section 5, we prove the main result of this paper, namely that themonomial algebras satisfying the Fg conditions are exactly the Gorenstein ones.In Section 6, we offer a combinatorial characterisation of the Fg conditions, andrecord some interesting results on the structure of Hochschild cohomology. Conventions and notation.
We use the language of quivers; a quiver is a directedgraph Q = ( Q , Q , s, t ) with the set of vertices Q and the set of arrows Q ,where for any arrow α ∈ Q , we denote by t ( α ) , s ( α ) ∈ Q its target and sourcerespectively. All quivers are assumed finite and connected. We use the notation Q m for the set of paths in Q of length m , and P Q = Q ∪ Q ∪ Q ∪ · · · for the setof all paths in Q . We call a path in Q non-trivial if it has length at least one.All algebras in this article are quotients of the path algebras kQ := span k P Q , where k denotes the ground field. The product convention for paths in kQ is given byfunction composition, so that αβ = 0 unless s ( α ) = t ( β ). We denote by m theJacobson radical rad kQ ; it is spanned by all non-trivial paths. Unless specifiedotherwise, we use the notation Λ = kQ/I for a finite dimensional quotient of thepath algebra with I ⊆ m admissible (so that m n ⊆ I for some n ; this is automaticin the case of finite dimensional monomial algebras). We let k := kQ denote DOTSENKO, G´ELINAS, AND TAMAROFF the corresponding semisimple k -algebra, and use the notation r = rad Λ for theJacobson radical of Λ. This allows us treat Λ as an augmented algebra over k withaugmentation ε : Λ ։ Λ / r = k . All unadorned tensors are over k .All modules over finite dimensional algebras are taken to be right modules andfinitely generated; we simply call them finite. Elements of graded algebras andmodules are always taken to be homogeneous. We refer to the internal grading ofalgebras and modules as the weight grading. The (co)homology groups of gradedalgebras are naturally bi-graded; one grading is given by the weight, and the otherby (co)homological degree, which we simply refer to as degree. The weight anddegree of an element x are denoted by wt( x ) and | x | respectively.Unless otherwise specified, we focus on the case of monomial algebras with idealof relations generated by a set of paths which we assume minimal with respect toinclusion of paths.When dealing with some formulas for homology, we find it useful to utilise the stan-dard convention for homological suspension, implementing it via a formal symbol s of homological degree 1. Acknowledgments.
The second named author is supported by Simons Founda-tion (through a postdoctoral fellowship at Hamilton Mathematics Institute). Weare grateful to Ben Briggs for useful discussions. A crucial thank you is due to JoeChuang and Alastair King who kindly shared with us their unpublished manuscriptfrom fifteen years ago that highlights the importance of previous work of Gruenberg[Gru68] for computing higher structures on Ext algebras of monomial algebras; thebeautiful example of Section 1.3 is also coming from their unpublished note (andis reproduced here with their permission). Their work on higher structures ulti-mately led to a paper on functorial non-minimal resolutions of general associativealgebras [CK12], and their ideas related to the specific case of monomial algebrasnever made it to that paper. However, it was of utmost importance for the genesisof this paper, and we cannot thank them enough for sharing their work with us.1. Tor and
Ext of monomial algebras, and their higher structures
Two formulas for Tor groups.
In general, for an augmented algebra Λ, theTor groups Tor Λ • ( k , k ) of the trivial module can be computed as the homology ofthe bar construction B • (Λ) = { ( s r ) ⊗ n , d } . This formula is useful theoretically butin practice has a lot of limitations: the underlying space of bar construction is toobig. In this section, we shall discuss two formulas that produce the Tor groupson the nose; one is valid for any augmented algebra Λ presented by generatorsand relations, and the other only makes sense for monomial algebras. A few ofthe results of this paper arise from comparing those two formulas. To the best ofour knowledge, this has not been done explicitly before; the only sources we knowwhere the two formulas appear at the same time are the article [BK99] by Butlerand King, and the survey of Ufnarovski [Ufn95].The first formula for the Tor groups is as follows. Theorem 1.1.
Let
Λ = kQ/I with I ⊆ m . For p ∈ N we have that Tor Λ2 p ( k , k ) ∼ = I p ∩ m I p − m I p m + m I p , (1)Tor Λ2 p +1 ( k , k ) ∼ = I p m ∩ m I p I p +1 + m I p m . (2) INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 5
Similar formulas were first discovered in the case of integral homology of groupspresented by generators and relations by Gruenberg [Gru68], and then establishedin the case of associative algebras over a field by Govorov [Gov73]. The work ofGovorov remained largely unnoticed, and it seems that the first prominent appear-ance of this result for quiver algebras is in a paper of Bongartz [Bon83] who refersto a private communication from Butler inspired by work of Gruenberg. We shallrefer to these formulas for Tor-groups as the Govorov–Gruenberg formulas.The Govorov–Gruenberg formulas are way too general to be of computational rel-evance for arbitrary algebras. However, in the case of monomial algebras they canbe made completely combinatorial, if we note that all the vector spaces in thoseformulas have compatible combinatorial bases, e.g. the vector space m I p has a basisof paths obtained as concatenations of a nontrivial path with a path containing atleast p disjoint occurrences of the monomial relations. Once those compatible basesare identified, all intersections and quotients are computed on the level of combi-natorial bases by taking the intersections and set-theoretic differences respectively.The second formula for Tor-groups which is only valid for algebras with monomialrelations utilises combinatorics of the so called Anick chains . We shall use thatterminology, even though it is inaccurate historically; in the case of algebras withmonomial relations that combinatorics seems to have been first discovered by Green,Happel and Zacharia [GHZ85], but is better known from works of Anick [Ani86]and Anick and Green [AG87] where the case of monomial algebras was upgradedto the case of augmented algebras with a Gr¨obner basis of relations.Let us recall the definition of the set C n of (right) Anick n -chains and of a tail ofan Anick chain; this is done by induction on n ≥
0. We let C = Q be the set ofarrows, and any a ∈ C is its own tail. Suppose that C n − is defined, as well astails of ( n − n -chain is a path γ in P Q such that(i) we can write γ = γ ′ t with γ ′ ∈ C n − ,(ii) if t ′ is the tail of γ ′ , then t ′ t has a right divisor which is a monomial relation,(iii) no proper left divisor of γ satisfies both (i) and (ii).By definition, the tail of γ is the path t γ := t . Note that C is the set of the definingrelations of Λ, with the tail of each monomial relation given by the path obtainedafter removing the leftmost arrow in it. It is shown in [Ani86, AG87] that a path γ admits at most one structure of a chain: if γ is an n -chain with tail t γ , then n and t γ are uniquely determined.The following result is essentially due to Green, Happel, and Zacharia [GHZ85]. Theorem 1.2.
For each n ∈ N we have isomorphisms Tor Λ n ( k , k ) ∼ = kC n − . To prove this result, one uses the combinatorial definition above to prove a slightlystronger statement: the minimal free right module resolution of k over Λ can beproved to have the form · · · → kC n ⊗ k Λ d −→ kC n − ⊗ k Λ d −→ · · · d −→ kC ⊗ k Λ d −→ kC ⊗ k Λ → Λ → d ( γ ⊗
1) = γ ′ ⊗ t γ . (Of course applying the functor − ⊗ Λ k to this resolution,we immediately see that Tor Λ n ( k , k ) ∼ = kC n − , as required.)Similarly, one can define the set of left Anick chains C ′ n by considering heads insteadof tails, and thus obtain the minimal free left module resolution of k over Λ. Bardzell[Bar97] proved that C ′ n = C n for all n ≥
0, and so for any n -chain γ , we can write DOTSENKO, G´ELINAS, AND TAMAROFF γ = γ ′ t γ = h γ γ ′′ for unique ( n − γ ′ , γ ′′ , with tail t γ and head h γ . Anotherexplanation of this symmetry from the point of view of the Govorov–Gruenbergformulas was given by Butler and King, see [BK99, Th. 8.2, Remark 8.3].At this point, it is appropriate to make a remark on how the extra grading men-tioned earlier can be utilised. Since the path algebra kQ , as well as its ideals gen-erated by monomials, is graded by the category C ( Q ) generated by Q , that gradingdescends to all monomial quotients of kQ , and to bar constructions of those quo-tients, where the boundary map manifestly respects the grading. Thus, every groupTor Λ n ( k , k ) is graded by C ( Q ). From the description of Tor via Anick chains, it isclear that for every path p ∈ P Q , the p -graded component of Tor is of dimensionat most one (and when it is of dimension one, the corresponding one-dimensionalspace has the Anick chain p as its basis). Since the Tor groups (together with their C ( Q )-gradings) are well defined, all the descriptions we outlined (via the bar con-struction, of Govorov and Gruenberg, and of Green–Happel–Zacharia) must givethe same result. The first immediate consequence of that is the result of Bardzellon the coincidence of left and right Anick chains mentioned in the previous para-graph: both of those index nonzero graded components in the homology of the barconstruction. Another interesting observation is that the above property of Torstating that for each path p ∈ P Q , the p -graded component of Tor is of dimensionat most one is precisely the key result of the paper of Iyudu [Iyu16] who suggeststhat it was conjectured by Kontsevich in 2015.1.2. Higher structures of Tor and Ext.
Recall that an A ∞ -algebra over k isa graded k -bimodule E equipped with multilinear operations m n : E ⊗ n → E ofdegree 2 − n satisfying the Stasheff identity X r + s + t = n ( − rs + t m r +1+ t ◦ (id ⊗ r ⊗ m s ⊗ id ⊗ t ) = 0for each n ≥
1. These identities imply that m is a differential for E and that m is associative up to coboundary given by m , and we call ( E, { m n } ) minimal when m = 0. In this case, ( E, m ) is an honest graded augmented k -algebra with someextra data given by the higher products { m n } n ≥ . We refer the reader to [LV12,Chapter 9] for further information on A ∞ -algebras.The A ∞ -algebra E is said to be strictly unital if it contains an element 1 for which m (1 , x ) = x = m ( x,
1) and m n ( . . . , , . . . ) = 0 for n ≥ . The A ∞ -algebra E is augmented over k if it is equipped with degree zero morphisms ε : E ⇆ k : η satisfying εη = id, so that it is strictly unital for 1 := η (1), and theproducts preserve the augmentation ideal E := ker( ε ). We always take E to beaugmented.In particular, the Yoneda algebra E = Ext ∗ Λ ( k , k ) of any augmented dg k -algebraadmits a minimal augmented A ∞ -algebra structure making it A ∞ -equivalent tothe dg algebra RHom Λ ( k , k ). Such a structure is unique up to A ∞ -isomorphism; itrecovers Λ up to an A ∞ -quasi-isomorphism. For the rest of this article Ext ∗ Λ ( k , k )will always be considered equipped with such an A ∞ -structure, which we shall callcanonical. (Dually, there is a notion of an A ∞ -coalgebra; for each augmented k -algebra Λ, the graded space Tor Λ ∗ ( k , k ) has an A ∞ -coalgebra structure which is A ∞ -equivalent to the bar construction B ∗ (Λ), regarded as a dg coalgebra with thedeconcatenation coproduct.) INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 7
For a monomial algebra Λ, the canonical A ∞ -structure on Ext was determined bythe third named author in [Tam18]; this result generalises the classical formula ofGreen and Zacharia [GZ94] for the Yoneda product. By Theorem 1.2, if we denoteby ( − ) ∨ the k -dual, we have identificationsExt ∗ Λ ( k , k ) = ( k ⊕ kC ∗− ) ∨ = k ⊕ kC ∨∗− , where we set C − = ∅ and moreover abuse notation and let C ∨ r ⊆ P Q op consist ofthe paths in C r with opposite orientation. More generally we identify the sets ofpaths P Q and P Q op via the notation p ↔ p ∨ which reverses orientation, so that( pq ) ∨ = q ∨ p ∨ .Consider the Yoneda algebra E = Ext ∗ Λ ( k , k ) = k ⊕ kC ∨∗− . For each n ≥
2, let usdefine morphisms m n : E ⊗ n → E on (duals of) Anick chains γ i ∈ C ∨ r i by m n ( γ , . . . , γ n ) = ( ( − N γ . . . γ n if γ . . . γ n ∈ C ∨ r + ··· + r n +1 , N = P i These maps model the canonical A ∞ -algebra structure on Ext ∗ Λ ( k , k ) .Proof. This is analogous to [Tam18, Th. 4.2] but uses left Anick chains insteadof right ones; one just has to remark that in this case the only non-vanishing treeappearing in the homotopy transfer formula for the higher products is the left comb,and that no extra signs arise, as opposed to what happens for the right comb. (cid:3) Example of non-vanishing higher products. At this point we alreadyhave enough information to present an interesting example due to Chuang andKing of a very simple monomial quiver algebra which however has highly non-trivial higher structure on its Yoneda algebra. The only available source in theliterature where an example of a similar nature appears is a paper of Conner andGoetz [CG11]; however, in that case the authors show that a certain canonical A ∞ -structure is highly nontrivial, and do not establish that it holds for every canonical A ∞ -structure. Hence we believe that this example should be of interest to experts.Let us consider the cyclic quiver C with the arrows a, b, c, d, e in the cyclic order. ••• • • ea b cd The algebra CK is defined by generators and relations as kC / ( abcd, bcde, deab ), which is a finite-dimensional alge-bra. We will prove that a canonical A ∞ -structure on itsYoneda Ext-algebra cannot be “too simple”. Theorem 1.4. The Yoneda algebra B = Ext ∗ CK ( k , k ) has infinitely many non-vanishing higher products, for anycanonical A ∞ -structure.Proof. The Anick 1-chains of the algebra CK are the re-lations abcd , bcde , and deab , and the Anick 2-chains are abcde , bcdeab , deabcd .Furthermore, it turns out that for each n ≥ n = 2 s − 1, the monomials bcde ( abcde ) s − abcd and de ( abcde ) s − ab, and for n = 2 s , the monomials bcde ( abcde ) s − ab and de ( abcde ) s − abcd. DOTSENKO, G´ELINAS, AND TAMAROFF This can be easily proved by induction on n . We identify these chains with theirduals in the Yoneda algebra B .We prove the statement of the theorem in a very concrete way. Namely, we demon-strate that for the element ω n = d ⊗ e ⊗ ( abcde ) ⊗ ( n − ⊗ a ⊗ b ∈ B ⊗ n , and for any canonical A ∞ -structure { ν n } on B , we have ν n ( ω n ) = 0.First, we note that the total degree of ω n is 1 + 1 + 3( n − 4) + 1 + 1 = 3 n − 8, andthe concatenation de ( abcde ) n − ab is a chain of length 2( n − − 1, so an elementof homological degree 2( n − 3) = 2 n − 6. Since 2 n − n − − n , thehomological degrees match, so for the canonical structure { µ n } of Tamaroff, wehave µ n ( ω n ) = 0.Suppose that ν n is another canonical structure, and { f n } is an ∞ -equivalence be-tween them; without loss of generality, f = id. Let us denote by Φ n the set of all n -fold products of chains γ ⊗ · · · ⊗ γ n in B ⊗ n with the following properties:(i) all degrees | γ i | are odd,(ii) if | γ i | = 1, then i ∈ { , , n − , n } ,(iii) if | γ | = | γ | = 1, then γ = d , γ = e ,(iv) if | γ n − | = | γ n | = 1, then γ n − = a , γ n = b ,(v) | γ i | > i .By direct inspection, every connected r -fold subtensor ψ of φ ∈ Φ n is in Φ r \ { ω r } .Also, for the canonical structure of Tamaroff, if µ n ( φ ) = 0 for some φ ∈ Φ n , then φ = ω n , which easily follows from the observation that concatenations of pairs ofelements of degree greater than 2 cannot be found as divisors of any Anick chain.Using these observations, we prove by induction on n that for any minimal canonical A ∞ -structure { ν m } and for any φ ∈ Φ n we have ν n ( φ ) = µ n ( φ ). The morphismidentity states that X i + j + k = n ± f i +1+ k (1 ⊗ i ⊗ µ j ⊗ ⊗ k )( φ ) = X i + ... + i r = n ± ν r ( f i ⊗ · · · ⊗ f i r )( φ ) . By induction and above observations, the only element on the left that does notvanish is f ( µ n ), and the only element on the right that does not vanish is ν n ( f ⊗ n )(since for r < n , in order for ν r to not vanish, ( f i ⊗ · · · ⊗ f i r )( φ ) must create ω n with a nonzero coefficient, which is impossible for degree reasons), so we aredone. (cid:3) Vanishing patterns for higher products. Let us now demonstrate howone can combine the two formulas for Tor groups to prove some nontrivial results. Proposition 1.5. Consider the canonical A ∞ -algebra structure on Ext ∗ Λ ( k , k ) fromTheorem 1.3. Suppose γ , . . . , γ n are elements of the Ext algebra.(i) If at least three of them are of even degree, then m n ( γ , . . . , γ n ) = 0 .(ii) If exactly two of them are of even degree, then m n ( γ , . . . , γ n ) = 0 unlessthose elements are γ and γ n .(iii) If exactly one element is of even degree, then m n ( γ , . . . , γ n ) = 0 unless thatelement is γ or γ n .Proof. By linearity, it is enough to prove this result in the case of all those elementsbeing (duals of) Anick chains. Suppose that among them, we have k elements of even INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 9 degrees 2 d , . . . , 2 d k , and ℓ elements of odd degrees 2 e + 1, . . . , 2 e ℓ + 1; of course, n = k + ℓ . We once again use the fact that Tor groups and their combinatorial basesare well defined, and identify those elements with the elements of the Govorov–Gruenberg spaces; from Formulas (1) and (2) it follows in particular that thoseelements are represented by paths in I d , . . . , I d k , I e m ∩ m I e , . . . , I e ℓ m ∩ m I e ℓ respectively. By Formula (3), the value of the operation µ n on these elements isproportional to the concatenation of the corresponding paths.Note that the concatenation of paths from I d , . . . , I d k , I e m ∩ m I e , . . . , I e ℓ m ∩ m I e ℓ (in any order) is in I d + ··· + d k + e + ··· + e ℓ . At the same time, the result of applying µ n to such words has homological degree2( d + · · · + d k + e + · · · + e ℓ ) + ℓ + 2 − n = 2( d + · · · + d k + e + · · · + e ℓ ) + 2 − k. In particular, for k > 2, the degree of µ n ( γ , . . . , γ n ) is less than d = 2( d + · · · + d k + e + · · · + e ℓ ) . Applying Formulas (1) and (2) again, we conclude that the concatenation of all ourelements is in the zero coset of the corresponding Govorov–Gruenberg space.Similarly, for k = 2, the degree of µ n ( γ , . . . , γ n ) is the even number2( d + · · · + d k + e + · · · + e ℓ ) . If we assume that at least one of the two elements γ , γ n is of odd degree, weobserve that the concatenation of all our elements is in m I d + I d m , which is in thezero coset of the corresponding Govorov–Gruenberg space.Finally, for k = 1, the degree of µ n ( γ , . . . , γ n ) is the odd number2( d + · · · + d k + e + · · · + e ℓ ) + 1 . If we assume that both elements γ , γ n are of odd degree, we observe that theconcatenation of all our elements is in m I d m , which is in the zero coset of thecorresponding Govorov–Gruenberg space. (cid:3) As a first consequence, one may simplify the signs arising in the higher products,at least to some extent. Corollary 1.6. Formula (3) can be simplified to m n ( γ , . . . , γ n ) = ( ( − nr + r r n + r + r n γ . . . γ n if γ . . . γ n ∈ C ∨ r + ··· + r n +1 , Application: Bardzell’s resolution. In [Bar97], Bardzell constructed ex-plicitly the minimal bimodule resolution of the diagonal bimodule over any algebraΛ with monomial relations. By an intricate study of combinatorics of Anick chains,he established that the shape of the differential of such resolution alternates, de-pending on the homological degree. Let us explain how this phenomenon is animmediate consequence of Proposition 1.5.Repeating the argument of [Her18b, Th. 4.2] mutatis mutandis for the case of quiveralgebras, we see that the canonical A ∞ -algebra structure on Ext ∗ Λ ( k , k ) gives riseto a free bimodule resolution of the diagonal bimodule. The corresponding twistingcochain τ : Tor Λ ∗ ( k , k ) → Λ is extremely easy to describe for a monomial quiveralgebra Λ = kQ/I : it annihilates elements of homological degree different fromone, and on elements of degree one it is given by the identity map under theidentification Tor Λ1 ( k , k ) = kQ ⊂ Λ. Now, the only thing that remains is to examine carefully the dual formulas of 3.We shall use the variant of that formula which describes the boundary map in theminimal model of Λ, and can be chosen in the form similar to [Tam18, Th. 4.1],but with a different choice of signs arising from considering left Anick chains: b ( s − γ ) = X n ≥ ( − | s − γ | s − γ ⊗ · · · ⊗ s − γ n , where the sum ranges through all decompositions of an Anick chain γ . This shape ofthe formula is particularly convenient for computing the differential of the bimoduleresolution of the diagonal module, since the twisting cochain τ is precisely theoperator γ s − γ on γ of degree 1. Thus, to obtain the answer, we need toclassify all possible decompositions γ = γ · · · γ n with | s − γ | = | s − γ | + · · · + | s − γ n | + 1 and where in addition all s − γ i butone are of homological degree 0. By Proposition 1.5, this depends on the parityof | γ | . If | γ | is odd, then for the only element s − γ i of positive homological degree, | γ i | is even, and so i = 1 or i = n . On the other hand, if | γ | is even, then forthe only element s − γ i of positive homological degree, | γ i | is odd, so there are noconstraints on the position of this element. Thus, Formula 1.5 suggests that theimage of the generator e t ( γ ) ⊗ γ ⊗ e s ( γ ) of the bimodule resolution Λ ⊗ kC ∗ ⊗ Λunder the differential of that resolution is given by the formula ( h γ ⊗ γ ′′ ⊗ e s ( γ ) − e t ( γ ) ⊗ γ ′ ⊗ t γ , | γ | ≡ , X α ⊗ ˜ γ ⊗ β, | γ | ≡ , where γ = h γ γ ′′ = γ ′ t γ are the decompositions of c as the left chain and the rightchain that factor out the head and the tail, and the sum is over all decompositions c = α ˜ γ β with ˜ γ a chain of homological degree one less than γ . This is precisely theresult of [Bar97, Th. 4.1], the main theorem of that paper.2. Perfect paths and Gorenstein algebras Gorenstein-projective modules and related combinatorics. Recall thata module M ∈ mod Λ is called Gorenstein-projective if it satisfies the followingconditions:i) There exists an acyclic complex of finite projectives C ∗ : · · · → C d −→ C d −→ C d −→ C d −→ C − d −→ C − → · · · such that coker( C d −→ C ) ∼ = M .ii) The dual complex Hom Λ ( C ∗ , Λ) is also acyclic.In this case the non-negative truncation of C ∗ resolves M , and we call C ∗ a completeresolution of M . Equivalently [Buc86, Chp. 4], M is Gorenstein-projective if andonly if Ext i Λ ( M, Λ) = 0 and Ext i Λ op ( M ∗ , Λ op ) = 0 for all i > M ∼ = M ∗∗ is reflexive, where M ∗ = Hom Λ ( M, Λ) is the dual module.In [CSZ18], Chen–Shen–Zhou classified the indecomposable Gorenstein-projectivesover monomial algebras. Let us recall the combinatorial notions that underpin that INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 11 classification. Note that for a monomial algebra Λ, the coset of a path p ∈ P Q iszero in Λ if and only if it is divisible by a relation. We let P Λ = { p ∈ P Q | p is not divisible by any relation } be the subset of nonzero paths in Λ, and the canonical surjection kQ ։ Λ identifies P Λ with a basis for Λ.Given any path p ∈ P Λ , we define minimal left zero cofactors of p to be thosenon-trivial paths q ∈ P Λ with s ( q ) = t ( p ), qp = 0, and for which no proper endsegment q ′ has the same property: if q = q ′′ q ′ , and q ′ p = 0, then q = q ′ . Similarlyone defines minimal right zero cofactors. We let L ( p ) = { q ∈ P Λ | q is a minimal left zero cofactors of p } R ( p ) = { q ∈ P Λ | q is a minimal right zero cofactors of p } . Definition 2.1. A pair of non-trivial paths ( p, q ) in Λ with s ( p ) = t ( q ) is a perfectpair if R ( p ) = { q } and L ( q ) = { p } . Definition 2.2. Let p = ( p , p , . . . , p r − ) be a sequence of non-trivial paths in Λsuch that s ( p i ) = t ( p i +1 ) for all i ∈ Z /r Z . We say that p is a perfect cycle if allpairs ( p i , p i +1 ) are perfect; in such case we call the paths p i perfect as well. If r isthe minimal integer with this property, we call r the period of p .Given a perfect cycle p = ( p , p , . . . , p r − ), one may extend it by periodicity to p = ( p i ) i ∈ Z by setting p i + r = p i . We then attach an unbounded, periodic complexof finite right projectives Q ( p ) ∗ , with terms Q ( p ) i = e s ( p i ) Λ and differentials given byleft multiplication by the p i : Q ( p ) ∗ : · · · → e s ( p i +3 ) Λ l pi +3 −−−→ e s ( p i +2 ) Λ l pi +2 −−−→ e s ( p i +1 ) Λ l pi +1 −−−→ e s ( p i ) Λ → · · · That this is a complex follows from the zero cofactor relation p i p i +1 = 0 in Λ.Moreover, Q ( p ) ∗ is acyclic as each pair ( p i , p i +1 ) is perfect [CSZ18, Prop. 4.4].Let GP (Λ) be the set of isomorphism classes of non-projective, indecomposableGorenstein-projective right modules over Λ. Theorem 2.3 ([CSZ18, Th. 4.1]) . Let Λ be a monomial algebra. Then the map p p Λ is a bijection { perfect paths in P Λ } ↔ GP (Λ) . Moreover, given a perfect path p = p in a perfect cycle p = ( p , p , . . . , p r − ) , thecomplex Q ( p ) ∗ is a complete resolution of p Λ , and in particular its non-negativetruncation · · · → e s ( p ) Λ l p −−→ e s ( p ) Λ l p −−→ e s ( p ) Λ l p −−→ e s ( p ) Λ l p −−→ p Λ → is a minimal projective resolution of p Λ = p Λ . A monomial algebra Λ may contain no perfect path and so may admit no non-trivial Gorenstein-projective module at all. At the opposite end of the spectrum,Gorenstein monomial algebras (of infinite global dimension) have an abundance ofnon-trivial Gorenstein-projective modules, and so have a large supply of perfectpaths in view of the classification result of Chen–Shen–Zhou.Recall that Λ is Gorenstein if the two-sided injective dimensions injdim Λ Λ < ∞ andinjdimΛ Λ < ∞ are finite, in which case they are equal by a theorem of Zaks [Zak69]. In this case, we denote this common number by dim Λ and call it the Gorensteindimension of Λ. When Λ is Gorenstein, [Buc86, 4.2] shows that Ext i Λ ( M, Λ) = 0 forall i > M to be Gorenstein-projective and so the n -th syzygy Ω n N of any module N ∈ mod Λ is Gorenstein-projective for n ≥ dim Λ.For finite dimensional algebras, by work of Bergh–Jorgensen–Oppermann [BOJ15]one can in fact characterise the Gorenstein property for Λ in terms of the syzygiesof k . We will use this in Section 3 to characterise Gorenstein monomial algebras interms of Anick chains. The following easy consequence of [BOJ15, Th. 4.1] will bethe relevant result for us. Proposition 2.4. Let Λ = kQ/I be a finite-dimensional path algebra. Then Λ isGorenstein if and only if Ω n k is Gorenstein-projective for some n ≥ . In this case,the minimal such n equals the Gorenstein dimension of Λ .Proof. If Λ is Gorenstein then Ω n k is Gorenstein-projective whenever n ≥ dim Λ.Conversely, if Ω n k is Gorenstein-projective for some n ≥ Gorenstein defect category (that is, the singularity category of Λ mod-ulo Gorenstein-projectives, see [BOJ15]) vanishes by Jordan-H¨older filtration argu-ments, and this characterises the Gorenstein property by [BOJ15, Th. 4.1]. Forthe last statement, let Λ be Gorenstein and let n be the minimal such integer. Itis immediate that n ≤ dim Λ. Since Ω n k is Gorenstein-projective, we haveExt n + i Λ ( k , Λ) = Ext i Λ (Ω n k , Λ) = 0 for all i > . From Jordan-H¨older filtrations one obtains that Ext n + i Λ ( N, Λ) = 0 for all i > N ∈ mod Λ, and so dim Λ ≤ n . (cid:3) Anick chains over Gorenstein algebras. We now know how to test theGorenstein property for a monomial algebra Λ in terms of the syzygy modulesΩ n k via Proposition 2.4. In this section, we translate this into the combinatoriallanguage of Anick chains, and deduce normal forms for chains of sufficiently largedegree over Gorenstein monomial algebras.Up to now we have taken modules without further qualifiers to mean right modules.However, certain results below will need to be stated for both left and right modules,and similarly results on the structure of Anick chains will need to be stated forboth right and left Anick chains, which correspond to the right and left minimalprojective resolution of k = Λ / r , respectively. In what follows we will continueworking with right modules and right Anick chains, and simply note that all dualstatements for left modules and chains can be obtained formally by passing to theopposite algebra Λ Λ op .We start by relating the syzygy modules to Anick chains. Proposition 2.5. Let Λ be a monomial algebra and n ≥ . We have isomorphismsof right and left modules Ω n ( k Λ ) ∼ = M γ ∈ C n − t γ ΛΩ n ( Λ k ) ∼ = M γ ∈ C n − Λ h γ . INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 13 Proof. Recall that the minimal resolution of ( P ∗ , ∂ ) ∼ −→ k is given by the Anickresolution · · · → kC n − ⊗ k Λ ∂ n − −−−→ kC n − ⊗ k Λ ∂ n − −−−→ · · · ∂ −→ kC ⊗ k Λ ∂ −→ Λ → n -th term P n = kC n − ⊗ k Λ and differential ∂ n − ( γ ) = γ ′ ⊗ t γ , and wecan identify Ω n k = Im( ∂ n − ) explicitly. Consider the identification of projectivemodules kC n − ⊗ k Λ = M γ ∈ C n − e s ( γ ) Λgiven by sending γ ⊗ e s ( γ ) . The image of ∂ n − then takes the formIm( ∂ n − ) = X γ ∈ C n − t γ Λ ⊆ M γ ′ ∈ C n − e s ( γ ′ ) Λwhere by abuse of notation we let γ ′ ∈ C n − run over all ( n − n − γ = γ ′ t γ so that t γ Λ ⊆ e s ( γ ′ ) Λ sits as a submodule in thecorresponding copy of e s ( γ ′ ) Λ. Moreover, here we must assume that n ≥ 2, but thecase n = 1 is treated similarly by replacing the righthand side by Λ.Now note that this sum is in fact direct. Assume that γ = γ ′ t γ and γ = γ ′ t γ are ( n − t γ Λ, t γ Λ are sent into the same summand e s ( γ ′ ) Λ, meaningthat γ ′ = γ ′ = γ ′ . If t γ Λ ∩ t γ Λ = 0, then there is a path left divisible by bothtails t γ and t γ , and so one tail must divide the other. As γ ′ = γ ′ , one of γ , γ then divides the other, contradicting minimality. Hence we obtain:Im( ∂ n − ) = M γ ∈ C n − t γ Λ ⊆ M γ ′ ∈ C n − e s ( γ ′ ) Λ , as required. The second claim is dual. (cid:3) Our next step is to characterise Gorenstein monomial algebras in terms of Anickchains. We first need two auxiliary statements. The first of them collects severalresults of [CSZ18]. Lemma 2.6 ([CSZ18, Lem. 3.1, Prop. 4.6]) . Let Λ be a monomial algebra and p anon-trivial path in Λ . Then:i) p Λ is Gorenstein-projective and non-projective if and only if R ( p ) = { q } for q a perfect path.ii) Λ q is Gorenstein-projective and non-projective if and only if L ( q ) = { p } for p a perfect path.iii) p Λ is projective if and only if R ( p ) is empty.iv) Λ q is projective if and only if L ( q ) is empty.Remark . This lemma is “best possible” in that Chen-Shen-Zhou construct[CSZ18, Ex. 4.5] an example of a non-perfect path p with Λ p Gorenstein-projective,along with an isomorphism Λ p ∼ = Λ p ′ for some perfect path p ′ .The second result we need is as follows. Lemma 2.8. Let Λ be a Gorenstein algebra, and let ( P ∗ , ∂ ) ∼ −→ k be the minimalprojective resolution. If Ω n k = Im( ∂ : P n → P n − ) has a non-trivial projectivesummand, then n ≤ dim Λ . Proof. Decompose Ω n k = N ⊕ P with P a non-trivial projective summand. Con-sider the short exact sequence0 → Ω n k ι −→ P n − ∂ −→ Ω n − k → . If n > dim Λ, we show that the embedding ι | P : P ֒ → P n − splits, and this willcontradict minimality of the resolution. Consider the commutative diagram of shortexact sequences 0 0 N s (cid:21) (cid:21) πιs / / O O M < < ①①①①①①①①① (cid:15) (cid:15) / / Ω n k (cid:21) (cid:21) O O ι / / P n − / / π : : ✉✉✉✉✉✉✉✉✉ Ω n − k / / P O O ι | P ; ; ①①①①①①①①① = = ④④④④④④④④ O O A diagram chase shows that we have an induced exact sequence0 → N → M → Ω n − k → . Now, Gorenstein-projectives are closed under summands and extensions, and mustbe projective the moment they have finite projective dimension. If n > dim Λ, thenΩ n − k and Ω n k , and therefore also N , are Gorenstein-projective, and thus so is theextension M . But the diagonal short exact sequence shows that projdim M < ∞ and so M must be projective. Hence the map ι | P : P ֒ → P n − is split as claimed,contradicting minimality of the resolution. (cid:3) Results of Lu–Zhu [LZ17, Rem. 4.12] give a simple description of the self-injectivemonomial algebras: these are precisely the algebras Λ = Λ m,n whose quiver is asimple oriented cycle on m vertices with relations consisting of all paths of length n . We now show how to characterise the Gorenstein algebras of dimension at most d + 1 in terms of the structure of d -chains; note that this forces 0 ≤ d < ∞ , and soour results complement the case of Gorenstein dimension zero of Lu and Zhu. Theirpaper also characterises monomial algebras of Gorenstein dimension at most 1; weleave it to the reader to compare [LZ17, Th. 5.4] with the corresponding particularcase of the result below. Theorem 2.9. Let Λ be a monomial algebra. The following are equivalent, for d ≥ :i) Λ is Gorenstein of dimension at most d + 1 .ii) For every d -chain γ , either R ( t γ ) is empty or equal to { p } with p perfect.iii) For every d -chain γ , either L ( h γ ) is empty or equal to { q } with q perfect.Moreover if dim Λ < d + 1 , then t γ and h γ themselves are perfect, and in particularthe sets above are never empty. INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 15 Proof. Prop. 2.4 and Prop. 2.5 show that Λ is Gorenstein of dimension dim Λ ≤ d + 1 if and only if Ω d +1 k = L γ ∈ C d t γ Λ is Gorenstein-projective, and by Lemma2.6 this occurs if and only if R ( t γ ) is empty or consists of a single perfect path.This shows the equivalence i)-ii), with i)-iii) formally dual.For the last claim assume that actually dim Λ ≤ d . First assume that d ≥ 1. Let γ ∈ C d , and write it as γ = γ ′ t γ for γ ′ ∈ C d − . By the above we have R ( t γ ′ ) = ∅ or R ( t γ ′ ) = { p } for p perfect. But t γ ∈ R ( t γ ′ ) = ∅ , which shows that t γ = p isperfect. Dually h γ is perfect.For the case d = 0 we have γ = t γ , which must then be an arrow. Lemma 2.8shows that t γ Λ cannot be projective and so R ( t γ ) = ∅ , which forces R ( t γ ) = { p } for p perfect. Letting L ( p ) = { q } , we see that t γ p = 0 forces q to be a right divisorof t γ . But t γ is an arrow and so t γ = q is perfect, as claimed. The case of h γ isdual as before. (cid:3) Ignoring the precise dimension, we obtain a cleaner characterisation of Gorensteinmonomial algebras. Corollary 2.10. Let Λ be a monomial algebra. The following are equivalent:i) Λ is Gorenstein.ii) There exists an n ∈ N such that every n -chain γ for n ≥ n has tail t γ = p given by a perfect path.iii) There exists an n ∈ N such that every n -chain γ for n ≥ n has head h γ = q given by a perfect path. As an example of how our results can be applied, let us give a characterisationof local Gorenstein monomial algebras. By a local monomial algebra, we meanalgebras Λ of the form Λ = k h S i /I with S a non-empty finite set and I ⊆ ( S ) generated by monomial relations. (We still assume Λ finite dimensional.) Proposition 2.11. Let Λ = k h S i /I be a local monomial algebra. Then Λ is Goren-stein if and only if Λ ∼ = k [ t ] / ( t n ) for some n ≥ .Proof. We show that | S | ≥ s = t ∈ S bedistinct elements, and note that s m , t n ∈ R for some minimal m, n ≥ t, t n , t n +1 , t n , . . . , t kn , t kn +1 , . . . are all Anickchains and in particular t kn +1 has tail t . If Λ is Gorenstein then t kn +1 has perfecttail for k ≫ t must be a perfect path. Similarly s is perfect.We have ( ts ) l ∈ I for l ≫ tstst . . . or ststs . . . , say the first one without loss of generality. Theperfect pair ( t, t n − ) then shows that t n − left divides stst . . . , a contradiction. (cid:3) We were not able to locate a precise reference for Proposition 2.11 in the literature,although we have no doubts that it is known to experts. In particular, it admits thefollowing short proof not relying on combinatorics of Anick chains. A local Goren-stein algebra A must be self-injective, as follows from the Auslander–BuchsbaumFormula for noncommutative local rings of Wu-Zhang (see [WZ01, Th. 0.3], usingHom k ( A, k ) for the pre-balanced dualizing complex). Applying [LZ17, Rem. 4.12],we see that A is a local Gorenstein monomial algebra if and only if A ∼ = k [ t ] / ( t n )for some n . Example of a Gorenstein monomial algebra. Let d ≥ 2. Let Λ d = kQ d /I d , with quiver Q d as shown in the picture and relations I d = ( R d ) given by R d = { β β , β β , α i α i +1 | i d − } . Then Λ d is Gorenstein of dimension d . • • ••• β β • • • δ δ α α α d Indeed all Anick ( d − β β β . . . or β β β . . . with tails β i perfect, or bythe path α α . . . α d whose tail α d satisfies R ( α d ) = ∅ , and so Λ d is Gorenstein of dimension ≤ d by The-orem 2.9. However α . . . α d − is an ( d − α d − and R ( α d − ) = { α d } is neither emptynor is α d perfect, and so Λ d is not Gorenstein of di-mension ≤ d − 1. Finally, note that α d Λ d is a non-trivial projective summand in Ω d k , and that suchprojective summands do not occur in higher degreesas all further Anick chains are given by β β β . . . and β β β . . . with tails β i .2.4. Perfect walks. In analysing Anick chains of sufficiently high degree overGorenstein algebras, it will turn out useful to introduce paths which consist ofwalking along a perfect cycle. Definition 2.12. A sequence of paths ( p , p , . . . , p l − ) is called a perfect walk ifit can be extended to a perfect cycle p = ( p , p , . . . , p l − , p l , . . . , p s − ). We call l ≥ R ( p i ) = { p i +1 } and L ( p i ) = { p i − } , and in particular any perfect path p = p can be extended in both directions to a perfect walk of any length . . . , p − , p − , p , p , p , . . . By abuse of notation we will often say that a path α = p p . . . p l is given bya perfect walk if ( p , p , . . . , p l ) forms a perfect walk. For instance, the paths β β β . . . and β β β . . . of example in Section 2.3 are given by perfect walks. Wenote that the decomposition of a path as a perfect walk is not in general unique. Example 2.13. Let Λ = k [ t ] / ( t n ) for n ≥ 2. Then t kn can be written as a perfectwalk in two ways: t kn = t · t n − · · · t · t n − = t n − · t · · · t n − · t corresponding to the two perfect walks ( t, t n − , . . . , t, t n − ) and ( t n − , t, . . . , t n − , t ).In the above example, the path t kn forms a (2 k − Example 2.14. Let Λ = k [ t ] / ( t n ) for n ≥ 3. Consider the perfect cycle ( t n − , t ).Then ( t n − , t, t n − ) is a perfect walk as it can be extended to ( t n − , t, t n − , t ) but t n − = t n − · t · t n − is not an Anick chain.We will be particularly interested in Anick chains that can be (partially) written asperfect walks, and so we will derive a few properties of such chains. The followinglemma easily follows from the definition, we leave the proof to the reader. Lemma 2.15 (Extending Anick chains) . Let γ n be an n -chain. Then:i) γ n +1 = γ n p is an ( n + 1) -chain with tail t γ n +1 = p if and only if p ∈ R ( t γ n ) . INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 17 ii) γ n +1 = q γ n is an ( n + 1) -chain with head h γ n +1 = q if and only if q ∈ L ( h γ n ) . Proposition 2.16 (Weak unique extension property) . Let γ n be an n -chain. Then:i) If t γ n = p is perfect, then γ n +1 = γ n p is the unique right extension of γ asan ( n + 1) -chain, then with tail p .ii) If h γ n = q is perfect, then q − γ n is the unique left extension of γ as an ( n + 1) -chain, then with head q − .Proof. This follows from Lemma 2.15 as R ( p ) = { p } and L ( q ) = { q − } . (cid:3) Corollary 2.17 (Unique extension property) . Let γ be an n -chain and α somenon-trivial path in Q .i) If t γ = p is perfect, then γ α is an Anick ( n + k ) -chain for some k ≥ if andonly if α = p . . . p k , in which case t γ α = p k .ii) If h γ = q is perfect, then α γ is an Anick ( n + k ) -chain for some k ≥ if andonly if α = q − k . . . q − , in which case h α γ = q − k .Proof. We only prove the first claim as the proof of the second one is similar. Theif direction follows from iterating Prop. 2.16, which shows α = p . . . p k has therequired property. We prove the converse by induction starting with k = 1. Clearlyif γ α is an ( n + 1)-chain then α = p by Prop. 2.16, and then t γ α = p .For k > γ α = ηt γ α for an ( n + k − η . Then one of γ and η divide the other, which forces η = γ α ′ as n < n + k − α ′ = p . . . p k − and t γ α ′ = p k − . Then γ α = ( γ α ′ ) t γ α is an( n + k )-chain extending the ( n + k − γ α ′ = γ p . . . p k − and so t γ α = p n + k by the base case. Thus α = α ′ t γ α = p . . . p k and t γ α = p k as claimed. (cid:3) From this, we obtain a description of Anick chains of high degree over a Gorensteinmonomial algebra. Corollary 2.18. Let Λ be a Gorenstein monomial algebra, let d = dim Λ and let γ n be an n -chain for n ≥ d . Then γ n is extended from a ( d − -chain by a perfectwalk. More precisely:i) There is a ( d − -chain γ d − and a perfect walk ( p d , p d +1 , . . . , p n ) such that γ d − p d p d +1 . . . p i is an i -chain with tail p i for i = d, d + 1 , . . . , n and γ n = γ d − p d p d +1 . . . p n .ii) There is a ( d − -chain γ d − and a perfect walk ( q − n , q − n +1 , . . . , q − d ) suchthat q − i q − i +1 . . . q − d γ d − is an i -chain with head q − i for i = d, d + 1 , . . . , n and γ n = q − n q − n +1 . . . q d γ d − .Proof. We prove i) as ii) is dual. Write γ n = γ n − t γ n = · · · = γ d − t γ d t γ d +1 . . . t γ n for a ( d − γ d − and a suitable sequence of tails. Now the d -chain γ d musthave a perfect tail by Theorem 2.9 since d + 1 > d = dim Λ and so we may set t γ d = p d . The unique extension property then gives t γ d +1 = p d +1 , . . . , t γ n = p n . (cid:3) In other words, this result shows that the structure of n -chains over Gorensteinmonomial algebras eventually stabilises for n ≥ dim Λ and becomes predictable,with the subchain γ d − of γ n consisting of noise which we will be able to ignore. The Fg conditions and higher structures Hochschild cohomology and the Fg conditions. Let Λ e := Λ op ⊗ k Λ bethe enveloping algebra of Λ, and consider Λ ∈ mod Λ e with its natural modulestructure. Recall that the Hochschild cohomology ring HH ∗ (Λ , Λ) := Ext ∗ Λ e (Λ , Λ)is a graded-commutative algebra in that x ⌣ y = ( − | x || y | y ⌣ x for all elements x, y ∈ HH ∗ (Λ , Λ), and we let HH ev (Λ , Λ) denote the subalgebra of elements of evendegree.For any M ∈ mod Λ, we have an algebra homomorphism ϕ M := M ⊗ Λ − : HH ∗ (Λ , Λ) → Ext ∗ Λ ( M, M ) . The map ϕ M induces on Ext ∗ Λ ( M, N ) for any N ∈ mod Λ the structure of a rightgraded module over Hochschild cohomology; moreover this is compatible with theleft action coming from ϕ N as these satisfy ϕ N ( x ) · θ = ( − | x || θ | θ · ϕ M ( x ) for allelements θ ∈ Ext ∗ Λ ( M, N ) and x ∈ HH ∗ (Λ , Λ). This further restricts to a rightmodule structure over any subalgebra H ⊆ HH ∗ (Λ , Λ). The finite generation ( Fg )conditions of Snashall–Solberg are then stated as follows: Fg 1. HH ev (Λ , Λ) contains a Noetherian graded subalgebra H with H = HH (Λ , Λ). Fg 2. Ext ∗ Λ ( M, N ) is a finitely generated H-module for all M, N ∈ mod Λ.We say that Λ satisfies Fg if both conditions above hold. The Fg conditions implythat HH ∗ (Λ , Λ) is module-finite over H, and so that HH ∗ (Λ , Λ) is itself a Noetherianalgebra. Moreover, in the presence of Fg 1, by Jordan-H¨older filtration argumentsit is enough to establish Fg M = N = k .3.2. A ∞ -centres of minimal A ∞ -algebras. In [BG17] the image of the charac-teristic homomorphism ϕ M : HH ∗ (Λ , Λ) → Ext ∗ Λ ( M, M )was related for any M ∈ mod Λ to the A ∞ -structure on Ext ∗ Λ ( M, M ), where it wasshown to consist of A ∞ -central classes. Recall that a class a ∈ E = ( E, m ) ina graded algebra is graded-central if m ( a, x ) = ( − | a || x | m ( x, a ) for all x ∈ E .Equivalently, the inner derivation ad a vanishes:ad a ( x ) = [ a, x ] = m ( a, x ) − ( − | a || x | m ( x, a ) = 0 . We denote by Z gr E = { a ∈ E | ad a = 0 } the graded centre.Generalising to E = ( E, { m n } n ≥ ) a minimal A ∞ -algebra, given a ∈ E , for each n ≥ n -ary higher commutator[ a ; x , . . . , x n ] ,n := n X i =0 ( − i ( − | a | ( | x | + ··· + | x i | ) m n +1 ( x , . . . , x i , a, x i +1 , . . . , x n ) . In other words we apply m n +1 to the (signed) shuffle product a x ( x ⊗· · ·⊗ x n ). Wethen define the homotopy inner derivation ad a = { ad a,n } n ≥ to be a collection of n -ary operations ad a,n : E ⊗ n → E given by ad a,n ( x , . . . , x n ) := [ a ; x , . . . , x n ] ,n .The collection ad a forms a cocycle in the complex of homotopy derivations ad a ∈ hoder( E ) := (cid:16) Hom k ( ⊕ n ≥ E ⊗ n , E ) , ∂ (cid:17) which is a subcomplex of the Hochschild cochain complex C ∗ ( E, E ) of the A ∞ -algebra E , see [BG17] for details. INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 19 Definition 3.1. The A ∞ -centre of E is the space Z ∞ E := { a ∈ E | [ad a ] = 0 in H ∗ (hoder( E )) } . The A ∞ -centre Z ∞ E ⊆ E is a graded subalgebra of the underlying graded algebra E = ( E, m ), and A ∞ -central classes are always graded-central, so that we havecontainment Z ∞ E ⊆ Z gr E with equality whenever m n = 0 for all n ≥ ϕ k : HH ∗ (Λ , Λ) → Ext ∗ Λ ( k , k ) isalways contained in the graded centreim( ϕ k ) ⊆ Z gr Ext ∗ Λ ( k , k )with equality in the case of Λ Koszul by a theorem of Buchweitz–Green–Snashall–Solberg, but with proper inclusion in general. Since the Koszul case correspondsto the situation where one can take m n = 0 for all n ≥ A ∞ -structure onExt ∗ Λ ( k , k ), this result was refined in [BG17] as follows: Theorem 3.2 ([BG17]) . The image of the characteristic homomorphism is pre-cisely the A ∞ -centre: im( ϕ k ) = Z ∞ Ext ∗ Λ ( k , k ) . Reading off the A ∞ -central condition [ad a ] = 0 in the cohomology H ∗ (hoder( E, E ))of E = Ext ∗ Λ ( k , k ) can be subtle, and so in practice one may focus on the simplersufficient condition that ad a vanishes on the nose. Equivalently, for a ∈ Ext ∗ Λ ( k , k ),we are interested in the vanishing of higher commutators n X i =0 ( − i ( − | a | ( | x | + ··· + | x i | ) m n +1 ( x , . . . , x i , a, x i +1 , . . . , x n ) = 0for all x , . . . , x n ∈ Ext ∗ Λ ( k , k ) and all n ≥ 1. We record this as a corollary: Corollary 3.3. Let a ∈ Ext ∗ Λ ( k , k ) be such that all higher commutators (3.2) van-ish. Then a is in the image of ϕ k : HH ∗ (Λ , Λ) → Ext ∗ Λ ( k , k ) . Finally, we can recast the Fg conditions in terms of Z ∞ Ext ∗ Λ ( k , k ). For that, weintroduce another set of conditions Fg ′ : Fg ′ Z ∞ := Z ∞ Ext ∗ Λ ( k , k ) is a Noetherian algebra. Fg ′ 2. Ext ∗ Λ ( k , k ) is module-finite over Z ∞ . Proposition 3.4. The conditions Fg and Fg ′ are equivalent.Proof. If Λ satisfies Fg with regards to H ⊆ HH ∗ (Λ , Λ) then ϕ k (H) ⊆ Z ∞ is aNoetherian algebra over which Ext ∗ Λ ( k , k ) is module-finite and so Fg ′ holds.Conversely if Λ satisfies Fg ′ , then Z ∞ is Noetherian and must be finitely generatedas it is positively graded. Moreover, so must be its even subalgebra Z ev ∞ . Pickinglifts of said generators, we can form a Noetherian subalgebra H ⊆ HH ev (Λ , Λ)satisfying Fg 1, possibly after adding H = HH (Λ , Λ). Since ϕ k (H) = Z ev ∞ , onesees from Fg ′ ∗ Λ ( k , k ) is module-finite over H and the same then holds forExt ∗ Λ ( M, N ) for all M, N ∈ mod Λ by Jordan–H¨older filtration arguments. Hence Fg holds. (cid:3) Let us remark that this approach to studying the Fg condition is not new, see[ES11, Th. 1.3] for the Koszul case where Z gr Ext ∗ Λ ( k , k ) = Z ∞ Ext ∗ Λ ( k , k ). Combinatorics of the A ∞ -centre. In the case of monomial algebras, it ispossible to use Proposition 1.5 to simplify the general formula for higher commu-tators. Lemma 3.5. Suppose a ∈ Ext ∗ Λ ( k , k ) is of even degree. Then for any x , . . . , x n in Ext ∗ Λ ( k , k ) , the higher commutators simplify to [ a ; x , . . . , x n ] ,n = m n +1 ( a, x , . . . , x n ) + ( − n m n +1 ( x , . . . , x n , a ) . Proof. Indeed, if the degree of a is even, we have m n +1 ( x , . . . , x i , a, x i +1 , . . . , x n ) =0 for 0 < i < n . (cid:3) Using this lemma, we obtain a combinatorial criterion for A ∞ -centrality. Givenelements a , . . . , a m with a i ∈ Ext r i +1Λ ( k , k ) = kC ∨ r i with r i ≥ 0, we interpret theexpression a . . . a m ∈ kC ∨ r + ··· + r m +1 as denoting the sum of terms of a . . . a m ∈ kQ op which are in kC ∨ r + ··· + r m +1 , that is as the projection on the natural summand.Next, we say that an element a ∈ Ext ∗ Λ ( k , k ) is symmetric if a · e = e · a for all e ∈ k , i.e. a is a linear combination of closed oriented cycles in the quiver for Ext.The combinatorial criterion then states: Proposition 3.6 (Combinatorial criterion) . Suppose a ∈ Ext ∗ Λ ( k , k ) is a symmetricelement of even degree | a | = r + 1 ≥ . Assume that for each tuple of chains c , . . . , c n with c i ∈ C ∨ r i , we have equalities ac . . . c n = c . . . c n a in kC ∨ r + r + ··· + r n +1 . Then a is ∞ -central and so lies in the image of Hochschildcohomology.Proof. By Corollary 3.3 it’s enough to show that [ a ; x , . . . , x n ] ,n = 0 for all x , . . . , x n in Ext ∗ Λ ( k , k ), and by linearity it’s enough to do this for chains c , . . . , c n .Since ac . . . c n = c . . . c n a , we deduce that m n +1 ( a, c , . . . , c n ) = ± m n +1 ( c , . . . , c n , a )and so the two products are zero or non-zero simultaneously. If both are zero we aredone by Lemma 3.5, and if both are nonzero the elements c , . . . , c n ∈ Ext ∗ Λ ( k , k )have odd degree by Proposition 1.5. Since kC ∨ r i = Ext r i +1Λ ( k , k ), we deduce that r , . . . , r n are even. We can then compute the sign precisely: m n +1 ( a, c , . . . , c n ) = ( − ( n +1) r + r r n + r + r n ac . . . c n = ( − ( n +1) r + r ac . . . c n = ( − ( n +1)+ r c . . . c n a = ( − n +1 ( − r c . . . c n a = ( − n +1 ( − ( n +1) r + r r + r + r c . . . c n a = ( − n +1 m n +1 ( c , . . . , c n , a ) . Lemma 3.5 then gives[ a ; c , . . . , c n ] ,n = m n +1 ( a, c , . . . , c n ) + ( − n m n +1 ( c , . . . , c n , a ) = 0 , which is what we wanted. (cid:3) INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 21 Periodicity operators for Gorenstein algebras We now turn to the construction of A ∞ -central operators χ γ ∈ Ext ∗ Λ ( k , k ) asso-ciated to special Anick chains γ which we will call stable relation cycles . In thissubsection we will always assume that Λ is Gorenstein.We define the period of a Gorenstein monomial algebra Λ as follows. The algebraΛ contains finitely many perfect paths and so there are finitely many indecom-posable Gorenstein-projectives M over Λ, each with a periodic minimal projectiveresolution. We let p M denote the minimal period of this resolution. Definition 4.1. The period of Λ is the least common multiple of all periods p M ,as M runs over indecomposable non-projective Gorenstein-projectives. We denotethat number by ℓ .One notes that this is unchanged if we replace right modules by left modules as theduality M M ∗ preserves the period of Gorenstein-projectives. Definition 4.2 (Stable relation cycles) . Let γ be an ( s − γ is a stable relation cycle if:i) s ≥ dim Λ + 1 and s is even.ii) s is a multiple of ℓ .iii) γ = p p . . . p s − for a perfect cycle p = ( p , p , . . . , p s − ) with tail t γ = p s − .iv) γ = q q . . . q s − for a perfect cycle q = ( q , q , . . . , q s − ) with head h γ = q .Let us unpack the definition.Properties iii) and iv) show that stable relation cycles enjoy the unique extensionproperty of Corollary 2.17. Specifically, for any k ≥ 1, the path γ p s . . . p s − k is the unique extension of γ to the right as an ( s − k )-chain, then with tail t γ p s ...p s − k = p s − k , and dually q − k . . . q − γ is the unique extension to the left asan ( s − k )-chain, then with head h q − k ...q − γ = q − k . Note that the decompositioniii) and iv) can differ as seen in Example 2.13.The unique extension property of stable relation cycles γ will give us control overmultiplication against γ ∨ in Ext ∗ Λ ( k , k ). However γ ∨ is not a central class in general,and the additional constraints i-ii) will allow us to “rotate” γ without breaking theconstraints iii)-iv), to obtain new stable relation cycles whose total sum will forman A ∞ -central class χ γ in Ext ∗ Λ ( k , k ).Green, Snashall and Solberg have introduced in [GSS06] a closely related notionof stability for relation cycles which is weaker than what we consider here; theirsis concerned with the stability of tails and heads under taking power, which alsofollows from our definition. However the precise decomposition in terms of perfectpaths is what gives us the additional leverage needed to study higher commuta-tors and the finite generation conditions. The whole apparatus requires a carefulbalancing act, and so we begin with studying properties of perfect walks.4.1. Perfect walks of even length. Now, in general if ( w , w , . . . , w n − ) is aperfect walk of even length n , then the path w w . . . w n − w n − is a concatena-tion of relations r = w w , r = w w , . . . , r n − = w n − w n − , and so we have w w . . . w n − = r r . . . r n − . In particular stable relation cycles are concatena-tions of relations. Our next aim will be to recognise stable relation cycles amongstperfect walks of even length. This will come through a series of lemmas. We remarkthat Lemma 4.3 below is closely related to [GSS06, Prop. 2.3]. Lemma 4.3. Let ( w , w , . . . , w n − ) be a perfect walk of even length n . Then γ = w w . . . w n − is an ( n − -chain and w n − is a right divisor of t γ .Proof. Write n = 2 k , we work by induction on k ≥ 1. For k = 1, γ = w w ∈ R = C is a relation and w w = γ = at γ for some arrow a , so that w rightdivides t γ for length reasons. Next, consider γ ′ = w w . . . w k − which is a chainby induction, and for which w k − right divides t γ ′ . This gives t γ ′ w k − = 0 in Λ,so there is a left divisor w ′ of w k − = w ′ w ′′ such that γ ′ w ′ is a chain with tail w ′ .Consider w ′′ w k , and note that w ′ w ′′ w k = w k − w k = 0 in Λ. Hence we can finda left divisor u of w ′′ w k such that w ′ u = 0 in Λ and such that γ ′ w ′ u forms a chain.Since w ′ w ′′ = w k − = 0 in Λ, we have u = w ′′ u ′ and in particular u ′ left divides w k . But we have w k − u ′ = w ′ w ′′ u ′ = w ′ u = 0, which shows that w k left divides u ′ , thus forcing w k = u ′ .Putting it together, this gives a chain γ ′ w ′ u = γ ′ w ′ w ′′ u ′ = γ ′ w k − w k with tail u right divisible by w k , as claimed. (cid:3) Lemma 4.4. Let γ = w w . . . w n − be an ( n − -chain given by a perfect walk ofeven length n . Then:i) The relations r , r , . . . , r n − are independent of the choice of perfect walk ( w , w , . . . , w n − ) with γ = w w . . . w n − .ii) If γ ′ = w ′ w ′ . . . w ′ n − is another perfect walk of even length n with r ′ = r or r ′ n − = r n − , then γ ′ = γ .Proof. i). If γ = r r . . . r s − = e r e r . . . e r s − is written in two ways as a concate-nation of relations, then one of r , e r divides the other and so r = e r . Applyingthe same argument inductively to the substrings r k . . . r s − = e r k . . . e r s − gives theresult.ii). We assume that r ′ n − = r n − , the case r ′ = r is similar. We will show thatthe previous relation in a perfect walk of even length is uniquely determined.The hypothesis gives w ′ n − w ′ n − = r ′ n − = r n − = w n − w n − , so one of w ′ n − , w n − left divides the other, say w n − = w ′ n − x . Then w ′ n − w n − = w ′ n − w ′ n − x = 0 inΛ. As w ′ n − = 0 in Λ, we must have w ′ n − = yw n − since ( w n − , w n − ) formsa perfect pair. Continuing, we have w ′ n − w ′ n − = w ′ n − yw n − = 0 in Λ and so w ′ n − y = zw n − . We obtain the equality w ′ n − w ′ n − = w ′ n − yw n − = zw n − w n − ,so that z is the empty word and we have equality of relations r ′ n − = w ′ n − w ′ n − = w n − w n − = r n − . Iterating, we get r ′ k = r k for all k and so γ ′ = γ . (cid:3) The next result allows us to rewrite perfect walks of even length in normal forms. Lemma 4.5 (Perfect rewriting) . Let ( w , w , . . . , w n − ) be a perfect walk of evenlength n and let γ = w w . . . w n − .i) If t γ is perfect, then γ = p . . . p n − is given by the perfect walk with p n − = t γ .ii) If h γ is perfect, then γ = q . . . q n − is given by the perfect walk with q = h γ .Proof. We prove i) as ii) is dual. Assuming t γ = p n − perfect, since both p n − and w n − right divide γ , one must divide the other, say p n − = xw n − for some x (theother case is similar). Then p n − p n − = p n − xw n − = 0 in Λ. Since p n − x = 0 inΛ as it is a proper divisor of p n − p n − , we must have p n − x = yw n − for some y ,which gives p n − p n − = p n − xw n − = yw n − w n − . It follows that y is the emptyword and p n − p n − = w n − w n − .Now γ ′ = p p . . . p n − p n − and γ = w w . . . w n − w n − share the last relation p n − p n − = w n − w n − , and so γ ′ = γ by Lemma 4.4 ii). (cid:3) INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 23 We now obtain a recognition lemma for stable relation cycles. Lemma 4.6 (Recognition Lemma) . Let ( w , w , . . . , w n − ) be a perfect walk, andassume that n ≥ dim Λ + 1 , n is even and divisible by ℓ . Then γ = w w . . . w n − is a stable relation cycle.Proof. First γ is an ( n − n − ≥ dim Λ we seethat p n − := t γ and q := h γ are perfect by Theorem 2.9, and Lemma 4.5 showsthat γ = p p . . . p n − = q q . . . q n − . Lastly, since n is a multiple of the period ℓ ,it is then a multiple of the minimal period of every perfect cycle and so we see that p = ( p , p , . . . , p n − ) and q = ( q , q , . . . , q n − ) are in fact perfect cycles. (cid:3) Corollary 4.7. Let p be a perfect path in Λ . Then p = p ( resp. p = p s − ) can beextended to a stable relation cycle γ = p p . . . p s − .Proof. Simply take the perfect walk ( p , p , . . . , p s − ) of even length s , satisfying s ≥ dim Λ + 1 and with s some multiple of ℓ . (cid:3) Rotating stable relation cycles. Now let γ = a a . . . a n − ∈ C s − be astable relation cycle, which we write as a path of length n with a i ∈ Q . We nowwant to “rotate” γ to produce new stable relation cycles, also of path length n . Wecan think of γ as a substring of a periodic string, infinite in both directions: a : . . . a a . . . a n − ( a a . . . a n − ) a a . . . a n − . . . One can recover this string by periodicity from any length n substring. Setting γ = γ , we define the set S = { γ , γ , . . . , γ n γ − } by the property that:i) γ i = a j i a j i +1 . . . a j i + n − is an ( s − j < j < · · · < j n γ − ≤ n − γ i = a j i a j i +1 . . . a j i + n − ∈ S , we have γ i = a j a j +1 . . . a j + n − if 0 ≤ j < j i .iv) { j , j , . . . , j n γ − } ⊆ { , , . . . , n − } is maximal with these property.We think of n γ as the number of possible distinct rotations of γ as a stable relationcycle. Moreover, all γ i are length n substrings of a , and it is clear that the same setis produced by this construction starting from any other γ i ∈ { γ , γ , . . . , γ n γ − } .We call this set S the associated set of γ , which is also the equivalence class for anequivalence relation on stable relation cycles which we denote γ ∼ γ ′ . Lemma 4.8 (Completeness Lemma) . Let γ ∈ C s − be a stable relation cycle, andwrite γ = p p . . . p s − = q q . . . q s − for the corresponding perfect walks. Then forany k ∈ Z , both p k p k +1 . . . p k + s − and q k q k +1 . . . q k + s − are in { γ , γ , . . . , γ n γ − } .Proof. Both strings are substrings of a , and the periodicities p i + s = p i and q i + s = q i shows that if γ is a substring of length n in a , then so are p k p k +1 . . . p k + s − and q k q k +1 . . . q k + s − as these differ by cyclic shifts, which preserves path length. Finallyboth strings are given by perfect walks of even lengths s satisfying the hypothesesof Lemma 4.6, and so p k p k +1 . . . p k + s − and q k q k +1 . . . q k + s − are both relationcycles. (cid:3) Lemma 4.9 (Rigidity Lemma) . Let α be a path in Q and let k ≥ . Then γ i α (resp. α γ i ) is an ( s − k ) -chain for at most one choice of i ∈ { , , . . . , n γ − } .Proof. Let γ i , γ i ′ be two stable relation cycles in the associated set, and write γ i = p p . . . p s − with tail t γ i = p s − for the corresponding perfect cycle (resp. γ i ′ = p ′ p ′ . . . p ′ s − with tail t γ i ′ = p ′ s − ). If γ i α and γ i ′ α are both ( s − k )-chains, then the unique extension property shows that α = p s p s +1 . . . p s − k = p ′ s p ′ s +1 . . . p ′ s − k , and by s -periodicity we can rewrite this as α = p p . . . p k − = p ′ p ′ . . . p ′ k − .If k = 1 we are done as p = α = p ′ , and the perfect walks ( p , p , . . . , p s − ) and( p ′ , p ′ , . . . , p ′ s − ) are equal the moment p k = p ′ k for any k , so that γ i = γ i ′ .If k ≥ 2, then the equality α = p p . . . p k − = p ′ p ′ . . . p ′ k − shows that one of thetwo relations r = p p , r ′ = p ′ p ′ divides the other, so that r = r ′ . Since stablerelation cycles are determined by their first relation (Lemma 4.4) then γ i = γ i ′ . (cid:3) Periodicity operators. We can now construct the operators χ γ ∈ Ext ∗ Λ ( k , k )associated to stable relation cycles. Definition 4.10 (Periodicity operators) . Let γ ∈ C s − be a stable relation cycleand { γ , γ , . . . , γ n γ − } the associated set. Then χ γ = n γ − X i =0 γ ∨ i is the periodicity operator χ γ ∈ Ext s Λ ( k , k ) attached to γ . Example 4.11. Let Λ = k [ t ] / ( t n ) for n ≥ 2. Then γ = t n ∈ C is the unique stablerelation cycle of minimal length. The periodicity operator χ γ = ( t n ) ∨ ∈ Ext ( k , k )is the classical periodicity operator of Gulliksen [Gul74].Finally, we can show that the operators χ γ ∈ Ext ∗ Λ ( k , k ) lift to Hochschild coho-mology. Proposition 4.12. Let Λ be a Gorenstein monomial algebra and let γ ∈ C s − bea stable relation cycle. Then χ γ is ∞ -central, and therefore lies in the image ofHochschild cohomology.Proof. By the combinatorial criterion of Proposition 3.6, it’s enough to show theequality χ γ c . . . c n = c . . . c n χ γ in kC ∨ N for all tuples c , c , . . . , c n with c j ∈ C ∨ k j and N = ( s − 1) + k + · · · + k n + 1.Let us write c j = b ∨ j for the dual of a chain b j ∈ C k j . Since γ ∨ i b ∨ j = ( b j γ i ) ∨ , we let χ ∨ γ = P n γ − i =0 γ i in kC s − and prove the dual equality χ ∨ γ b n . . . b = b n . . . b χ ∨ γ in kC N . Let α = b n . . . b , considered as a path in Q .If χ ∨ γ α = 0 in kC N then γ i α is an N -chain for some i , in which case i is uniqueby rigidity (Lemma 4.9). Writing γ i = p p . . . p s − for the perfect walk with t γ i = p s − , the unique extension property gives α = p s . . . p N . We then rotate thestring γ i α : γ i α = ( p p . . . p s − ) p s . . . p N = ( p p . . . p s − ) p . . . p N − s = p p . . . p N − s ( p N − s +1 . . . . . . p N − s )= α ( p N − s +1 . . . . . . p N − s ) . INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 25 By completeness (Lemma 4.8) we have p N − s +1 . . . p N − s = γ j for some γ j ∈{ γ , γ , . . . , γ n γ − } , and as α γ j = γ i α = 0 in kC N , rigidity again gives γ j asthe unique stable relation cycle with α γ j = 0 in kC N . Putting this together gives χ ∨ γ α = γ i α = α γ j = αχ ∨ γ in kC N . More formally, this argument shows that χ ∨ γ α = 0 in kC N implies αχ ∨ γ = 0in kC N , in which case χ ∨ γ α = αχ ∨ γ in kC N . As the argument is symmetrical (usingthe dual properties established instead), we also see that αχ ∨ γ = 0 in kC N implies χ ∨ γ α = 0 in kC N , in which case αχ ∨ γ = χ ∨ γ α in kC N , and so the claim holds. (cid:3) Remark . We thus see that periodicity operators lift to Hochschild cohomology,and any two lifts differ by an element of ker( ϕ k ), which is nilpotent. However ourproof shows the stronger statement that all operators χ γ ∈ Ext ∗ Λ ( k , k ) have anunambiguous lift to an operator χ γ ∈ HH ∗ (Λ , Λ), which we denote by the sameletter by abuse of notation.This lift is described as follows [BG17]: the invariance of Hochschild cohomologyunder Koszul duality gives an isomorphism HH ∗ (Λ , Λ) ∼ = HH ∗ ( E, E ), with E =Ext ∗ Λ ( k , k ) treated as an A ∞ -algebra. Consider the cochain in C ∗ ( E, E ) given by1 χ γ and sending everything else to zero. The condition ad χ γ = 0 is preciselythe Hochschild cocycle condition and so this cocycle gives rise to a cohomologyclass χ γ ∈ HH ∗ ( E, E ) ∼ = HH ∗ (Λ , Λ), which provides a section for the characteristicmorphism. Hence to any stable relation cycle γ we attach a well-defined class χ γ ∈ HH ∗ (Λ , Λ).4.4. The ring of periodicity operators. We now turn to the study of the mul-tiplicative properties of the periodicity operators { χ γ } γ ⊆ Ext ∗ Λ ( k , k ). In thissubsection we assume that Λ is a Gorenstein monomial algebra of infinite globaldimension. In this case Λ always has stable relation cycles by Corollary 4.7.Let R := k h χ γ | γ i ⊆ Ext ∗ Λ ( k , k ) be the A ∞ -central graded subalgebra generatedby the classes { χ γ } γ . It will turn out that the graded connected algebra R isNoetherian, reduced and of Krull dimension one, and the structure theory of suchrings dictates that R embeds in a finite product of polynomial algebras R ⊆ b Y i =1 k [ t i ]with k [ t i ] the graded normalisation of R / p i for p i the i -th minimal homogeneousprime. Composing with the i -th projection gives a map π i : R → k [ t i ], and sofor each χ γ ∈ R we may write π i ( χ γ ) = t n i i for some n i ∈ N ∪ {−∞} (where weinterpret t −∞ i = 0). Denoting this exponent by n i = ord i ( γ ), it is clear that thering structure of R is determined by the values b and ord i ( γ ) as i = 1 , , . . . , b and γ ranges over all stable relation cycles. Turning this around, we will firstgive a combinatorial definition of b and ord i ( γ ) which we will use to establish thestructure of R as claimed in the above paragraph. In fact we will obtain a slightlymore precise description of the structure of R , see Prop. 4.23 below. Branch equivalence relation. Let γ , γ be stable relation cycles, and recall that γ ∼ γ if they have the same associated set (in particular then χ γ = χ γ ). Itis easy to see that powers of stable relation cycles are also stable relation cyclesand that γ ∼ γ implies γ n ∼ γ n for any n ≥ 2, and so this equivalence relationdetermines another coarser equivalence relation. Definition 4.14 (Branch equivalence relation) . The stable relation cycles γ , γ are branch equivalent if there exists n , n ≥ γ n ∼ γ n . We denotethe branch equivalence relation by γ ≈ γ . Lemma 4.15. Let γ , γ be stable relation cycles. If t γ = t γ or h γ = h γ , then γ ≈ γ .Proof. We assume that t γ = t γ = p , the other case is dual. We can write γ = p p . . . p s − γ = p ′ p ′ . . . p ′ s ′ − for perfect cycles with p s − = p ′ s ′ − = p . This implies p = p ′ , p = p ′ , . . . , andso on, so that p = ( p , p , . . . , p s − ) and p ′ = ( p ′ , p ′ , . . . , p ′ s ′ − ) are both powers ofthe same perfect cycle of minimal length s ′′ | s, s ′ . Taking a common power, we seethat there are n , n ≥ γ n = γ n , and so γ ≈ γ . (cid:3) Lemma 4.16. There are finitely many branch equivalence classes { Γ i } i ∈ I of stablerelation cycles.Proof. The map S i ∈ I Γ i → { perfect paths } sending γ to its tail t γ sends distinctequivalence classes into disjoint non-empty subsets by Lemma 4.15, and so | I | isbounded above by the number of perfect paths. (cid:3) It follows that we can identify I = { , , . . . , b } and so list the branch equivalenceclasses as Γ , Γ , . . . , Γ b for some b ∈ N . We call b the number of branches.Fixing an equivalence class Γ i , consider the set of path lengths of elements of Γ i which we denote g N Γ i = { len ( γ ) | γ ∈ Γ i } ⊆ N . Letting gcd i := gcd( g N Γ i ), wenormalise this to N Γ i := 1gcd i g N Γ i ⊆ N . Definition 4.17. We define the function ord i : { γ } → N ∪ {−∞} on the set ofstable relation cycles by ord i ( γ ) := ( i len ( γ ) if γ ∈ Γ i −∞ if γ / ∈ Γ i . Note in particular that N Γ i = { ord i ( γ ) | γ ∈ Γ i } ⊆ N . Lemma 4.18. Let γ , γ ∈ Γ i . Then γ ∼ γ if and only if ord i ( γ ) = ord i ( γ ) .Proof. Equivalently we show that γ ∼ γ if and only if len ( γ ) = len ( γ ). Sincestable relation cycles in the same associated set have the same path length, thenecessary implication holds by definition. For the converse, assume that len ( γ ) = len ( γ ) = n . Write γ = a a . . . a n − and γ = a ′ a ′ . . . a ′ n − with a i , a ′ i ∈ Q .Since γ ≈ γ , there are n , n ≥ γ n ∼ γ n , and so the two infiniteperiodic strings a : . . . ( a a . . . a n − ) n ( a a . . . a n − ) n ( a a . . . a n − ) n . . .a ′ : . . . ( a ′ a ′ . . . a ′ n − ) n ( a ′ a ′ . . . a ′ n − ) n ( a ′ a ′ . . . a ′ n − ) n . . . must coincide up to some translation. But then γ , γ are two length n substringsof the same infinite periodic string . . . a a . . . a n − ( a a . . . a n − ) a a . . . a n − . . . INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 27 and so γ ∼ γ by contruction of the associated set. (cid:3) Multiplicative properties. We next establish the multiplicative properties of { χ γ } γ .We aim to prove the following: Proposition (Prop. 4.22) . Let γ and γ , γ , γ denote stable relation cycles.i) If χ γ · χ γ = 0 , then γ ≈ γ .ii) Assume that γ ≈ γ . Then there is a γ ≈ γ , γ such that χ γ · χ γ = χ γ ,and then ord i ( γ ) + ord i ( γ ) = ord i ( γ ) .iii) We have χ n γ = χ γ n for any n ≥ . The proof will be done in a series of steps. Let us first collect some comments.By property i) it is enough to understand the product of χ γ , χ γ for γ , γ ∈ Γ i within the same branch equivalence class. Property ii) then tells us that { χ γ } γ ∈ Γ i forms a multiplicative basis for the subalgebra R i := k h χ γ | γ ∈ Γ i i ⊆ R . Toanalyse the ring structure, we may extend the notation ord i ( χ γ ) := ord i ( γ ) to theclass χ γ , which is independent of choice of representative γ by Lemma 4.18, andin fact Lemma 4.18 tells us that any value n ∈ N Γ i can be represented by a uniqueclass χ γ for γ ∈ Γ i . Hence the function ord i sets up a bijection { χ γ } γ ∈ Γ i ↔ N Γ i .Since ord i ( − ) is additive on products, that the set { χ γ } γ ∈ Γ i forms a multiplicativebasis for R i by property ii) translates into N Γ i ⊆ N being a subsemigroup, and wesee that the above bijection induces an algebra isomorphism onto the semigroupalgebra R i = k h χ γ | γ ∈ Γ i i ∼ = −→ k [ t N Γ i ] := k h t n | n ∈ N Γ i i ⊆ k [ t ]sending χ γ t ord i ( γ ) .Keeping this in mind, we see that many conditions on the behavior of ord i ( − ) and { χ γ } γ have to be verified. First, assuming the above isomorphism we see that thering R i ∼ = k [ t N Γ i ] is bigraded by ord i ( χ γ ) and cohomological degree | χ γ | , and eitherdegree determines the other. This translates into the next simple combinatoriallemma.If γ is an n -chain, let us call n the Anick degree of γ . Lemma 4.19. Let γ , γ ∈ Γ i . Then γ , γ have the same Anick degree if andonly if γ ∼ γ .Proof. If γ ∼ γ then the implication holds by definition. Conversely assumethat γ , γ are both n -chains. Since γ ≈ γ , there exists n , n ≥ γ n ∼ γ n . The relation ∼ preserves both the Anick degree and the path length,and so we have n n = n n and n len ( γ ) = n len ( γ ). Cancellation first gives n = n and then len ( γ ) = len ( γ ). Thus ord i ( γ ) = ord i ( γ ) and so γ ∼ γ byLemma 4.18. (cid:3) Next, understanding the product χ γ · χ γ will quickly reduce to understandingtriples of stable relation cycles γ , γ , γ such that s ( γ ) = t ( γ ) and γ = γ γ .In particular establishing identities of the form χ γ · χ γ = χ γ will require under-standing the behavior of associated sets under products of stable relation cycles.This is done in the next two lemma. Lemma 4.20 (Product lemma) . Let γ , γ be stable relation cycles of Anick degrees s − and s − , respectively. Assume that s ( γ ) = t ( γ ) and γ γ is also a stablerelation cycle. i) We have γ ≈ γ ≈ γ γ .ii) If e γ ∼ γ and e γ ∼ γ are equivalent stable relation cycles with s ( e γ ) = t ( e γ ) and e γ e γ also a stable relation cycle, then e γ e γ ∼ γ γ .Proof. i). Since γ , γ and γ γ are stable relation cycles, the unique extensionproperty (Prop. 2.17) shows that t γ γ = t γ and h γ γ = h γ . Lemma 4.15 thengives γ ≈ γ γ ≈ γ .ii). Applying i) to e γ , e γ gives e γ e γ ≈ e γ ∼ γ ≈ γ γ , and so e γ e γ ≈ γ γ .Since the stable relation cycles e γ e γ and γ γ both have Anick degree s − s = s + s , Lemma 4.19 shows that e γ e γ ∼ γ γ . (cid:3) Lemma 4.21 (Product decompositions for associated sets) . Let γ , γ , γ , γ denotestable relation cycles and assume that γ = γ γ . Let S i = { e γ i | e γ i ∼ γ i } be theassociated sets of γ i , i = 1 , , .i) The associated set S decomposes as a product of elements of S , S , in that:a) For every e γ ∈ S , there is a unique e γ ∈ S such that e γ e γ ∈ S .b) For every e γ ∈ S , there is a unique e γ ∈ S such that e γ e γ ∈ S .c) Every e γ ∈ S is of the form e γ = e γ e γ for unique e γ ∈ S , e γ ∈ S .Hence we have S = S · S := { e γ e γ | e γ i ∈ S i and ( e γ , e γ ) compatible } , withthe compatibility in the sense of a)-b).ii) For any n ≥ the associated set of γ n is { e γ n | e γ ∼ γ } .Proof. i). Let s i − γ i , so that s = s + s . We prove 1). Let e γ ∈ S and write e γ = p p . . . p s − for the perfect cycle with t e γ = p s − , andextend this to a perfect walk p p . . . p s − p p . . . p s − of length s + s = s . Set e γ := p p . . . p s − and write e γ := e γ e γ . Note that e γ , e γ are given by perfectwalks of appropriate length and so are stable relation cycles by the recognitionlemma (Lemma 4.6).Part i) of the product lemma (Lemma 4.20) gives e γ ≈ e γ ∼ γ ≈ γ , and so e γ ≈ γ . Since e γ and γ are both ( s − e γ ∼ γ ,and finally part ii) gives e γ = e γ e γ ∼ γ γ = γ . Finally, the uniqueness followsfrom the unique extension property (Prop. 2.17), and 1) follows. The proof of 2)is dual.For 3), let e γ ∈ S and write e γ = p p . . . p s − for the perfect cycle with tail t e γ = p s − . Since s = s + s and both s , s are multiples of ℓ , and thereforeof the minimal period of this perfect cycle, we can break it down further as e γ = p p . . . p s − = ( p p . . . p s − )( p p . . . p s − ). Letting e γ i := p p . . . p s i − , sincethe length s i is appropriate the recognition lemma (Lemma 4.6) again shows that e γ i are stable relation cycles. Part i) of the product lemma (Lemma 4.20) then gives e γ i ≈ e γ ∼ γ ≈ γ i for i = 1 , 2, so that e γ i ≈ γ i , and Lemma 4.19 then gives e γ i ∼ γ i .Hence e γ = e γ e γ with e γ ∈ S , e γ ∈ S as claimed. Finally, to see uniquenessassume that e γ = e γ e γ = e γ ′ e γ ′ for some possibly different e γ ′ ∈ S , e γ ′ ∈ S . Theunique extension property (Prop. 2.17) implies that t e γ = t e γ = t e γ ′ . Lemma 4.15then gives e γ ′ ≈ e γ , and since they have the same Anick degree Lemma 4.19 gives e γ ′ ∼ e γ . In particular len ( e γ ′ ) = len ( e γ ), and so e γ ′ = e γ . The equality e γ ′ = e γ then follows from 2), and we have shown 3).ii). This follows from i) by setting γ = γ and γ = γ n − and induction on n . (cid:3) We can now prove the claim. INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 29 Proposition 4.22. Let γ and γ , γ , γ denote stable relation cycles.i) If χ γ · χ γ = 0 , then γ ≈ γ .ii) Assume that γ ≈ γ . Then there is a γ ≈ γ , γ such that χ γ · χ γ = χ γ ,and then ord i ( γ ) + ord i ( γ ) = ord i ( γ ) .iii) We have χ n γ = χ γ n for any n ≥ .Proof. i). If χ γ · χ γ = 0 then there exists e γ ∼ γ such that e γ ∨ · γ ∨ = ( γ e γ ) ∨ = 0.In this case the Anick chain γ e γ must be given by a perfect walk by the uniqueextension property (Prop. 2.17), and γ e γ is then a stable relation cycle by therecognition lemma (Lemma 4.6) as this perfect walk has the required length. Theproduct lemma (Lemma 4.20) then gives γ ∼ e γ ≈ γ and so γ ≈ γ .ii). Let n , n ≥ γ n ∼ γ n . Then χ γ n = χ γ n and so χ n γ · χ n γ = χ γ n · χ γ n = χ γ n · χ γ n = χ γ n n = 0 . Hence χ γ · χ γ = 0. Possibly replacing γ by e γ ∼ γ , we can assume that γ ∨ γ ∨ = ( γ γ ) ∨ = 0; in particular γ γ is an Anick chain, and the recognitionlemma again shows that γ := γ γ is a stable relation cycle. We are then in thesetting of Lemma 4.21.The decomposition of the associated set S = S · S of Lemma 4.21 i) then gives χ γ = X e γ ∈ S e γ ∨ = X e γ e γ ∈ S ( e γ e γ ) ∨ = X e γ e γ ∈ S , e γ ∨ e γ ∨ = X e γ ∈ S e γ ∨ · X e γ ∈ S e γ ∨ = χ γ · χ γ where the second equality writes e γ = e γ e γ in terms of the unique decompositionof Lemma 4.21 and the fourth equality follows from the Rigidity Lemma (Lemma4.9). This proves the main claim, and ord i ( γ ) + ord i ( γ ) = ord i ( γ ) simply followsfrom additivity of path lengths.iii). For n = 1 there is nothing to prove and n ≥ γ = γ n − and γ = γ , giving γ = γ n . (cid:3) We can finally describe the ring structure of R = k h χ γ | γ i ⊆ Ext ∗ Λ ( k , k ). First, weobserved at the beginning of this subsection that Prop. 4.22 implies that N Γ i ⊆ N is a subsemigroup for each branch equivalence class Γ i . Since gcd( N Γ i ) = 1 byconstruction, we see that | N \ N Γ i | < ∞ by [RGS09, Lemma 2.1], so that N Γ i is a numerical semigroup. More importantly for us, N Γ i is a finitely generatedsemigroup [RGS09, Corollary 2.8]. It follows that the (semigroup) algebra k [ t N Γ i i ] = k h t ni | n ∈ N Γ i i ⊆ k [ t i ] . is a finitely generated k -algebra, and in particular is Noetherian.Let us write ε : k [ t N Γ i i ] → k for the augmentation with ε ( t ni ) = 0 for all n ∈ N Γ i .Lastly, we grade the algebra above by setting | t i | = gcd {| χ γ | | γ ∈ Γ i } . Proposition 4.23. The graded algebra R = k h χ γ | γ i is Noetherian, reduced andof Krull dimension one. There is an embedding of graded algebras φ : R ֒ → b Y i =1 k [ t i ] inducing an isomorphism onto the fibre product R ∼ = k [ t N Γ1 ] ε × ε . . . ε × ε k [ t N Γ b b ] .Proof. It is clear that the graded algebra k [ t N Γ1 ] ε × ε . . . ε × ε k [ t N Γ b b ] is Noetherian,reduced and of Krull dimension one, and so it suffices to prove the second claim.Define the k -algebra morphism φ : R → Q bi =1 k [ t i ] on generators by φ ( χ γ ) = (0 , . . . , , t ord i ( γ ) i , , . . . , 0) for γ ∈ Γ i where one may interpret the zero coefficients as t ord j ( γ ) j = t −∞ j = 0 for j = i .Then φ is a well-defined k -algebra morphism by Lemma 4.18 and Prop. 4.22.Moreover Prop. 4.22 shows that { χ γ } γ is a multiplicative basis for R , which φ sendsbijectively onto a multiplicative basis for k [ t N Γ1 ] ε × ε . . . ε × ε k [ t N Γ b b ] ⊆ Q bi =1 k [ t i ],and so φ induces an algebra isomorphism of R onto its image.Finally, the grading on R induces a grading onto k [ t N Γ1 ] ε × ε . . . ε × ε k [ t N Γ b b ] bytransport of structure, and since N Γ i ⊆ N are numerical semigroups (in particulargcd( N Γ i ) = 1) this forces | t i | = gcd {| χ γ | | γ ∈ Γ i } . Since this is the grading weimposed on the larger algebra Q bi =1 k [ t i ], we see that φ was in fact an embeddingof graded subalgebras and we are done. (cid:3) Computation of a ring of periodicity operators. Let Λ = kQ/I for thequiver given by an oriented cycle on seven vertices with arrows a, b, c, d, e, f, g andrelations abcd, bcde, def, ef g, f gab, gabc . This example was considered by Green, •••• • • • gab c d ef Snashall and Solberg in [GSS06, Example 4.1, 7.5].The algebra Λ is Gorenstein of dimension 6, and hasa single branch Γ with numerical semigroup N Γ = { , , . . . } . We then have R ∼ = k [ t , t ] ⊂ k [ t ] with | t | = 4, with t , t corresponding to χ γ , χ γ where γ = ( abcdef g ) and γ = ( abcdef g ) . In [GSS06,Example 7.5] the authors compute the Hochschild co-homology ring modulo nilpotents as HH ∗ (Λ , Λ) / N ∼ = k [ t ] with | t | = 4, and the embedding above is pre-cisely the natural map R ֒ → HH ∗ (Λ , Λ) / N , withagreement in all degrees past the Gorenstein dimen-sion. Note that the ring R does not contain elementsof lower degree by design. 5. Main theorem Using the machinery developed in the previous sections we are now able to charac-terise the monomial algebras satisfying the Fg conditions of Snashall-Solberg. Webegin with some minor setup.Consider a Gorenstein monomial algebra Λ, and we may assume that gldim Λ = ∞ as all results below will immediately reduce to this case. To prove that Fg holdsfor Λ, it is enough to show that Ext ∗ Λ ( k , k ) is module-finite over its A ∞ -central INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 31 subalgebra R ⊆ Ext ∗ Λ ( k , k ). We will prove a slightly stronger statement with respectto a Noether normalisation of R .Let Γ , Γ , . . . , Γ b be all branch equivalence classes of stable relation cycles, withaccompanying numerical semigroups N Γ , N Γ , . . . , N Γ b . We have seen in Prop. 4.23that R depends only on the structure of these numerical semigroups, and this makesit easy to construct a Noether normalisation.Let n i := min N Γ i and let γ i ∈ Γ i be a stable relation cycle with ord i ( γ i ) = n i .Consider the class χ i := χ γ i , which is independent of the choice of γ i above byLemma 4.18. Introduce the following class χ ∈ R : χ := b X i =1 χ m i i (4)where m i ≥ m i | χ i | = m j | χ j | for all i, j and gcd { m i } = 1. FromProp. 4.23 it is clear that the polynomial subalgebra k [ χ ] ⊆ R is a Noether nor-malisation, and we let p := | χ | = m i | χ i | denote its degree. We note that p is amultiple of the period ℓ of Λ.Next, recall that by Corollary 4.7, any perfect path w ∈ Λ over a Gorensteinmonomial algebra can be extended to a perfect cycle ( w , w , . . . , w s − ) with w = w s − such that γ = w w . . . w s − is a stable relation cycle. Lemma 5.1. Let Λ be a Gorenstein monomial algebra and w ∈ Λ be a perfectpath. Let γ = w w . . . w s − be a stable relation cycle with w = w s − , and γ hasminimal path length amongst stable relation cycles with this property. If γ ∈ Γ i ,then ord i ( γ ) ∈ N Γ i takes on the minimal value.Proof. Let γ i ∈ Γ i be a class with ord i ( γ i ) minimal; we will show that γ ∼ γ i , sothat ord i ( γ ) = ord i ( γ i ). Since γ ≈ γ i , there are n, n i ≥ γ n ∼ γ n i i . Wehave seen in Lemma 4.21 that the associated set of γ n i i is of the form { e γ n i i | e γ i ∼ γ i } ,and so γ n = e γ n i i for some e γ i ∼ γ i . The unique extension property (Prop. 2.17)then gives t γ = t γ n = t e γ nii = t e γ i , and so γ , e γ i are stable relation cycles with thesame tails.Let p = t γ = t e γ i be the common tail, and note that we may have p = w s − as wedid not assume w s − was the tail of γ = w w . . . w s − . We can then rewrite the( s − γ as γ = p p . . . p s − for the perfect cycle ( p , p , . . . , p s − ) endingin p = p s − . While we may have w i = p i in general, note that the relations w w = p p , . . . , w s − w s − = p s − p s − are unique by Lemma 4.4, and thereforewe have equalities of proper right subtrings γ k of γ : γ k := p k p k +1 . . . p s − p s − = w k w k +1 . . . w s − w s − . Our assumption on γ shows that γ k is not a stable relation cycle for any k > γ also has the smallest path length amongst stable relation cycleswith tail t γ = p . Since e γ i has minimal value of ord i ( e γ i ) = i len ( e γ i ) in N Γ i andtail t e γ i = p , we conclude that γ = e γ i and so finally γ ∼ γ i . (cid:3) By the lemma we see that any perfect path w in Λ can be extended to a stablerelation cycle γ = w w . . . w s − , not necessarily with t γ = w s − , such that χ γ = χ i agrees with a term of χ in (4), where γ ∈ Γ i belongs to the i -th branch equivalenceclass. This will let us control multiplication against χ ∈ Ext ∗ Λ ( k , k ), as in the nextproposition: Proposition 5.2. Let Λ be a Gorenstein monomial algebra and χ be the classdefined in (4) . Then left multiplication by χχ · − : Ext n Λ ( k , k ) → Ext n + p Λ ( k , k ) is an isomorphism for all n ≥ dim Λ + 1 and an epimorphism for n = dim Λ .Proof. If gldim Λ < ∞ the claim is trivial and so assume that gldim Λ = ∞ . Weconsider the action of χ · − on the basis of chains kC ∨ n − ∼ = Ext n Λ ( k , k ) and provethat χ · − is injective for n ≥ dim Λ + 1 and surjective for n ≥ dim Λ.Let n ≥ dim Λ + 1. Then by Theorem 2.9 any γ n − ∈ C n − can be written as γ n − = γ n − p for p = t γ n − perfect (using n > dim Λ). Writing p = p s − , welet p = ( p , p , . . . , p s − ) be the perfect cycle of minimal length such that γ = p p . . . p s − is a stable relation cycle. Letting Γ i be the branch equivalence classof γ , Lemma 5.1 shows that ord i ( γ ) ∈ N Γ i attains the minimal value. In particular γ ∼ γ i for any other γ i ∈ Γ i with ord i ( γ i ) also attaining the minimum value.Writing χ = P bi =1 χ m i i as in (4) where χ i := χ γ i for γ i ∈ Γ i a class with ord i ( γ i )minimal, this shows that each ( n − γ n − has tail p given by a perfect pathoccuring as tail of a stable relation cycle γ with γ ∼ γ i for some i . The uniqueextension property (Prop. 2.17) and the rigiditiy lemma (Lemma 4.9) then give χ · γ ∨ n − = χ m i i · γ ∨ n − = ( γ m i ) ∨ · γ ∨ n − = ( γ n − γ m i ) ∨ where γ n − γ m i is the unique right extension of γ n − as an ( n − p )-chain. Since γ n − can be recovered from the ( n − p )-chain γ n − γ m i by successively removingtails, this shows that χ · − : kC ∨ n − → kC ∨ n − p is injective.Now let n ≥ dim Λ. We prove that χ · − : kC ∨ n − → kC ∨ n − p is surjective analo-gously. Corollary 2.18 shows that every ( n − p )-chain is of the form γ n − w n w n +1 . . . w n − p for a perfect walk ( w n , w n +1 , . . . , w n − p ) and an ( n − γ n − . Since theperfect walk ( w n , w n +1 , . . . , w n − p ) has length p ≥ dim Λ + 1, with p even and amultiple of ℓ , the recognition lemma (Lemma 4.6) shows that w n w n +1 . . . w n − p isa stable relation cycle, and we may write w n w n +1 . . . w n − p = γ m for some stablerelation cycle γ of minimal path length and some m ≥ 1. Letting Γ i again be theclass of γ , Lemma 5.1 shows that ord i ( γ ) takes on minimal value so that γ ∼ γ i ;in particular χ γ = χ γ i and the equality m | χ γ | = p = m i | χ γ i | forces m = m i .It follows that every ( n − p )-chain has the form γ n − γ m i , and so every basiselement of kC ∨ n − p can be written as( γ n − γ m i ) ∨ = χ · γ ∨ n − for some γ ∨ n − ∈ C ∨ n − . This proves surjectivity, and we are done. (cid:3) Remark . The bound n ≥ dim Λ+1 cannot be improved to n ≥ dim Λ in general.This is clear if gldim Λ < ∞ as χ = 0 and Ext dim ΛΛ ( k , k ) = 0 then, but even whengldim Λ = ∞ this can fail due to non-trivial projective summands in Ω dim Λ k asseen in Section 2.3.We now obtain the main result of this paper. INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 33 Theorem 5.4. Let Λ be a monomial algebra. Then Λ satisfies Fg if and only if Λ is Gorenstein.Proof. As we mentioned before, the necessary implication is due to Erdmann-Holloway-Snashall-Solberg-Taillefer in [EHT + 04, Th. 1.5]. Conversely assume thatΛ is Gorenstein, and that gldim Λ = ∞ as the result is trivial otherwise.In this case Proposition 5.2 shows that Ext ∗ Λ ( k , k ), as a module over its polynomialsubalgebra k [ χ ] ⊆ R ⊆ Z ∞ Ext ∗ Λ ( k , k ), is generated by elements of degree at mostdim Λ + p − 1. It follows that Ext ∗ Λ ( k , k ) and Z ∞ = Z ∞ Ext ∗ Λ ( k , k ) ⊆ Ext ∗ Λ ( k , k ) aremodule-finite over k [ χ ]. In particular Z ∞ is Noetherian and Ext ∗ Λ ( k , k ) is module-finite over Z ∞ , so that Fg holds by Proposition 3.4. (cid:3) Further results Combinatorial characterisation of finite generation. Let Λ be a mono-mial algebra. We have just established that Λ satisfies Fg if and only if it isGorenstein, and we would like to have a simple combinatorial criterion for testingthe Gorenstein property of Λ starting from the quiver and relations.For any d with 1 ≤ d < ∞ , we have seen in Theorem 2.9 a combinatorial characteri-sation for the condition that Λ is Gorenstein of dimension dim Λ ≤ d in terms of thestructure of ( d − n -chains for all n ≥ d , in that all subsequent chains areobtained by appending the next path in a perfect cycle, which are periodic.Starting with an arbitrary monomial algebra Λ, this eventual periodicity of n -chains, if it occurs, could a priori begin in arbitrarily large degree n ≫ 0, and so itis not clear that one can rule out the Fg property for Λ by inspecting finitely manychains.As it turns out, there is a simple upper bound n Λ such that this periodicity eitheroccurs for n -chains for all n ≥ n Λ or never occurs at all. Recall that the (little)right finitistic dimension of Λ is defined asrfindim Λ := sup { projdim M | M ∈ mod Λ and projdim M < ∞} . The left finitistic dimension is defined analogously via left modules, or simply vialfindim Λ = rfindim Λ op . From the work of Igusa, Zacharia [IZ90] and Green,Kirkman and Kuzmanovich [GKK91], one knows that the finitistic dimension of amonomial algebra is always finite. Moreover, if Λ is any Gorenstein algebra then thefinitistic dimension is also known to be finite, and in fact agrees with the Gorensteindimension rfindim Λ = dim Λ by [Iwa80] (and then lfindim Λ = dim Λ op = dim Λ =rfindim Λ). Letting n Λ be any upper bound for the finitistic dimension of a mono-mial algebra Λ, it follows that Λ is either Gorenstein of dimension dim Λ ≤ n Λ ornot Gorenstein at all.Simple upper bounds are known by work of Igusa, Zacharia [IZ90] and Zimmermann-Huisgen [ZH91]. For instance, let K ≥ r K +1 = 0and set n Λ := min { dim k r , dim k Λ / r K } . Then rfindim Λ ≤ n Λ by [ZH91, Section 3]. Using this upper bound, we obtain thefollowing decidable combinatorial criterion for Fg : Theorem 6.1. Let Λ be a monomial algebra. The following are equivalent:(1) Λ satisfies Fg . (2) Λ is Gorenstein.(3) Let n = rfindim Λ . Then every n -chain γ has a perfect path t γ for tail.(4) Let n = lfindim Λ . Then every n -chain γ has a perfect path h γ for head.(5) Let n = n Λ . Then every n -chain γ has a perfect path t γ for tail.(6) Let n = n Λ . Then every n -chain γ has a perfect path h γ for head.Proof. The equivalence of (1) and (2) follows from Theorem 5.4. Next, note that(3)-(4) and (5)-(6) are dual under Λ ↔ Λ op , and as (2) is self-dual it is enough toshow that (3) is equivalent to (2) and (5) is equivalent to (2).Set N Λ = rfindim Λ. Assuming (3), Theorem 2.9 shows that Λ is Gorenstein ofdimension dim Λ ≤ N Λ + 1, in particular Λ is Gorenstein and (2) holds. Conversely,by the remarks above Λ is Gorenstein if and only if Λ is Gorenstein of dimensiondim Λ ≤ N Λ . Assuming (2), we obtain that dim Λ < N Λ + 1 and so (3) follows fromTheorem 2.9. Finally the equivalence of (2) and (5) follows verbatim by setting N Λ = n Λ instead in the previous argument. (cid:3) Remark . In practice the bound n Λ appears not to be very sharp and so period-icity may occur much faster; in any practical implementation better upper boundsshould be used, see [ZH91] for further details. Our choice of bound n Λ was mainlyfor its simplicity to state.6.2. The structure of Hochschild cohomology. We conclude with some inter-esting corollaries on the structure of Hochschild cohomology. Let χ ∈ Ext p Λ ( k , k )be the A ∞ -central class defined in (4), and we also write χ ∈ HH p (Λ , Λ) for thecorresponding lift. Theorem 6.3. Let Λ be a Gorenstein monomial algebra. Then HH ∗ (Λ , Λ) is even-tually periodic; more precisely, taking cup product with χ gives an isomorphism χ ⌣ − : HH n (Λ , Λ) ∼ = −→ HH n + p (Λ , Λ) for all n ≥ dim Λ + 1 . Proof. We give a quantitative version of a standard filtration argument [EHT + M, N ∈ mod Λ there are natural isomorphismsHH ∗ (Λ , Hom k ( M, N )) ∼ = Ext ∗ Λ ( M, N ) . Moreover one has Hom k ( k , k ) ∼ = D( k ) ⊗ k k ∼ = Λ e / rad Λ e for D = Hom k ( − , k ).For any bimodule B ∈ mod Λ e the vector space HH ∗ (Λ , B ) is a right-module overHH ∗ (Λ , Λ), and this module structure on HH ∗ (Λ , Hom k ( k , k )) ∼ = Ext ∗ Λ ( k , k ) agreeswith that given by the characteristic morphism ϕ k . Now, consider a Jordan-H¨olderfiltration of Λ in mod Λ e B K +1 ⊆ B K ⊆ · · · ⊆ B ⊆ B ⊆ B = Λwith B i /B i +1 ∈ add Λ e (Hom k ( k , k )) for 0 ≤ i ≤ K . The long exact sequence ofHH ∗ (Λ , B ) is compatible with the module structure over HH ∗ (Λ , Λ) and so gives This follows from binaturality of Ext ∗ Λ e (Λ , M ) making it a right module over Ext ∗ Λ e (Λ , Λ);note that a priori this only makes the map − ⌣ χ commute with the long exact sequence, but weuse χ ⌣ − for consistency with the theorem’s claim and use graded-commutativity. INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 35 rise to commutative diagrams of the form (writing HH n ( B ) := HH n (Λ , B ) for short)HH n − ( B i /B i +1 ) χ⌣ − (cid:15) (cid:15) / / HH n ( B i +1 ) χ⌣ − (cid:15) (cid:15) / / HH n ( B i ) χ⌣ − (cid:15) (cid:15) / / HH n ( B i /B i +1 ) χ⌣ − (cid:15) (cid:15) / / HH n +1 ( B i +1 ) χ⌣ − (cid:15) (cid:15) HH n − ( B i /B i +1 ) / / HH n ( B i +1 ) / / HH n ( B i ) / / HH n ( B i /B i +1 ) / / HH n +1 ( B i +1 )Since B i /B i +1 ∈ add Λ e (Hom k ( k , k )), from the above paragraph and Prop. 5.2 wesee that the fourth column map is an isomorphism for n ≥ dim Λ + 1, and thefirst column map is an isomorphism for n ≥ dim Λ + 2 and an epimorphism for n = dim Λ + 1. The 5-Lemma then shows that if the second and fifth columnmaps are isomorphism for n ≥ dim Λ + 1, then so is the third column map. Theresult for B = Λ then follows by induction from the base case B K = B K /B K +1 ∈ add Λ e (Hom k ( k , k )), which follows from Prop. 5.2. (cid:3) Finally, recall that for Λ Gorenstein the enveloping algebra Λ e = Λ op ⊗ k Λ isalso Gorenstein, then of dimension dim Λ e = 2 dim Λ, see [BIKP19, Prop. 6.1].Following Buchweitz, we define the Tate-Hochschild cohomology as the Ext algebraof the bimodule Λ ∈ D sg (Λ e ) in the singularity category of Λ e : d HH ∗ (Λ , Λ) := Ext ∗ D sg (Λ e ) (Λ , Λ) . Corollary 6.4. Let Λ be a Gorenstein monomial algebra. Then the Tate-Hochschildcohomology ring is given by periodic Hochschild cohomology: d HH ∗ (Λ , Λ) ∼ = HH ∗ (Λ , Λ)[ χ − ] . Proof. We first find a good model for d HH ∗ (Λ , Λ) by constructing an appropriatecomplete resolution of the diagonal. Let P ∗ ∼ −→ Λ Λ Λ be a minimal projective reso-lution of the diagonal bimodule Λ. We may represent the class χ ∈ HH p (Λ , Λ) bya chain map χ : P ∗ + p → P ∗ The corresponding class ϕ k ( χ ) = χ ∈ Ext p Λ ( k , k ) is then represented by k ⊗ Λ χ : k ⊗ Λ P ∗ + p → k ⊗ Λ P ∗ . Minimality of P ∗ means that P ∗ ⊗ Λ e Λ e / rad Λ e ∼ = k ⊗ Λ P ∗ ⊗ Λ k has trivial differential,and so k ⊗ Λ P ∗ is a minimal right resolution of k over Λ. Right multiplication by χ then corresponds toExt n Λ ( k , k ) −· χ / / Ext n + p Λ ( k , k )Hom Λ ( k ⊗ Λ P n , k ) −◦ χ / / Hom Λ ( k ⊗ Λ P n + p , k )Hom k ( k ⊗ Λ P n ⊗ Λ k , k ) −◦ χ / / Hom k ( k ⊗ Λ P n + p ⊗ Λ k , k ) . The first row is an isomorphism in degree n ≥ dim Λ+1, and the third row is k -dualto the map χ tensored down χ ⊗ Λ e Λ e / rad Λ e : P n + p ⊗ Λ e Λ e / rad Λ e → P n ⊗ Λ e Λ e / rad Λ e which must then also be an isomorphism for n ≥ dim Λ + 1. By the NakayamaLemma we conclude that χ : P n + p → P n is an isomorphism in the same range. Inparticular Λ has an eventually periodic minimal projective bimodule resolution.Let C ∗ be the unique periodic, unbounded, acyclic complex of finite Λ e projectiveswhich agrees with P ∗ in degree n ≥ dim Λ + 1; the complex C ∗ is then a completeresolution of (the Gorenstein-projective approximation of) Λ over Λ e . By Buch-weitz’s theorem [Buc86] we may compute the Tate-Hochschild cohomology algebraas d HH ∗ (Λ , Λ) ∼ = H ∗ (Hom Λ e ( C ∗ , C ∗ ))The chain-map χ : P ∗ + p → P ∗ extends by periodicity to χ : C ∗ + p → C ∗ , giving riseto a class χ ∈ d HH p (Λ , Λ) which is the image of χ ∈ HH p (Λ , Λ) under the naturalalgebra map ψ : HH ∗ (Λ , Λ) → d HH ∗ (Λ , Λ). More importantly, χ : C ∗ + p → C ∗ represents the periodicity isomorphism by construction and therefore admits aninverse χ − : C ∗− p → C ∗ , with χ − ∈ \ HH − p (Λ , Λ). The natural map then factorsthrough the localisation as ψ = e ψ ◦ u HH ∗ (Λ , Λ) u −→ HH ∗ (Λ , Λ)[ χ − ] e ψ −→ d HH ∗ (Λ , Λ) . By [Buc86, 6.3.5] the map ψ : HH n (Λ , Λ) → [ HH n (Λ , Λ) is an isomorphism forall n ≥ dim Λ e + 1 = 2 dim Λ + 1, and so u is injective in the same degrees. Sincemultiplication by χ on HH ≥ n (Λ , Λ) acts by periodicity for n ≥ dim Λ + 1 by the lasttheorem, the map u is surjective in degree n ≥ dim Λ + 1. Hence u , and therefore e ψ , is an isomorphism in degree n ≥ e ψ commutes withthe action of the periodicity operators χ ± we see that e ψ is an isomorphism in alldegrees. (cid:3) References [AG87] David J. Anick and Edward L. Green. On the homology of quotients of path algebras. Comm. Algebra , 15(1-2):309–341, 1987. 5[Ani86] David J. Anick. On the homology of associative algebras. Trans. Amer. Math. Soc. ,296(2):641–659, 1986. 5[Bar97] Michael J. Bardzell. The alternating syzygy behavior of monomial algebras. J. Algebra ,188(1):69–89, 1997. 5, 9, 10[BG17] Benjamin Briggs and Vincent Gelinas. The A-infinity Centre of the Yoneda Algebraand the Characteristic Action of Hochschild Cohomology on the Derived Category. arXiv e-prints , page arXiv:1702.00721, Feb 2017. 2, 18, 19, 25[BIKP19] Dave Benson, Srikanth B. Iyengar, Henning Krause, and Julia Pevtsova. Local dualityfor the singularity category of a finite dimensional Gorenstein algebra. arXiv e-prints ,page arXiv:1905.01506, May 2019. 35[BK99] M. C. R. Butler and A. D. King. Minimal resolutions of algebras. J. Algebra ,212(1):323–362, 1999. 4, 6[BOJ15] Petter Andreas Bergh, Steffen Oppermann, and David A. Jorgensen. The Gorensteindefect category. Q. J. Math. , 66(2):459–471, 2015. 12[Bon83] Klaus Bongartz. Algebras and quadratic forms. J. London Math. Soc. (2) , 28(3):461–469, 1983. 5[Buc86] Ragnar-Olaf Buchweitz. Maximal Cohen-Macaulay Modules andTate-Cohomology Over Gorenstein Rings. Manuscript available athttps://tspace.library.utoronto.ca/handle/1807/16682 , 1986. 10, 12, 36[Car83] Jon F. Carlson. The varieties and the cohomology ring of a module. J. Algebra ,85(1):104–143, 1983. 1 INITE GENERATION FOR GORENSTEIN MONOMIAL ALGEBRAS 37 [CG11] Andrew Conner and Pete Goetz. A ∞ -algebra structures associated to K algebras. J.Algebra , 337:63–81, 2011. 7[CK12] Joseph Chuang and Alastair King. Free resolutions of algebras. arXiv e-prints , pagearXiv:1210.5438, Oct 2012. 4[CSZ18] Xiao-Wu Chen, Dawei Shen, and Guodong Zhou. The Gorenstein-projective modulesover a monomial algebra. Proc. Roy. Soc. Edinburgh Sect. A , 148(6):1115–1134, 2018.2, 10, 11, 13[EHT + 04] Karin Erdmann, Miles Holloway, Rachel Taillefer, Nicole Snashall, and Øyvind Solberg.Support varieties for selfinjective algebras. K -Theory , 33(1):67–87, 2004. 1, 33, 34[ES11] Karin Erdmann and Øyvind Solberg. Radical cube zero weakly symmetric algebrasand support varieties. J. Pure Appl. Algebra , 215(2):185–200, 2011. 19[Eve61] Leonard Evens. The cohomology ring of a finite group. Trans. Amer. Math. Soc. ,101:224–239, 1961. 1[FS11] Takahiko Furuya and Nicole Snashall. Support varieties for modules over stacked mono-mial algebras. Comm. Algebra , 39(8):2926–2942, 2011. 1[GHZ85] E. L. Green, D. Happel, and D. Zacharia. Projective resolutions over Artin algebraswith zero relations. Illinois J. Math. , 29(1):180–190, 1985. 5[GKK91] Edward L. Green, Ellen Kirkman, and James Kuzmanovich. Finitistic dimensions offinite-dimensional monomial algebras. J. Algebra , 136(1):37–50, 1991. 33[Gol59] E. Golod. The cohomology ring of a finite p -group. Dokl. Akad. Nauk SSSR , 125:703–706, 1959. 1[Gov73] V. E. Govorov. The global dimension of algebras. Mat. Zametki , 14:399–406, 1973. 5[Gru68] K. W. Gruenberg. The universal coefficient theorem in the cohomology of groups. J.London Math. Soc. , 43:239–241, 1968. 4, 5[GSS06] Edward L. Green, Nicole Snashall, and Øyvind Solberg. The Hochschild cohomologyring modulo nilpotence of a monomial algebra. J. Algebra Appl. , 5(2):153–192, 2006.2, 21, 30[Gul74] Tor H. Gulliksen. A change of ring theorem with applications to Poincar´e series andintersection multiplicity. Math. Scand. , 34:167–183, 1974. 24[GZ94] E. L. Green and D. Zacharia. The cohomology ring of a monomial algebra. ManuscriptaMath. , 85(1):11–23, 1994. 7[Her18a] Estanislao Herscovich. A simple note on the Yoneda (co)algebra of a monomial algebra. Preprint , 2018. 2[Her18b] Estanislao Herscovich. Using torsion theory to compute the algebraic structure ofHochschild (co)homology. Homology Homotopy Appl. , 20(1):117–139, 2018. 9[HL05] Ji-Wei He and Di-Ming Lu. Higher Koszul algebras and A -infinity algebras. J. Algebra ,293(2):335–362, 2005. 2[Iwa80] Yasuo Iwanaga. On rings with finite self-injective dimension. II. Tsukuba J. Math. ,4(1):107–113, 1980. 33[Iyu16] Natalia Iyudu. On the proof of the homology conjecture for monomial non-unitalalgebras. , IHES/M/16/15, 2016. 6[IZ90] Kiyoshi Igusa and Dan Zacharia. Syzygy pairs in a monomial algebra. Proc. Amer.Math. Soc. , 108(3):601–604, 1990. 33[LPWZ04] D. M. Lu, J. H. Palmieri, Q. S. Wu, and J. J. Zhang. A ∞ -algebras for ring theorists.In Proceedings of the International Conference on Algebra , volume 11, pages 91–128,2004. 2[LPWZ09] D.-M. Lu, J. H. Palmieri, Q.-S. Wu, and J. J. Zhang. A -infinity structure on Ext-algebras. J. Pure Appl. Algebra , 213(11):2017–2037, 2009. 2[LV12] Jean-Louis Loday and Bruno Vallette. Algebraic operads , volume 346 of Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sci-ences] . Springer, Heidelberg, 2012. 6[LZ17] Ming Lu and Bin Zhu. Singularity categories of gorenstein monomial algebras. arXive-prints , page arXiv:1708.00311, August 2017. 14, 15[Nag11] Hiroshi Nagase. Hochschild cohomology and Gorenstein Nakayama algebras. In Pro-ceedings of the 43rd Symposium on Ring Theory and Representation Theory , pages37–41. Symp. Ring Theory Represent. Theory Organ. Comm., Soja, 2011. 2 [Pro11] Alain Prout´e. A ∞ -structures. Mod`eles minimaux de Baues-Lemaire et Kadeishvili ethomologie des fibrations. Repr. Theory Appl. Categ. , (21):1–99, 2011. Reprint of the1986 original, With a preface to the reprint by Jean-Louis Loday. 2[RGS09] J. C. Rosales and P. A. Garc´ıa-S´anchez. Numerical semigroups , volume 20 of Devel-opments in Mathematics . Springer, New York, 2009. 29[SS04] Nicole Snashall and Øyvind Solberg. Support varieties and Hochschild cohomologyrings. Proc. London Math. Soc. (3) , 88(3):705–732, 2004. 1[Tam18] Pedro Tamaroff. Minimal models for monomial algebras. arXiv e-prints , pagearXiv:1804.01435, April 2018. 2, 7, 10[Ufn95] V. A. Ufnarovskij. Combinatorial and asymptotic methods in algebra [ MR1060321(92h:16024)]. In Algebra, VI , volume 57 of Encyclopaedia Math. Sci. , pages 1–196.Springer, Berlin, 1995. 4[Ven59] B. B. Venkov. Cohomology algebras for some classifying spaces. Dokl. Akad. NaukSSSR , 127:943–944, 1959. 1[WZ01] Q.-S. Wu and J. J. Zhang. Dualizing complexes over noncommutative local rings. J.Algebra , 239(2):513–548, 2001. 15[Zak69] Abraham Zaks. Injective dimension of semi-primary rings. J. Algebra , 13:73–86, 1969.11[ZH91] Birge Zimmermann-Huisgen. Predicting syzygies over monomial relations algebras. Manuscripta Math. , 70(2):157–182, 1991. 33, 34 School of Mathematics, Trinity College, Dublin 2, Ireland E-mail address : [email protected] School of Mathematics, Trinity College, Dublin 2, Ireland E-mail address : [email protected] School of Mathematics, Trinity College, Dublin 2, Ireland E-mail address ::