Linear Resolutions Over Koszul Complexes and Koszul Homology

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See, for example, the survey [13] and the monograph [23] for expositions on the noncommutative side, and the survey [10] directed at applications in commutative algebra. On the other hand, what we mean by “linearity” of the resolutions over K and H requires further explanation. As we will recall below, there is a DG version of TorK (k, k) which can be computed via a semifree resolution of k over K. The grading on R induces an extra internal degree on TorK (k, k) in addition to its usual homological degree, and we then say that K is a Koszul DG algebra when the “non-diagonal” part of TorK (k, k) vanishes, in complete analogy with the classical definition of Koszulness. This definition follows in the spirit of (but is not exactly identical to) the definition offered by He and Wu in [18] who deal with connected cochain DG algebras instead of chain algebras, which is where our focus lies. The differential accompanying the former type of algebra raises homological degree, while the one accompanying the latter lowers homological degree. The relationship between Koszulness of the algebras R and K is then satisfyingly simple and should not surprise specialists in this field. Theorem A. Let R be a standard graded commutative algebra over a field k and let K be its Koszul complex. The algebras R and K are Koszul simultaneously. Shifting attention to the homology algebra H, the grading on R also induces an internal degree on H giving it the structure of bigraded k-algebra. It is easy to show that Hij 6= 0 implies i = j =0 or j − i> 0, a property that we describe concisely by saying H lives in positive strands. This shape makes it possible to perform a totalization process, which we call strand totalization, to produce a connected ′ ′ graded k-algebra H with Hn = j−i=n Hij for each n. Then, when we speak of k having a linear free resolution over H, we mean that H′ is a Koszul algebra in the L classical sense, in which case we say H is strand-Koszul. The complete relationship between the three notions of Koszulness is expressed in Theorem B. Let R be a standard graded commutative algebra over a field k, let K be its Koszul complex, and let H be the homology algebra of K. The following are equivalent. (1) The algebra H is strand-Koszul. (2) The algebra K is Koszul and quasi-formal. (3) The algebra R is Koszul and K is quasi-formal. (4) The algebra R is Koszul and the minimal graded free resolution of k over R has an Eilenberg-Moore filtration. Quasi-formality and the existence of an Eilenberg-Moore filtration are two tech- nical conditions which facilitate a clean transfer of homological properties from R and K to H. The former condition, introduced and studied by Positselski in [24], concerns the degeneracy of a spectral sequence which connects H to K, while an Eilenberg-Moore filtration allows one to “build” the minimal free resolution of k over H starting from the minimal free resolution of k over R. LINEARRESOLUTIONSOVERKOSZULCOMPLEXES 3

Theorem B thus shows that strand-Koszulness of H is stronger than Koszulness of R and K. For example, the Koszul algebra k[x,y,z,u] R = , (x2, xy, xz + u2,xu,y2 + z2,zu) which is listed as “isotope 63ne” in Roos’ catalog [26, Appendix A], has a Koszul homology algebra which is not strand-Koszul (verification by Macaulay2 [16] using the DGAlgebras package written by Frank Moore). We note that out of the 104 algebras of embedding dimension 4 that Roos catalogs, there are only two Koszul algebras which have Koszul homology algebras that are not strand-Koszul. Thus, in combination with our next result, we have that strand-Koszulness of H is “nearly” equivalent to Koszulness of R, as long as R has embedding dimension ≤ 4. Theorem C. Let R be a standard graded commutative algebra over a field k, let H be the Koszul homology algebra of R, and suppose that one of the following statements is true. (1) The algebra R is Koszul with embedding dimension ≤ 3. (2) The algebra R is Koszul and Golod. (3) The algebra R is a quadratic complete intersection. (4) The algebra R is artinian Gorenstein of socle degree 2, and either k does not have characteristic 2 or the embedding dimension of R is odd. (5) The algebra R is Koszul and the defining ideal of R is minimally generated by three elements. (6) The defining ideal of R is the edge ideal of a path on ≥ 3 vertices. Then H is strand-Koszul. The proof uses a mixture of techniques. Previously established theory makes quick work of the implications “(1)-(3) ⇒ H is strand-Koszul,” while our proof of the implications “(4)-(6) ⇒ H is strand-Koszul” is based on (noncommutative) Gröbner basis techniques and computations of presentations of H as a quotient of a free algebra. We point out that the strand totalizations H′ are not new, except perhaps for the name. For example, a question posed by Boocher, D’Al`ı, Grifo, Montaño, and Sammartano [7] asks whether or not the algebra H′ is generated by degree- 1 elements when R is a Koszul integral domain (this replaces the same question posed earlier by Avramov without the hypothesis that R is an integral domain — the answer to that question turned out to be negative, see [7, Remark 3.2]). In [14], Fröberg and Löfwal study H′ and uncover a connection between this algebra and (part of) the homotopy Lie algebra of R. The relation between strand totalizations and Koszulness of R was also considered by Croll et al. [11] who showed that if ′ ′ ′ there is an element ζ ∈ H1,2 ⊆ H1 such that H · ζ = H>1, then R is Koszul. Section 1 of this paper sets up definitions and notations and proves some prelim- inary results, while section 2 is dedicated to the proofs of Theorems A and B and also contains some results on relationships between Poincaré-Betti series, Hilbert series, and low-degree Betti numbers. The third and final section contains the proof of Theorem C.

1. Definitions and preliminaries Throughout this paper k denotes a ﬁeld. Elements of graded objects will always be assumed homogeneous and thus all elements of such objects have degrees. 4 JOHN MYERS

We assume that the reader is familiar with the basic terminology and theory of connected Z-graded k-algebras; everything that we need can be found in sections 0 and 4 in Chapter 1 of [23], or the Appendice in [20]. We single out just a few definitions and terms from this theory that are of particular importance to us. First, we make Definition 1.0.1. Let A be a connected Z-graded k-algebra. We will say A is A Koszul if βij (k) 6= 0 implies i = j for all i, j ≥ 0. Here A A βij (k) = dimk Tori (k, k)j is the (i, j)-th Betti number of k. All of these numbers are finite, and the generating function built from them is denoted A A i j Pk (s,t)= βij (k)s t i,j X and is called the bigraded Poincaréseries of k. A Z-graded module M over a connected N-graded k-algebra is said to be finite if Mj = 0 for all j ≪ 0 and each Mj is finite-dimensional over k. The q-th shift of M is defined to be the Z-graded module

M(q)= M(q)j with M(q)j = Mq+j . Z Mj∈ 1.1. Connected Z2-graded algebras. This paper is concerned with both Z- and Z2-graded objects; hereafter, the former objects will simply be called graded, while the latter objects will be called bigraded.

A bigraded k-algebra A = i,j∈Z Aij is said to be connected when A00 = k; Aij 6= 0 implies i, j ≥ 0; and each homogeneous component Aij is ﬁnite-dimensional. L Given an element a ∈ Aij , the integers i and j, the pair (i, j), and the sum i+j will be called, respectively, the homological degree, internal degree, bidegree, and total degree of a. The homological degree of a will be denoted |a|, its internal degree denoted deg a, and its bidegree denoted bideg a. We will write Ai for the graded k-space Ai,∗. Let A be a connected bigraded k-algebra and M a bigraded left A-module. The p-th homological shift of M is the bigraded module

p p p Σ M = (Σ M)ij , with (Σ M)ij = Mi−p,j . Z i,jM∈ The q-th internal shift of M is the bigraded module

M(q)= M(q)ij , with M(q)ij = Mi,q+j . Z i,jM∈ The module M is said to be finite if Mi = 0 for all i ≪ 0 and each Mi is finite as an A0-module (as above, we write Mi for Mi,∗). The field k can be viewed as a bigraded k-algebra concentrated in bidegree (0, 0). The natural map A → k (called the augmentation) is a morphism of bigraded algebras and gives k a bigraded A-module structure. We define the augmentation ideal of A, denoted mA, to be the ideal ker (A → k). If the algebra A is understood, we will write m in place of mA. LINEARRESOLUTIONSOVERKOSZULCOMPLEXES 5

If M is ﬁnite, then M has a bigraded minimal free resolution of the form

i β0 i β1 (1.1.1) 0 ← M ← Σ A(−j) ,i,j ← Σ A(−j) ,i,j ← · · · , Z Z i,jM∈ i,jM∈ A where the integers βpij (M) = βpij are uniquely determined by M and are called the trigraded Betti numbers of M. They are all finite, and they can be measured via the dimensions of certain Tor-spaces: A A βpij (M) = dimk Torp (k,M)ij . We define the trigraded Poincaréseries of M to be the formal series A A p i j PM (r,s,t)= βpij (M)r s t . p,i,j X For later applications, we note the following proposition which shows that the “shape” of the augmentation ideal m is reflected in the “shape” of the trigraded Betti numbers of k. Proposition 1.1.2. Let A be a connected bigraded k-algebra. If

mij 6=0 ⇒ i, j − i> 0, then A βpij (k) 6=0 ⇒ i, j − i ≥ p.

1.2. Strand totalizations. Let V = i,j∈Z Vij be a bigraded k-vector space. We define an exact functor V 7→ V ′ from the category of Z2- to Z-graded vector spaces L via the formulas ′ ′ ′ V = Vn, Vn = Vij . Z j−i=n Mn∈ M The Z-graded vector space V ′ is called the strand totalization of V , and we define ′ the strand degree of an element v ∈ Vij to be the difference j − i. The functor (−) carries a bigraded k-algebra A to the graded k-algebra A′, and it sends a bigraded A-module M to the graded A′-module M ′. Its action on shifts is explained in the formula p ′ ′ (Σ M(q)) = M (p + q). We will say that a connected bigraded k-algebra A lives in positive strands if its strand totalization A′ is connected (as a Z-graded algebra). If M is a finite bigraded module over such an algebra and M ′ is a finite A′-module, then applying (−)′ to the bigraded minimal free resolution (1.1.1) produces the exact sequence 0 ← M ′ ← A′(i − j)β0,i,j ← A′(i − j)β1,i,j ← · · · , Z Z i,jM∈ i,jM∈ which is a graded minimal free resolution of M ′ over A′. This yields the following formula relating the bigraded Betti numbers of M ′ and the trigraded Betti numbers of M: ′ A ′ A (1.2.1) βpq (M )= βpij (M). j−i=q X This leads us to Definition 1.2.2. Let A be a connected bigraded k-algebra. We will say A is strand-Koszul if it lives in positive strands and A′ is Koszul as a graded algebra. 6 JOHN MYERS

Then, in view of equation (1.2.1), we have Proposition 1.2.3. Let A be a connected bigraded k-algebra which lives in positive A strands. The algebra A is strand-Koszul if and only if βpij (k) 6=0 implies p = j − i. 1.3. DG algebras and modules. A DG k-algebra is a bigraded k-algebra A equipped with a k-linear endomorphism ∂A : A → A of bidegree (−1, 0), called the differential, such that ∂A ◦ ∂A = 0, ∂A(1) = 0, and which satisfies the Leibniz rule: ∂(ab)= ∂(a)b + (−1)|a|a∂(b) for all a,b ∈ A. We write A♮ for the underlying bigraded k-algebra obtained by forgetting the differential of A. If no confusion will arise, we will omit the superscript from ∂A and write ∂ in its place. The Leibniz rule and the condition ∂A(1) = 0 imply that the space Z(A) = ker ∂A of cycles is a bigraded k-subalgebra of A. The space B(A) = im ∂A of boundaries is an ideal of Z(A), and hence the homology H(A) = Z(A)/ B(A) is a bigraded k-algebra. Letting A and B be two DG algebras, a function ϕ : A → B is called a morphism of DG algebras if it induces a morphism A♮ → B♮ of the underlying bigraded algebras and if ∂B ◦ ϕ = ϕ ◦ ∂A. The morphism ϕ is called a quasi-isomorphism if the induced map H(ϕ):H(A) → H(B) of bigraded algebras is an isomorphism. Let A be a DG k-algebra. A (left) DG A-module is a bigraded left A♮-module M equipped with a k-linear endomorphism ∂M : M → M of bidegree (−1, 0), called a differential, such that ∂M ◦ ∂M = 0 and which satisfies the Leibniz rule: ∂M (am)= ∂A(a)m + (−1)|a|a∂M (m) for all a ∈ A, m ∈ M. We write M ♮ for the underlying bigraded A♮-module. Right DG A-modules are defined analogously. The homology H(M) = ker ∂M / im ∂M is a module over H(A); the DG module M will be called homologically finite if the homology H(M) is finite as a bigraded H(A)-module. The field k can view viewed as a DG k-algebra concentrated in bidegree (0, 0) with trivial differential. If M is a right DG A-module and N a left DG A-module, their tensor product, denoted M ⊗A N, has underlying bigraded k-module ♮ ♮ ♮ (M ⊗A N) = M ⊗A♮ N and differential defined as ∂(m ⊗ n) = ∂m ⊗ n + (−1)|m|m ⊗ ∂n. Tensor product op yields an additive bifunctor −⊗A − : M(A) × M(A) → M(k) where M(A) and M(k) are the abelian categories of bigraded DG A- and k-modules, respectively. Letting M and N be two DG A-modules, a function α : M → N is called a morphism of DG modules if it induces an A♮-linear map M ♮ → N ♮ and if ∂N ◦ α = α ◦ ∂M . The morphism α is called a quasi-isomorphism if the induced H(A)-linear map H(α) : H(M) → H(N) is an isomorphism. We will say that A is connected if the bigraded algebra A♮ is connected and m m there is an inclusion ∂A1 ⊆ A0 , where A0 denotes the augmentation ideal of the connected Z-graded algebra A0. For such a DG algebra, we define the augmentation ideal of A to be m m A = A0 ⊕ A1 ⊕ A2 ⊕ · · · . LINEARRESOLUTIONSOVERKOSZULCOMPLEXES 7

If A is connected, the natural map A → k (called the augmentation) is a morphism of DG k-algebras and gives k a DG A-module structure.

1.4. Semifree resolutions and the derived tensor product. Let A be a DG k-algebra. A DG module F is said to be semifree if there exists an exhaustive filtration 0= F −1 ⊆ F 0 ⊆···⊆ F p−1 ⊆ F p ⊆···⊆ F by DG submodules such that for each p ≥ 0 the A♮-module (F p)♮/(F p−1)♮ is free on a basis of cycles. Such a filtration is called a semifree filtration of F . For each DG A-module M, there exists a quasi-isomorphism ε : F → M with F a semifree DG A-module, and this property determines F up to homotopy equivalence; see [12, Proposition 6.6] for proofs. Such a quasi-isomorphism is called a semifree resolution of M, though we shall often refer to just F as a semifree resolution of M and omit mention of the map ε. After choosing a semifree resolution F (M) for every DG module M, the left derived functor of −⊗A − is defined to be

L M ⊗A N = F (M) ⊗A F (N). We set A L Tori (M,N)j = Hij (M ⊗A N) for each i and j. By [12, Proposition 6.7], we have isomorphisms

A H(F (M) ⊗A N) =∼ Tor (M,N) =∼ H(M ⊗A F (N)) of bigraded k-modules. Let A be a connected DG k-algebra and M a homologically ﬁnite DG A-module. Via a process similar to the familiar one used to construct minimal graded free resolutions over a connected algebra, one can construct a semifree resolution F of ♮ ♮ M such that F is a ﬁnite A -module and ∂F ⊆ mAF . Such a resolution is called a minimal semifree resolution of M and is determined by M up to isomorphism of DG A-modules. We have that

A Tor (k,M) =∼ k ⊗A F if F is a minimal semifree resolution of M. We define the (i, j)-th bigraded Betti number of M to be A A βij (M) = dimk Tori (k,M)j , and we define the bigraded Poincaréseries of M to be the formal series

A A i j PM (s,t)= βij (M)s t . i,j X We now offer a definition of Koszulness which applies to connected DG algebras; it is an analog of the definition given by He and Wu in [18].

Deﬁnition 1.4.1. Let A be a connected DG algebra over a ﬁeld k. We say that A A is a Koszul DG k-algebra if βij (k) 6= 0 implies i = j. 8 JOHN MYERS

1.5. Standard graded commutative algebras and Koszul complexes. A standard graded k-algebra is a connected graded algebra R such that there exists an isomorphism R =∼ Q/J of k-algebras where J is a homogeneous ideal in a polynomial ring Q = k[X1,...,Xn] generated by variables of degree 1. We as- m2 sume that J is contained in Q, and in this case the embedding dimension of R is the integer n. This integer does not depend on the presentation R =∼ Q/J, since m m2 n = dimk ( R/ R). Let R be a standard graded k-algebra of embedding dimension n, and choose a minimal generating set {x1,...,xn} of the ideal mR. We view R as a bigraded algebra concentrated in homological degree 0. Let t1,...,tn be a set of exterior variables of bidegree (1, 1), and define the Koszul complex of R to be the exterior R R algebra K = ΛR(t1,...,tn). We define a differential on K by setting ∂ti = xi for each i =1,...,n, and extending to all of KR via the Leibniz rule. Equipped with this differential, the algebra KR is a connected DG k-algebra; it is independent (up to isomorphism of DG algebras) of the choice of minimal generating set of mR. We define the Koszul homology algebra of R to be the homology algebra HR = H(KR); it is a connected bigraded k-algebra which lives in positive strands. If the algebra R under consideration is understood and no confusion is likely to arise, we will write K and H in place of KR and HR.

2. Koszulness of R, KR, and HR Throughout the rest of this paper we let R denote a standard graded k-algebra of embedding dimension n. We write K and H for its Koszul complex and Koszul homology algebra. We set as our ﬁrst task to show that the graded minimal free resolution of k over R and the minimal semifree resolution of k over K are “the same.” We begin by (very) brieﬂy recalling a construction due to Tate; for more details, see the original paper [28], or [4, §6].

2.1. Tate resolutions. The construction begins with the Koszul complex K. If T1 = {t1,...,tn} is the set of exterior variables generating K, we set the notation

R hT1i = K.

Now choose cycles ζ1,...,ζc ∈ Z1(R hT1i) whose homology classes minimally generate the module H1(R hT1i). Letting T2 = {tn+1,...,tn+c} be a set of divided- power variables with bideg(tn+i)=(2, deg ζi) for each i =1,...,c, we set Γ (j) (j−1) R hT1,T2i = R hT1i⊗R R(T2), ∂tn+i = ζit , and extend the diﬀerential from R hT1i to all of R hT1,T2i via the Leibniz rule. By construction we have

H1(R hT1,T2i)=0.

One then adjoins to R hT1,T2i a set T3 of exterior variables in homological degree 3 in order to “kill” cycles whose homology classes minimally generate H2(R hT1,T2i): the result is the DG algebra

R hT1,T2,T3i = R hT1,T2i⊗R ΛR(T3) with H2(R hT1,T2,T3i) = 0. This process continues, alternating between adjoining sets T2i of divided-power variables (in even homological degree) and sets T2i+1 of LINEARRESOLUTIONSOVERKOSZULCOMPLEXES 9 exterior variables (in odd homological degree) to produce a DG algebra

R hT i ,T = i≥1 Ti which is a graded free resolution of k over RS. But even more, Gulliksen [17] and Schoeller [27] proved that this construction yields the minimal graded free resolution.

2.2. A proof of Theorem A. Using Tate’s construction, the proof of the next proposition goes quickly. Proposition 2.2.1. Let F be a minimal graded free resolution of k over R constructed according to Tate’s method; see subsection 2.1. (1) The resolution F has a DG K-module structure making it the minimal semifree resolution of k over K. (2) There are equalities R n K n K Pk (s,t)=(1+ st) Pk (s,t) and HR(t)(1 − t) Pk (−1,t)=1, i where HR(t)= i≥0(dim Ri)t is the Hilbert series of R. Proof. We carry over theP notation introduced in subsection 2.1, supposing that F = R hT i. (1): The inclusion K ⊆ R hT i exhibits K as a DG subalgebra of R hT i, and thus gives R hT i a DG K-module structure. However, we have R hT i = K ⊗R R hT≥2i, and since R hT≥2i is a free bigraded R-module, we conclude that R hT i is a semifree DG K-module. Minimality of R hT i over K follows immediately from minimality of R hT i over R. (2): We have that R Tor (k, k) =∼ k ⊗R R hT i =∼ k hT i and K Tor (k, k) =∼ k ⊗K R hT i =∼ k ⊗K (K ⊗R R hT≥2i) =∼ k hT≥2i . Thus we have R K Tor (k, k) =∼ Λk(T1) ⊗k Tor (k, k) as bigraded vector spaces, from which the ﬁrst equality follows. The second then R follows from the ﬁrst in view of the well-known equality HR(t)Pk (−1,t) = 1; see, for example, [23, Chapter 2, Proposition 2.1]. As a corollary, we obtain Theorem A from the introduction: Corollary 2.2.2. The algebra R is Koszul if and only if K is Koszul.

Proof. The algebra R is Koszul if and only if

R R i Pk (s,t)= βii (k)(st) , Xi≥0 while K is Koszul if and only if

K K i Pk (s,t)= βii (k)(st) . Xi≥0 The corollary thus follows from the ﬁrst equation in Proposition 2.2.1(2). 10 JOHN MYERS

Having addressed the relationship between Koszulness of R and K, we bring the homology H into the picture. The main vehicle for transporting information from H to R and K will be a pair of spectral sequences. 2.3. Eilenberg-Moore spectral sequences and quasi-formality. Let ε : F ≃ k be a semifree resolution of k over K; recall that this means ε : F → k is a quasi-isomorphism and F is a semifree K-module. Using a semifree filtration of F i (see subsection 1.4), one can prove the existence of a collection {V }i≥0 of bigraded k-spaces such that for every p ≥ 0 we have p ♮ ∼ ♮ p i p p−1 (F ) = K ⊗k i=0 V and ∂(V ) ⊆ F . Thus the spectral sequence arisingL from the semifree filtration has 1 ∼ −p p Epqj = (H ⊗k Σ V )qj . The differentials 1 1 1 dpqj : Epqj → Ep−1,q,j and the augmentation ε assemble together to yield a chain complex

H(ε) 0 d1 −1 1 d2 −2 2 0 ← k = H(k) ←−−− H ⊗k V ←− H ⊗k Σ V ←− H ⊗k Σ V ← · · · of free bigraded H-modules; if this chain complex is exact, then the semifree resolution ε : F ≃ k is called an Eilenberg-Moore resolution of k over K, and the p filtration {F }p≥−1 is called an Eilenberg-Moore filtration. These always exist; see, for example, [12, Proposition 20.11]. From the existence of an Eilenberg-Moore resolution of k over K, one can derive the corresponding Eilenberg-Moore spectral sequence, which reads as 2 H K Epqj = Torp (k, k)qj ⇒ Torp+q (k, k)j and with differentials acting as r r r dpqj : Epqj → Ep−r,q+r−1,j . See [12, §20], or, alternatively, one can derive this spectral sequence by filtering the bar construction of K by the “number of bars.” Adopting Positselski’s terminology in [24], we shall say the DG algebra K is quasi-formal if the Eilenberg-Moore ≥2 spectral sequence degenerates on the second page, i.e., all differentials dpqj vanish. The terminology is justified by noting that formal DG algebras (i.e., ones which are connected to their homology algebras via a string of quasi-isomorphisms) are quasi-formal; see [24, Proposition 2.1]. 2.4. Avramov spectral sequences. In addition to the Eilenberg-Moore spectral sequence which links H and K, we will use a second spectral sequence which goes directly from H to R. To describe it, we first recall that R can be presented as a quotient Q/J where Q is a polynomial ring of embedding dimension n. Then, using the classical “external homology product” (see [21]), one can give TorQ (k, k) an H-module structure; see also [4, §2.3]. Our second spectral sequence then reads 2 H Q R Epqj = Torp (k, Tor (k, k))qj ⇒ Torp+q (k, k)j and has differentials acting as r r r dpqj : Epqj → Ep−r,q+r−1,j . We call this the Avramov spectral sequence, though we note that this name is not standard. For example, Roos [26] refers to the Eilenberg-Moore spectral sequence LINEAR RESOLUTIONS OVER KOSZUL COMPLEXES 11 above as the Avramov spectral sequence. See [3] for Avramov’s original derivation of the spectral sequence, and also Iyengar’s alternate derivation in [19]. Since Q is a polynomial ring, the bigraded k-space TorQ (k, k) is “concentrated Q n Q along the diagonal,” i.e., Tor (k, k)= i=0 Tori (k, k)i. In view of this fact, the proof of the following result is straightforward. L Proposition 2.4.1. There is an isomorphism n n Q ∼ i ( i ) Tor (k, k) = i=0 Σ k(−i) of left bigraded H-modules. In particular,L there are k-linear isomorphisms n H Q ∼ n H ( i ) Torp (k, Tor (k, k))qj = i=0 Torp (k, k)q−i,j−i for all p,q,j and an equality L H n H PTorQ (k,k)(s,s,t)=(1+ st) Pk (s,s,t). 2 2.5. A proof of Theorem B. A convergent spectral sequence Epqj ⇒ E of trigraded vector spaces generates inequalities of sums of dimensions. Indeed, for all m, j we have 2 ∞ (2.5.1) dimk Epqj ≥ dimk Epqj = dimk Emj . p+q=m p+q=m X X The following proposition follows. Proposition 2.5.2. (1) The Eilenberg-Moore spectral sequence yields a coefficient-wise inequality K H Pk (s,t) ≤ Pk (s,s,t), and equality holds if and only if K is quasi-formal. (2) The Avramov spectral sequence yields a coefficient-wise inequality R H Pk (s,t) ≤ PTorQ (k,k)(s,s,t), and equality holds if and only if the spectral sequence degenerates on the second page. As a corollary, we get that the Eilenberg-Moore and Avramov spectral sequences degenerate on the second page simultaneously: Corollary 2.5.3. The algebra K is quasi-formal if and only if the Avramov spectral sequence degenerates on the second page. Proof. Propositions 2.2.1, 2.4.1, and 2.5.2 give n K R H n H (1 + st) Pk (s,t)=Pk (s,t) ≤ PTorQ (k,k)(s,s,t)=(1+ st) Pk (s,s,t). Now apply Proposition 2.5.2 one more time. We now arrive at the central theoretical result of the paper, Theorem B from the introduction, renamed here as Theorem 2.5.4. The following statements are equivalent. (1) The algebra H is strand-Koszul. (2) The algebra K is Koszul and quasi-formal. (3) The algebra R is Koszul and K is quasi-formal. 12 JOHN MYERS

(4) The algebra R is Koszul and the minimal graded free resolution of k over R has an Eilenberg-Moore filtration. Furthermore, if one of the above statements is true (and hence all are), then there are equalities K H R n H Pk (s,t)=Pk (s,s,t) and Pk (s,t)=(1+ st) Pk (s,s,t). Proof. (1) ⇒ (2): We shall first prove that K is quasi-formal. To do so, we observe that if the differential 2 2 2 dpqj : Epqj → Ep−2,q+1,j in the Eilenberg-Moore spectral sequence were nonzero, then necessarily H 2 H 2 Torp (k, k)qj = Epqj 6=0 and Torp−2 (k, k)q+1,j = Ep−2,q+1,j 6=0. But H is strand-Koszul, and thus Proposition 1.2.3 implies both p = j − q and 2 p − 2= j − (q + 1). This is absurd, which means that we must have dpqj = 0 and therefore E2 = E∞. Now, the coefficient-wise inequality in Proposition 2.5.2(1) turns into an equality, and it yields equalities

K H (∗) βij (k)= βpqj (k) p+q=i X K for each i and j. Thus if βij (k) 6= 0, then there is a pair p, q with p + q = i and H βpqj (k) 6= 0. Proposition 1.2.3 then gives p = j − q, and hence i = j. This proves K is Koszul. (2) ⇒ (1): From Proposition 2.5.2 and degeneracy of the spectral sequence, we H K still have the equalities (∗). Thus if βpqj (k) 6= 0 for some p,q,j, then βp+q,j (k) 6= 0. Since K is Koszul, this means that p = j − q, and by Proposition 1.2.3, this means H is strand-Koszul. (2) ⇔ (3): Immediate from Corollaries 2.2.2 and 2.5.3. (1) ⇒ (4): One can begin from any bigraded free resolution 0 1 0 ← k ← H ⊗k W ← H ⊗k W ← · · · of k over H (where the W p’s are bigraded k-spaces) and produce an Eilenberg- Moore resolution F of k over K; see [12, Proposition 20.11]. Indeed, one sets ∞ p p ♮ ♮ W = p=0 Σ W and F = K ⊗k W, and defines the differentialL on F by induction. If H is strand-Koszul, then we may as well begin with the minimal bigraded free resolution of k over H, in which case p p (Σ W )p+i,j 6= 0 implies p + i = j for all p,i,j (see Proposition 1.2.3). This means that the basis of F ♮ is concentrated along the diagonal, and thus F is the minimal semifree resolution of k over K. p (4) ⇒ (1): Let F be the minimal semifree resolution of k over K and let {F }p≥−1 be an Eilenberg-Moore filtration. Since K is Koszul, in the notation of subsection i 2.3 the bigraded vector spaces {V }i≥0 are concentrated along the diagonal. But then the resulting free resolution 0 −1 1 −2 2 0 ← k ← H ⊗k V ← H ⊗k Σ V ← H ⊗k Σ V ← · · · −p p of k over K has the property that (Σ V )ij 6= 0 implies p = j − i for all p,i,j. Hence H is strand-Koszul, by Proposition 1.2.3. LINEAR RESOLUTIONS OVER KOSZUL COMPLEXES 13

Example 2.5.5. One class of algebras for which we can explicitly write down Eilenberg-Moore ﬁltrations is the class of quadratic complete intersections. If R is such an algebra, then H is strand-Koszul; this follows, for example, from a (graded version of a) result of Tate [28], which states that H is a bigraded exterior algebra ′ generated by H1,2; hence the stand totalization H (recall the deﬁnition in section 1.2) is an exterior algebra generated in degree 1, which is well-known to be Koszul. The details are as follows. Assume R is a quadratic complete intersection of embedding dimension n and codimension c; this means that

k[X1,...,Xn] R =∼ (G1,...,Gc) where G1,...,Gc is a regular sequence of degree-2 homogeneous polynomials. Write xi for the image of Xi in R, so that the Koszul complex is K = ΛR(t1,...,tn) with ∂ti = xi for each i. If we write

Gh = λhij XiXj , λhij ∈ k 1≤Xi≤j≤n for each h =1,...,c, then each zh = 1≤i≤j≤n λhij xitj is a cycle in K. In fact, if we write ζh for the homology class of zh, then Tate’s result mentioned above shows that P

H = Λk(ζ1,...,ζc). As always, a minimal graded free resolution F of k over R can be constructed according to Tate’s method; see subsection 2.1. However, since R is a complete intersection, Tate’s theory shows that the construction stops after adjoining divided- power variables in homological and internal degree 2; the result is that

F = K ⊗R ΓR(s1,...,sc), ∂sh = zh, where s1,...,sc are divided-power variables of bidegree (2, 2). Now, for each p ≥ 0, set

p (i1) (ic) p (i1) (ic) F = Ks1 ··· sc and V = ks1 ··· sc . 1 i1+···+i =p i +···X+ic≤p Xc p One can check that {F }p≥0 is an increasing filtration of F by DG K-submodules 1 1 1 which is semifree. In the resulting spectral sequence the differential dp : Ep → Ep−1 can be identified with the mapping −p p −p+1 p−1 H ⊗k Σ V → H ⊗k Σ V defined on a basis element by c −p (i1) (ic) −p+1 (i1) (ij −1) (ic) 1 ⊗ Σ s1 ··· sc 7→ ζj ⊗ Σ s1 ··· sj ··· sc j=1 X (ij −1) where we set s = 0 if ij − 1 < 0. Thus

0 d1 −1 1 d2 −2 2 0 ← H ⊗k V ←− H ⊗k Σ V ←− H ⊗k Σ V ← · · · is the minimal bigraded free resolution of k over H; one can verify this, for example, by using Koszul duality theory (see [23, Chapter 2, §3]). We conclude that the ﬁltration {F p} is an Eilenberg-Moore ﬁltration. 14 JOHN MYERS

We finish this section by noting that even when the Avramov spectral sequence does not degenerate predictably, it is still possible to obtain relations from it between the low-degree Betti numbers of k over R and k over H. These relations are presented in Theorem 2.5.6. The following equalities hold: n βR (k)= + βH (k), 2,2 2 1,1,2 R H β2,j (k)= β1,1,j(k) for j > 2, R H H β3,j (k)= β1,2,j(k)+ nβ1,1,j−1(k) for j > 3, n βR (k)= βH (k)+ nβH (k)+ βH (k)+ βH (k) for j > 4. 4,j 1,3,j 1,2,j−1 2 1,1,j−2 2,2,j Proof. Combining the Avramov spectral sequence with Proposition 2.4.1 produces a spectral sequence n 2 n H ( i ) R Epqj = i=0 Torp (k, k)q−i,j−i ⇒ Torp+q (k, k)j with differentials on theL second page acting on tridegrees as 2 2 2 dpqj : Epqj → Ep−2,q+1,j . 2 The algebra H satisfies the hypotheses of Proposition 1.1.2; thus if Epqj 6= 0, then H there is an i with 0 ≤ i ≤ n such that βp,q−i,j−i(k) 6= 0. Hence q − i ≥ p, so that necessarily q ≥ p. Thus from (2.5.1) we have R ∞ βmj (k)= dimk Epqj p+q=m Xq≥p for all m and j. In particular, we have R ∞ ∞ β2,j (k) = dimk E0,2,j + dimk E1,1,j . 2 However, according to the description of the differentials dpqj given above, we must ∞ 2 ∞ 2 have E0,2,j = E0,2,j and E1,1,j = E1,1,j . Thus n n n n βR (k)= βH (k)+ βH (k) 2,j i 0,2−i,j−i i 1,1−i,j−i i=0 i=0 X X n = βH (k)+ βH (k), 2 0,0,j−2 1,1,j R from which the equations for β2,j (k) follow. Similarly, we have R ∞ ∞ β3,j (k) = dimk E0,3,j + dimk E1,2,j ∞ 2 = dimk E0,3,j + dimk E1,2,j ∞ H H = dimk E0,3,j + β1,2,j (k)+ nβ1,1,j−1(k). But n n n 2 H ( i ) H (3) E0,3,j = Tor0 (k, k)3−i,j−i = Tor0 (k, k)0,j−3, i=0 M LINEAR RESOLUTIONS OVER KOSZUL COMPLEXES 15

∞ R so that E0,3,j = 0 if j > 3. The equations for β3,j (k) follow. Finally, we have R ∞ ∞ ∞ β4,j (k) = dimk E0,4,j + dimk E1,3,j + dimk E2,2,j ∞ 2 2 = dimk E0,4,j + dimk E1,3,j + dimk ker d2,2,j . ∞ 2 2 One checks just as above that E0,4,j = 0 and ker d2,2,j = E2,2,j when j > 4; the R equations for β4,j (k) then follow.

R Remark 2.5.7. The ﬁrst two equations for β2,j (k) reﬁne well-known existing relations; see, for example, [9, Theorem 2.3.2].

3. Classes of Koszul homology algebras which are strand-Koszul In this section we describe several classes of standard graded algebras that possess Koszul homology algebras which are strand-Koszul. Our goal is to systematically work through the proof of Theorem C from the introduction, noting that part (3) of that theorem has already been established in Example 2.5.5. Throughout this section R denotes a standard graded algebra written as a quotient of a polynomial ring Q; see subsection 1.5. 3.1. Small embedding codepth and Golod algebras. We deﬁne the depth of R, denoted depth R, to be the maximal length of a (homogeneous) regular sequence in the augmentation ideal of R. Proposition 3.1.1. Let R be a standard graded algebra of embedding dimension n and let H be its Koszul homology algebra. If n − depth R ≤ 3, then the algebra R is Koszul if and only if H is strand-Koszul. Proof. Avramov proved in [2, Theorem 5.9] that the Koszul complex K is quasi- formal when n − depth R ≤ 3. Now apply Theorem 2.5.4. By using bar constructions, Iyengar [19] gave an alternate construction of the Avramov spectral sequence (see subsection 2.4) and described its ﬁrst page as

1 ⊗p Q Epqj = H ⊗k Tor (k, k) , qj where H is the cokernel of the unit map k → H. We can thus add a series to the inequality in Proposition 2.5.2(2) to produce n R H (1 + st) (3.1.2) Pk (s,t) ≤ PTorQ (k,k)(s,s,t) ≤ Q . 1 − t(PR(s,t) − 1) If the series at the two ends are equal, then R is called a Golod algebra. These rings have been intensively studied in the homology theory of commutative local rings (see [4] for an overview). Proposition 3.1.3. Let R be a standard graded algebra and let H be its Koszul homology algebra. If R is Koszul and Golod, then H is strand-Koszul. Proof. If R is a Golod algebra, then the inequalities in (3.1.2) are equalities, and by Proposition 2.5.2 and Corollary 2.5.3 this means the Koszul complex K is quasi- formal. If R is Koszul as well, we can apply Theorem 2.5.4 to obtain the desired conclusion. 16 JOHN MYERS

In the rest of this section, the technique we use for establishing strand-Koszulness comes from the theory of noncommutative Gröbner bases. Our main reference is [6], but we will collect here the main definitions and results for the reader’s convenience. 3.2. Noncommutative Gröbner bases. Let k be a field and write

T = k hζ1,...,ζni for the polynomial algebra generated by noncommuting variables ζ1,...,ζn of degree 1. We consider the degree-lexicographic ordering on the monomials of T with the variables ordered ζ1 < ··· < ζn.

Let G = {γ1,...,γt} be a collection of elements of T which generate an ideal I. Multiplying through by scalars if necessary, for each i = 1,...,t we can write γi = µi + αi where µi is the leading monomial of γi (with coefficient 1) and αi is a k-linear combination of monomials which are all <µi. A nonzero monomial µ ∈ T is said to be reduced (with respect to G) if it does not contain any of the µi’s as a sub-monomial; we then say G forms a Gröbner basis of I if the set of images in T/I of the reduced monomials form a k-basis for the quotient algebra. These images always linearly span the quotient algebra, so to verify that a set of generators of I is a Gröbner basis we need only check linear independence. The relevance of Gröbner bases is explained by the following fact: Ifa quadratic Gröbner basis of I exists, then the quotient T/I is necessarily Koszul (in commutative algebra such algebras are called G-quadratic). Indeed, this follows from a filtration argument; see, for example, [23, Chapter 4, Theorem 7.1]. 3.3. Short Gorenstein algebras. Suppose R is artinian Gorenstein of socle degree 2 and let H be its Koszul homology algebra. We then have that Hn,n+2 =∼ k and Hn,j = 0 for all j 6= n + 2. But in fact, more is true: Avramov and Golod proved in [5] that the Gorenstein condition implies H is a Poincaréalgebra, which means that for all i, j there are k-linear isomorphisms

Hij → Homk (Hn−i,n+2−j ,Hn,n+2) given by h 7→ λh where λh is (left) multiplication by h. In particular, the Betti table of R over Q looks like 0 1 2 ··· n − 1 n 0 1 −−··· − − 1 − b1 b2 ··· bn−1 − 2 − − − ··· − 1 where bi = bn−i for each i =1, 2,..., ⌊n/2⌋. Theorem 3.3.1. Let R be an artinian Gorenstein standard graded k-algebra of socle degree 2 and embedding dimension n, and let H be the Koszul homology algebra of R. If k does not have characteristic 2, or if n is odd, then H is strand-Koszul. Proof. Suppose first that n is odd, set c = ⌊n/2⌋, and consider the multiplication maps µi : Hi,i+1 ⊗k Hn−i,n−i+1 → Hn,n+2 for i = 1, 2,...,c. As can be seen from the Betti table above, these are the only nonzero multiplication maps besides the ones that involve H0,0. Since Hn,n+2 is 1-dimensional, fixing a nonzero element σ ∈ Hn,n+2 allows us to identify Hn,n+2 LINEAR RESOLUTIONS OVER KOSZUL COMPLEXES 17 with k. Then, by the Poincarécondition on H, for each i = 1, 2,...,c there are ordered bases ζi,1,...,ζi,bi of Hi,i+1 and ηn−i,1,...,ηn−i,bi of Hn−i,n−i+1 such that 0 : j 6= ℓ, (3.3.2) ζi,j ηn−i,ℓ = (σ : j = ℓ, for all 1 ≤ j, ℓ ≤ bi. c We set S = i=1{ζi,1,...,ζi,bi , ηn−i,1,...,ηn−i,bi } and we consider the noncommutative polynomial ring k hSi generated by S. There is an evident surjective k-algebra map S ϕ : k hSi ։ H which sends the elements of S to themselves and which preserves bidegrees (and hence also strand degrees). Therefore, if we set I = ker ϕ and write H′ for the strand totalization of H, then we have an isomorphism k hSi /I =∼ H′ of graded algebras. Thus to prove H is strand-Koszul it will suffice to show that I has a quadratic (with respect to strand degree) Gröbner basis. Let G be the subset of k hSi consisting of the following four types of elements:

(1) all monomials of strand degree 2, except for monomials of the form ζi,j ηn−i,j and ηn−i,j ζi,j for i =1,...,c and j =1,...,bi, i(n−i) (2) all commutators ηn−i,j ζi,j − (−1) ζi,j ηn−i,j for i = 1,...,c and j = 1,...,bi, (3) all elements of the form ζi,j ηn−i,j − ζ1,1ηn−1,1 for i = 2,...,c and j = 1,...,bi, and (4) all elements of the form ζ1,j ηn−1,j − ζ1,1ηn−1,1 for j =2,...,bi. If we use the degree-lexicographic ordering on the monomials in k hSi with

ζ1,1 < ··· < ζ1,b1 < ··· < ζc,1 < ··· < ζc,bc < ηn−c,1 < ···

··· < ηn−c,bc < ··· < ηn−1,1 < ··· < ηn−1,b1 , then the only reduced monomial (with respect to G) of strand degree ≥ 2 is ζ1,1ηn−1,1. Since this monomial is not in the ideal (G), we conclude that G forms a Gr¨obner basis of (G). Using the Betti diagram of R over Q, equation (3.3.2), and the fact H is a quotient of an exterior algebra, one can show that G ⊆ I. Thus we have a surjection k hSi ։ H (G) induced by ϕ. Finally, since we have dim (k hSi /(G))ij = dim Hij for all i, j, we have (G)= I. Hence I has a quadratic Gr¨obner basis. If n is even, the same argument as above goes through as long as k does not have characteristic 2; indeed, the “middle” multiplication map

Hn/2,n/2+1 ⊗ Hn/2,n/2+1 → Hn,n+2 can then be diagonalized.

3.4. Koszul algebras deﬁned by three relations. Suppose R is Koszul and that J (the deﬁning ideal of R; see subsection 1.5) is minimally generated by three elements. In [8] it is shown that H is generated by its linear strand and that the Betti table of R over Q must be one of the following four: 18 JOHN MYERS

0 1 2 3 0 1 2 3 0 1 − − − 0 1 − − − 1 − 3 2 − 1 − 3 3 1

0 1 2 3 0 1 2 3 0 1 − − − 0 1 − − − 1 − 3 − − 1 − 3 1 − 2 − − 3 − 2 − − 2 1 3 − − − 1 If the Betti table of R is one of the two in the top row, then H is obviously strand- Koszul. The bottom-left Betti table corresponds to an H which is an exterior algebra generated by H1,2, and hence H is again strand-Koszul. Thus in the proof of the next theorem we may assume that the Betti table of R is equal to the bottom-right table. Theorem 3.4.1. Let R be a Koszul standard graded algebra. If the deﬁning ideal of R is minimally generated by three elements, then the Koszul homology algebra of R is strand-Koszul. Proof. As we just noted in the paragraph immediately preceding the theorem, the algebra H is generated by its linear strand. Using this fact along with the form of the Betti table of R, we can choose elements ζ1, ζ2, ζ3,η,σ ∈ H such that each ζ1, ζ2, ζ3 has bidegree (1, 2), η has bidegree (2, 3), σ has bidegree (3, 5), and

ζ1η = σ, ζ2η = ζ3η =0.

Furthermore, since the 2-dimensional vector space H2,4 is spanned by the three monomials ζ1ζ2, ζ1ζ3, ζ2ζ3, there must be a,b,c ∈ k, not all zero, such that

aζ1ζ2 + bζ1ζ3 + cζ2ζ3 =0. We have an evident surjective k-algebra map

ϕ : k hζ1, ζ2, ζ3, ηi ։ H. This map preserves bidegrees and strand degrees, so we have a graded k-algebra ′ isomorphism k hζ1, ζ2, ζ3, ηi /I =∼ H where I = ker ϕ. As in the proof of Theorem 3.3.1, we will show H is strand-Koszul by proving I has a quadratic (with respect to strand degree) Gr¨obner basis. Let G be the subset of k hζ1, ζ2, ζ3, ηi consisting of the following four types of elements:

(1) all commutators ηζi − ζiη for 1 ≤ i ≤ 3, (2) all skew commutators ζj ζi + ζiζj for 1 ≤ i < j ≤ 3, 2 2 2 2 (3) all squares ζ1 , ζ2 , ζ3 , η and ζ2η, ζ3η, (4) a linear combination aζ1ζ2 + bζ1ζ3 + cζ2ζ3 where not all a, b, c, ∈ k are zero. If c 6= 0, then with the degree-lexicographic ordering and

ζ1 < ζ2 < ζ3 < η, the reduced monomials of degree ≥ 2 are ζ1η, ζ1ζ2, and ζ1ζ3. Since the images of these monomials in the quotient k hζ1, ζ2, ζ3, ηi /(G) are linearly independent, we conclude that G forms a Gr¨obner basis of (G) and hence the quotient k hζ1, ζ2, ζ3, ηi /(G) is Koszul. LINEAR RESOLUTIONS OVER KOSZUL COMPLEXES 19

If c = 0 and a,b 6= 0, then the reduced monomials of degree ≥ 2 are ζ1η, ζ1ζ2, and ζ2ζ3. These form a Gröbner basis of (G), and the quotient k hζ1, ζ2, ζ3, ηi /(G) is again Koszul. If c = 0 and one of a or b is also 0, then k hζ1, ζ2, ζ3, ηi /(G) is the quotient of a skew-polynomial algebra by an ideal generated by quadratic monomials; such algebras are Koszul (see [23, Chapter 4, Theorem 8.1]). It is not difficult to prove that G ⊆ I, and hence we have a surjection k hζ , ζ , ζ , ηi 1 2 3 ։ H (G) induced by ϕ. Since the dimensions of k hζ1, ζ2, ζ3, ηi /(G) and H are equal in each bidegree, we have that (G)= I. Hence H is strand-Koszul. 3.5. Koszul algebras defined by edge ideals of paths. Finally, we turn our attention to algebras of the form k[X ,...,X ] R = 1 n , n ≥ 3. (X1X2,X2X3,...,Xn−1Xn) Again letting H denote the Koszul homology algebra of R, we shall prove H is strand-Koszul via the same Gröbner basis techniques used above and the main input to our proof will again be the dimensions of the homogeneous components of H. However, this time our R has extra structure that will prove to be useful; namely, the defining ideal of R is generated by monomials and hence R has an Zn-grading which refines its usual Z-grading. This finer grading induces a Zn+1- grading on H which, in turn, refines its usual Z2-grading. Thus our first goal is to set up some notation and terminology to describe these finer gradings.

n 3.5.1. Multidegrees. Elements of Z will be called multidegrees. If v = (v1,...,vn) is a multidegree, we set |v| = v1 + ··· + vn and we deﬁne the support of v to be the set

supp (v)= {i | vi 6=0}.

We call v squarefree if every component vi is either 0 or 1. Write

e1 = (1, 0,..., 0), e2 = (0, 1, 0,..., 0),..., en = (0,..., 0, 1). Then, for 1 ≤ i ≤ n and 1 ≤ r ≤ n − i +1, set n pi,r = ei + ··· + ei+r−1 ∈ Z . Every squarefree multidegree v ∈ Zn (not equal to (0,..., 0)) can be decomposed uniquely into a sum of pi,r’s with maximal support; for example, v = (1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1) ∈ Z12 decomposes as v = p1,3 +p5,2 +p9,4. Such a decomposition will be called a complete decomposition of v. n The polynomial ring Q = k[X1,...,Xn] is Z -graded by assigning the variable Xi the multidegree ei. The deﬁning ideal (X1X2,X2X3,...,Xn−1Xn) of R is homogeneous with respect to this Zn-grading, and hence R inherits a Zn-grading from Q. 20 JOHN MYERS

Write xi for the image of Xi in R and let K be the Koszul complex of R gen- n erated by exterior variables t1,t2,...,tn. Given multidegrees v, w ∈ Z , we deﬁne monomials v v1 vn w w1 wn x = x1 ··· xn ∈ R and t = t1 ··· tn ∈ K. The monomial xvtw ∈ K is then assigned the multigraded bidegree (|w|, v + w) ∈ Zn+1, where |w| is the usual homological degree of the monomial and v + w is called the multigraded internal degree. The diﬀerential on K preserves this latter degree and hence the homology H inherits a Zn+1-grading from K. For each i ∈ Z n and u ∈ Z , we let Ki,u and Hi,u be the k-subspaces of K and H, respectively, which are spanned by all homogeneous elements of multigraded bidegree (i, u); for each j ∈ Z we then have decompositions

Kij = Ki,u and Hij = Hi,u. u∈Zn u∈Zn |Mu|=j |Mu|=j Hence the multigraded bidegrees on K and H reﬁne the usual bidegrees. The strand degree of a homogeneous element of multigraded bidegree (i, u) is |u|− i. We now turn our attention to the result which serves as the basis for our study of H: Proposition 3.5.2 (Boocher, D’Al`ı,Grifo, Monta˜no, Sammartano [7]).

(1) The algebra H is generated by the subspaces H1,2 and H2,3. (2) Let u ∈ Zn be a squarefree multidegree with complete decomposition u = m p=1 pip,rp . Then

P 1 : rp 6≡ 1 (mod 3) for each p, dimk H∗,u = (0 :otherwise. m In the former case, dimk Hi,u =1 exactly when i = p=1 ⌊2rp/3⌋.

Remark 3.5.3. We note that H∗,u = 0 if u is not squarefreeP (see, for example, [22, Corollary 1.40]), so Proposition 3.5.2 describes the dimensions of all nonzero Hi,u.

Now, for all i = 1,...,n − 1 and j = 1,...,n − 2, the monomials xi+1ti and xj+1tj tj+2 are cycles in the Koszul complex K, and hence we may deﬁne the homology classes

ζi,i+1 = cls (xi+1ti) ∈ H1,2 and ηj,j+1,j+2 = cls (xj+1tj tj+2) ∈ H2,3.

The homology classes ζi,i+1 and ηj,j+1,j+2 have multigraded bidegrees (1, pi,2) and (2, pj,j+1,j+2), respectively, and thus they all have strand degree 1.

Proposition 3.5.4. The sets {ζ1,2,...,ζn−1,n} and {η1,2,3,...,ηn−2,n−1,n} form k-bases for the spaces H1,2 and H2,3, respectively.

Proof. That the ﬁrst set is a basis of H1,2 follows from [9, Theorem 2.3.2], for example. To prove the second set is a basis of H2,3, we ﬁrst note that

H2,3 = H2,u. u∈Zn |Mu|=3 LINEAR RESOLUTIONS OVER KOSZUL COMPLEXES 21

We claim, in fact, that n−2

(∗) H2,3 = H2,pj,3 . j=1 M To see this, suppose that H2,u 6= 0 and |u| = 3. By Remark 3.5.3, we have that u must be squarefree. But then Proposition 3.5.2(2) shows that u = pj,3 for some j ∈ {1, 2,...,n − 2}, which establishes the equation (∗). But the same proposition shows that each H2,pj,3 is 1-dimensional, and thus to prove that the set {η1,2,3,...,ηn−2,n−1,n} is a basis of H2,3 it will suﬃce to show that ηj−2,j−1,j is nonzero (since it has multigraded bidegree (2, pj,3)).

To do this, ﬁrst note that K2,pj,3 and K3,pj,3 are 3- and 1-dimensional, respectively, with bases

{xj+1tj tj+2, xj tj+1tj+2, xj+2tj tj+1} and {tj tj+1tj+2}. Then, writing ∂ for the diﬀerential of K, since

∂(tj tj+1tj+2)= xj tj+1tj+2 − xj+1tj tj+2 + xj+2tj tj+1 we deduce that there is no a ∈ k such that ∂(atjtj+1tj+2) = xj+1tj tj+2. Hence xj+1tj tj+2 is not a boundary in K, and so ηj,j+1,j+2 is not zero in H2,pj,3 . Setting T = k hζ1,2,...,ζn−1,n, η1,2,3,...,ηn−2,n−1,ni , there is a natural homomorphism ϕ : T → H of Zn+1-graded algebras which sends the ζi,i+1’s and ηj,j+1,j+2’s to themselves. By Propositions 3.5.2(1) and 3.5.4, it is surjective. Setting I = ker ϕ, we thus have an isomorphism T/I =∼ H′ of graded algebras, and to prove H is strand-Koszul it will suﬃce to show I has a quadratic (with respect to strand degree) Gr¨obner basis. Let G be the subset of T consisting of the following ten types of elements. First, certain commutators and skew-commutators:

(1) ηj,j+1,j+2ηi,i+1,i+2 − ηi,i+1,i+2ηj,j+1,j+2 for all 1 ≤ i < j ≤ n − 2 (if n ≥ 4), (2) ηj,j+1,j+2ζi,i+1 − ζi,i+1ηj,j+1,j+2 for all 1 ≤ i ≤ j ≤ n − 2, (3) ζi,i+1ηj,j+1,j+2 − ηj,j+1,j+2ζi,i+1 for all 1 ≤ j

(5) ηi,i+1,i+2ζi+3,i+4 − ζi,i+1ηi+2,i+3,i+4 for all 1 ≤ i ≤ n − 4 (if n ≥ 5), (6) ζi,i+1ζi+2,i+3 for all 1 ≤ i ≤ n − 3 (if n ≥ 4). Finally, the “overlap” elements:

(7) ζi,i+1ζj,j+1 for each i = j − 1, j with 1 ≤ i ≤ j ≤ n − 1, (8) ζi,i+1ηj,j+1,j+2 for each i = j − 1, j, j +1, j + 2 with 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n − 2, (9) ηj,j+1,j+2ζi,i+1 for each i = j − 1, j, j +1, j + 2 with 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n − 2, and (10) ηi,i+1,i+2ηj,j+1,j+2 for each i = j − 2, j − 1, j with 1 ≤ i ≤ j ≤ n − 2. Some elementary (if slightly tedious) computations show (G) ⊆ I. We then consider the degree-lexicographic ordering on the monomials of T with respect to the ordering

ζ1,2 < η1,2,3 < ζ2,3 < η2,3,4 < ··· < ζn−2,n−1 < ηn−2,n−1,n < ζn−1,n. 22 JOHN MYERS

The proof of the next lemma then easily follows. Lemma 3.5.5. Suppose 1 ≤ i, j ≤ n − 1 and 1 ≤ ℓ,m ≤ n − 2. With respect to G,

(1) the monomial ζi,i+1ζj,j+1 is reduced if and only if i +1 ≤ j − 2; (2) the monomial ζi,i+1ηℓ,ℓ+1,ℓ+2 is reduced if and only if i +1 ≤ ℓ − 1; (3) the monomial ηℓ,ℓ+1,ℓ+2ζi,i+1 is reduced if and only if ℓ +2 ≤ i − 2; (4) the monomial ηℓ,ℓ+1,ℓ+2ηm,m+1,m+2 is reduced if and only if ℓ +2 ≤ m − 1. Now, suppose 1 ≤ i ≤ n, 1 ≤ r ≤ n − i + 1, and r 6≡ 1 (mod 3). Deﬁne

ζi,i+1 : r =2, µi,r = ζi,i+1ηi+2,i+3,i+4ηi+5,i+6,i+7 ··· ηi+r−3,i+r−2,i+r−1 : r ≡ 2 (mod 3), r> 2, ηi,i+1,i+2ηi+3,i+4,i+5 ··· ηi+r−3,i+r−2,i+r−1 : r ≡ 0 (mod 3).

By Lemma 3.5.5 each µi,r is reduced, and the following proposition shows that all reduced monomials in T are products of these monomials. Proposition 3.5.6. Let µ ∈ T be a reduced monomial (not equal to 1) of multigraded internal degree u ∈ Zn. (1) The multidegree u is squarefree. (2) If the complete decomposition of u is

u = pi1,r1 + ··· + pim,rm (i1 < ···

then rp 6≡ 1 (mod 3) for each p =1,...,m and µ = µi1,r1 ··· µim,rm .

In particular, there is exactly one reduced monomial in T∗,u. Proof. We shall prove both statements simultaneously by inducing on the internal degree |u|. Note that each ζi,i+1 has internal degree 2 while each ηj,j+1,j+2 has internal degree 3. Hence we must have |u| ≥ 2, and if equality holds, then necessarily µ = ζi,i+1 = µi,2 for some i ∈{1, 2,...,n−1}. Similarly, if |u| = 3, then necessarily µ = ηi,i+1,i+2 = µi,3 for some i ∈{1, 2,...,n − 2}. Now assume |u| > 3 and that statements (1) and (2) both hold for reduced monomials of internal degree < |u|. There are two cases to consider: either

(a) µ = νζi,i+1 or (b) µ = νηi,i+1,i+2 where ν is a submonomial of µ (not equal to 1). In both cases since µ is reduced so is ν, and our inductive hypotheses then imply that the multidegree v of ν is squarefree and that if v has complete decomposition

v = pj1,s1 + ··· + pjℓ,sℓ (j1 < ··· < jℓ), then sq 6≡ 1 (mod 3) for each q = 1,...,ℓ and ν = µj1,s1 ··· µjℓ,sℓ . In case (a), since µ is reduced Lemma 3.5.5 implies i ≥ jℓ + sℓ + 1, which in turn implies that u (which is equal to v + ei + ei+1) is squarefree with complete decomposition

u = pj1,s1 + ··· + pjℓ,sℓ + pi,2.

Since also µ = µj1,s1 ··· µjℓ,sℓ µi,2, we see that both statements (1) and (2) hold for u and µ. In case (b), Lemma 3.5.5 implies that i ≥ jℓ + sℓ. If i = jℓ + sℓ, then u is squarefree with complete decomposition

u = pj1,s1 + ··· + pjℓ,sℓ+3 LINEAR RESOLUTIONS OVER KOSZUL COMPLEXES 23

and µ = µj1,s1 ··· µjℓ,sℓ+3. If i > jℓ + sℓ, then u is again squarefree with complete decomposition

u = pj1,s1 + ··· + pjℓ,sℓ + pi,3 and µ = µ = µj1,s1 ··· µjℓ,sℓ µi,3. In either case, statements (1) and (2) both hold for u and µ. We now have enough to prove the main result. Theorem 3.5.7. Let n be an integer ≥ 3 and set k[X ,X ,...,X ] R = 1 2 n . (X1X2,X2X3,...,Xn−1Xn) The Koszul homology algebra H of R is strand-Koszul. Proof. We will prove that the natural surjection ϕ : T ։ H induces an isomorphism T/(G) =∼ H, having already shown (G) ⊆ I where I = ker ϕ. For a ﬁxed multidegree u ∈ Zn, the map ϕ induces a surjection

(T/(G))∗,u ։ H∗,u of k-spaces. To show that the surjection is an isomorphism we will prove that

(∗) dimk (T/(G))∗,u = dimk H∗,u. m Let u have complete decomposition p=1 pip,rp . We recall that the images of the reduced monomials linearly span the quotient P algebra T/(G). Thus if (T/(G))∗,u 6= 0, then by Proposition 3.5.6 we have that dimk (T/(G))∗,u = 1, the multidegree u is squarefree, and rp 6≡ 1 (mod 3) for each p =1,...,m. By Proposition 3.5.2, we then have dimk H∗,u = 1. This establishes (∗). Now, the isomorphism T/(G) =∼ H and Propositions 3.5.2 and 3.5.6 show that the images of the reduced monomials in T/(G) form a basis, and hence the generators (1)-(10) of (G) form a Gr¨obner basis. Each of these generators has strand degree 2, and hence H is strand-Koszul. Remark 3.5.8. In addition to algebras cut out by edge ideals of paths, the paper [7] also contains results on algebras whose deﬁning ideals are edge ideals of cycles. If R denotes one of these latter types of algebras, then its Koszul homology algebra H has a more complicated structure than what we’ve just seen above; indeed, Theorem 3.15 of [7] states that H is no longer generated in its linear strand if n ≡ 1 (mod 3) and n> 4 (where n is the number of vertices in the cycle). However, if n 6≡ 1 (mod 3), then just as we saw above, H is generated by the subspaces H1,2 and H2,3; so, could it be that H is strand-Koszul? Not always. Indeed, if n = 9, then running the following Macaulay2 [16] code H shows that β2,6,9(k) = 1, so that H is not even strand quadratic: needsPackage "DGAlgebras" Q = QQ[x1..x9] I = ideal(x1*x2,x2*x3,x3*x4,x4*x5,x5*x6,x6*x7,x7*x8,x8*x9,x9*x1) R = Q/I H = HH koszulComplexDGA R k = coker vars H F = res(k,LengthLimit=>2) peek betti F 24 JOHN MYERS

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Department of Mathematics, SUNY Oswego, Oswego, New York, USA Email address: [email protected]