Canonical Resolutions over Koszul Algebras
Eleonore Faber, Martina Juhnke-Kubitzke, Haydee Lindo, Claudia Miller, Rebecca R.G., Alexandra Seceleanu
aa r X i v : . [ m a t h . A C ] N ov CANONICAL RESOLUTIONS OVER KOSZUL ALGEBRAS
ELEONORE FABER, MARTINA JUHNKE-KUBITZKE, HAYDEE LINDO,CLAUDIA MILLER, REBECCA R.G., AND ALEXANDRA SECELEANU
Abstract.
We generalize Buchsbaum and Eisenbud’s resolutions forthe powers of the maximal ideal of a polynomial ring to resolve powersof the homogeneous maximal ideal over graded Koszul algebras.
Contents
1. Introduction 12. Koszul algebras and the Priddy resolution 33. Resolutions via the enveloping algebra 44. The case of Koszul algebras 65. Minimal resolutions for powers of the maximal ideal 86. Examples 13References 161.
Introduction
Koszul algebras show up naturally and abundantly in algebra and topol-ogy. They were first introduced by Priddy in 1970 as algebras for whichthe bar resolution, which is normally far from minimal, admits a reductionto a comparatively small subcomplex; see [Pri70]. Priddy’s work explainedcontemporaneous ideas on restricted Lie algebras in the work of May and, sep-arately, that of Bousfield, Curtis, Kan, Quillen, Rector, and Schlesinger; see[May66, BCK + A has a dual algebra A ! which is also a Koszul algebra (seeSection 2). The prototypical example of such a Koszul pair is a polynomial Date : November 26, 2020.2020
Mathematics Subject Classification.
Primary: 16E05, Secondary: 13D02.
Key words and phrases.
Koszul algebra, minimal free resolution, Betti numbers. algebra S over a field, together with the corresponding exterior algebra Λ .The associated theory of Koszul duality is a generalization of the duality un-derlying the Bernstein–Gelfand–Gelfand correspondence [BGG78] describingcoherent sheaves on projective space in terms of modules over the exterioralgebra. This exemplifies the philosophy that facts relating the symmetricand exterior algebras often have Koszul duality counterparts.In this paper we extend Priddy’s methods of constructing free resolutionsover Koszul algebras and generalize Buchsbaum and Eisenbud’s resolutionsin [BE75] to resolve powers of the homogeneous maximal ideal over Koszulalgebras. In particular, in [Pri70] Priddy exploits a natural differential on A ⊗ k A ! to give an explicit construction for the linear minimal graded freeresolution of the residue field of a graded Koszul algebra; see Definition 2.5.In this paper, we extend this construction to a family of acyclic complexesthat yields highly structured resolutions of the powers of the homogeneousmaximal ideal over Koszul algebras; see Definition 4.1. Since these com-plexes are typically not minimal, we also seek to determine their minimalcounterparts. To achieve this, we take inspiration from results that describestructured resolutions over a polynomial ring S constructed starting fromthe Koszul complex (exterior algebra) Λ and its generalizations; see [BE75].We provide analogs of these results using any pair of Koszul dual algebras, A and A ! , instead of S and Λ .Our main result generalizes the canonical resolutions for the powers ofthe homogeneous maximal ideal, constructed over a polynomial ring S byBuchsbaum and Eisenbud in [BE75], to obtain minimal free resolutions forpowers of the homogeneous maximal ideal of a graded Koszul algebra. Incontrast to the situation over S , these are in general infinite resolutions. Thisallows us to obtain an explicit formula for the graded Betti numbers definedby β i,j ( m a ) = dim k Tor i ( m a , k ) j . The following is a combination of Theorem5.3 and Corollary 5.5. Theorem. If A is a graded Koszul algebra with homogeneous maximal ideal m , the complexes L Aa : · · · → L An,a ∂ ′ n −→ L An − ,a ∂ ′ n − −−−→ . . . ∂ ′ −→ L A ,a ε a −→ m a → , defined in equation (5.3) with the augmentation map ε a defined in equation (5.4) are minimal free resolutions of the powers m a with a ≥ .The nonzero graded Betti numbers of the powers of m are given by β An,n + a ( m a ) = a X i =1 ( − i +1 dim k ( A ! n + i ) dim k ( A a − i ) . In particular, the minimal graded resolution of m a is a -linear. The paper is structured as follows. In Section 2, we provide backgroundon Koszul algebras and the Priddy complex. In Section 3, we explain howto obtain a free solution of a module M from a resolution of the ring over ANONICAL RESOLUTIONS OVER KOSZUL ALGEBRAS 3 its enveloping algebra. In Section 4, we rewrite this resolution as the total-ization of a double complex, in the case that the module is a power of thehomogeneous maximal ideal and the ring is a Koszul algebra. In Section 5,we give the minimal resolution and Betti numbers for the powers of the ho-mogeneous maximal ideal over a Koszul algebra. In Section 6 we apply ourconstruction to several specific Koszul algebras A to obtain explicit formulasfor the Betti numbers of m a .2. Koszul algebras and the Priddy resolution
Throughout k is a field and A is a not necessarily commutative graded k -algebra having finite-dimensional graded components with A i = 0 for i < and A = k . Definition 2.1 ([Pri70, Chapter 2]) . We say that A is Koszul if k admits alinear graded free resolution over A , i.e., a graded free resolution P • in which P i is generated in degree i .Classes of graded Koszul algebras arise from: quadratic complete intersec-tions [Tat57], quotients of a polynomial ring by quadratic monomial ideals[Frö99], quotients of a polynomial ring by homogeneous ideals which havea quadratic Gröbner basis, and from Koszul filtrations [CTV01]; see, forexample, the survey paper by Conca [Con14].We now focus on quadratic algebras in order to define Koszul duality. Definition 2.2 ([PP05, Chapter 1, Section 2]) . Let A be a graded k -algebra.We say that A is quadratic if A = T ( V ) /Q , where V is a k -vector space, T ( V ) is the tensor algebra of V , and Q is a quadratic ideal of T ( V ) .If A is a quadratic algebra, its quadratic dual algebra is defined by A ! = T ( V ∗ ) Q ⊥ where V ∗ = Hom k ( V, k ) and Q ⊥ is the quadratic ideal generated by theorthogonal complement to Q in T ( V ∗ ) = V ∗ ⊗ k V ∗ with respect to thenatural pairing between V ⊗ V and V ∗ ⊗ V ∗ given by h v ⊗ v , v ∗ ⊗ v ∗ i = h v , v ∗ ih v , v ∗ i . Choosing dual bases x , . . . , x d and x ∗ , . . . , x ∗ d for V and V ∗ respectivelyyields that T ( V ) = k h x , . . . , x d i and T ( V ∗ ) = k h x ∗ , . . . , x ∗ d i are polynomialrings in noncommuting variables. This allows one to compute Q ⊥ given aquadratic ideal Q ⊆ T ( V ) using linear algebra, as described for example in[MP15, Section 8].Graded Koszul algebras are quadratic (see, for example, [PP05, Chapter2, Definition 1]) and the duality of quadratic algebras restricts well to theclass of Koszul algebras since A and A ! are Koszul simultaneously [PP05,Chapter 2, Corollary 3.2 ]. FABER, JUHNKE-KUBITZKE, LINDO, MILLER, R.G., AND SECELEANU
Example 2.3.
The main example of Koszul dual algebras is given by thesymmetric algebra on a vector space VS = k [ x , . . . , x d ] = k h x , . . . , x d i ( x i x j − x j x i , ≤ i < j ≤ d ) and the exterior algebra on V ∗ S ! = Λ = k h x ∗ , . . . , x ∗ d i (( x ∗ i ) , x ∗ i x ∗ j + x ∗ j x ∗ i , ≤ i ≤ j ≤ d ) . Example 2.4.
For the following commutative Koszul algebra A = k [ x, y, z ]( x , xy, y ) = k h x, y, z i ( x , xy, y , xz − zx, xy − yx, yz − zy ) , the dual algebra is given by A ! = k h x ∗ , y ∗ , z ∗ i (( z ∗ ) , x ∗ z ∗ + z ∗ x ∗ , y ∗ z ∗ + z ∗ y ∗ ) . This pair of algebras are further discussed in Example 6.1.
Definition 2.5.
The
Priddy complex [Pri70] of a quadratic algebra A is thecomplex P A • whose i -th term is given by P Ai = A ⊗ k ( A ! ) ∗ i , and the differential is defined by right multiplication by P di =0 x i ⊗ x ∗ i , wheremultiplication by x ∗ i ∈ A ! on ( A ! ) ∗ is defined as the dual of multiplication by x ∗ i on A ! . The importance of the Priddy complex lies in the fact that P A • is acyclicif and only if A is Koszul; see [PP05, Chapter 2, Corollary 3.2]. Moreover,when A is Koszul the Priddy complex, also called the generalized Koszulresolution, is a minimal free resolution of the residue field k of A . Thiswill be the base case in the proof that our construction in Section 4 is aresolution. 3. Resolutions via the enveloping algebra
In this section, we review how one obtains a free resolution of any A -module M from a resolution of A over its enveloping algebra. In general, oneobtains a resolution that is far from minimal. We remedy this in Sections 4and 5 over Koszul algebras A for the modules m a (and hence A/ m a ). Given a k -algebra A , its enveloping algebra is given by A e = A ⊗ k A op .A left A e -module structure is equivalent to an A - A -bimodule structure via ( a ⊗ b ) · m = a · m · b We consider A as an A e -module via the multiplication map A e = A ⊗ k A op ε −→ A. ANONICAL RESOLUTIONS OVER KOSZUL ALGEBRAS 5
Consider a graded free resolution of A over A e and note that any free left A e -module F can be rewritten as F = A e ⊗ k V = A ⊗ k A op ⊗ k V ∼ = A ⊗ k V ⊗ k A for some vector space V , where the rightmost expression is thought of as an A - A -bimodule via the outside two factors. Thus the resolution will be of theform(3.1) · · · → A ⊗ k V ⊗ k A → A ⊗ k V ⊗ k A → A ⊗ k A ε −→ A → , where the augmentation ε from A ⊗ k A to A is given by multiplication acrossthe tensor.We observe that A ⊗ k k ⊗ k A ∼ = A ⊗ k A . Thus setting V = k , we maywrite the resolution as a quasi-isomorphism of A e -modules A ⊗ k V. ⊗ k A ≃ −→ A Next we show how to construct an A -free resolution for arbitrary A -modules M using (3.1). This is well known; we include it because theconstruction is the basis of our next step in Section 4. Proposition 3.2. If M is a graded A -module and A ⊗ k V. ⊗ k A ≃ −→ A is afree resolution of A over A e , then A ⊗ k V. ⊗ k M ≃ −→ M is a graded A -freeresolution of M .Proof. Note that the complex A ⊗ k V. ⊗ k A and the trivial complex A bothconsist of free A -modules (although the latter is not free as an A e -module), sothe quasi-isomorphism A ⊗ k V. ⊗ k A ≃ −→ A is actually a homotopy equivalenceover A and hence remains so after tensoring over A with arbitrary A -modules.To see this, note that the augmented complex of free A -modules (3.1) iscontractible, that is, homotopy equivalent to 0 (equivalently, it is split exactover A ). But this complex is the mapping cone of the chain map A ⊗ k V. ⊗ k A → A .Therefore, upon tensoring (3.1) on the right over A with a left A -module M , one obtains a quasi-isomorphism of left A -modules ( A ⊗ k V. ⊗ k A ) ⊗ A M ≃ −→ A ⊗ A MA ⊗ k V. ⊗ k M ≃ −→ M Viewing V. ⊗ k M as a (rather large) k -vector space, one sees that the complexon the left consists of free A -modules, giving a free A -resolution of M . (cid:3) Remark 3.3.
Suppose A is local (or standard graded) k -algebra with (ho-mogeneous) maximal ideal m . The resolutions obtained in Proposition 3.2are in general not minimal (respectively, minimal graded) resolutions evenwhen one starts with a minimal (respectively, minimal graded) resolution of A over A e . One can always take the bar resolution of A over its enveloping algebra, but that isusually far from minimal. FABER, JUHNKE-KUBITZKE, LINDO, MILLER, R.G., AND SECELEANU
This is because the modules in the resolution A ⊗ k V. ⊗ k M are consideredas (free) A -modules via the first factor and elements m of m M are part ofthe k -basis of M . More explicitly, given k -bases B V of V and B M of M , theelements ⊗ v ⊗ m with v ∈ B V and m ∈ B M form a basis for A ⊗ k V. ⊗ k M .For example, for the explicit resolution X Aa given in Corollary 4.4, although ∂ ′ is minimal, ∂ ′′ is clearly not.4. The case of Koszul algebras
In this section, under the further assumption that A is a Koszul k -algebra,we write the A -free resolution of A/ m a obtained in the previous section asthe totalization of a certain double complex.First we recall the minimal graded resolution of A over A e following thepresentation in [VdB94, Section 3]. It is a symmetrization of the resolutionof k over A found by Priddy in [Pri70], which is presented in Definition 2.5. Definition 4.1.
Let A be a Koszul k -algebra with dual Koszul algebra A ! ,where multiplication by x ∗ i ∈ A ! on ( A ! ) ∗ is defined as the dual of multipli-cation by x ∗ i on A ! .Define free A -modules F n = A ⊗ k ( A ! ) ∗ n ⊗ k A and differential maps ∂ n = ( ∂ ′ ) n + ( − n ( ∂ ′′ ) n where ∂ ′ = right multiplication by d X i =0 x i ⊗ x ∗ i ⊗ (which will form our vertical maps) and ∂ ′′ = left multiplication by d X i =0 ⊗ x ∗ i ⊗ x i (which will form our horizontal maps) . Then the complex(4.1) F A : · · · → F n ∂ n −→ F n − → · · · F ε −→ A → augmented by the multiplication map ε from F = A ⊗ k k ⊗ k A ∼ = A ⊗ k A to A is the minimal graded free resolution of A over A e [VdB94, Proposition3.1]. Remark 4.2.
Here is the explicit connection with Priddy’s resolution: ten-soring (4.1) on the right over A with k gives Priddy’s minimal resolutionof k as a left A -module in Definition 2.5, also called the generalized Koszulresolution. Tensoring (4.1) on the left gives the minimal resolution of k as aright A -module. There is a misprint in [VdB94] with regards to this map.
ANONICAL RESOLUTIONS OVER KOSZUL ALGEBRAS 7 ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ a ⊗ k A ∂ ′ (cid:15) (cid:15) ( − a ∂ ′′ / / A ⊗ k ( A ! ) ∗ a − ⊗ k A ∂ ′ (cid:15) (cid:15) ( − a − ∂ ′′ / / A ⊗ k ( A ! ) ∗ a − ⊗ k A ∂ ′ (cid:15) (cid:15) ( − a − ∂ ′′ / / · · · A ⊗ k ( A ! ) ∗ a − ⊗ k A (cid:15) (cid:15) ( − a − ∂ ′′ / / A ⊗ k ( A ! ) ∗ a − ⊗ k A (cid:15) (cid:15) ( − a − ∂ ′′ / / A ⊗ k ( A ! ) ∗ a − ⊗ k A (cid:15) (cid:15) ( − a − ∂ ′′ / / · · · ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ ⊗ k A ∂ ′′ / / ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ ⊗ k A − ∂ ′′ / / ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ ⊗ k A A ⊗ k ( A ! ) ∗ ⊗ k A − ∂ ′′ / / ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ ⊗ k A A ⊗ k ( A ! ) ∗ ⊗ k A Figure 1.
The minimal resolution of a Koszul algebra A over A e . Considering the graded strands of (4.1), one can write this complex asa totalization of an anticommutative double complex, which we also call F A ,of free A -modules given by the free A -modules(4.2) F ij = A ⊗ k ( A ! ) ∗ i ⊗ k A j where we are using the first tensor factor as “coefficients” and the maps ∂ ′ and ± ∂ ′′ of Definition 4.1 become the vertical and horizontal maps, respectivelyin the diagram (4). Note that i is the homological degree in the complex F A .Next we apply the discussion from Section 3 to the A -module A/ m a andarrange its resolution into a double complex similarly to the one above. Corollary 4.4.
Totalization of the truncation of the bicomplex (4.2) obtainedby removing the columns with index j ≥ a ≥ gives a graded A -free resolution (4.3) X Aa = A ⊗ k ( A ! ) ∗ . ⊗ k A ≤ a − ≃ −→ A/ m a . Proof.
Applying Proposition 3.2 by tensoring the resolution of A over A e onthe right over A with A/ m a gives a graded A -free resolution X Aa = A ⊗ k ( A ! ) ∗ . ⊗ k A/ m a ≃ −→ A/ m a Since A/ m a = A ≤ a − the resolution becomes X Aa = A ⊗ k ( A ! ) ∗ . ⊗ k A ≤ a − ≃ −→ A/ m a FABER, JUHNKE-KUBITZKE, LINDO, MILLER, R.G., AND SECELEANU
Viewing graded strands, one can write this as a totalization of an anticom-mutative double complex of free A -modules given by the terms F ij = A ⊗ k ( A ! ) ∗ i ⊗ k A j with i ≥ , ≤ j ≤ a − of (4.2) with the differentials inheritedfrom those described in Definition 4.1. (cid:3) We display the diagram for the A -free graded resolution X Aa of A/ m a obtained in Corollary 4.4 below.... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ a ⊗ k A ∂ ′ (cid:15) (cid:15) ( − a ∂ ′′ / / A ⊗ k ( A ! ) ∗ a − ⊗ k A ∂ ′ (cid:15) (cid:15) ( − a − ∂ ′′ / / · · · ∂ ′′ / / A ⊗ k ( A ! ) ∗ ⊗ k A a − ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ a − ⊗ k A (cid:15) (cid:15) ( − a − ∂ ′′ / / A ⊗ k ( A ! ) ∗ a − ⊗ k A (cid:15) (cid:15) ( − a − ∂ ′′ / / · · · − ∂ ′′ / / A ⊗ k ( A ! ) ∗ ⊗ k A a − ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ ⊗ k A − ∂ ′′ / / ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ ⊗ k A A ⊗ k ( A ! ) ∗ ⊗ k A Figure 2.
The the A -free graded resolution X Aa of A/ m a . Remark 4.5.
The resolution X Aa is minimal for a = 1 in which case X Aa recovers the Priddy complex without its first term. However for a ≥ thisresolution is typically non minimal as the rows are split acyclic; see 5.1. Thegoal of Section 5 is to produce a minimal free resolution for A/ m a using X Aa .5. Minimal resolutions for powers of the maximal ideal
We introduce complexes L Aa inspired by work of Buchsbaum and Eisenbud[BE75]. These will turn out to be the minimal resolutions for the powers ofthe homogeneous maximal ideal of a graded Koszul algebra. We define free A -modules analogous to the Schur modules used byBuchsbaum and Eisenbud in their resolutions (the case where A is a polyno-mial ring) in [BE75]. First note that the rows of the double complex (4.2)except the bottom one are exact; in fact, they can be viewed as the result ofapplying the exact base change A ⊗ k − to the strands of the dual Priddy com-plex, all of which are exact except the one whose homology is k (in the casewhere A is a polynomial ring, it is applied to the strands of the tautologicalKoszul complex; see [MR18, Section 1.4]). These complexes are contractible,as they consist of free A -modules, and so all kernels, images, and cokernelsof the differentials are free as well. ANONICAL RESOLUTIONS OVER KOSZUL ALGEBRAS 9
Define for a > the following free A -modules L An,a = im (cid:18) A ⊗ k A ! ∗ n +1 ⊗ k A a − − n +1 ∂ ′′ −−−−−−−→ A ⊗ k A ! ∗ n ⊗ k A a (cid:19) (5.1) = ker (cid:18) A ⊗ k A ! ∗ n ⊗ k A a ( − n ∂ ′′ −−−−−→ A ⊗ k A ! ∗ n − ⊗ k A a +1 (cid:19) . (5.2)The vertical differentials ∂ ′ in the diagram 4 induce maps on these modules,which we again denote by ∂ ′ , to yield a complex(5.3) L Aa : · · · → L An,a ∂ ′ n −→ L An − ,a ∂ ′ n − −−−→ . . . ∂ ′ −→ L A ,a . This complex is minimal in the sense that ∂ ′ ( L An,a ) ⊆ m L An − ,a for all n sincethe same property holds for the columns of Diagram 4 viewed as complexeswith differential ∂ ′ .... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ n +1 (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ n + a ⊗ k A ∂ ′ (cid:15) (cid:15) / / A ⊗ k ( A ! ) ∗ n + a − ⊗ k A ∂ ′ (cid:15) (cid:15) / / · · · / / A ⊗ k ( A ! ) ∗ n +1 ⊗ k A a − ∂ ′ (cid:15) (cid:15) / / L An,a∂ ′ n (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ a ⊗ k A ∂ ′ (cid:15) (cid:15) / / A ⊗ k ( A ! ) ∗ a − ⊗ k A ∂ ′ (cid:15) (cid:15) / / · · · / / A ⊗ k ( A ! ) ∗ ⊗ k A a − ∂ ′ (cid:15) (cid:15) / / L A ,a A ⊗ k ( A ! ) ∗ a − ⊗ k A (cid:15) (cid:15) / / A ⊗ k ( A ! ) ∗ a − ⊗ k A (cid:15) (cid:15) / / · · · / / A ⊗ k ( A ! ) ∗ ⊗ k A a − ... ∂ ′ (cid:15) (cid:15) ... ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ ⊗ k A / / ∂ ′ (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ ⊗ k A A ⊗ k ( A ! ) ∗ ⊗ k A Figure 3.
The construction of the complex L Aa . Lemma 5.2.
The complex L Aa can be augmented by the evaluation map (5.4) ε a : L A ,a = A ⊗ k A ! ∗ ⊗ k A a → m a which is the restriction of the multiplication map ε : A ⊗ k A ! ∗ ⊗ k A → A sending r ⊗ v ⊗ s rvs. Proof.
As stated in Definition 4.1, ε is an augmentation map F A • → A . Weverify explicitly that ε satisfies the required property ε ◦ ( ∂ ′ − ∂ ′′ ) = 0 below: ε ◦ ( ∂ ′ − ∂ ′′ )( r ⊗ v ⊗ s ) = ε ( rv ⊗ ⊗ s − r ⊗ ⊗ vs ) = rvs − rvs = 0 . Since ∂ ′′ | L A ,a = 0 it follows from the computation above that ε ◦ ∂ ′ ( L A ,a ) = 0 ,hence the complex L Aa can be augmented to · · · → L An,a ∂ ′ n −→ L An − ,a ∂ ′ n − −−−→ . . . ∂ ′ −→ L A ,a ε a −→ m a → . (cid:3) The following is the main result of our paper. The case when A is anexterior algebra has appeared previously in [EFS03, Corollary 5.3]. Theproof therein uses the self-injectivity of the exterior algebra in a crucialmanner, and therefore does not seem to extend to all Koszul algebras. Theorem 5.3. If A is a Koszul algebra, the complexes L Aa defined in equa-tion (5.3) with the augmentation map ε a defined in (5.4) are minimal freeresolutions for the powers m a of the maximal ideal with a ≥ .Proof. The complexes L Aa are minimal by the discussion in 5.1. The proofof the remaining claims is by induction on a ≥ .The definition of L n, in (5.1) shows that there are isomorphisms L An, ∼ = A ⊗ k A ! ∗ n +1 ⊗ k k ∼ = A ⊗ k A ! ∗ n +1 , since the map ∂ ′′ is injective on F An +1 , for n ≥ , the leftmost column ofthe double complex in Diagram 4 (this column considered by itself is in fact X A ). Therefore there is an isomorphism of complexes L A ∼ = ( X A ) ≥ [ − ,where ( X A ) ≥ denotes the truncation of the complex X A by removing thehomological degree 0 component. Since X A = P A • is just the Priddy complex(upon noting that ∂ ′ : ( X A ) → ( X A ) agrees with ε under the identification ( X A ) = A ⊗ k A ! ∗ ⊗ k A ∼ = A ), we see that ( X A ) ≥ ε −→ m is a resolution of m by 2.6 and the base case that L A ε −→ m is a minimal resolution of m follows.For arbitrary a ≥ , (5.1) gives a short exact sequence of complexes → L Aa − → P A • ⊗ k A a − → L Aa [ − → , where P An ⊗ k A a − = A ⊗ k A ! ∗ n ⊗ k A a − is the ( a − -st column of the doublecomplex (4). The notation signifies that this column can be viewed as thePriddy complex P A • tensored with A a − . From the long exact sequence inhomology induced by the short exact sequence of complexes displayed abovewe deduce H i ( L Aa ) = ( i ≥ (cid:0) H ( L Aa − ) → H ( P A • ⊗ k A a − ) (cid:1) i = 0 . It remains to show that H ( L Aa ) = m a . Indeed, the induced map in homology H ( L Aa − ) → H ( P A • ⊗ k A a − ) ∼ = H ( P A • ) ⊗ k A a − ANONICAL RESOLUTIONS OVER KOSZUL ALGEBRAS 11 can be recovered as the bottom map in the following commutative diagram L A ,a − ε a − (cid:15) (cid:15) A ⊗ k ( A ! ) ∗ ⊗ k A a − ε ⊗ id Aa − (cid:15) (cid:15) m a − / / k ⊗ k A a − ∼ = m a − / m a . Commutativity of the diagram yields that for r ⊗ s ⊗ v ∈ L A ,a the inducedmap in homology is given by ε a ( r ⊗ s ⊗ v ) = rsv rsv, where r is the coset of r in k = A/ m . Thus we obtain the desired identifica-tion ker (cid:0) H ( L Aa − ) → H ( P A • ⊗ k A a − ) (cid:1) = ε a (Span { r ⊗ s ⊗ v | r ∈ m , s ∈ k, v ∈ A a − } ) ∼ = m a . (cid:3) Remark 5.4.
Recall that rows of the double complex (4.2) except the bot-tom one are exact; in fact, they can be viewed as the result of applying abase change to the strands of the dual Priddy complex; see 5.1. These com-plexes are contractible, as they consist of free R -modules. Hence for n ≥ the ( n + a ) -th row of (4), counting from the bottom (as the 0th row), isquasi-isomorphic to L An,a and the lower rows (numbered 1 through a ) aresplit exact. The acyclic assembly lemma [Wei94, Lemma 2.7.3] yields quasi-isomorphisms ( X Aa ) ≥ [ − ≃ −→ L Aa for a ≥ . As shown in Corollary 4.4 thereare quasi-isomorphisms X Aa ≃ −→ A/ m a , hence also ( X Aa ) ≥ ≃ −→ m a . Transitivityyields a quasi-isomorphism L Aa ≃ −→ m a .This approach gives an alternate proof for our main result, but it onlydetermines the augmentation map up to an isomorphism on its target. Weprefer the more explicit approach of Theorem 5.3, which specifies the aug-mentation map ε a .The following corollary of Theorem 5.3 gives an explicit formula for theBetti numbers of powers of the maximal ideal of a Koszul algebra. That m a has an a -linear minimal free resolution also follows from [Ş01, Theorem 3.2].Once the linearity of this resolution has been established, [PP05, Chapter 2,Corollary 3.2 (iiiM)] gives an alternate interpretation for the Betti numbersof m a in terms of the graded components of a quadratic dual module forthe A -module m a . However this description seems less amenable to explicitcomputations than our methods. Corollary 5.5. If ( A, m ) is a Koszul algebra, the nonzero graded Betti num-bers of the powers of m are given by β An,n + a ( m a ) = a X i =1 ( − i +1 dim k ( A ! n + i ) dim k ( A a − i ) . In particular, the minimal graded resolution of m a is a -linear.Consequently, the nonzero Betti numbers of A/ m a are given by β An,j ( A/ m a ) = (P ai =1 ( − i +1 dim k ( A ! n + i − ) dim k ( A a − i ) n > , j = n + a n = j = 0 . Proof.
The fact that minimal free resolution of m a is a -linear follows fromthe Theorem 5.3 and the description of the differential ∂ ′ of the complex(5.3) in view of the fact that there is a splitting of the map ∂ ′′ to each L n,a identifying a basis of it with part of a basis of the last column of X Aa . Considerthe rows of the truncated complex X Aa when augmented to the relevant L n,a as follows. −→ A ⊗ k ( A ! ) ∗ n + a ⊗ k A −→ · · · −→ A ⊗ k ( A ! ) ∗ n +1 ⊗ k A a − −→ L An,a −→ The exactness of this complex, as explained in Remark 5.4, yields the iden-tities β An,n + a ( m a ) = rank A ( L An,a ) = a X i =1 ( − i +1 rank A (cid:16) A ⊗ k A ! ∗ n + i ⊗ k A a − i (cid:17) = a X i =1 ( − i +1 dim k ( A ! n + i ) dim k ( A a − i ) and the vanishing of the remaining Betti numbers is due to the fact that theminimal resolution in Theorem 5.3 is a -linear. (cid:3) In contrast to Theorem 5.3, for non Koszul algebras the A -free resolutionof A/ m a afforded by Corollary 4.4 cannot be minimized by the procedurepresented in this section. We illustrate the obstructions by means of thefollowing example. Example 5.6.
Let A = k [ x ] / ( x ) , which is a non Koszul (also non quadratic)algebra. The enveloping algebra is A e = A ⊗ k A = k [ x ] / ( x ) ⊗ k k [ y ] / ( y ) ∼ = k [ x, y ] / ( x , y ) and the A e -module structure induced on A by the (surjective) multiplicationmap A e = A ⊗ k A ε −→ A yields the isomorphism A ∼ = A e / ( x − y ) . Therefore A has the following two-periodic resolution over the complete intersection A e · · · → A e x − y −−→ A e x + xy + y −−−−−−→ A e x − y −−→ A e ε −→ A → . Rewriting this complex in the form of Section 3 gives · · · → A ⊗ k V ⊗ k A ∂ −→ A ⊗ k V ⊗ k A ∂ −→ A ⊗ k V ⊗ k A ε −→ A → , ANONICAL RESOLUTIONS OVER KOSZUL ALGEBRAS 13 where each V i is a one dimensional vector space with basis { e i } and for i > ∂ (1 ⊗ e i ⊗
1) = ( x ⊗ e i − ⊗ − ⊗ e i − ⊗ y for i odd x ⊗ e i − ⊗ x ⊗ e i − ⊗ y + 1 ⊗ e i − ⊗ y for i even . The conclusion of Corollary 4.4 still holds and indicates that the truncatedcomplexes X Aa are (non minimal) free resolutions for A/ m a . But by contrastto the Koszul case, we see that arranging by grading as in (4.2) yields adiagram that is not a bicomplex and whose rows are no longer exact (oreven complexes!), and so in the truncated complex (4.3) the rows are nolonger acyclic. Correspondingly, the modules L n,a one could define are nolonger free. Thus there is no clear way to minimize the complex X Aa in asimilar manner to the technique used in this section, except for the case a = 1 where X Aa is already minimal.6. Examples
In this section we provide examples which illustrate our constructions forcertain Koszul algebras. For simplicity, all our examples are commutativealgebras defined by quadratic monomial ideals, but of course there are plentyof noncommutative examples as well. This class is known to yield Koszulalgebras by [Frö99].
Example 6.1.
Consider the following pair of dual Koszul algebras fromExample 2.4 A = k [ x, y, z ]( x , xy, y ) and A ! = k h x ∗ , y ∗ , z ∗ i (( z ∗ ) , x ∗ z ∗ + z ∗ x ∗ , y ∗ z ∗ + z ∗ y ∗ ) . The graded pieces ( A ! ) n are spanned by the words of length n on the alphabet { x ∗ , y ∗ , z ∗ } where the first letter is x ∗ , y ∗ or z ∗ and the other n − are x ∗ or y ∗ , whence dim k ( A ! ) n = 3 · n − for n ≥ . For n ≥ , A n is spanned bymonomials of the form ( z ∗ ) n , x ∗ ( z ∗ ) n − , and y ∗ ( z ∗ ) n − so that dim k A n = 3 .Thus in this case both A and A ! are infinite dimensional k -algebras.The Priddy complex P A • (2.5) consists of terms of the form P = A ⊗ k kP n = A ⊗ k k · n − for n ≥ and the resolution of A over A e viewed as a double complex (4.2) has terms F i,j = A ⊗ k k · i − ⊗ k A j = ( A · i − ⊗ A A i ≥ , j = 0 A · i − ⊗ A A i ≥ , j ≥ . Corollary 5.5 reveals that the Betti numbers of m a are independent of a .Indeed for a ≥ and n ≥ we have β n,n + a ( m a ) = a − X i =1 ( − i +1 · · n + i − + ( − a +1 · · n + a − = 9 · n · − ( − a − − a − · · a + n − = 3 · n . Example 6.2.
Consider the following complete intersection Koszul algebra A = k [ x, y, z ]( x , y ) = k h x, y, z i ( x , y , xz − zx, xy − yx, yz − zy ) . The Koszul dual algebra is given by A ! = k h x ∗ , y ∗ , z ∗ i (( z ∗ ) , x ∗ y ∗ + y ∗ x ∗ , x ∗ z ∗ + z ∗ x ∗ , y ∗ z ∗ + z ∗ y ∗ ) . In this case we find that F i,j = A ⊗ k A ! ∗ i ⊗ k A j = A i +1 , j = 0 A i +1) , j = 1 A i +1) , j ≥ and therefore the ranks of the modules L n,a can be computed by noting that rank L n, = − rank F n +2 , + rank F n +1 , = 4( n − and for a ≥ we have rank L n,a +1 − rank L n,a = a +2 X i =1 ( − i +1 rank F n + i,a +1 − i − a X i =1 ( − i +1 rank F n + i,a − i = ± [rank F a + n +1 , − rank F a + n, + rank F a + n − , − F a + n, − rank F a + n − , ]= 0 . Thus the Betti numbers of m a are independent of a for a ≥ and given by β An,n +1 ( m ) = rank L n, = rank F n +1 , = 2 n + 3 β An,n + a ( m a ) = rank L n,a = 4 n + 4 for a ≥ . Example 6.3.
Consider the following commutative Koszul algebra A = k [ x, y, z ]( xy, xz ) = k h x, y, z i ( xy, xz, xz − zx, xy − yx, yz − zy ) . The Koszul dual algebra is given by A ! = k h x ∗ , y ∗ , z ∗ i (( x ∗ ) , ( y ∗ ) , ( z ∗ ) , y ∗ z ∗ + z ∗ y ∗ ) ANONICAL RESOLUTIONS OVER KOSZUL ALGEBRAS 15 and its Hilbert function satisfies the Fibonacci recurrence dim A n +2 = dim A n +dim A n +1 . Indeed, setting u ( n ) to be the number of monomials in A ! of de-gree n ending in x and v ( n ) to be the number of monomials in A ! of degree n not ending in x ∗ , yields u ( n ) = v ( n − and v ( n ) = 2 u ( n −
1) + u ( n − .The second expression follows because the number of monomials ending in y ∗ or z ∗ where the previous letter is x ∗ is u ( n − and the number of mono-mials ending in y ∗ z ∗ (or equivalently, z ∗ y ∗ ) where the previous letter is x is u ( n − . Thus this leads to dim A n +2 = u ( n + 2) + v ( n + 2) = v ( n + 1) + 2 u ( n + 1) + u ( n )= v ( n + 1) + u ( n + 1) + v ( n ) + u ( n ) = dim A n +1 + dim A n . The identity above in turn implies that the terms of the double complex aswell as the free modules in the resolution of m a satisfy similar recurrences rank F n +2 ,a = rank F n +1 ,a +rank F n,a , rank L n +2 ,a = rank L n +1 ,a +rank L n,a . We conclude that the Fibonacci recurrence holds for Betti numbers β An +2 ,n +2+ a ( m a ) = β An +1 ,n +1+ a ( m a ) + β An,n + a ( m a ) for a ≥ , n ≥ subject to the initial conditions β A ( m a ) = a + 2 and β A ( m a ) = 2 a + 3 .Solving the above recurrence yields closed formulas for these Betti numbersas follows β An,n + a ( m a ) = (cid:18) a + 22 + 3 a + 42 √ (cid:19) √ ! n + (cid:18) a + 22 − a + 42 √ (cid:19) − √ ! n . We now give an infinite resolution counterpart to a family of square-freemonomial ideals that have appeared as ideals of the polynomial ring in workof Galetto [Gal20].
Example 6.4.
Consider the dual pair of Koszul algebras A = k [ x , . . . , x d ]( x , . . . , x d ) , and A ! = k h x ∗ , . . . , x ∗ d i ( x ∗ i x ∗ j + x ∗ j x ∗ i , ≤ i < j ≤ d ) , where dim k ( A j ) = (cid:0) dj (cid:1) and dim k ( A ! i ) = (cid:0) i + d − d − (cid:1) . Thus the terms in thedouble complex (4.2) are F i,j = A ⊗ k k ( i + d − d − ) ⊗ k k ( dj ) . Notice that for a ≤ d the ideal m a of A can be described as the ideal generatedby all square-free monomials of degree a in A , while for a > d we have m a = 0 .We compute the Betti numbers of this family of ideals using Corollary 5.5as follows(6.1) β n,n + a ( m a ) = a X i =1 ( − i +1 (cid:18) n + i + d − d − (cid:19)(cid:18) da − i (cid:19) Note that ( − i +1 (cid:18) n + i + d − d − (cid:19) is equal to ( − n − times the coefficientof t n + i in the Taylor expansion of the rational function t ) d around 0.Similarly, (cid:18) da − i (cid:19) is the coefficient of t a − i in the binomial expansion of (1 + t ) d . Since t ) d · (1 + t ) d = 1 , for n + a > , the coefficient of t n + a intheir product is 0, i.e. a X i = − n ( − n + i (cid:18) n + i + d − d − (cid:19)(cid:18) da − i (cid:19) = 0 . However, when i < a − d , the second binomial coefficient is 0, so this can berestated as a X i = a − d ( − n + i (cid:18) n + i + d − d − (cid:19)(cid:18) da − i (cid:19) = 0 . Combined with (6.1), the identity above leads to the more compact formula β n,n + a ( m a ) = (P i = a − d ( − i (cid:0) n + i + d − d − (cid:1)(cid:0) da − i (cid:1) ≤ a ≤ d a ≥ d + 1 . This is consistent with m a = 0 for a > d and can be easier to evaluate than(6.1) for some values of a . For example, setting a = d yields β n,n + d ( m d ) = (cid:18) n + d − d − (cid:19)(cid:18) da (cid:19) . Acknowledgements.
Our work started at the 2019 workshop "Womenin Commutative Algebra" hosted by Banff International Research Station.We thank the organizers of this workshop for bringing our team together.We acknowledge the excellent working conditions provided by BIRS and thesupport of the National Science Foundation for travel through grant DMS-1934391. We thank the Association for Women in Mathematics for fundingfrom grant NSF-HRD 1500481.In addition, we have the following individual acknowledgements for sup-port: Faber was supported by the European Union’s Horizon 2020 researchand innovation programme under the Marie Skłodowska-Curie grant agree-ment No 789580. Miller was partially supported by the NSF DMS-1003384.R.G.’s travel was partially supported by an AMS-Simons Travel Grant. Se-celeanu was partially supported by NSF DMS-1601024.We thank Liana Şega for helpful comments and for bringing [Ş01] to our at-tention and Ben Briggs for answering a question and pointing us to [VdB94].
References [BCK +
66] A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G. Quillen, D. L. Rector, andJ. W. Schlesinger,
The mod − p lower central series and the Adams spectralsequence , Topology (1966), 331–342. MR 199862 ANONICAL RESOLUTIONS OVER KOSZUL ALGEBRAS 17 [BE75] David A. Buchsbaum and David Eisenbud,
Generic free resolutions and afamily of generically perfect ideals , Advances in Math. (1975), no. 3, 245–301. MR 396528[BGG78] I. N. Bernšte˘ın, I. M. Gel’fand, and S. I. Gel’fand, Algebraic vector bundles on P n and problems of linear algebra , Funktsional. Anal. i Prilozhen. (1978),no. 3, 66–67. MR 509387[BGS96] Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul dualitypatterns in representation theory , J. Amer. Math. Soc. (1996), no. 2, 473–527.MR 1322847[Con14] Aldo Conca, Koszul algebras and their syzygies , Combinatorial algebraic ge-ometry, Lecture Notes in Math., vol. 2108, Springer, Cham, 2014, pp. 1–31.MR 3329085[Ş01] Liana M. Şega,
Homological properties of powers of the maximal ideal of a localring , J. Algebra (2001), no. 2, 827–858. MR 1843329[CTV01] Aldo Conca, Ngô Viêt Trung, and Giuseppe Valla,
Koszul property for pointsin projective spaces , Math. Scand. (2001), no. 2, 201–216. MR 1868173[EFS03] David Eisenbud, Gunnar Fløystad, and Frank-Olaf Schreyer, Sheaf cohomologyand free resolutions over exterior algebras , Trans. Amer. Math. Soc. (2003),no. 11, 4397–4426. MR 1990756[Frö99] R. Fröberg,
Koszul algebras , Advances in commutative ring theory (Fez, 1997),Lecture Notes in Pure and Appl. Math., vol. 205, Dekker, New York, 1999,pp. 337–350.[Gal20] Federico Galetto,
On the ideal generated by all squarefree monomials of a givendegree , J. Commut. Algebra (2020), no. 2, 199–215. MR 4105544[GKM98] Mark Goresky, Robert Kottwitz, and Robert MacPherson, Equivariant coho-mology, Koszul duality, and the localization theorem , Invent. Math. (1998),no. 1, 25–83. MR 1489894[Man88] Yu. I. Manin,
Quantum groups and noncommutative geometry , Universitéde Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1988.MR 1016381[May66] J. P. May,
The cohomology of restricted Lie algebras and of Hopf algebras , J.Algebra (1966), 123–146. MR 193126[MP15] Jason McCullough and Irena Peeva, Infinite graded free resolutions , Commu-tative algebra and noncommutative algebraic geometry. Vol. I, Math. Sci. Res.Inst. Publ., vol. 67, Cambridge Univ. Press, New York, 2015, pp. 215–257.MR 3525473[MR18] Claudia Miller and Hamidreza Rahmati,
Free resolutions of Artinian com-pressed algebras , J. Algebra (2018), 270–301. MR 3743182[Pos14] Leonid Positselski,
Galois cohomology of a number field is Koszul , J. NumberTheory (2014), 126–152. MR 3253297[PP05] Alexander Polishchuk and Leonid Positselski,
Quadratic algebras , UniversityLecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005.MR 2177131[Pri70] Stewart B. Priddy,
Koszul resolutions , Trans. Amer. Math. Soc. (1970),39–60. MR 265437[Tat57] John Tate,
Homology of Noetherian rings and local rings , Illinois J. Math. (1957), 14–27. MR 86072[VdB94] Michel Van den Bergh, Noncommutative homology of some three-dimensionalquantum spaces , Proceedings of Conference on Algebraic Geometry and RingTheory in honor of Michael Artin, Part III (Antwerp, 1992), vol. 8, 1994,pp. 213–230. MR 1291019 [Wei94] Charles A. Weibel,
An introduction to homological algebra , Cambridge Studiesin Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge,1994. MR 1269324
School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK
Email address : [email protected] Department of Mathematics, University of Osnabrück, Germany
Email address : [email protected] Department of Mathematics, Harvey Mudd College, Claremont, CA 91711
Email address : [email protected] Department of Mathematics, Syracuse University, Syracuse, NY 13244
Email address : [email protected] Department of Mathematical Sciences, George Mason University, Fairfax,VA 22030
Email address : [email protected] Department of Mathematics, University of Nebraska–Lincoln, Lincoln,NE 68588
Email address ::