Higher resonance schemes and Koszul modules of simplicial complexes

Each connected graded, graded-commutative algebra A of finite type over a field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Bbbk $$\end{document}k of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=\Bbbk \langle \Delta \rangle $$\end{document}A=k⟨Δ⟩ is the exterior Stanley–Reisner algebra associated to a finite simplicial complex \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}Δ, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.


Introduction and statement of results
Koszul modules are graded modules over a symmetric algebra that are constructed from the classical Koszul complex.They emerged from geometric group theory and topology [2,12] and found applications in other fields such as algebraic geometry.One prominent instance is [1], where the effective vanishing in high degrees of some Koszul modules led to a new proof of the celebrated Green's Conjecture on syzygies of generic canonical curves.The argument relies on a connection between the graded pieces of those particular Koszul modules and the Koszul cohomology of the tangent developable surface of a rational normal curve.The non-trivial vehicle that permits the passage in [1] from symmetric powers (Koszul modules) to exterior powers (Koszul cohomology) is an explicit version of the Hermite reciprocity formula.
It is the aim of this paper to describe a completely new instance where the passage from Koszul modules to Koszul cohomology of some homogeneous coordinate ring is still possible.The setup is however simpler and more elementary than the one involved with Green's Conjecture.
For a ground field k of characteristic 0, a classical construction of Stanley and Reisner associates to every simplicial complex ∆ on n vertices a graded, graded-commutative algebra kx∆y " E{J ∆ , where E " Ź k pe 1 , . . ., e n q is the exterior algebra over k and J ∆ is the ideal generated by all the monomials e σ " e j 1 ^¨¨¨^e js corresponding to simplices σ " pj 1 , . . ., j s q with 1 ď j 1 ă ¨¨¨ă j s ď n which do not belong to ∆.
Let S :" krx 1 , . . ., x n s be the polynomial ring in n variables over k, and consider the cochain complex pkx∆y ‚ b k S, δq of free, finitely generated, graded S-modules obtained by applying the BGG correspondence to the finitely generated, graded E-module kx∆y ‚ .The Fitting ideals of this complex define the jump resonance loci of our simplicial complex, where β i`1 is the number of faces of dimension i in ∆.It was shown in [10] that the irreducible components of R i p∆q are coordinate subspaces of kx∆y 1 " k n , given explicitly in terms of the (simplicial) homology groups of certain subcomplexes of ∆.Now let `kx∆y ‚ b k S, B ˘be the dual chain complex, and define the Koszul modules (in weight i) of the simplicial complex ∆ to be the homology S-modules of this complex, W i p∆q :" H i `kx∆y ‚ b k S, B ˘. (1.2) An alternate definition of resonance is given by the support loci of these modules, These varieties, called the support resonance loci, are again finite unions of coordinate subspaces.Though they do not coincide in general with the previously defined sets R i p∆q, it is known that R 1 p∆q " R 1 p∆q (away from 0) and Ť jďi R j p∆q " Ť jďi R j p∆q for all i ě 1.A notable property of the higher Koszul modules associated to simplicial complexes is that they are multigraded as opposed to the general case when they are only graded modules.Using the general theory of multi-graded square-free modules, we prove that the multi-graded pieces of the Koszul modules can be described as multi-graded pieces of some Tor's over symmetric algebras.It is known (see for example [12]) that the graded pieces of weight-one Koszul modules are graded pieces of Tor's over exterior algebras; however, their relations with Tor's over symmetric algebras is quite rare in general.
Theorem 1.1.For any i ě 1 and any square-free multi-index b, there is a natural isomorphism of vector spaces, where kr∆s is the polynomial Stanley-Reisner ring of ∆.
We refer to Section 3.1 for a quick review of multi-graded square-free modules.This multigraded structure of the Koszul modules is captured in the Hilbert series.Theorem 1.2.For every simplicial complex ∆, the multigraded Hilbert series of the Koszul modules W i p∆q are given by ÿ In Section 5, we give a precise description of the irreducible components of the support resonance loci.In each weight i, they correspond to maximal subcomplexes with non-vanishing reduced homology in degree i ´1.
Theorem 1.3.For every simplicial complex ∆ and every i ě 1, the scheme structure on the support resonance R i p∆q is reduced.Moreover, the decomposition in irreducible components is given by Particularly interesting is the case when ∆ is 1-dimensional, that is, it may be viewed as a finite simple graph Γ.It was shown in [9] that all the irreducible components of R 1 pΓq are coordinate subspaces, which correspond to the maximally disconnected full subgraphs of Γ.This result comes as a direct consequence of our analysis.The statement concerning the reducedness of R i p∆q can be compared with the detailed study performed in [3] on the scheme structure of (support) resonance varieties associated to classical Koszul modules.

Graded algebras, Koszul modules, and higher resonance
We start in a more general context (adapted from the setup in [11,13]), that will be used throughout the paper.

2.1.
Chain complexes associated to graded algebras.Let A ‚ be a graded, graded-commutative algebra over a field k of characteristic 0, with multiplication maps A i b k A j Ñ A i`j .We will assume that A is connected (that is, A 0 " k) and of finite-type (that is, dim k A i ă 8, for all i ą 0), and we will write β i pAq " dim k A i .To avoid trivialities, we always assume that β 1 pAq ‰ 0.
For each a P A 1 , graded commutativity of multiplication yields a 2 " 0, therefore, we have a cochain complex with differentials δ i a puq " a ¨u, for all u P A i .The resonance varieties of A are the jump loci for the cohomology groups of this complex: for each i ě 0, we put Clearly, these are homogeneous subsets of the affine space A 1 .Since A 0 is 1-dimensional, generated by 1 P k, and since δ a p1q " a for each a P A 1 , it follows that R 0 pAq " t0u.The most studied is the first resonance variety, which can be described as the set where K K denotes the kernel of the multiplication map Let us now fix a k-basis te 1 , . . ., e n u of A 1 and let tx 1 , . . ., x n u be the dual basis of the dual k-vector space A 1 " pA 1 q _ .This allows us to identify the symmetric algebra SympA 1 q with the polynomial ring S " krx 1 , . . ., x n s, the coordinate ring of the affine space A 1k β 1 pAq .
Viewing A ‚ as a graded module over the exterior algebra E ‚ " Ź A 1 , the BGG correspondence [7] yields a cochain complex of finitely generated, free S-modules, whose coboundary maps are the S-linear maps given by δ i A pu b sq " ř n j"1 e j u b sx j for u P A i and s P S. It is readily seen that this cochain complex is independent of the choice of basis for A 1 and that, moreover, the specialization of pA b k S, δ A q at an element a P A 1 coincides with the complex pA, δ a q defined by (2.1).
Remark 2.1.Typically, the elements in the exterior algebra E ‚ are given negative degrees, see for instance [7].However, we prefer to work here with the positive grading, which amounts to negating the standard action of the torus pk ˚qn on E ‚ as well.This convention carries over to the exterior Stanley-Reisner rings A ‚ " kx∆y in Section 3.2 and is compatible with the notation of [4], as recalled in Proposition 3.8.
It follows directly from the definition (2.2) that a point a P A 1 belongs to R i pAq if and only if rank δ i´1 a `rank δ i a ă β i pAq.Therefore, where ψ 1 ' ψ 2 denotes the block sum of two matrices, Fitt r pψq denotes the ideal of minors of size n´r of a matrix ψ : S m Ñ S n , and V pIq denotes the zero-set of an ideal I Ă S. This shows that the sets R i pAq are algebraic subvarieties of the affine space A 1 called jump resonance loci.

2.2.
Koszul modules and their support loci.Set A i :" pA i q _ and B A i :" pδ i´1 A q _ and consider the chain complex of finitely generated S-modules We define the Koszul modules (in weight i) of the algebra A as the homology S-modules of this chain complex, that is, Clearly, these are finitely generated, graded S-modules.The degree d component of the Koszul module W i pAq is computed by the homology of the complex where we recall that S " SympA1 q.It follows straight from the definitions that W 0 pAq " k is the trivial S-module.
Setting E ‚ :" Ź A 1 , the first Koszul module also has the following presentation where ι The resonance schemes of the graded algebra A are defined by the annihilator ideals of the Koszul modules of A, R i pAq :" Spec `S{ Ann W i pAq ˘. (2.10) By slightly abusing notation, we also denote by R i pAq " Supp W i pAq the underlying sets and call them support resonance loci.Note that the algebra structure on A ‚ is not essential in the discussion above, as the definitions of Koszul modules and support resonance loci only use the E-module structure.In particular, the constructions apply for finitely-generated graded E-modules, as well.

2.3.
Quotients of exterior algebras through ideals generated in fixed degree.We now discuss a particularly interesting case of this general construction.Fix integers d ě 1 and n ě 3. Let V be an n-dimensional vector space over the field k and let K Ď Ź d`1 V be a subspace.Set S :" SympV q and E ‚ :" Ź V _ , and then consider the linear subspace

.12)
Letting A ‚ :" E ‚ {xK K y be the quotient of the exterior algebra E ‚ by the (homogeneous) ideal generated by K K , we clearly have K " A d`1 .Conversely, if J Ď E ‚ is a homogeneous ideal generated in degree d `1 and we take K K :" J d`1 , then the algebra A ‚ " E ‚ {J is obtained as above.Denote by j the inclusion of the dual algebra is the Koszul differential, we have the following characterization.
Proposition 2.2.The Koszul modules W i pV, Kq " W i pAq satisfy the following properties: (1) W i pAq " 0 for i ď d ´1. (2) Proof.The first part is quite straightforward, as J i " 0 for i ď d ´1 and hence A i " E i for i ď d ´1.For the second part, first note that the d-th Koszul module is in this case the middle homology of the complex and hence ) .
Applying now the Snake Lemma to the diagram establishes the claim.
Remark 2.3.Note that the Snake Lemma also applies to the diagram leading to the simpler presentation (2.17) 3,12] and elsewhere.However, note the shift by two in degrees, that is, W pV, Kq " W 1 pAqp2q.
Example 2.4.Let X be a smooth complex projective variety, and consider a vector bundle E on X of rank ě r `1, for some integer r ě 1.We consider the determinant maps and take K K r :" kerpd r q.Then the above construction applies, producing for each r a series of Koszul modules W r pX, Eq :" W r `H0 pX, Eq _ , K r ˘.
As in the case d " 1 (see [1,12]), we have a geometric characterization of vanishing resonance, in which case the corresponding Koszul module is of finite length and hence vanishes in high degrees.For an element ω P Ź d`1 V _ , we denote by ϕpωq : Consider the projective variety parameterizing the decomposable elements, Standard multilinear algebra proves the following proposition.
Proof.Recall that a P V _ divides ω if and only if a ^ω " 0. Therefore, R d pAq ‰ t0u if and only if there exist a P V _ and b P Ź d V _ such that 0 ‰ a ^b P K K .This is equivalent to the existence of a non-zero element a P V _ and of a non-zero element ω P K K such that a ^ω " 0, i.e., 0 ‰ a P kerpϕpωqq, and hence rωs P Σ d .
The case d " 1 is special, since ϕpωq non-injective implies its kernel is at least 2-dimensional.Indeed, if ω " a ^b ‰ 0 then kerpϕpωqq is generated by a and b.In this case, Σ 1 is the Grassmann variety Gr 2 pV _ q Ď P `Ź2 V _ ˘.
Remark 2.6.For the Koszul module W r pX, Eq considered in Example 2.4, we have that the resonance R r pX, Eq :" Supp W r pX, Eq is non trivial if and only if there exists a section 0 ‰ s P H 0 pX, Eq such that the determinant map d s : Ź r H 0 pX, Eq Ñ Ź r`1 H 0 pX, Eq given by ω Þ Ñ d r`1 ps ^ωq is not injective.

Simplicial complexes and their Koszul modules
3.1.Square-free modules.We start this section with some algebraic preliminaries regarding square-free modules.We recall from [14] some basic facts about this type of modules, which will be needed in Sections 4.2, 5, and 6.1.
Let V be a k-vector space of dimension n, and identify the symmetric algebra SympV q with the polynomial ring S " krx 1 , . . ., x n s.We consider the standard N n -multigrading on S, defined by degpx i q " e i P N n , where e i " p0, . . ., 1, . . ., 0q is the multi-index with 1 placed in the i-th position.Given a multi-index a " pa 1 , . . ., a n q P N, its support is defined as the set Supppaq :" ti | a i ą 0u.Definition 3.1.An N n -graded S-module M is said to be square-free if for any a P N n and any i P Supppaq, the multiplication map This definition is a direct generalization of the case of ideals.Indeed, an ideal I Ď S is a square-free module if and only if it is a square-free monomial ideal, and this is also equivalent to S{I being a square-free module.
Note that a free N n -graded S-module is square-free if and only it is generated in square-free multidegrees.
Proposition 3.2.If f : M Ñ N is a morphism of N n -graded S-modules, and M and N are square-free modules, then kerpf q and cokerpf q are also square-free.Moreover, if is an exact sequence of N n -graded S-modules, and M 1 and M 2 are square-free, then so is M .
Proposition 3.2 has a few interesting consequences.
Corollary 3.3.Let M be an N n -graded square-free S-module.Then all the modules in the minimal free N n -graded resolution of M are square-free.
Corollary 3.4.If F is a bounded complex of free square-free S-modules, then the homology modules of F are also square-free.
The following result will be of particular interest for us.
Theorem 3.5.If M is an N n -graded, square-free S-module, then its annihilator is a squarefree monomial ideal.In particular, the annihilator of M is a radical ideal.
Proof.Since M is an N n -graded S-module, the annihilator AnnpM q Ď S is also N n -graded, that is, it is a monomial ideal.Let m " x a 1 1 ¨¨¨x an n P AnnpM q be a monomial annihilating M , and assume a k ą 1 for some k.Then the multiplication map is zero for all b P N n .We have k P Supppb `degpmq ´ek q, and so, by hypothesis, the map is an isomorphism.Therefore, is the zero map for all b P N n , and thus x a 1 1 ¨¨¨x a k ´1 k ¨¨¨x an n P AnnpM q.By repeating the argument, we see that AnnpM q is a square-free monomial ideal.
Finally, we note that Theorem 3.5 and [14, Lemma 2.2] give the following.Proposition 3.6.Let M be a finitely-generated N n -graded, square-free S-module.Then the annihilator scheme structure on the support of M is reduced.Moreover, the decomposition of the support in irreducible components is given by Proposition 3.6 will be essential for describing the components of the support resonance loci of a simplicial complex in the next section.
We end this section with the following definition: Definition 3.7.For an N n -graded vector space M , the square-free part of M is the subspace sqfpM q Ď M concentrated in square-free multidegrees.

3.2.
Stanley-Reisner rings.Let S " krx 1 , . . ., x n s be the polynomial ring in n variables over a field k of characteristic 0. Given a simplicial complex ∆ on n vertices, we let kr∆s :" S{I ∆ be the (polynomial) Stanley-Reisner ring of ∆, where I ∆ is the ideal generated by the (squarefree) monomials x σ " x i 1 ¨¨¨x is for all simplices σ " pi 1 , . . ., i s q with 1 ď i 1 ă ¨¨¨ă i s ď n not in ∆.Similarly, we define the exterior Stanley-Reisner ring of ∆ as kx∆y :" E{J ∆ , where E " Ź pe 1 , . . ., e n q is the exterior algebra in n variables over k and J ∆ is the ideal generated by the monomials e σ " e i 1 ^¨¨¨^e is for all simplices σ R ∆.
Consider the graded, graded-commutative k-algebra A ‚ :" kx∆y.As mentioned in Remark 2.1, this algebra is given the positive grading.In each degree d, the vector space A d is spanned by multivectors e σ , where σ is a pd´1q-dimensional face of ∆.Indeed, since σ " pi 1 , . . ., i s q R ∆ implies pi 1 , . . ., i s , jq R ∆ for all j R Suppp∆q, it follows that in each degree d, the vector space J ∆,d is spanned by the multivectors e σ with σ R ∆ of dimension d ´1.With the notation of the previous sections, the dual A d is generated by the vectors v σ with σ P ∆ being of dimension d ´1.
For an element a " ř n i"1 λ i e i P A 1 , let pA ‚ , δ a q be the cochain complex from (2.1).As shown in [4, Proposition 4.3] (see also [10,Lemma 3.4]), this complex depends only on Supppaq :" ti | λ i ‰ 0u; more precisely, pA ‚ , δ a q is isomorphic to pA ‚ , δ āq, where ā " ř iPSupppaq e i .The following Hochster-type formula from [4, Proposition 4.3], suitably interpreted and corrected in [10, Proposition 3.6], describes the cohomology groups of the cochain complexes pA ‚ , δ a q.Proposition 3.8 ( [4,10]).Let ∆ be a finite simplicial complex on vertex set V " rns and a P A 1 as above.Writing V 1 " Supppaq, we have The range of summation in the above formula includes the empty simplex, with the convention that |H| " 0 and r H ´1pH; kq " k.
3.3.Koszul modules of a simplicial complex.Fix a basis v 1 , . . ., v n of the k-vector space V .Let K ‚ denote the Koszul complex of x 1 , . . ., x n , whose i-th free module is and set degpv i q " e i P N n .Then K ‚ is a complex of N n -graded square-free S-modules.
A simplicial complex ∆ on vertex set rns " t1, . . ., nu determines a subcomplex K ∆ ‚ of K ‚ , whose i-th module K ∆ i is the free S-module generated by the exterior monomials v j 1 ^¨¨¨^v j i such that tj 1 , . . ., j i u is a face of ∆.Applying (2.7), the i-th Koszul module W i p∆q defined as the i-th Koszul module of the exterior Stanley-Reisner ring of ∆ is the i-th homology H i pK ∆ ‚ q.Proposition 3.9.For every simplicial complex ∆ on n vertices and for every i, the Koszul module W i p∆q is an N n -graded square-free S-module.
Proof.The subcomplex K ∆ ‚ is a complex of N n -graded square-free S-modules.By Corollary 3.4, it follows that each homology vector space W i p∆q is a square-free S-module.

Hilbert series for Koszul modules of simplicial complexes
We fix some notation first.For a multidegree b, we denote the sum of its entries by |b|.For a square-free multidegree b P N n , we denote by ∆ b the restriction of the simplicial complex ∆ to the subset of the vertices Supppbq Ď t1, . . ., nu.
We denote by hi p´; kq and hi p´; kq the dimensions of the simplicial homology groups Hi p´; kq and of the reduced cohomology groups Hi p´; kq -Hi p´; kq _ with coefficients in k, respectively.4.1.Koszul modules vs. Koszul (co)homology.We establish a duality result between the Koszul modules associated to a simplicial complex and Koszul (co)homology of the symmetric Stanley-Reisner algebra.Proof.For the square-free multidegree b, we denote j :" |b| ´i.
We start by proving the first isomorphism.We use the notation from Section 3.2.Let A d Ď Ź d V be the subspace generated by the exterior monomials v σ such that σ is a face of ∆.Denote by S d the graded component of S " SympV q of total degree d.Then, the vector space rW i p∆qs b is the middle homology of the complex of vector spaces By Proposition 3.9, this complex is the same as Upon identifying sqfpS d q " Ź d V _ via the k-linear map given by this chain complex may be written as which, by dualization gives As usual, all these identifications are made without changing the positivity of the grading.After having identified A d " sqfpS{I ∆ q d , the sequence (4.3) may be written as Since b is a square-free multidegree, this complex is the same as By the properties of the Koszul complex, the middle cohomology of this complex is isomorphic to " Tor S j pk, kr∆sq ‰ b , and this concludes the proof of the first isomorphism.
For the second isomorphism, note that " Tor S j pk, kr∆sq Indeed, for j ě 2, this is clear, whereas for j " 1 this follows from the fact the I ∆ is contained in the ideal generated by the variables, and hence kk b S kr∆s.From [8, Proof of Theorem 8.1.1]we obtain an isomorphism " Tor S j´1 pk, as soon as |b| ´i ě 1.
Remark 4.2.The isomorphism in the statement of Theorem 4.1 does not necessarily hold if we drop the hypothesis that b is square-free.Indeed, . On the other hand, since the square-free multi-indices are finitely many, the vanishing of rW i p∆qs b for all b that is not square-free implies W i p∆q is of finite length.
An alternate, less explicit proof of the above theorem can be obtained by applying the Bernstein-Gelfand-Gelfand correspondence to express rW i p∆qs b as the (duals) of some Tor spaces over the exterior algebra, and then apply a theorem of Aramova, Avramov, and Herzog [4], see [8, Corollary 7.5.2].More precisely, we have the following result.Proposition 4.3.For any i ě 1 and any square-free multi-index b, there is a natural isomorphism of vector spaces The proof of the proposition follows from an adaptation to the multi-graded context [5] of the classical BGG correspondence, as described in [7].4.2.Multigraded Hilbert series.Our next goal is to determine the Hilbert series of the Koszul modules W i p∆q associated to a simplicial complex ∆.For a multidegree a " pa 1 , . . ., a n q P N n , we will write t a :" t a 1  1 ¨¨¨t an n .Theorem 4.4.For every simplicial complex ∆ and every i ą 0, the N n -graded Hilbert series of the Koszul module W i p∆q is given by ÿ Proof.We begin by observing that, by the definition of a square-free module, we have ÿ Thus, it suffices to determine dimrW i p∆qs b when b is a square-free multidegree.Fix a squarefree multidegree b, and let j " |b| ´i.From Theorem 4.1, we know that Therefore, ÿ and this completes the proof.
Specializing to the single N-grading, the above theorem yields the following formula for the Hilbert series of the Koszul module W i p∆q: In the particular case when ∆ has dimension at most 1, that is, when ∆ is equal to a (simplicial) graph Γ, we recover the Hilbert series of the module W Γ :" W 1 pΓqp2q, as computed in [9, Theorem 4.1].

Corollary 4.5 ([9]
).For a graph Γ on vertex set V, we have where Q Γ ptq " ř jě2 c j pΓqt j and c j pΓq " The significance of the above formula is that it gives the Chen ranks of the right-angled Artin group G Γ associated to the graph Γ.

Resonance varieties of a simplicial complex
Given an (abstract) simplicial complex ∆ on vertex set V, we define its resonance varieties as those of the corresponding exterior Stanley-Reisner ring.That is, we put R i p∆q :" R i pkx∆yq for the jump resonance and R i p∆q :" R i pkx∆yq for the support resonance varieties, respectively.
Using Proposition 3.8, a precise description of the varieties R i p∆q was given in [10, Theorem 3.8], as follows.
Proposition 5.1.For each i ě 1, the decomposition in irreducible components of the jump resonance variety is given by (5.1) Here k V 1 denotes the coordinate subspace of k V " k n (where n " |V|) spanned by the vectors te i | i P V 1 u.On the other hand, for the support resonance defined in (2.10), the situation is different in degrees i ą 1.
Theorem 5.2.For each i ě 1, the scheme structure on the support resonance locus R i p∆q is reduced.Moreover, the decomposition in irreducible components is given by Proof.The first claim follows from Proposition 3.9 and Theorem 3.5.The precise structure of the decomposition in irreducible components is governed by the multi-graded structure detailed in Theorem 4.4 and Proposition 3.6.Observe that (5.2) corresponds to the primary decomposition of the ideal AnnpW i p∆qq.
Notice the difference at the set level between (5.1) and (5.2); in particular, observe that the support resonance loci are easier to describe.Furthermore, whereas Theorem 5.2 guarantees that the support resonance schemes R i p∆q are always reduced, the corresponding jump resonance loci R i p∆q are not necessarily reduced (with the Fitting scheme structure), even in weight one, as the following example illustrates.
A simplicial complex ∆ of dimension d is said to be a Cohen-Macaulay complex over k if r H ‚ plkpσq; kq is concentrated in degree d ´|σ|, for all σ P ∆.As shown in [6], the jump resonance varieties of such a simplicial complex propagate; that is, (5.3) For arbitrary simplicial complexes, though, the resonance varieties do not always propagate.This phenomenon, first identified in [10], happens even for graphs.
Example 5.4 ( [10]).Let ∆ be the disjoint union of two edges.Then R 1 p∆q " k 4 , whereas R 2 p∆q " k 2 Y k 2 , the union of two transversal coordinate planes.Thus, R 1 p∆q Ę R 2 p∆q.
When ∆ is Cohen-Macaulay, propagation and formula (2.11) give R i p∆q " Ť jďi R j p∆q.But it is not known whether the support resonance varieties R i p∆q propagate when ∆ is Cohen-Macaulay, or, equivalently, whether R i p∆q " R i p∆q in this case.In general, though, we can use the previous example to settle the latter question in the negative.
Example 5.5.Let ∆ be the disjoint union of two edges.Then R 1 p∆q " R 1 p∆q " k 4 but R 2 p∆q " H whereas, as we saw before, R 2 p∆q " k 2 Y k 2 .Thus, R 2 p∆q ‰ R 2 p∆q.

Regularity and projective dimension for Koszul modules of simplicial complexes
6.1.General bounds.We start this section with an upper bound on the Castelnuovo-Mumford regularity and projective dimension of the Koszul modules.
Proposition 6.1.For every simplicial complex ∆ on n vertices and every i ą 0, the Koszul module W i p∆q has regularity at most n and projective dimension at most n ´i ´1.
Proof.By definition, the Koszul module W i p∆q is a sub-quotient of the module of i-th cycles in the Koszul complex of x 1 , . . ., x n .Since Z i is generated in degree i`1, it follows that the degree of any of the generators of W i p∆q is at least i `1.Let F ‚ denote the minimal free resolution of W i p∆q.By Proposition 3.9 and Corollary 3.3, F ‚ is a complex of N n -graded S-modules generated in square-free multidegrees, hence, the total degree of the generators of each F h is at most n.The statement on the regularity follows immediately.Since the least degree of the generators of F h`1 is strictly larger than the least degree of the generators of F h , it follows that F h " 0 for h ą n ´i ´1.
6.2.Regularity of Koszul modules for simplicial complexes of special type.We fix integers n ě 4 and 1 ď d ď n ´3 and assume ∆ is a simplicial complex of dimension d on n vertices whose pd ´1q-skeleton coincides with that of the full simplex, that is, ∆ pd´1q " `2rns ˘pd´1q .(6.1) For instance, if d " 1, then ∆ is simply a (simplicial) graph on n vertices.If d " 2, then ∆ is obtained from the complete graph on n vertices by filling in some triangles.For this type of simplicial complexes that generalize graphs, the nature of the Koszul modules can be made more precise, as follows.
Proposition 6.2.For a simplicial complex ∆ as above, the following hold.
Proof.Recall that W i p∆q " H i pA ‚ b k S, Bq, where A " kx∆y.By condition (6.1), the graded algebra A " E{J ∆ satisfies the hypothesis from the beginning of Section 2.3.Hence, Proposition 2.2 applies, showing that W i p∆q " 0 for i ă d.Moreover, since dimp∆q " d, we have that A i " 0 for i ą d `1, and so W i p∆q also vanishes in that range, thereby proving Part (1).
Using the explicit presentation of W d p∆q from part (2), we can improve the bound on regularity from Proposition 6.1.and this completes the proof.
If d ď 1, that is, if ∆ " Γ is a graph on n vertices, taking into account the degree shift, we obtain the bound reg W 1 pΓq ď n ´4.(6.4) Example 6.4.If Γ " C n is the cycle on n ě 4 vertices, then the regularity of W 1 pΓq attains the above bound: reg W 1 pΓq " n ´4 and pdim W 1 pΓq " n ´2.This follows from (6.2), since in this case the module D has only one syzygy, of degree n.Remark 6.5.For the Koszul module W d p∆q, the simplified presentation (2.17) has the following nice interpretation.Let r ∆ be the simplicial complex which is maximal (with respect to inclusion) among all simplicial complexes that share the same d-skeleton with ∆; for instance, if d " 1, then r ∆ is the flag complex of the graph Γ " ∆.Denote by V i the k-vector space with basis the set of i-dimensional missing faces of r ∆.We then have an exact sequence, Using Proposition 6.2, together with formula (5.1), we obtain the following immediate corollary.Corollary 6.6.With ∆ as above, we have: (1) R i p∆q " R i p∆q for all i ‰ d `1.
(2) R d`1 p∆q is equal to either H or k n .
As already mentioned before, the loci R d`1 p∆q and R d`1 p∆q can be different, in general.For example, if we take the graph Γ on four vertices with edges p1, 2q and p3, 4q as in Example 5.4, then R 2 pΓq " H while R 2 pΓq " V px 1 , x 2 q Y V px 3 , x 4 q.

Theorem 4 . 1 .
For any i ě 1 and any square-free multi-index b, there are natural isomorphisms of vector spaces rW i p∆qs b -" Tor S |b|´i pk, kr∆sq ı _ b -r H i´1 p∆ b ; kq _ -r H i´1 p∆ b ; kq.(4.1)

Proposition 6 . 3 .
With notation as above, we have reg W d p∆q ď n ´2.Proof.We have a Z d Ď Ź d V b k S is the module of Koszul d-cycles, and D is the image of K b k S under the Koszul differential.Both Z d and D are generated in degree d `1.The module Z d has a linear free resolution, consisting of the truncated Koszul complex, so reg Z d " d `1.Since the module D is square-free, its syzygy modules are also square-free, and hence, they are generated in degrees at most n.This implies that reg D ď n ´1, since D does not have generators of degree n.Applying the long exact sequence of Torp´, kq to (6.2), we obtain reg W d p∆q ď maxpreg Z d , reg D ´1q " maxpd `1, n ´2q " n ´2.(6.3) Aprodu was supported by the PNRR grant CF 44/14.11.2022Cohomological Hall algebras of smooth surfaces and applications.Farkas supported by the DFG Grant Syzygien und Moduli and by the ERC Advanced Grant SYZYGY.This project has received funding from the European Research Council (ERC) under the EU Horizon 2020 program (grant agreement No. 834172).Raicu was supported by the NSF Grant No. 2302341.Sammartano was supported by the grant PRIN 2020355B8Y Square-free Gröbner degenerations, special varieties and related topics.Suciu was supported by Simons Foundation Collaboration Grant for Mathematicians No. 693825.