An equivariant Hochster's formula for \mathfrak S_n-invariant monomial ideals
aa r X i v : . [ m a t h . A C ] D ec AN EQUIVARIANT HOCHSTER’S FORMULAFOR S n -INVARIANT MONOMIAL IDEALS SATOSHI MURAI AND CLAUDIU RAICU
Abstract.
Let R = k [ x , . . . , x n ] be a polynomial ring over a field k and let I ⊂ R be a monomialideal preserved by the natural action of the symmetric group S n on R . We give a combinatorialmethod to determine the S n -module structure of Tor i ( I, k ). Our formula shows that Tor i ( I, k ) isbuilt from induced representations of tensor products of Specht modules associated to hook partitions,and their multiplicities are determined by topological Betti numbers of certain simplicial complexes.This result can be viewed as an S n -equivariant analogue of Hochster’s formula for Betti numbers ofmonomial ideals. We apply our results to determine extremal Betti numbers of S n -invariant monomialideals, and in particular recover formulas for their Castelnuovo–Mumford regularity and projectivedimension. We also give a concrete recipe for how the Betti numbers change as we increase the numberof variables, and in characteristic zero (or > n ) we compute the S n -invariant part of Tor i ( I, k ) interms of Tor groups of the unsymmetrization of I . Introduction
Let R = k [ x , . . . , x n ] be a polynomial ring over a field k . The study of the graded Betti numbers β ij ( I ) = dim k Tor i ( I, k ) j of a homogeneous ideal I is one of the central research topics in commutative algebra. When I is amonomial ideal, this is closely related to combinatorial topology, and is the subject of a vast literature[6, Section 5], [15, Part II], [20, Part I], [23, Part III]. One of the most famous results on this topicis Hochster’s formula [6, § I that are invariantunder the action of the symmetric group S n by coordinate permutations.In general, if G ⊆ GL n ( k ) is a subgroup and I ⊆ R is a G -invariant ideal ( g ( I ) = I for all g ∈ G ),then each Tor i ( I, k ) acquires a G -module structure. The representation theory of G dictates thepossible building blocks that make up Tor i ( I, k ), reducing the problem of understanding Tor i ( I, k ) tothat of identifying the multiplicity of each building block. When G = k × n is the n -torus, the buildingblocks are 1-dimensional, given by the torus characters, and the calculation of their multiplicity (thedimension of the multigraded components of Tor i ( I, k )) is the content of Hochster’s formula. When G = k × n ⋊ S n extends the torus action by that of the symmetric group, our work will show that anatural set of building blocks arises via induction from Specht modules associated to hook partitionsfor smaller symmetric groups. We then identify the multiplicities of each block with topological Bettinumbers of associated simplicial complexes, which yields an S n -equivariant Hochster’s formula .To state our results, we first introduce some notation. We let P n = { ( λ , . . . , λ n ) ∈ Z n | λ ≥ · · · ≥ λ n ≥ } be the set of partitions consisting of n non-negative integers. For a vector a = ( a , . . . , a n ) ∈ Z n ≥ ,we write x a = x a · · · x a n n , and write part( a ) ∈ P n for the unique partition which is a rearrangementof a , . . . , a n . For example, we have part(2 , , ,
2) = (3 , , , I will denotean S n -invariant monomial ideal in R . For such I , we let P ( I ) = { λ ∈ P n | x λ ∈ I } , and think of P ( I ) informally as the set of partitions in I . For partitions λ , λ , . . . , λ r ∈ P n , we define(1.1) h λ , . . . , λ r i S n = P rk =1 ( σ ( x λ k ) | σ ∈ S n ) ⊂ k [ x , . . . , x n ]and call it the S n -invariant monomial ideal generated by λ , . . . , λ r . For instance, we have(1.2) I = h (4 , , , (5 , , i S = ( x x x , x x x , x x x , x x , x x , x x , x x , x x , x x ) . If M is a Z n -graded R -module and a ∈ Z n , then M a denotes the a -th graded component of M . Let M h λ i = M a ∈ Z n part( a )= λ M a for λ ∈ P n . We note that Tor i ( I, k ) h λ i is fixed by the S n -action, so we have a decompositionTor i ( I, k ) = M λ ∈ P n Tor i ( I, k ) h λ i as S n -modules. Therefore, we may focus on the S n -module structure of each Tor i ( I, k ) h λ i .It will be convenient to use the abbreviation( a p , a p , . . . , a p s s ) = ( a , . . . , a , a , . . . , a , . . . , a s , . . . , a s )where each a k appears p k times on the right side, and to identify each partition µ with its Youngdiagram. For example, (2 ,
1) will be identified with . For a partition µ = ( d p , . . . , d p s s , p s +1 ) ∈ P n , (1.3)where d > · · · > d s > p , . . . , p s > p s +1 ≥
0, we define(1.4) p ( µ ) = ( p − , . . . , p s −
1) and s ( µ ) = s. For a vector c = ( c , . . . , c s ) ∈ Z s ≥ with c ≤ ( p , . . . , p s ), we define(1.5) µ \ c = ( d p − c , ( d − c , d p − c , ( d − c , . . . , d p s − c s s , ( d s − c s , p s +1 ) ∈ P n . Example 1.1.
Suppose that µ = (5 , , ) and n = 6. We have s = 3, p = p = p = 2, and p = 0, hence p ( µ ) = (1 , , c = (1 , ,
1) then(5 , , ) \ (1 , ,
1) = (5 , , , . As illustrated below, we can think of (5 , , ,
1) as being obtained from (5 , , ) by removing onebox from the fifth column, two boxes from the third column, and one box from the second column:(5 , , ) −→ ×××× (5 , , , QUIVARIANT HOCHSTER’S FORMULA 3
We let e , . . . , e s be the standard vectors of Z s and write e F = P i ∈ F e i for F ⊂ [ s ] = { , . . . , s } .For c ≤ p ( µ ), we define the simplicial complex (see Section 2.2 for some background)(1.6) ∆ µ, c ( I ) = { F ⊂ [ s ] | µ \ ( c + e F ) ∈ P ( I ) } , and define the numbers γ µ, c i ( I ) by γ µ, c i ( I ) = dim k (cid:16) e H i − (∆ µ, c ( I )) (cid:17) , where e H • (∆) denote the reduced homology groups of a simplicial complex ∆, with coefficients in k . Example 1.2.
Let I be as in (1.2), let µ = (5 , ,
1) and let c = (0 , , µ, c ( I ) can be identified with the intersection of the interval [ µ \ c , µ \ ( c + e [ s ] )] = [(5 , , , (4 , , P ( I ) in the poset P n (with the reversed order). The faces of the complex and the correspondingpartitions are colored in red and are pictured below. { , , } ❏❏❏❏❏❏❏❏❏❏❏ttttttttttt { , } ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ { , } ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ttttttttttttttt { , } ttttttttttttttt { } { } { } ∅ ❏❏❏❏❏❏❏❏❏❏❏❏ tttttttttttt ❋❋❋❋❋❋❋❋❋❋❋①①①①①①①①①①①❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋ ❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①❊❊❊❊❊❊❊❊❊❊❊ ②②②②②②②②②②② It follows that ∆ µ, c ( I ) = (cid:8) { , } , { } , { } , { } , ∅ (cid:9) , whose only non-vanishing reduced homology group is e H (∆ µ, c ( I )), of dimension γ µ, c ( I ) = 1.To introduce the final ingredient of our main result, we let S λ denote the Specht module associatedwith a partition λ (see Section 2.4 for some background), which is a module over S | λ | . We say that λ is a hook partition if λ i ≤ i > λ ). For a sequence π = (( p , q ) , . . . , ( p r , q r ))of hook partitions with P rk =1 ( p k + q k ) = n , we write S π = Ind S n S p q ×···× S pr + qr (cid:0) S ( p , q ) ⊠ · · · ⊠ S ( p r , qr ) (cid:1) , (1.7)where ⊠ denotes the (external) tensor product of representations, and Ind denotes the inducedrepresentation. We are now ready to state our main theorem. SATOSHI MURAI AND CLAUDIU RAICU
Theorem 1.3.
Let µ = ( d p , . . . , d p s s , p s +1 ) ∈ P n with d > · · · > d s > and let I be an S n -invariantmonomial ideal of R . We have an isomorphism of k -vector spaces (1.8) Tor i ( I , k ) h µ i ∼ = M ≤ c =( c ,...,c s ) ≤ p ( µ ) (cid:16) S (( p − c , c ) ,..., ( p s − c s , cs ) , ( p s +1 )) (cid:17) γ µ, c i −| c | ( I ) . Moreover, if char( k ) = 0 or char( k ) > n , then (1.8) is an isomorphism of S n -modules. In fact, our proof will show that (in arbitrary characteristic) Tor i ( I, k ) h µ i has a filtration by S n -submodules, whose associated graded is isomorphic to the right side of (1.8). Since representationsof S n are semisimple if char( k ) = 0 or char( k ) > n , it follows that (1.8) is an isomorphism of S n -modules in these cases (we note that a similar issue arises in the calculation of Ext modules in [24,Main Theorem]). Theorem 1.3 clarifies our earlier assertion that the building blocks of Tor i ( I, k )are induced representations of tensor products of Specht modules, and that their multiplicities aretopological Betti numbers. We now illustrate Theorem 1.3 with an example. Example 1.4. If I is as in (1.2) then using Macaulay2 one can see that the Betti table of I is and that Tor ( I, k ) = Tor ( I, k ) h (4 , , i ⊕ Tor ( I, k ) h (5 , , i , Tor ( I, k ) = Tor ( I, k ) h (4 , , i ⊕ Tor ( I, k ) h (5 , , i ⊕ Tor ( I, k ) h (5 , , i , Tor ( I, k ) = Tor ( I, k ) h (4 , , i ⊕ Tor ( I, k ) h (5 , , i . To identify the S -module structure, one first computes the relevant complexes ∆ µ, c ( I ): ∆ (4 , , , (0 , ( I ) = ∆ (5 , , , (0 , ( I ) = { ∅ } , ∆ (4 , , , (1 , ( I ) = ∆ (5 , , , (1) ( I ) = { ∅ } , ∆ (5 , , , (0 , , = (cid:8) { , } , { } , { } , { } , ∅ (cid:9) , ∆ (4 , , , (2) ( I ) = { ∅ } and ∆ (5 , , , (1 , = (cid:8) { } , { } , ∅ (cid:9) . We leave the details of this calculation to the reader, noting that the description of ∆ (5 , , , (0 , , ( I )was explained in Example 1.2. We then have γ (4 , , , (0 , ( I ) = γ (5 , , , (0 , ( I ) = 1 ,γ (4 , , , (1 , ( I ) = γ (5 , , , (0 , , ( I ) = γ (5 , , , (1)0 ( I ) = 1 , and γ (4 , , , (2)0 ( I ) = γ (5 , , , (1 , ( I ) = 1 . QUIVARIANT HOCHSTER’S FORMULA 5
Theorem 1.3 implies based on these computations that we haveTor ( I, k ) h (4 , , i ∼ = Ind S S × S (cid:0) S ⊠ S (cid:1) , Tor ( I, k ) h (5 , , i ∼ = Ind S S × S × S (cid:0) S ⊠ S ⊠ S (cid:1) , Tor ( I, k ) h (4 , , i ∼ = Ind S S × S (cid:16) S ⊠ S (cid:17) , Tor ( I, k ) h (5 , , i ∼ = Ind S S × S × S (cid:0) S ⊠ S ⊠ S (cid:1) , Tor ( I, k ) h (5 , , i ∼ = Ind S S × S (cid:16) S ⊠ S (cid:17) , Tor ( I, k ) h (4 , , i ∼ = S ,
Tor ( I, k ) h (5 , , i ∼ = Ind S S × S (cid:16) S ⊠ S (cid:17) . As a first application of Theorem 1.3, we explain how to determine the S n -invariant part ofTor i ( I, k ). We express I as in (1.1), and define the unsymmetrization of I to be the ideal J = ( x λ , · · · , x λ r ) ⊆ R. The relationship between the Tor groups of I and J is given by the following. Theorem 1.5. If char( k ) = 0 or char( k ) > n , then for partitions λ , · · · , λ r ∈ P n we have Tor i (cid:0) h λ , · · · , λ r i S n , k (cid:1) S n ∼ = Tor i (cid:0) ( x λ , · · · , x λ r ) , k (cid:1) for all i. The proof of Theorem 1.5 is explained in Section 4 as an application of Theorem 1.3 and the NerveTheorem. Here, we illustrate Theorem 1.5 with an example.
Example 1.6. If I is as in (1.2), then its unsymmetrization is the ideal J = ( x x x , x x ). Sincethe generators of J have a unique syzygy (coming from their lcm x x x ), we getTor ( J, k ) = Tor ( J, k ) (4 , , ⊕ Tor ( J, k ) (5 , , ∼ = k ⊕ k and Tor ( J, k ) = Tor ( J, k ) (5 , , ∼ = k . Theorem 4.1 implies thatTor ( I, k ) S = Tor ( I, k ) S h (4 , , i ⊕ Tor ( I, k ) S h (5 , , i ∼ = k ⊕ k and Tor ( I, k ) S = Tor ( I, k ) S h (5 , , i ∼ = k , which can be checked (based on Pieri’s rule) using the computations in Example 1.4: the trivial S -module S appears as a summand in S π if and only if each of the partitions in π has a singlerow (in which case the multiplicity of S is one). Remark 1.7.
Theorem 4.1 implies that if h λ , . . . , λ r i S n has a linear resolution, then so does( x λ , . . . , x λ r ). A combinatorial characterization of S n -invariant monomial ideals having a linearresolution was given in [24]. These are exactly the symmetric shifted ideals (generated in a singledegree) defined in [4]. It follows that the unsymmetrizations of these ideals have a linear resolution. SATOSHI MURAI AND CLAUDIU RAICU
Theorem 1.3 gives a concrete recipe for computing the multigraded Betti numbers β i, a ( I ) = dim Tor i ( I, k ) a , but in practice, the difficulty of the calculation depends on the complexity of evaluating the homologyof ∆ µ, c . As shown by example in [21, Section 5], the numbers β i, a ( I ) may depend on the characteristicof k . However, the shape of the Betti table, as measured by the Castelnuovo–Mumford regularityreg( I ), and by the projective dimension pdim( I ), does not depend on char( k )! This was first shownin [24], and in Section 5 we give an equivalent (but somewhat simpler) recipe for computing reg( I )and pdim( I ). We also extend the notion of extremal Betti numbers from [2] to our context, andcompute the extremal Betti numbers of I in Theorem 5.1.The results of our work relate to the broader context of the study of finiteness properties of idealsin an infinite polynomial ring, which are invariant under a large group of symmetries. A significantbody of research has been performed in recent years on finite generation statements, most oftenunder the designation Noetherianity up to symmetry or representation stability , and has had impor-tant applications including two (of several) recent proofs of Stillman’s conjecture on the projectivedimension (and regularity) of polynomial ideals [9, 11]. In the case of the infinite polynomial ring R ∞ = k [ x , x , · · · ], with the action of the infinite symmetric group S ∞ by coordinate permutations,ideals I ∞ ⊆ R ∞ that are S ∞ -invariant are generated by finitely many S ∞ -orbits (this is a classicalresult due to Cohen [8], rediscovered more recently in [1, 16]). It is then natural to explore finitenessbeyond the set of generators, and to understand how it is reflected in other homological invariants.To that end, we let f , . . . , f r ∈ k [ x , . . . , x n ] ⊂ R ∞ be polynomials whose S ∞ -orbits generate I ∞ ,and consider the sequence of ideals(1.9) I m = ( σ ( f i ) | ≤ i ≤ r, σ ∈ S m ) ⊆ k [ x , . . . , x m ] for m ≥ n. One can guess that the finiteness properties of I ∞ are reflected by uniform behaviors of homologicalinvariants of the ideals I m . For instance, it is shown in [19, Corollary 3.12] that the (co)dimensionof the ideals I m is computed by a linear function when m ≫
0, and it is conjectured in [19, Con-jecture 1.3] that the same result is true for the projective dimension pdim( I m ). Similarly, it isconjectured in [18, Conjecture 1.1] that reg( I m ) is a linear function when m ≫
0. In the case when f , · · · , f r are monomials, the linearity of pdim( I m ) and reg( I m ) is established by the authors in [21,Corollary 1.2] and [24, Theorem 6.1]. In Section 6, we extend these results to each of the Bettinumbers of the ideals I m , by providing a concrete recipe of how these numbers change as we vary m .Exhibiting a uniform behavior for the Betti numbers of I m , when f , · · · , f r are no longer assumedto be monomials, remains a significant open problem, and we hope that our work will inspire furtherinvestigations in this direction. Organization.
In Section 2 we introduce the necessary notation and preliminary results regardingpartitions, simplicial complexes, Betti numbers, Specht modules, and multidimensional chain com-plexes. In Section 3 we prove Theorem 1.3, and in Section 4 we explain the proof of Theorem 1.5. InSection 5 we discuss the primary decomposition, extremal Betti numbers, regularity and projectivedimension of S n -invariant monomial ideals. Finally, in Section 6 we explain how the Betti numberschange as we increase the number of variables.2. Preliminaries
In this section, we introduce some basic notation which will be used in the paper.
QUIVARIANT HOCHSTER’S FORMULA 7
Some remarks on partitions and multidegrees.
Let I be an S n -invariant monomial idealof R . Recall that P ( I ) = { λ ∈ P n | x λ ∈ I } . Since x a ∈ I if and only if part( a ) ∈ P ( I ), the set P ( I )determines the ideal I . We regard P n as a poset with the relation defined by ( a , . . . , a n ) ≥ ( b , . . . , b n )if a i ≥ b i for all i = 1 , , . . . , n . Let Λ( I ) be the set of minimal elements in P ( I ). Identifying partitionswith the corresponding monomials in R , we have that up to the action of S n , the set Λ( I ) forms aminimal set of generators of I : we have I = h λ | λ ∈ Λ( I ) i S n and no proper subset of Λ( I ) generates I . In particular, λ ∈ P n is contained in P ( I ) if and only if there is µ ∈ Λ( I ) such that λ ≥ µ .We sometimes regard a partition λ as an element of Z n . To avoid confusion, for a partition λ = ( λ , . . . , λ n ), when we denote the graded component of a module M of degree ( λ , . . . , λ n ) ∈ Z n ,we write it as M λ instead of M λ . Also, when we write partitions, we sometimes ignore “0” and identify( λ , . . . , λ n ) and ( λ , . . . , λ n , , . . . , a = ( a , · · · , a n ) ∈ Z n we write | a | = a + a + · · · + a n for the size of a .We often consider both Z n and Z s . We write e , . . . , e s for the standard vectors of Z s and write e , . . . , e n for the standard vectors of Z n . Also, for subsets F ⊂ [ s ] and G ⊂ [ n ], we write e F = P i ∈ F e i and e G = P i ∈ G e i .2.2. Simplicial complexes and their homology groups.
A simplicial complex on a finite set V is a collection ∆ of subsets of V satisfying the condition that F ∈ ∆ and G ⊂ F imply G ∈ ∆.Elements of ∆ are called faces and maximal elements of ∆ are called facets . We distinguish theempty simplicial complex ∅ from the simplicial complex { ∅ } consisting only of the empty face.If ∆ is a simplicial complex on V = { v , . . . , v n } , we write e C • : 0 ←− C − (∆) ∂ ←− C (∆) ∂ ←− C (∆) ∂ ←− · · · for the (reduced) simplicial chain complex of ∆ over a field k . Here, each C k (∆) is the k -vector spacespanned by the symbols { α F | F ∈ ∆ , | F | = k + 1 } . If we consider the total order v < · · · < v n on V then the boundary map is given by ∂ ( α F ) = X v ∈ F ǫ v ( F ) · α F \{ v } where ǫ v ( F ) = ( − |{ u ∈ F | u ≤ v }| . The homology e H i (∆) = H i ( e C • (∆)) is called the i -th reduced homol-ogy group of ∆. We note that e H − ( { ∅ } ) ∼ = k while e H i ( ∅ ) = 0 for all i .2.3. Betti numbers of monomial ideals.
It is known that when I is a monomial ideal, the Z n -graded components of Tor i ( I, k ) can be identified with reduced homology groups of certain simplicialcomplexes. We quickly recall this fact.Let I ⊂ R be a monomial ideal. Let K R • = K R • ( x , . . . , x n ) be the Koszul complex w.r.t. thevariables x , . . . , x n and K • ( I ) = K R • ( x , . . . , x n ) ⊗ R I . We have that K Ri is the free R -module whosebasis is the set { e a ∧ · · · ∧ e a i | ≤ a < · · · < a i ≤ n } , where e a ∧ · · · ∧ e a i is an element of theexterior algebra generated by e , . . . , e n . For a = ( a , . . . , a n ) ∈ Z n ≥ , we define the simplicial complex∆ I a = { F ⊂ [ n ] | x a − e F ∈ I } . One has an identification K • ( I ) a ∼ = e C • +1 (∆ I a )given by the correspondence x a − e { i ,...,ik } e i ∧ · · · ∧ e i k → α { i ,...,i k } , for i < · · · < i k . Since Tor i ( I, k ) ∼ = H i ( K • ( I )), we obtain the following formula. SATOSHI MURAI AND CLAUDIU RAICU
Theorem 2.1 ([7, Proposition 1.1]) . For any monomial ideal I ⊂ S and a ∈ Z n ≥ , Tor i ( I, k ) a ∼ = e H i − (∆ I a ) . For square-free monomial ideals, the above formula coincides via Alexander duality with Hochster’sformula (see [7, § Specht modules of hook partitions.
Here we explain a few basic facts on Specht modules.We only consider those modules for hook partitions since these are the only cases which we need.We refer the readers to [17, 25] for a general theory.A hook partition is a partition of the form ( p, q ) with p ≥ q ≥
0. Let λ = (1 + t, s − )be a hook partition with s ≥ , t ≥
0. A (Young) tableau of shape λ is an assignment of distinctpositive integers to each box in λ . We consider the following relation ∼ (extended linearly) on thevector space spanned by tableaux of shape λ .(I) For any permutation σ on [ s ] with σ ( k ) = p k and for any permutation τ on [ t ] with τ ( k ) = q k ,we have a b · · · b t a ... a s ∼ sgn( σ ) · a p b q · · · b q t a p ... a p s . (II) For any sequence of integers a , a , . . . , a s , b , . . . , b t , we have s X i =0 ( − i · a a i b · · · b t ... a i − a i +1 ... a s ∼ . Let Tab( λ ) be the set of tableaux of shape λ with entries 1 , , . . . , s + t . The quotient space S λ =(span k (Tab( λ ))) / ∼ is called the Specht module of shape λ . We say that a tableau a b · · · b t a ... a s is standard if a < · · · < a s and a < b < · · · < b t . It is well-known that standard tableaux inTab( λ ) form a basis of S λ .2.5. Multi-dimensional complexes.
We define a Z s -complex of k -vector spaces to be a complex( K • , ∂ ) where each term has a decomposition(2.1) K l = M c + ··· + c s = l K ( c , ··· ,c s ) QUIVARIANT HOCHSTER’S FORMULA 9 and the differential ∂ sends ∂ ( K ( c , ··· ,c s ) ) ⊆ s M i =1 K ( c , ··· ,c i − , ··· ,c s ) . All our complexes are finite, that is, K c = 0 for finitely many c ∈ Z s , and the spaces K c are finitedimensional. We have a decomposition ∂ = L ∂ i c with 1 ≤ i ≤ s and c ∈ Z s , where ∂ i c : K c −→ K c − e i , and the condition that ∂ is a differential translates into ∂ i c − e i ◦ ∂ i c = 0 , and ∂ j c − e i ◦ ∂ i c + ∂ i c − e j ◦ ∂ j c = 0 for i = j. We write ∂ K and ∂ K,i c when we want to emphasize the complex that ∂ is a differential of. We definea morphism of Z s -complexes f : K −→ K ′ to be a morphism of complexes which is compatible with(2.1). Equivalently, for each c ∈ Z s we have a k -linear map f c : K c −→ K ′ c , satisfying ∂ K ′ ,i c ◦ f c = f c − e i ◦ ∂ K,i c for all 1 ≤ i ≤ s, c ∈ Z s . We write com s for the category of Z s -complexes, identify com with the category vec of k -vectorspaces, and note that com is the usual category of complexes associated to vec . If we write E = V • ( ∂ , · · · , ∂ s ) for the exterior algebra on ∂ , · · · , ∂ s , then E has a Z s -grading with deg( ∂ i ) = − e i ,and the notion of a Z s -complex is equivalent to that of a finitely generated Z s -graded E -module. Wewill write grmod E for the category of such modules, and use freely the equivalence between com s and grmod E .We define the support of a Z s -complex to be the set(2.2) supp( K ) = { c ∈ Z s : K c = 0 } . For d ∈ Z s , we define the shifted complex K [ d ] by K c = K c + d , with differentials shifted accordingly.We can think of a vector space W as a Z s -complex supported at (0 s ), and we write W [ d ] for thecorresponding shift (which is supported at − d ). Example 2.2.
For a vector space W , we define the Z s -complex E = E ( W ) by E c = ( W if c ∈ { , } s ;0 otherwise . For c ∈ supp( E ) with c i = 1, we define ∂ i c : W −→ W to be multiplication by ( − c + ··· + c i . It iseasy to see that E is an exact complex (isomorphic up to shift to the tensor product of W withthe reduced chain complex of a simplex). As an object of grmod E , E can be identified with the freemodule W ⊗ k E ( − s ), with generators W in degree (1 s ). As such, E is a projective object of com s (it is also injective by [10, Proposition 7.19]).We define a Boolean Z s -complex to be a Z s -complex K which is isomorphic to E [ d ], where d ∈ Z s and E is as in Example 2.2. This is equivalent to the fact that supp( K ) = − d + { , } × n and ∂ i c : K c −→ K c − e i is an isomorphism whenever c , c − e i ∈ supp( K ). Whenever we want to emphasizethe relation between the Boolean complex K and the reduced chain complex of a simplex, we willwrite for each subset F ⊆ [ s ](2.3) K F = K − d + e F . It follows from Example 2.2 that Boolean complexes are projective (and injective) objects in com s ,which has the following useful consequence. Corollary 2.3.
Suppose that K is a Z s -complex with a filtration whose composition factors E , · · · , E r are Boolean Z s -complexes. We have that K ≃ E ⊕ · · · ⊕ E r . If K ∈ com s and L ∈ com t , then the (external) tensor product K ⊠ L is defined to be the complexin com s + t with ( K ⊠ L ) c = M d + e = c K d ⊗ k L e , and differential ∂ K ⊠ L c ( u ⊗ v ) = ∂ K ( u ) ⊗ v + ( − | d | u ⊗ ∂ K ( v ) for u ∈ K d , v ∈ L e . We note that the complex in Example 2.2 is the tensor product of the Z -complex W ≃ W with( s −
1) copies of the Z -complex k ≃ k . In general, the tensor product of a Boolean s -complex witha Boolean t -complex is a Boolean ( s + t )-complex.If F • ( K ) is an decreasing filtration of K by Z s -subcomplexes, we writegr k ( K ) = F k ( K ) / F k +1 ( K ) and gr( K ) = M k gr k ( K ) . Given filtrations F • ( K ), F • ( L ), we get an induced filtration on K ⊠ L , with F k ( K ⊠ L ) = X i + j = k F i ( K ) ⊠ F j ( L ) and gr k ( K ⊠ L ) = M i + j = k gr i ( K ) ⊠ gr j ( L ) . If G is a group, we will be interested more generally in the category com Gs of finite complexesof finite k [ G ]-modules, or equivalently, the category grmod G E of finitely generated G -equivariant Z s -graded E -modules, where the action of G on E is trivial. For s = 0, com G is the category mod G of finite G -modules. If K ∈ com Gs and L ∈ com G ′ t then K ⊠ L ∈ com G × G ′ s + t , and the discussion offiltrations is analogous in the equivariant setting.Using the natural isomorphismsHom com Gs ( E ( W ) , K ) ≃ Hom grmod G E ( W ⊗ k E ( − s ) , K ) ≃ Hom mod G ( W, K (1 s ) )we can interpet the construction of the Boolean complex E ( W ) in Example 2.2 as a functor E : mod G −→ com Gs which is left-adjoint to the functor P : com Gs −→ mod G given by P ( K ) = K (1 s ) .Since P is exact, we have that E ( W ) is projective whenever W is a projective G -module. When G is a finite group and k has characteristic zero or coprime to | G | , we have that mod G is semi-simple,and in particular E ( W ) is a projective object of com Gs for every G -module W . We get the followingequivariant version of Corollary 2.3: Corollary 2.4.
Suppose that G is a finite group, k is a field of characteristic zero or coprime to | G | , and K ∈ com Gs has a filtration with composition factors E i = E ( W i ) , where W i ∈ mod G , for i = 1 , . . . , r . We have that K ≃ E ⊕ · · · ⊕ E r . QUIVARIANT HOCHSTER’S FORMULA 11
We end this section by explaining how Corollaries 2.3 and 2.4 will be applied in our work. Supposethat F ∈ com G is an exact complex supported in non-negative degrees:0 ←− F ∂ ←− F ←− · · · ∂ r ←− F r ←− . We let U l = Im( ∂ l +1 ) and D l = F l /U l , so that ∂ l establishes an isomorphism D l ≃ U l − . We have anatural filtration on F given by the canonical truncations F l ( F ) : 0 ←− U l ←− F l +1 ←− · · · ←− F r − ←− F r ←− , with gr l ( F ) = ( D l +1 ≃ U l ) a Boolean Z -complex. By Corollary 2.3, we have an isomorphism F ≃ gr( F ) in com , which by Corollary 2.4 can be taken to be G -equivariant (that is, in com G ) if G is finite and k has characteristic zero or coprime to | G | . Choosing (not necessarily G -equivariant)sections of the quotient maps F l ։ D l , we can picture the complex F as: U U U U L L L · · · D ≃ ∂ g g ◆◆◆◆◆◆◆◆◆◆◆ D ≃ ∂ g g ◆◆◆◆◆◆◆◆◆◆◆◆ D ≃ ∂ g g ◆◆◆◆◆◆◆◆◆◆◆◆ More generally, if F i ∈ com G i for i = 1 , . . . , s , then the canonical filtrations on each F i induce afiltration on the tensor product F = F ⊠ · · · ⊠ F s ∈ com Gs , where G = G × · · · × G s . We have anisomorphism F ≃ gr( F ) in com s , and if G is finite and k has characteristic zero or coprime to | G | ,then F ≃ gr( F ) in com Gs . 3. Proof of the main theorem
The goal of this section is to prove Theorem 1.3. We first study the complex(3.1) K µ • = M a ∈ Z n , part( a )= µ ( K R • ) a , where µ ∈ P n . We note that K µ • is a complex of S n -modules, and using the notation in Section 2.5,we will show that K µ • can be thought of as an object in com S n s for an appropriate value of s , andthat K µ • has a natural filtration with composition factors that are S n -equivariant Boolean complexes.Based on the discussion in Section 2.5, this gives a decomposition of K µ • into a direct sum of Booleancomplexes, which is S n -equivariant in characteristic zero or > n . This decomposition is then the keyingredient in the proof of Theorem 1.3. Step 1.
Suppose that µ = ( a n ) with a >
0. We have K µl = span k (cid:8) σ (cid:0) ( x a − · · · x a − l x al +1 · · · x an ) · ( e ∧ · · · ∧ e l ) (cid:1) | σ ∈ S n (cid:9) . Since K R • is exact except in degree (0 n ), it follows that K µ • is also exact. We can then define U l = Im( ∂ l +1 ) and D l = K µl /U l as in Section 2.5, to get a filtration of K µ • by Boolean S n -equivariantcomplexes. To determine the isomorphism type of each D l as an S n -module, we note that it does not depend on a , and hence we can take a = 1. The natural map that associates a b · · · b n − l a ... a l −→ x b · · · x b n − l · e a ∧ · · · ∧ e a l induces an isomorphism between the Specht module S ( n − l +1 , l − ) and D l (the relations (I) correspondto the skew-symmetric property of wedge products, while the relations (II) correspond to the gen-erators ∂ l +1 ( x b · · · x b n − l e a ∧ · · · ∧ e a l ) of U l ). Using that K µl ≃ Ind S n S l × S n − l (cid:16) S (1 l ) ⊠ S ( n − l ) (cid:17) (see [12,Section 4]) we recover a special case of the filtrations in [17, §
16] known as Pieri’s rule: there is anexact sequence 0 −→ S ( n − l, l ) −→ Ind S n S l × S n − l (cid:16) S (1 l ) ⊠ S ( n − l ) (cid:17) −→ S ( n − l +1 , l − ) −→ , given by the inclusion of U l ≃ D l +1 into K µl , followed by the projection onto D l . Step 2.
We now consider the general case, when µ is of the form µ = ( µ , . . . , µ n ) = ( d p , . . . , d p s s , p s +1 ) ∈ P n , where d > · · · > d s >
0. We set d s +1 = 0 and let(3.2) X k = { x i | µ i = d k } and k [ X k ] = k [ x i : x i ∈ X k ] , for k = 1 , · · · , s + 1 , noting that | X k | = p k . By Step 1 , we have that F k = ( K k [ X k ] • ) ( d pkk ) is an object in com S pk for1 ≤ k ≤ s , which admits a filtration with composition factorsgr l ( F k ) = ( D kl +1 ≃ U kl ) = E ( S ( p k − l, l ) )[ − l ] for 0 ≤ l ≤ p k − , where U kl ≃ S ( p k − l, l ) , D kl ≃ S ( p k − l +1 , l − ) . We think of ( K k [ X s +1 ] • ) (0 ps +1 ) = k as an object in com S ps +1 , and represent it by the Specht module S ( p s +1 ) . If we let S p = S p × · · · × S p s × S p s +1 then we have(3.3) ( K R • ) µ = F ⊠ · · · ⊠ F s ⊠ S ( p s +1 ) ∈ com S p s . For (0 s ) ≤ c = ( c , · · · , c s ) ≤ ( p − , · · · , p s −
1) we define(3.4) E µ, c = gr c ( F ) ⊠ · · · ⊠ gr c s ( F s ) ⊠ S ( p s +1 ) = E (cid:0) S ( p − c , c ) ⊠ · · · ⊠ S ( p s − c s , cs ) ⊠ S ( p s +1 ) (cid:1) [ − c ] ∈ com S p s . Using (3.3) and the discussion in Section 2.5, we have that ( K R • ) µ admits a filtration with compositionfactors gr l ( K R • ) µ = M | c | = l E µ, c , for 0 ≤ l ≤ ( p −
1) + · · · + ( p s − . QUIVARIANT HOCHSTER’S FORMULA 13
Using (3.1), we have K µ • = Ind S n S p (( K R • ) µ ), and since induction is an exact functor, we have that K µ • admits a filtration withgr l ( K µ • ) = M | c | = l Ind S n S p ( E µ, c ) = M | c | = l E (cid:0) S (( p − c , c ) ,..., ( p s − c s , cs ) , ( p s +1 )) (cid:1) [ − c ] , where the last equality uses (1.7), (3.4), and the fact the construction of Boolean complexes inExample 2.2 commutes with induction. If we define(3.5) L µ, c • = E (cid:0) S (( p − c , c ) ,..., ( p s − c s , cs ) , ( p s +1 )) (cid:1) [ − c ]for (0 s ) ≤ c ≤ ( p − , · · · , p s − K µ • = M (0 s ) ≤ c ≤ ( p − , ··· ,p s − L µ, c • which is S n -equivariant when k has characteristic zero or > n .Before explaining the proof of Theorem 1.3, it will be useful to analyze an example in order toillustrate the structure of K µ • . Example 3.1.
Suppose that n = 3 and µ = (5 , , s = 2, p = 2, p = 1, and p = 0.Then K µ • is Z -complex (see (2.1)) by (3.3), and using the Z -grading on K µ • , we can picture thecomplex as(3.7) K µ (2 , t t ❥❥❥❥❥❥❥ K µ (1 , t t ❥❥❥❥❥❥❥ K µ (2 , t t ❥❥❥❥❥❥❥ j j ❚❚❚❚❚❚❚ K µ • : 0 K µ (0 , o o K µ (1 , j j ❚❚❚❚❚❚❚ t t ❥❥❥❥❥❥❥ K µ (0 , j j ❚❚❚❚❚❚❚ From (3.6), we have a direct sum decomposition K µ • = L µ, (0 , • ⊕ L µ, (1 , • , where the summands are Boolean complexes. Using (2.3), we refine (3.7) to(3.8) L µ, (1 , { } w w ♦♦♦♦♦♦♦♦ L µ, (1 , ∅ ⊕ L µ, (0 , { } { { ✇✇✇✇✇✇✇✇✇ L µ, (1 , { , } { { ✇✇✇✇✇✇✇✇✇ g g ◆◆◆◆◆◆◆◆ K µ • : 0 L µ, (0 , ∅ o o L µ, (1 , { } ⊕ L µ, (0 , { , } d d ■■■■■■■■ w w ♦♦♦♦♦♦♦♦ L µ, (0 , { } f f ◆◆◆◆◆◆◆◆ where the blue terms come from L µ, (0 , • , and the red ones from L µ, (1 , • . Notice that L µ, c F is a summandof K µ b if and only if b = c + e F . Each of the complexes L µ, (0 , • and L µ, (1 , • is isomorphic up to a shift to some number of copies of the reduced simplicial complex of a 1-dimensional simplex. As an S -representation, each of the blue modules is isomorphic to S ((2) , (1)) , and each of the red modulesis isomorphic to S ((1 , , (1)) .We are now ready to prove our main result. Proof of Theorem 1.3.
Recall that Tor i ( I, k ) can be computed as the i -th homology group of thesubcomplex of K R • given by K • ( I ) = K R • ⊗ R I. If we fix µ ∈ P n then Tor i ( I, k ) h µ i is the homology of ( K • ( I )) h µ i , which is a subcomplex of K µ • . Wewill describe ( K • ( I )) h µ i in relation to the decomposition (3.6).Consider any b ∈ supp( K µ • ) (as defined in (2.2)), and note that 0 ≤ b k ≤ p k for all k = 1 , · · · , s .Using the notation (1.5) and (3.2), we write µ \ b = µ − e G ∪ G ∪···∪ G s , for subsets G k ⊆ X k with | G k | = b k . We then have(3.9) K µ b = span k (cid:8) σ (cid:0) x µ \ b · e G ∧ · · · ∧ e G s (cid:1) | σ ∈ S n (cid:9) , where e G = e g ∧ · · · ∧ e g m for G = { g , · · · , g m } ⊆ [ n ]. Since K R • ( I ) is the subcomplex of K R • with K Rl ( I ) = span k { x a · e i ∧ · · · ∧ e i l | x a ∈ I, { i , . . . , i l } ⊆ [ n ] } , the equation (3.9) tells us that an element of K µ b appears in K R • ( I ) if and only if µ \ b ∈ P ( I ), andin that case the whole K µ b is contained in K R • ( I ). This shows that K Rl ( I ) h µ i = M | b | = lµ \ b ∈ P ( I ) K µ b . Using (3.6) and the notation (2.3) as in Example 3.1, it follows that L µ, c F is a summand of K Rl ( I ) h µ i if and only if µ \ ( c + e F ) ∈ P ( I ), which by (1.6) is equivalent to F being a face of ∆ µ, c ( I ). If weconsider the subcomplex L µ, c • ( I ) ⊆ L µ, c • defined by L µ, c l + | c | ( I ) = M F ∈ ∆ µ, c ( I ) , | F | = l L µ, c F , then it follows that(3.10) ( K R • ( I )) h µ i = M ≤ c ≤ p ( µ ) L µ, c • ( I ) , and moreover, we have from (3.5) that(3.11) L µ, c • ( I ) ∼ = e C • +1+ | c | (∆ µ, c ( I )) ⊗ k S (( p − c , c ) ,..., ( p s − c s , cs ) , ( p s +1 )) . Combining (3.10) with (3.11) and taking homology yields the desired description of Tor i ( I, k ) h µ i ,concluding the proof. (cid:3) We end this section by illustrating the proof of Theorem 1.3 with an example.
QUIVARIANT HOCHSTER’S FORMULA 15
Example 3.2.
We continue with the notation in Example 3.1, and consider the ideal I = h (4 , , , (5 , , i .When considering the subcomplex ( K • ( I )) h µ i ⊆ K µ • , the term K µ (2 , disappears since µ \ (2 ,
1) =(4 , , P ( I ). We get from (3.8) L µ, (1 , { } ( I ) v v ❧❧❧❧❧❧❧❧ L µ, (1 , ∅ ( I ) ⊕ L µ, (0 , { } ( I ) y y ssssssssss K µ • ( I ) : 0 L µ, (0 , ∅ ( I ) o o L µ, (1 , { } ( I ) ⊕ L µ, (0 , { , } ( I ) f f ▲▲▲▲▲▲▲ v v ♠♠♠♠♠♠♠♠ L µ, (0 , { } ( I ) h h ◗◗◗◗◗◗◗◗◗ The red complex L µ, (1 , • ( I ) is then the S -module S ((1 , , (1)) tensored with the reduced chain complexof two points, while the blue complex L µ, (0 , • ( I ) = L µ, (0 , remains acyclic. It follows as noted in theintroduction that Tor i ( I, k ) h (5 , , i = ( S ((1 , , (1)) if i = 2;0 otherwise . The S n -invariant part of the Betti table The goal of this section is to give a quick application of Theorem 1.3 and the Nerve Theorem,computing the S n -invariant part of Tor i ( I, k ) when I is an S n -invariant monomial ideal, and k is afield of characteristic zero or > n . More precisely, we show the following. Theorem 4.1.
Let λ , . . . , λ r ∈ P n . Then, for any µ ∈ P n , one has γ µ, i ( h λ , · · · , λ r i S n ) = dim k Tor i (( x λ , · · · , x λ r ) , k ) µ for all i. (4.1) In particular, if char( k ) = 0 or char( k ) > n , then Tor i (cid:0) h λ , · · · , λ r i S n , k (cid:1) S n ∼ = Tor i (cid:0) ( x λ , · · · , x λ r ) , k (cid:1) for all i. (4.2) Proof.
Let I = h λ , · · · , λ r i S n and J = ( x λ , · · · , x λ r ). For a subset Λ ⊂ P n we define the partitionlcm(Λ) ∈ P n by lcm(Λ) i = max { λ i | λ ∈ Λ } . Also, for a subset G ⊆ [ r ] we write lcm( G ) = lcm( { λ i | i ∈ G } ) . Consider the simplicial complex X <µ = { G ⊆ [ r ] | lcm( G ) < µ } . It follows from [3, Theorem 1.11] (see also the proof of [13, Theorem 2.1]) thatTor i ( J, k ) µ ∼ = e H i − ( X <µ ) . Then it follows that in order to prove (4.1), it suffices to show that the complexes ∆ µ, ( I ) and X <µ are homotopy equivalent, which we do next.If µ = ( d p , . . . , d p s s , p s +1 ) ∈ P n with d > · · · > d s >
0, then we have λ < µ ⇐⇒ λ ≤ µ \ e i for some i = 1 , · · · , s. It follows that the facets of X <µ are G i = { j ∈ [ r ] | λ j ≤ µ \ e i } for i = 1 , · · · , s. The Nerve Theorem (see for instance [5, Theorem 10.6]) implies that X <µ is homotopy equivalent tothe nerve of G , · · · , G s , which is the simplicial complex N ( G , . . . , G s ) = ( F ⊂ [ s ] | \ i ∈ F G i = ∅ ) . We note that T i ∈ F G i = ∅ is equivalent to the fact that for some λ j we have λ j ≤ µ \ e i for all i ∈ F. This is further equivalent to λ j ≤ µ \ e F , which shows that \ i ∈ F G i = ∅ ⇐⇒ µ \ e F ∈ P ( I ) . It follows from (1.6) that N ( G , . . . , G s ) = ∆ µ, ( I ), so X <µ is homotopy equivalent to ∆ µ, ( I ),proving (4.1).We now assume that char( k ) = 0 or char( k ) > n and prove (4.2). Using the Taylor resolutionof J [15, § i ( J, k ) a = 0 for some a ∈ Z n ≥ then a = lcm(Λ) for some subsetΛ ⊆ { λ , . . . , λ r } , and in particular a ∈ P n . Thus, to prove (4.2), it is then enough to show thatTor i ( I, k ) S n h µ i ∼ = Tor i ( J, k ) µ for all µ ∈ P n . It follows from the Littlewood–Richardson rule (see e.g., [25, Theorem 4.9.14]) that (cid:0) S (( p , q ) ,..., ( p s , qs )) (cid:1) S n ∼ = ( k if q = · · · = q s = 0;0 otherwise.Thus Theorem 1.3 and (4.1) imply the desired isomorphismTor i ( I, k ) S n h µ i ∼ = e H i − (∆ µ, ( I )) ∼ = Tor i ( J, k ) µ . (cid:3) Primary decomposition and extremal Betti numbers
The goal of this section is to describe a primary decomposition for any S n -invariant monomial ideal I , and to study the extremal Betti numbers of I . As an application, we recover using Theorem 1.3the formulas from [24] for the Castelnuovo–Mumford regularity, and for the projective dimensionof I . To formulate our results, we consider the set of extended partitions P ∞ n = { ( λ , . . . , λ n ) ∈ ( Z ≥ ∪ {∞} ) n | λ ≥ · · · ≥ λ n ≥ } where ∞ ≥ a for any a ∈ Z ≥ ∪ {∞} , for which a partial order is constructed in Section 5.3. InSection 5.1 we determine a finite subset Λ ∗ ( I ) ⊂ P ∞ n describing a natural primary decomposition QUIVARIANT HOCHSTER’S FORMULA 17 of I , and refer to Λ ∗ ( I ) as the set of dual generators of I . If ρ = ( ∞ p , d p , · · · , d p s s ) ∈ P ∞ n , with ∞ > d > · · · > d s ≥
0, then we write(5.1) ℓ ( ρ ) = p for the number of ∞ terms in ρ , and let(5.2) e ρ = (( d + 1) p + p , ( d + 1) p , · · · , ( d s + 1) p s ) . In analogy with the multigraded version of extremal Betti numbers from [2, p. 507], we say that apair ( i, λ ) ∈ [ n ] × P n is extremal in the Betti table of R/I if(a) Tor i ( R/I, k ) h λ i = 0, and(b) Tor j ( R/I, k ) h µ i = 0 for all j ≥ i and µ (cid:13) λ with | µ | − j ≥ | λ | − i .If ( i, λ ) is extremal, the extremal Betti number β i,λ ( I ) is dim k Tor i ( R/I, k ) λ (which is equal to β i,σ · λ ( I ) for all σ ∈ S n ). The main result of this section is the following. Theorem 5.1.
For any S n -invariant monomial ideal I , we have { ( i, λ ) ∈ [ n ] × P n | ( i, λ ) is an extremal pair in the Betti table of R/I } = { ( n − ℓ ( ρ ) , e ρ ) | ρ ∈ Λ ∗ ( I ) is a maximal dual generator of I } . Moreover, if ( i, λ ) is the extremal pair associated to ρ = ( ∞ p , d p , · · · , d p s s ) , then the correspondingextremal Betti number is β i,λ = (cid:0) p + p − p (cid:1) . We prove Theorem 5.1 in Section 5.4 using a reformulation of Theorem 1.3 via Alexander duality,which is explained in Section 5.2. In Section 5.5 we discuss the relationship between our results andthe combinatorics used in [24], and explain the relation of Theorem 5.1 to the study of Ext modules.5.1.
Primary decompositions of S n -invariant monomial ideals. We begin by recalling acanonical primary decomposition for a monomial ideal [15, Lemma 3.1].
Lemma 5.2.
Every monomial ideal I of R has a presentation I = Q ∩ Q ∩ · · · ∩ Q r , (5.3) where each Q i is an ideal of the form ( x a i , . . . , x a k i k ) . Moreover, such a presentation is unique if it isirredundant, i.e., if none of the ideals Q i can be omitted from (5.3) . To describe the irredundant presentation (5.3) for an S n -invariant monomial ideal, we define foreach µ = ( ∞ , . . . , ∞ , µ k , . . . , µ n ) ∈ P ∞ n with µ k < ∞ , the ideal Q µ = \ σ ∈ S n σ ( x µ k +1 k , . . . , x µ n +1 n ) . If I ⊂ R is an S n -invariant monomial ideal, its irredundant presentation (5.3) is preserved by the S n -action. Therefore, if σ ∈ S n then σ ( Q k ) = Q l for some 1 ≤ l ≤ r . This fact and Lemma 5.2imply the following. Lemma 5.3.
Let I ⊂ R be an S n -invariant monomial ideal. Then there are unique elements µ , · · · , µ t ∈ P ∞ n such that I = Q µ ∩ · · · ∩ Q µ t (5.4) and none of the ideals Q µ k can be omitted in the above presentation. We call (5.4) the irredundant decomposition of I , and call µ , . . . , µ t the dual generators of I .We write(5.5) Λ ∗ ( I ) = { µ , . . . , µ t } , and note that the condition that (5.4) is irredundant implies that(5.6) µ i µ j for 1 ≤ i = j ≤ t. Remark 5.4. If µ = ( ∞ p , d p , . . . , d p m m ) with ∞ > d > · · · > d m ≥
0, then Q µ = (cid:10)(cid:0) ( d + 1) p +1 (cid:1) , (cid:0) ( d + 1) p + p +1 (cid:1) , . . . , (cid:0) ( d m + 1) p + p + ··· + p m − +1 (cid:1)(cid:11) S n . The ideals Q µ are therefore the S n -invariant ideals generated by a set of rectangular partitions.Combining the formula for Q µ with h ( d p , . . . , d p s s ) i S n = h ( d p ) i S n ∩ h ( d p + p ) i S n ∩ · · · ∩ h ( d p + p + ··· + p s s ) i S n provides a way to compute the presentation (5.4). For example, we have h (4 , , , (5 , , i S = h (4 , , , (5 , , i S ∩ h (1 , , , (5 , , i S = h (4 , , i S ∩ h (1 , , , (5 , , i S ∩ h (1 , , , (2 , , i S = Q (3 , , ∩ Q (4 , , ∩ Q ( ∞ , , , where for the first two equalities we used h (4 , , i S = h (4 , , i S ∩ h (1 , , i S and h (5 , , i S = h (5 , , i S ∩ h (2 , , i S . We conclude that Λ ∗ ( h (4 , , , (5 , , i S ) = { (3 , , , (4 , , , ( ∞ , , } .To shed more light on the set Λ ∗ ( I ), we define O ( I ) = P n \ P ( I ) = { λ ∈ P n | x λ I } , which is the set of all partitions that are not in I . The irredundant decomposition (5.4) is related to O ( I ) as follows. For µ = ( ∞ , . . . , ∞ , µ k , . . . , µ n ) ∈ P ∞ n , let O µ = { λ ∈ P n | λ ≤ µ } . One can check that O µ = O ( Q µ ) , hence Lemma 5.3 implies that for any S n -invariant monomial ideal I ⊂ R , one has O ( I ) = [ µ ∈ Λ ∗ ( I ) O µ . (5.7)Moreover, the decomposition (5.7) is irredundant (that is, no O µ can be omitted). QUIVARIANT HOCHSTER’S FORMULA 19
A reformulation of Theorem 1.3 via Alexander duality.
With the notation in Section 3,we define for µ = ( d p , . . . , d p s s , p s +1 ) and c ≤ p ( µ ) the simplicial complex(5.8) Γ µ, c ( I ) = { F ⊆ [ s ] | µ \ ( c + e [ s ] \ F ) ∈ O ( I ) } . In other words, Γ µ, c ( I ) = { F ⊆ [ s ] | [ s ] \ F ∆ µ, c ( I ) } , that is, Γ µ, c ( I ) is the Alexander dual of ∆ µ, c .We have by [6, Lemma 5.5.3] that e H i − (∆ µ, c ( I )) ∼ = e H s − i − (Γ µ, c ( I )) . Using this isomorphism, Theorem 1.3 can be rewritten as follows.
Theorem 5.5.
Let µ = ( d p , . . . , d p s s , p s +1 ) ∈ P n and let I ⊂ R be an S n -invariant monomial ideal.We have an isomorphism of k -vector spaces Tor i ( R/I, k ) h µ i ∼ = M ≤ c ≤ p ( µ ) (cid:0) S (( p − c , c ) ,..., ( p s − c s , cs ) , ( p s +1 )) (cid:1) dim k e H s − i − | c | (Γ µ, c ( I )) , which is in addition an isomorphism of S n -modules when char( k ) = 0 or char( k ) > n . Maximal dual generators.
In what follows we introduce a partial order on the set of dualgenerators Λ ∗ ( I ) in (5.5), and explain how the maximal elements of Λ ∗ ( I ) contribute to the Bettinumbers of R/I . For µ = ( ∞ , . . . , ∞ , µ k , . . . , µ n ) ∈ P ∞ n with µ k = ∞ , we define ℓ ( µ ) = k − e µ as in (5.2), and let µ + = ( µ k + 1 , . . . , µ k + 1 , µ k , . . . , µ n ) . We note that µ + is obtained from µ by replacing ∞ with µ k + 1, and that e µ = µ + + e k + · · · + e n . We define the partial order on Λ ∗ ( I ) by µ ρ if(5.9) e µ ≤ e ρ and ℓ ( µ ) − ℓ ( ρ ) ≤ | e ρ | − | e µ | . Using the fact that(5.10) | µ + | = | e µ | − ( n − ℓ ( µ )) , we can rewrite the conditions (5.9) as(5.11) e µ ≤ e ρ and | µ + | ≤ | ρ + | . We write µ ≺ ρ if µ ρ and µ = ρ . We let Λ ∗ max ( I ) ⊆ Λ ∗ ( I ) denote the subset of maximal elementswith respect to , and call them maximal dual generators of I . Lemma 5.6.
Let I be an S n -invariant monomial ideal. (i) If µ, ρ ∈ Λ ∗ ( I ) satisfy µ ≺ ρ then ℓ ( µ ) > ℓ ( ρ ) . (ii) If ρ ∈ Λ ∗ max ( I ) then ρ + O µ for any µ ∈ Λ ∗ ( I ) \ { ρ } .Proof. (i) Write µ = ( ∞ , . . . , ∞ , µ k , . . . , µ n ) ≺ ρ = ( ∞ , . . . , ∞ , ρ l , . . . , ρ n ), and suppose by contra-diction that ℓ ( µ ) ≤ ℓ ( ρ ), or equivalently, that k ≤ l . The condition e µ ≤ e ρ implies µ m ≤ ρ m for all l ≤ m ≤ n . Since for 1 ≤ m < l we have µ m ≤ ∞ = ρ m , this shows that µ ≤ ρ , contradicting (5.6).(ii) Let µ = ( ∞ , . . . , ∞ , µ k , . . . , µ n ), ρ = ( ∞ , . . . , ∞ , ρ l , . . . , ρ n ), and suppose that ρ ∈ Λ ∗ max ( I ) and µ ∈ Λ ∗ ( I ) \ { ρ } . If µ m < ρ m for some l ≤ m ≤ n , then ρ + µ , hence ρ + O µ , as desired. We maytherefore assume that µ m ≥ ρ m for all l ≤ m ≤ n . By (5.6) we have ρ µ , hence µ l − = ∞ and thus k < l . If µ l − ≥ ρ l + 1 then we have µ + ≥ ρ + and e µ ≥ e ρ , which implies by (5.11) that µ ≻ ρ ,contradicting the maximality of ρ . It follows that µ l − < ρ l + 1, so ρ + µ , concluding the proof. (cid:3) Maximal dual generators have the following contributions to Betti numbers.
Lemma 5.7.
Let ρ = ( ∞ p , d p , . . . , d p s s ) ∈ Λ ∗ max ( I ) , with ∞ > d > · · · > d s ≥ , and let c =( p − , . . . , p s − . We have Γ e ρ, c = { ∅ } , and β n − ℓ ( ρ ) , e ρ ( R/I ) = (cid:18) p + p − p (cid:19) . Proof.
We observe that e ρ is as in (5.2), and e ρ \ ( c + e [ s ] ) = (cid:0) ( d + 1) p , d p , d p . . . , d p s s (cid:1) = ρ + . Since ρ + ≤ ρ , it follows from (5.7) that ρ + ∈ O ( I ), so ∅ ∈ Γ e ρ, c ( I ) by (5.8). To prove that Γ e ρ, c = { ∅ } ,it then suffices to check that { i } 6∈ Γ e ρ, c ( I ) for i ∈ [ s ].We fix i ∈ [ s ] and note that e ρ \ ( c + e [ s ] \{ i } ) = ρ + + e k for some k > p , hence e ρ \ ( c + e [ s ] \{ i } ) O ρ . Since ρ is maximal, we have by Lemma 5.6(ii) that ρ + O µ for all µ ∈ Λ ∗ ( I ) with µ = ρ . We get from (5.7) that e ρ \ ( c + e [ s ] \{ i } ) O ( I ), hence { i } 6∈ Γ e ρ, c ( I ) by (5.8),as desired.If we let i = n − ℓ ( ρ ) = n − p and µ = e ρ in Theorem 5.5, then we have s − i − | c | = s − ( n − p ) − n − p − s ) = − , and dim k e H s − i − | c | (Γ µ, c ( I )) = dim k e H − ( { ∅ } ) = 1 . It follows from Theorem 5.5 thatTor n − ℓ ( ρ ) ( R/I, k ) h e ρ i = S (( p +1 , p − ) , (1 p ) , ··· , (1 ps )) Restricting to the multidegree e ρ , and using the fact that each of the Specht modules S (1 pi ) hasdimension 1, it follows that β n − ℓ ( ρ ) , e ρ ( R/I ) = dim k S ( p +1 , p − ) = (cid:18) p + p − p (cid:19) , where the last equality follows from the Hook Length Formula [17, §
20] (or a direct count of thestandard tableaux in Tab(( p + 1 , p − ))). (cid:3) Example 5.8.
If we let I = h (4 , , , (5 , , i S , then as seen in Remark 5.4, we have Λ ∗ ( I ) = { (3 , , , (4 , , , ( ∞ , , } . The maximal dual generators of I are then (3 , ,
3) and (4 , , ℓ ((3 , , ℓ ((4 , , ( R/I, k ) h (4 , , i andTor ( R/I, k ) h (5 , , i respectively. More precisely, we have β , (4 , , ( R/I ) = β , (5 , , ( R/I ) = 1 , and Tor ( R/I, k ) h (4 , , i = Tor ( R/I, k ) (4 , , is 1-dimensional, whileTor ( R/I, k ) h (5 , , i = Tor ( R/I, k ) (5 , , ⊕ Tor ( R/I, k ) (5 , , ⊕ Tor ( R/I, k ) (1 , , QUIVARIANT HOCHSTER’S FORMULA 21 is 3-dimensional (see Example 1.4). We note that (3 , (4 , , , (5 , , R/I .5.4.
Extremal Betti numbers, regularity and projective dimension.
Our next goal is to givea proof of Theorem 5.1, and to derive formulas for the regularity and projective dimension of an S n -invariant monomial ideal I . We begin by establishing some preliminary results.We say that a simplicial complex ∆ is a cone with apex v if for every face F ∈ ∆, one has F ∪ { v } ∈ ∆. If ∆ is a cone then it is contractible, and e H i (∆) = 0 for all i . We will need thefollowing slight generalization of this fact. Lemma 5.9. If ∆ is a simplicial complex on the set [ n ] , with e H i − (∆) = 0 , then there exists a facet F of ∆ with F and | F | ≥ i .Proof. Let ∆ ′ be the simplicial complex whose facets are the facets of ∆ of dimension ≥ ( i − ′ and ∆ have the same faces of dimension ≥ ( i − e H i − (∆ ′ ) ∼ = e H i − (∆) = 0, andin particular ∆ ′ is not a cone with apex 1. We get that ∆ ′ contains a facet F with 1 F , which isthen a facet of ∆ with | F | ≥ i . (cid:3) We next record two more technical statements before proving Theorem 5.1.
Lemma 5.10.
Let µ = ( d p , . . . , d p s s , p s +1 ) , and define p ( µ ) as in (1.4) , and Γ µ, c ( I ) as in (5.8) , forsome c ≤ p ( µ ) . If F is a facet of Γ µ, c ( I ) with F , we let µ ′ = µ \ ( c + e [ s ] \ F ) = ( µ ′ , . . . , µ ′ n ) , and define h = min { i | µ ′ i = µ } . If ρ = ( ∞ , . . . , ∞ , ρ l , . . . , ρ n ) ∈ Λ ∗ ( I ) with ρ l = ∞ and µ ′ ≤ ρ , then we have h ≥ l and µ ≤ ρ l + 1 .Proof. Since 1 F , the first entry of c + e [ s ] \ F is positive. By (1.5), we have µ ′ h + 1 = µ h = µ . If h < l then ρ h = ∞ , and using µ ′ ≤ ρ we obtain µ \ ( c + e [ s ] \ ( F ∪{ } ) ) = µ \ ( c + e [ s ] \ F − e ) = µ ′ + e h ≤ ρ. This shows that µ \ ( c + e [ s ] \ ( F ∪{ } ) ) ∈ O ρ ⊂ O ( I ), so F ∪ { } ∈ Γ µ, c ( I ) by (5.8), contradicting thefact that F was a facet. It follows that h ≥ l , and since µ ′ ≤ ρ , we have µ ′ h ≤ µ ′ l ≤ ρ l . We get µ = µ ′ h + 1 ≤ ρ l + 1, concluding the proof. (cid:3) For the next result we recall the definition of s ( µ ) from (1.4). Lemma 5.11. If e H i − (Γ µ, c ( I )) = 0 for some µ ∈ P n and c ≤ p ( µ ) , then there exists ρ ∈ Λ ∗ max ( I ) such that (i) | c | + s ( µ ) − i ≤ n − ℓ ( ρ ) , (ii) | µ | − ( | c | + s ( µ ) − i ) ≤ | ρ + | , and (iii) µ ≤ e ρ .Proof. We note that each of n − ℓ ( ρ ), | ρ + | , e ρ , increases as ρ increases with respect to the order byLemma 5.6(i). It follows that it is enough to find ρ ∈ Λ ∗ ( I ) satisfying (i)–(iii).By Lemma 5.9, there is a facet F of Γ µ, c ( I ) such that 1 F and | F | ≥ i . We let µ =( d p , . . . , d p s s , p s +1 ), so that s = s ( µ ), and define µ ′ and h as in Lemma 5.10. Since F ∈ Γ µ, c ( I ), we have µ ′ ∈ O ( I ), so there exists an element ρ ∈ Λ ∗ ( I ) such that µ ′ ≤ ρ . We prove that ρ satisfiesconditions (i)–(iii).We first note that h = p − c , and that n = p + · · · + p s + p s +1 . Combining these observationswith the assumption c ≤ p ( µ ), we get n − | c + e [ s ] | ≥ s X j =1 ( p j − c j − ≥ p − c − h − . Since µ ′ ≤ ρ , we know from Lemma 5.10 that h ≥ ℓ ( ρ ) + 1, so | c | + s − i ≤ | c + e [ s ] | ≤ n − ( h − ≤ n − ℓ ( ρ ) , proving (i). The conclusion of Lemma 5.10 implies that µ ′ ≤ ρ + , hence | µ | − ( | c | + s − | F | ) = | µ ′ | ≤ | ρ + | , which proves (ii). Finally, if we write ρ = ( ∞ , . . . , ∞ , ρ l , . . . , ρ n ) with ρ l = ∞ , then Lemma 5.10implies µ l − ≤ · · · ≤ µ ≤ ρ l + 1 = e ρ = · · · = e ρ l − . Moreover, since µ ′ ≤ ρ , we get µ m ≤ µ ′ m + 1 ≤ ρ m + 1 = e ρ m for l ≤ m ≤ n. This shows that µ ≤ e ρ , proving (iii) and concluding our argument. (cid:3) We are now in the position to prove the main result of the section.
Proof of Theorem 5.1.
We know from Lemma 5.7 that for each ρ ∈ Λ ∗ max ( I ) we haveTor n − ℓ ( ρ ) ( R/I, k ) h e ρ i = 0 . To prove that every extremal pair for
R/I is of the form ( n − ℓ ( ρ ) , e ρ ), ρ ∈ Λ ∗ max ( I ), it then sufficesto check that for every pair ( j, µ ) with Tor j ( R/I, k ) h µ i = 0, there exists ρ ∈ Λ ∗ max ( I ) such that(5.12) j ≤ n − ℓ ( ρ ) , µ ≤ e ρ and | µ | − j ≤ | e ρ | − ( n − ℓ ( ρ )) . By Theorem 5.5, Tor j ( R/I, k ) h µ i = 0 implies that there is a c ≤ p ( µ ) such that e H s ( µ ) − j − | c | (Γ µ, c ( I )) = 0 . We apply Lemma 5.11 with i = s ( µ ) + | c | − j to find ρ ∈ Λ ∗ max ( I ) satisfying j = | c | + s ( µ ) − i ≤ n − ℓ ( ρ ) , | µ | − j = | µ | − ( | c | + s ( µ ) − i ) ≤ | ρ + | , and µ ≤ e ρ. Using the identity (5.10), these conditions are precisely the ones from (5.12).To conclude, we have to check that every pair ( n − ℓ ( ρ ) , e ρ ), with ρ ∈ Λ ∗ max ( I ), is extremal. Equiva-lently, we have to show that if (5.12) holds for ( j, µ ) = ( n − ℓ ( ρ ′ ) , e ρ ′ ) with ρ ′ ∈ Λ ∗ max ( I ), then ρ = ρ ′ .Indeed, in this case we can rewrite (5.12) as l ( ρ ) ≤ l ( ρ ′ ) , e ρ ′ ≤ e ρ, and l ( ρ ′ ) − l ( ρ ) ≤ | e ρ | − | e ρ ′ | , which implies ρ ′ ρ . Since ρ, ρ ′ ∈ Λ ∗ max ( I ), we must have ρ ′ = ρ , as desired. (cid:3) QUIVARIANT HOCHSTER’S FORMULA 23
For a quick application of Theorem 5.1, recall that for a homogeneous ideal I ⊂ R , the (Castelnuovo–Mumford) regularity of R/I isreg(
R/I ) = max { j − i | Tor i ( R/I, k ) j = 0 } and the projective dimension of R/I ispd(
R/I ) = max { i | Tor i ( R/I, k ) = 0 } . It follows that the extremal pairs in the Betti table of
R/I determine regularity and projectivedimension, and we get the following.
Corollary 5.12. If I ⊂ R is an S n -invariant monomial ideal then (i) reg( R/I ) = max {| µ + | | µ ∈ Λ ∗ ( I ) } . (ii) pd( R/I ) = max { n − ℓ ( µ ) | µ ∈ Λ ∗ ( I ) } .Proof. For µ, ρ ∈ Λ ∗ ( I ), the condition µ ρ implies that ℓ ( µ ) ≥ ℓ ( ρ ) by Lemma 5.6, and it implies | µ + | ≤ | ρ + | by (5.11). It follows that the right side of the equations (i) and (ii) only depends on themaximal dual generators. The conclusion then follows by combining Theorem 5.1 with (5.10). (cid:3) Extremal Betti numbers via
Ext modules.
The goal of this section is to explain how theextremal pairs and the extremal Betti numbers for
R/I can be recovered from the structure of themodules Ext • ( R/I, R ). Using results from [24], which determine the structure of Ext i ( I, R ) for any S n -invariant monomial ideal, we then give an alternative proof of Theorem 5.1, and we show howCorollary 5.12 is equivalent to [24, (1.3)].In analogy with the notion of extremal pair from the beginning of Section 5, we say that a pair( i, λ ) ∈ [ n ] × P n is Ext -extremal if(a) Ext i ( R/I, R ) h− λ i = 0, and(b) Ext j ( R/I, R ) h− µ i = 0 for all j ≥ i and µ (cid:13) λ with | µ | − j ≥ | λ | − i .We first show that the Ext -extremal pairs coincide with the extremal pairs, and moreover that theextremal Betti numbers can be computed via Ext modules as follows. Proposition 5.13.
We have that ( i, λ ) ∈ [ n ] × P n is extremal if and only if it is Ext -extremal.Moreover, for an extremal pair ( i, λ ) we have (5.13) β i,λ = dim k Ext i ( R/I, R ) − λ . Proposition 5.13 is a natural extension of the corresponding result in the standard-graded setting(see for instance [2, Proposition 1.1]). When I is a square-free monomial ideal (not necessarily S n -invariant), it follows from [22, Theorem 3.3] or [27, Theorem 2.6] that the multigraded components ofExt i ( R/I, R ) can be computed as Betti numbers of the Alexander dual I ∨ , in which case the equality(5.13) is equivalent to the one proved in [2, Theorem 2.8].To prove Proposition 5.13, we first establish a preliminary result. We write F • for the minimal freeresolution of R/I , so that F i = M c ∈ Z n R ( − c ) β i, c . We let F ∨• = Hom R ( F • , R ) be the dual of F • , so that Ext i ( R/I, R ) is the i -th cohomology module of F ∨• . We write ∂ i : F ∨ i −→ F ∨ i +1 for the differentials in F ∨• . Lemma 5.14. If Ext i ( R/I, R ) − λ = 0 then there exist k ≥ and µ ≥ λ , such that | µ | − | λ | ≥ k and F ∨ i + k has a minimal generator m of degree − µ , with ∂ i + k ( m ) = 0 . In particular, we have Ext i + k ( R/I, R ) − µ = 0 , and if ( i, λ ) is Ext -extremal, then every non-zero class in
Ext i ( R/I, R ) − λ isrepresented by a minimal generator of F ∨ i .Proof. By hypothesis, there exists f i ∈ ker( ∂ i ) with deg( f i ) = − λ , representing a non-zero elementof Ext i ( R/I, R ) − λ . For j ≥ = f i + j ∈ F ∨ i + j , we choose m i + j to be a minimalgenerator of F ∨ i + j of degree − µ j ≤ − deg( f i + j ), and set f i + j +1 = ∂ i + j ( m i + j ). Since F ∨• is a finitecomplex, this process ends after finitely many steps with a minimal generator m i + k of F ∨ i + k , satisfying ∂ i + k ( m i + k ) = 0. We take m = m i + k and show that µ = µ k satisfies µ ≥ λ and | µ | − | λ | ≥ k .We note that for each j = 0 , · · · , k −
1, deg( f i + j +1 ) = deg( m i + j ) = − µ j , since the differential ∂ i + j is degree-preserving. Moreover, since ∂ i + j is minimal, it follows that f i + j +1 is not a minimalgenerator of F ∨ i + j +1 , hence − µ j +1 < − µ j for j = 0 , · · · , k −
1. Starting with λ = deg( f i ), we get λ ≤ µ < µ < · · · < µ k = µ, which implies µ ≥ λ and | µ | − | λ | ≥ k , as desired.Since m is a minimal generator of F ∨ i + k , m is not a boundary, so it represents a non-zero elementof Ext i + k ( R/I, R ) − µ . If ( i, λ ) is Ext -extremal, this is only possible if µ = λ and k = 0. If f i wasnot a minimal generator of F ∨ i , then one can choose m i to be a minimal generator of F ∨ i of degree − µ < − λ , and the construction above yields a pair ( i + k, µ ) with Ext i + k ( R/I, R ) − µ = 0 and | µ | − | λ | ≥ k , contradicting the fact that ( i, λ ) was Ext -extremal, and concluding our proof. (cid:3) Proof of Proposition 5.13.
Suppose first that ( i, λ ) is an extremal pair, so F i has a minimal generatorof degree λ , and there is no pair ( j, µ ) with | µ | − j ≥ | λ | − i such that F j has a minimal generatorof degree µ . We show that every non-zero element m ∈ F ∨ i with deg( m ) = − λ represents a non-zero class in Ext i ( R/I, R ) − λ . Indeed, we know that m is not a boundary, since ∂ i − is minimal.Let f = ∂ i ( m ), and suppose by contradiction that f = 0. Since deg( f ) = deg( m ) = − λ and ∂ i is minimal, there exists a minimal generator of F ∨ i +1 of degree − µ < − λ . This corresponds to agenerator of F i +1 of degree µ > λ . This contradicts the fact that ( i, λ ) was extremal, since it implies | µ | − ( i + 1) ≥ | λ | − i . It follows that ∂ i ( m ) = 0, so m represents a non-zero class in Ext i ( R/I, R ) − λ ,as desired. By Lemma 5.14, every non-zero class in Ext i ( R/I, R ) − λ arises in this way, so (5.13) holds.To show that ( i, λ ) is Ext -extremal, suppose by contradiction that there exists a pair ( j, µ ) with µ (cid:13) λ , | µ | − j ≥ | λ | − i , and Ext j ( R/I, R ) − µ = 0. Applying Lemma 5.14 to ( j, λ ), we can find k ≥ δ ≥ µ with | δ | ≥ | µ | + k , and such that F ∨ j + k has a minimal generator of degree − δ . This showsthat F j + k has a minimal generator of degree δ , where | δ | − ( j + k ) ≥ | µ | − j ≥ | λ | − i, contradicting the fact that ( i, λ ) was extremal.Suppose now that ( i, λ ) is Ext -extremal. By Lemma 5.14, F ∨ i has a minimal generator of degree − λ , so Tor i ( R/I, k ) λ = 0. Suppose by contradiction that there exists a pair ( j, µ ) that satisfies µ (cid:13) λ , | µ | − j ≥ | λ | − i , and Tor j ( R/I, k ) µ = 0. We consider one such pair for which j is maximal, and let m denote a minimal generator of F ∨ j of degree − µ . If ∂ j ( m ) = 0 then m represents a non-zero classin Ext j ( R/I, R ) − µ (since it is not a boundary), contradicting the fact that ( i, λ ) was Ext -extremal.If ∂ j ( m ) = f = 0 then there exists a minimal generator of F ∨ j +1 of degree − δ < − µ . It follows that δ > µ (cid:13) λ , | δ | − ( j + 1) ≥ | µ | − j ≥ | λ | − i, QUIVARIANT HOCHSTER’S FORMULA 25 and Tor j +1 ( R/I, k ) δ = 0, so the pair ( j + 1 , δ ) contradicts the maximality of ( j, µ ), which concludesour proof. (cid:3) In order to apply Proposition 5.13, we recall some of the results and notation from [24]. Let z = ( z , . . . , z n ) ∈ P n and l ≥
0, with z = · · · = z l +1 . We define the module J z,l by (see [24, (2.5)])(5.14) J z,l = h z i S n / h λ | λ ≥ z and λ i > z i for some i > l i S n . (A) In [24, Corollary 2.13] it was shown that J z,l is a Cohen–Macaulay R -module of dimension l with an explicit description of Ext n − l ( J z,l , R ).(B) In [24, Main theorem] it was shown that for any S n -invariant monomial ideal I ⊆ R , thereis a finite set Z ( I ) ⊂ P n × Z (with the notation in [24, Definition 1.1], Z ( I ) = Z ( P ( I )))such that there exists a filtration of R/I whose composition factors are the modules J z,l with( z, l ) ∈ Z ( I ) and Ext i ( R/I, R ) ∼ = M ( z,l ) ∈Z ( I ) Ext i ( J z,l , R ) . We do not recall here the definition of Z ( I ), nor do we recall the description of Ext n − l ( J z,l , R ),since they are somewhat technical. We only record the following property which follows from [24,Corollary 2.13]. If we define(5.15) µ ( z, l ) = ( ∞ l , z l +1 , · · · , z n ) ∈ P ∞ n then we have(C) If z = ( z , . . . , z n ) with z = · · · = z p > z p +1 for some p > l , then λ = z + (1 n ) = ( z + 1 , · · · , z n + 1) = ^ µ ( z, l )is the unique minimal element in the set { λ ∈ P n | Ext n − l ( J z,l , R ) − λ = 0 } . Moreoverdim k Ext n − l ( J z,l , R ) − λ = (cid:0) p − l (cid:1) .We explain a relation between the set Z ( I ) and dual generators. LetΛ( J z,l ) = { λ ∈ P n | ( J z,l ) λ = 0 } (5.14) = { λ ∈ P n | λ ≥ z, λ i = z i for i ≥ l + 1 } . Since J z,l are composition factors of R/I , we have a partition O ( I ) = U ( z,l ) ∈Z ( I ) Λ( z, l ) . This allowsus to write O ( I ) = [ ( z,l ) ∈Z ( I ) O µ ( z,l ) , but this representation of O ( I ) is highly redundant.In analogy with Λ ∗ ( I ), we define the set of dual pairs (5.16) Z ∗ ( I ) = { ( z, l ) ∈ Z ( I ) | µ ( z, l ) µ ( y, u ) for ( z, l ) = ( y, u ) ∈ Z ( I ) } . We get an irredundant decomposition O ( I ) = [ ( z,l ) ∈Z ∗ ( I ) O µ ( z,l ) , and the formula (5.15) defines a bijection Z ∗ ( I ) −→ Λ ∗ ( I ). Example 5.15. If I = h (4 , , , (5 , , i S as in Remark 5.4 then we have Z ( I ) = ((0 , , , , ((1 , , , , ((2 , , , , ((3 , , , , ((4 , , , , ((3 , , , , ((4 , , , , ((4 , , , , ((1 , , , , ((2 , , , , ((3 , , , , ((2 , , , , ((3 , , , , ((3 , , , , ((2 , , , , ((3 , , , , ((3 , , , , ((3 , , , . The (significantly smaller) subset of dual pairs is Z ∗ ( I ) = { ((1 , , , , ((4 , , , , ((3 , , , } , corresponding to the dual generators of Iµ ((1 , , ,
1) = ( ∞ , , , µ ((4 , , ,
0) = (4 , , , µ ((3 , , ,
0) = (3 , , . To give another perspective on Z ∗ ( I ), we introduce a partial order on Z ( I ) by( z, l ) ≤ ( y, u ) ⇐⇒ l = u and z ≤ y. We have that ( z, l ) , ( y, u ) are incomparable if l = u , and ( z, l ) ≤ ( y, l ) if and only if µ ( z, l ) ≤ µ ( y, l ). For the proof of the next result, we assume that the reader has some familiarity with [24,Definition 1.1] and its implications, such as [24, Remark 2.3] (in particular, we use some notationfrom [24] which is not defined in this paper). Lemma 5.16.
We have that Z ∗ ( I ) = { maximal elements of Z ( I ) with respect to ≤} . Proof.
For the inclusion “ ⊆ ”, let ( z, l ) ∈ Z ∗ ( I ), and suppose by contradiction that there exists( y, l ) ∈ Z ( I ) with ( y, l ) > ( z, l ). This implies that µ ( y, l ) > µ ( z, l ), contradicting (5.16).For the reverse inclusion “ ⊇ ”, let ( z, l ) ∈ Z ( I ) be maximal with respect to ≤ , and suppose bycontradiction that ( z, l )
6∈ Z ∗ ( I ). By (5.16), there exists ( y, u ) ∈ Z ( I ) with µ ( z, l ) < µ ( y, u ), whichimplies l ≤ u . Moreover, if l = u then z < y , contradicting the maximality of ( z, l ). We thus have(5.17) l < u and z i ≤ y i for i ≥ u + 1 . We write c = z and d = y , and note that there exists x ∈ X ( I ) such that x ( c ) ≤ z and x ′ c +1 ≤ l +1.In particular, we must have x i ≤ c for all i > l + 1 (hence for i ≥ u + 1). Suppose first that c ≤ d .The condition x ( c ) ≤ z implies that x i = min( x i , c ) ≤ z i for i ≥ u + 1, which combined with (5.17)implies that x i ≤ y i for i ≥ u + 1. Since y i = d for i ≤ u + 1, this shows that x ( d ) ≤ y . Since( y, u ) ∈ Z ( I ), this forces x ′ d +1 ≥ u + 1, which contradicts (5.17) since it implies u + 1 ≤ x ′ d +1 ≤ x ′ c +1 ≤ l + 1 . Suppose now that c > d . We have that z i ≤ y i ≤ d < c for i ≥ u + 1, which combined with x ( c ) ≤ z implies that x i ≤ z i for i ≥ u + 1. Using (5.17), this shows that x i ≤ y i for i ≥ u + 1, hence x ( d ) ≤ y .Since ( y, u ) ∈ Z ( I ), we must have x ′ d +1 ≥ u + 1, hence x u +1 ≥ d + 1 > y u +1 ≥ z u +1 . Since x ( c ) ≤ z , this forces z u +1 = c . The above inequality implies d + 1 > z u +1 = c , contradictingthe fact that d < c and concluding the proof. (cid:3) QUIVARIANT HOCHSTER’S FORMULA 27
We can now explain how Corollary 5.12 is equivalent to the formulas [24, (1.3)], which assert thatreg(
R/I ) = max {| z | + l | ( z, l ) ∈ Z ( I ) } , pdim( R/I ) = max { n − l | ( z, l ) ∈ Z ( I ) } . Indeed, it follows from Lemma 5.16 that for every ( z, l ) ∈ Z ( I ) there exists ( y, l ) ∈ Z ( I ) with y ≥ z (and hence | y | ≥ | z | ), which then implies thatreg( R/I ) = max {| z | + l | ( z, l ) ∈ Z ∗ ( I ) } , pdim( R/I ) = max { n − l | ( z, l ) ∈ Z ∗ ( I ) } . Using the fact that if µ = µ ( z, l ) then | µ + | = | z | + l and ℓ ( µ ) = l , this shows that Corollary 5.12 isequivalent to the above formulas.We conclude this section with an alternative proof of Theorem 5.1. To that end, we consider theordering on Z ∗ ( I ) induced by the bijection with Λ ∗ ( I ). We have(5.18) ( z, l ) ( y, u ) ⇐⇒ z ≤ y and | z | + l ≤ | y | + u. Alternative proof of Theorem 5.1.
It follows from (B), (C) and (5.18) that ( i, λ ) is extremal if andonly if there exists a maximal ( z, l ) ∈ Z ∗ ( I ) with respect to such that i = n − l and λ = ^ µ ( z, l ).Since this is equivalent to the fact that µ ( z, l ) ∈ Λ ∗ max ( I ), and since ℓ ( µ ( z, l )) = l , this recovers thedesired description of the extremal pairs.Suppose that ( i, λ ) is an extremal pair with ( i, λ ) = ( n − l, ^ µ ( z, l )). If p is the unique intergersatisfying z = · · · = z p > z p +1 , then we have using Proposition 5.13 and (C) that(5.19) β i,λ = dim k Ext i ( J z,l , R ) − λ = (cid:18) p − l (cid:19) . If µ ( z, l ) = ( ∞ p , d p , · · · , d p s s ) with ∞ > d > · · · > d s ≥ l = p and p = p + p . This means that (5.19) agrees with the formula for the extremal Betti numbersfrom Lemma 5.7, concluding our proof. (cid:3) Varying the number of variables
In this section, we study how the multigraded Betti numbers of the ideals I m (defined in (1.9))vary with m , when f , · · · , f r are assumed to be monomials. This extends a result of the first authorfrom [21], that gives a simple recipe to determine for all m ≥ n all the multidegrees µ ∈ P m for whichTor i ( I m , k ) h µ i is non-zero. The recipe requires knowing the set { ( i, λ ) ∈ { , , . . . , n − } × P n | Tor i ( I n , k ) h λ i = 0 } , and is summarized in Theorem 6.1 below. The goal of this section is to explain how using Theorem 1.3we can determine not only which of the multigraded Betti numbers are non-zero, but to also computethem explicitly, and to describe the S m -module structure of Tor i ( I m , k ) h µ i for all m ≥ n and all µ ∈ P m . This is explained in Theorem 6.2, but before going into details we make some preliminaryconventions.Throughout this section, we identify ( a , . . . , a n ) ∈ P n and ( a , . . . , a n , m − n ) ∈ P m for m ≥ n .By this identification, if λ , . . . , λ r ∈ P n , we can regard them as elements of P m with m ≥ n , andconsider the ideals I m = h λ , . . . , λ r i S m ⊂ k [ x , . . . , x m ] . The (non-)vanishing of the multigraded Betti numbers of I m is characterized by the following theoremof the first author (see [21, Theorem 3.2]). Theorem 6.1.
Let λ , . . . , λ r ∈ P n and let I m = h λ , . . . , λ r i S m for m ∈ { n, n +1 } . For any ≤ i ≤ n and µ = ( µ , . . . , µ n , µ n +1 ) ∈ P n +1 , one has (i) if µ n +1 = 0 then Tor i ( I n +1 , k ) µ = 0 if and only if Tor i ( I n , k ) ( µ ,...,µ n ) = 0 , (ii) if < µ n +1 < µ n then Tor i ( I n +1 , k ) µ = 0 , (iii) if µ n +1 = µ n , then Tor i ( I n +1 , k ) µ = 0 if and only if Tor i − ( I n , k ) ( µ ,...,µ n ) = 0 . To analyze the dimensions of the non-vanishing Tor groups in Theorem 6.1, recall that for an S n -invariant monomial ideal I ⊂ k [ x , . . . , x n ], the numbers γ µ, c i ( I ) determine all the (multigraded)Betti numbers of I . In order to determine the Betti numbers for I m when m ≥ n , it is then enough tounderstand how each γ µ, c i ( I m ) changes when m increases. This is explained by the following simplerule. Theorem 6.2.
Let λ , . . . , λ r ∈ P n and I m = h λ , . . . , λ r i S m for m ∈ { n, n + 1 } . Let µ =( µ , . . . , µ n , µ n +1 ) ∈ P n +1 , s = s ( µ ) and c ≤ p ( µ ) . For every ≤ i ≤ n , we have (i) if µ n +1 = 0 , then γ µ, c i ( I n +1 ) = γ ( µ ,...,µ n ) , c i ( I n ) , (ii) if µ n +1 > , then γ µ, c i ( I n +1 ) = ( if c s = 0; γ ( µ ,...,µ n ) , c − e s i if c s > . We note that Theorem 6.1 can be recovered from Theorem 6.2 using Theorem 1.3. If µ n +1 > µ n then c s = 0, so Theorem 6.1(ii) follows from Theorem 6.2(ii). Proof.
We first observe that for any ρ = ( ρ , . . . , ρ n , ρ n +1 ) ∈ P n +1 , one has ρ ∈ P ( I n +1 ) ⇐⇒ ( ρ , . . . , ρ n ) ∈ P ( I n )(6.1)since both conditions in (6.1) are equivalent to the condition that ρ ≥ λ k for some k . Throughoutthe proof we will write ˆ µ = ( µ , . . . , µ n ).(i) If µ n +1 = 0 then it follows from (1.6) that ∆ µ, c ( I n +1 ) = ∆ ˆ µ, c ( I n ), and therefore γ µ, c i ( I n +1 ) = γ ˆ µ, c i ( I n ) for all i , proving (i).(ii) Suppose now that µ n +1 >
0. We first consider the case when c s = 0, and show that ∆ µ, c ( I n +1 )is a cone with apex s . Indeed, suppose that c s = 0, consider any face F ∈ ∆ µ, c ( I n +1 ) with s F ,and let µ \ ( c + e F ) = ( µ ′ , . . . , µ ′ n +1 ). We have µ \ ( c + e F ∪{ s } ) = µ \ ( c + e F ) − e n +1 = ( µ ′ , · · · , µ ′ n , µ ′ n +1 − c s = 0, so applying (6.1) twice we obtain µ \ ( c + e F ∪{ s } ) ∈ P ( I n +1 ) ⇐⇒ ( µ ′ , . . . , µ ′ n ) ∈ P ( I n ) ⇐⇒ µ \ ( c + e F ) ∈ P ( I n +1 ) . This proves that F ∪ { s } ∈ ∆ µ, c ( I n +1 ) and therefore ∆ µ, c ( I n +1 ) is a cone with apex s . We get e H i (∆ µ, c ( I n +1 )) = 0, and hence γ µ, c i ( I n +1 ) = 0, for all i .To conclude, we consider the case when c s >
0. Since c s ≤ p s − p s ≥ µ n = µ n +1 , and therefore p (ˆ µ ) = p ( µ ) − e s . We claim that(6.2) ∆ µ, c ( I n +1 ) = ∆ ˆ µ, c − e s ( I n ) . To prove (6.2), consider any subset F ⊂ [ s ] and write µ \ ( c + e F ) = ( µ ′ , . . . , µ ′ n +1 ) as before. Since p (ˆ µ ) = p ( µ ) − e s , it follows that ˆ µ \ ( c − e s + e F ) = ( µ ′ , . . . , µ ′ n ) . (6.3) QUIVARIANT HOCHSTER’S FORMULA 29
Using (6.1), we have µ \ ( c + e F ) ∈ P ( I n +1 ) ⇐⇒ ( µ ′ , . . . , µ ′ n ) ∈ P ( I n ) , which combined with (6.3) implies that F ∈ ∆ µ, c ( I n +1 ) if and only if F ∈ ∆ ˆ µ, c − e s ( I n ). This proves(6.2), showing that γ µ, c i ( I n +1 ) = γ ˆ µ, c − e s i for all i , as desired. (cid:3) Example 6.3.
Let I m = h (5 , , (2 , i S m for m ≥
2. If we apply Theorem 1.3 to I , we see that theonly numbers γ µ, c i that are non-zero are(6.4) γ (2 , , (0)0 ( I ) = γ (5 , , (0)0 ( I ) = γ (5 , , (0 , ( I ) = 1 . In particular, we haveTor ( I , k ) ∼ = Tor ( I , k ) h (2 , i ⊕ Tor ( I , k ) h (5 , i ∼ = S ⊕ S ( , ) and Tor ( I , k ) ∼ = Tor ( I , k ) h (5 , i ∼ = S ( , ) . Based on (6.4), Theorem 6.2 gives a recipe to compute all the numbers γ µ, c i for all the ideals I m with m ≥
2. For instance, when m = 4 we have γ (2 , , , , (0)0 ( I ) = γ (2 , , , , (1)0 ( I ) = γ (2 , , , , (2)0 ( I ) = 1 ,γ (5 , , , , (0 , ( I ) = γ (5 , , , , (0 , ( I ) = γ (5 , , , , (0 , ( I ) = 1 ,γ (5 , , , , (0 , ( I ) = γ (5 , , , , (0 , ( I ) = γ (5 , , , , (0 , ( I ) = 1 , and these are all the non-zero numbers γ µ, c i for I . By Theorem 1.3, the S -module structure ofTor i ( I , k ) is then computed as follows.Tor ( I, k ) ∼ = Tor ( I, k ) h (2 , , , i ⊕ Tor ( I, k ) h (5 , , , i ∼ = S ( , ) ⊕ S ( , , ) , Tor ( I, k ) ∼ = Tor ( I, k ) h (2 , , , i ⊕ Tor ( I, k ) h (5 , , , i ⊕ Tor ( I, k ) h (5 , , , i ∼ = S ( , ) ⊕ S ( , , ) ⊕ S ( , , ) , Tor ( I, k ) ∼ = Tor ( I, k ) h (2 , , , i ⊕ Tor ( I, k ) h (5 , , , i ⊕ Tor ( I, k ) h (5 , , , i ∼ = S ⊕ S ( , ) ⊕ S ( , , ) , and Tor ( I , k ) ∼ = Tor ( I , k ) h (5 , , , i ∼ = S ( , ) . By computing the dimensions of the relevant S -representations, we obtain the Betti tables of I (left) and I (right) below. 0 1total: 3 24: 1 .5: . .6: 2 2 0 1 2 3total: 18 32 19 44: 6 . . .5: . 8 . .6: 12 24 7 .7: . . 12 .8: . . . 4 Theorem 6.2(ii) tells that the representation of Tor i ( I ℓ , k ) and that of Tor i +1 ( I ℓ +1 , k ) are relatedwhen I m is generated by monomials. We expect that a similar phenomenon occurs even when I m isnot generated by monomials, and end this paper with the following question, which is inspired fromour result and a result given in [26] Question 6.4.
Let I m be as in (1.9) and let i be a sufficiently large integer. Suppose char( k ) = 0.Is is true that, for all ℓ > i , if S ( λ ,...,λ r ) is a summand of Tor i ( I ℓ , k ), then S ( λ ,...,λ r , is a summandof Tor i ( I ℓ +1 , k )? Acknowledgements
The second author would like to thank Eric Ramos and Steven Sam for helpful conversationsregarding this project. Experiments with the computer algebra software Macaulay2 [14] have providednumerous valuable insights. The first author acknowledges the support Waseda University GrantResearch Base Creation 2020C-147. The second author acknowledges the support of the NationalScience Foundation Grant No. 1901886.
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Satoshi Murai, Department of Mathematics Faculty of Education Waseda University, 1-6-1 Nishi-Waseda, Shinjuku, Tokyo 169-8050, Japan
Email address : [email protected] Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556, USAInstitute of Mathematics “Simion Stoilow” of the Romanian Academy
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