Klyachko Diagrams of Monomial Ideals

In this paper, we introduce the notion of a Klyachko diagram for a monomial ideal I in a certain multi-graded polynomial ring, namely the Cox ring R of a smooth complete toric variety, with irrelevant maximal ideal B. We present procedures to compute the Klyachko diagram of I from its monomial generators, and to retrieve the B −saturation Isat of I from its Klyachko diagram. We use this description to compute the first local cohomology module HB1(I)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${H}^{1}_{B}(I)$\end{document}. As an application, we find a formula for the Hilbert function of Isat, and a characterization of monomial ideals with constant Hilbert polynomial, in terms of their Klyachko diagram.


Introduction
Lying in the crossroads of commutative algebra and combinatorics, monomial ideals play a prominent role in the study of ideals in a polynomial ring R. Indeed, many properties of arbitrary ideals I ⊂ R are reduced to the monomial case, which can often be tackled using combinatorial tools.For instance, it is a classical result due to Macaulay in [19], that the Hilbert function of an ideal I ⊂ R coincides with the Hilbert function of its initial ideal in > (I), which is itself a monomial ideal (see for instance [8,Theorem 15.3]).Since the advent of combinatorial commutative algebra, the theory of monomial ideals has been linked with various topics in discrete mathematics, such as enumerative combinatorics, graph theory, simplicial geometry or lattice polytopes (see [10,12,1,23,13,11,7]).
The aim of this paper is to introduce the Klyachko diagram of a monomial ideal, which can be seen as a generalization of the classical staircase diagram, suited to study monomial ideals inside non-standard graded polynomial rings.More precisely, we focus on the polynomial ring R = C[x 1 , . . ., x r ] graded by the class group Cl(X) ∼ = Z ℓ of a smooth complete toric variety X.That is, r = |Σ(1)| is the number of rays of the fan Σ of X, and the degree of a variable x i is the class in Cl(X) of the torus-invariant Weil divisor D ρ i corresponding to the ray ρ i .The graded ring R can be considered as the Cox ring of the toric variety X, and it appears in the construction of a toric variety by a GIT quotient [4,Chapter 5].For instance, if we consider X = P r−1 , we recover the polynomial ring with its classical Z-grading.
Apart from the Cox ring being a generalization of the classical Z−graded polynomial ring, Cl(X)−graded R−modules correspond to quasi-coherent sheaves on X.In particular, a Cl(X)−graded ideal I ⊂ R corresponds to an ideal sheaf I on X such that H 0 (X, I(α)) ∼ = (I : where B is the irrelevant ideal of X.It is a monomial maximal ideal determined combinatorially by the fan Σ of the toric variety X.A Cl(X)−graded R−module E gives rise to an equivariant sheaf if and only if E is Z r −graded (also called fine-graded).In particular, equivariant ideal sheaves on X are in correspondence to monomial ideals in R.
In [17] and [18], Klyachko classified equivariant torsion-free sheaves on X in terms of filtered collections of vector spaces.These filtered collections, parameterized by the cones of the fan Σ, are often referred to in the literature as Klyachko filtrations (see Proposition 2.6).In [22], this device was formalized by Perling, who introduced the notion of a Σ−family, obtaining a general classification of equivariant quasi-coherent sheaves.From a geometrical point of view, these methods have been used in the last two decades to study equivariant vector bundles on toric varieties (see [15,21,5,6]).On the other hand, in [20], the present authors used the theory of Σ−families to study reflexive Cl(X)−graded R−modules from a commutative algebra perspective.
In this note, we use this construction to introduce the Klyachko diagram of a monomial ideal I ⊂ R: a family of staircase-like diagrams parametrized by the cones of Σ encoding algebraic properties of I (see for instance Example 3.5 and Figure 2).In particular, the Klyachko diagram is uniquely determined by the ideal I, up to B−saturation.We give procedures to compute the Klyachko diagram using the monomial generators of I as initial data and conversely, to determine the generators of a B−saturated ideal I sat from a given Klyachko diagram {(C σ I , ∆ σ I )} σ∈Σ .We also provide a method to compute the first local cohomology module H 1 B (I) with respect to B from the diagram {(C σ I , ∆ σ I )} σ∈Σ , which measures the saturatedness of I. We then use the Klyachko diagram to give a formula for the Cl(X)−graded Hilbert function of I sat in terms of lattice polytopes.Finally, we characterize monomial ideals I with constant Hilbert polynomial in terms of their Klyachko diagram.
Next we explain how this paper is organized.Section 2 contains all the preliminary results and definitions needed for the rest of this work, and it is divided in two parts.In Subsection 2.1, we recall the notation and basic results concerning toric varieties.In Subsection 2.2, we recall the theory of Klyachko filtrations.
The remaining two sections are the main body of the paper.In Section 3, we define the Klyachko diagram of a monomial ideal, and we establish its main properties.In Subsection 3.1, we present a procedure to obtain the Klyachko diagram from the generators of a given monomial ideal I, and we prove that it describes the collection of Klyachko filtrations of I (Proposition 3.4).As a corollary, we show how the Klyachko diagram of the sum of two monomial ideals can be computed.Conversely, in Subsection 3.2, we give a method to obtain a minimal set of generators of a B−saturated monomial ideal corresponding to a given Klyachko diagram.Finally in Subsection 3.3, we use our previous results to compute the first local cohomology module H 1 B (I) (Proposition 3.15) which measures how different I and I sat are.In the last part of this note, we give a formula for the Hilbert function of a B−saturated monomial ideal in terms of its Klyachko diagram (Proposition 4.1), and we finish characterizing the Klyachko diagram of monomial ideals with a constant Hilbert polynomial (Corollary 4.4).In particular, we characterize all one dimensional monomial ideals I ⊂ R in terms of the Klyachko diagram.

Preliminaries
In this section, we gather the basic notations, definitions and results about toric varieties needed in the sequel.We recall the notion of a Σ−family of an equivariant torsion-free sheaf, as introduced in [22], and we end specializing it to the setting of equivariant ideal sheaves.
2.1.Toric varieties.Let X be an n−dimensional smooth complete toric variety with torus T N ∼ = (C * ) n , associated to a fan Σ ⊂ N ⊗ R ∼ = R n , where N ∼ = Z n is the cocharacter lattice of T N .We denote by Σ(k) (respectively σ(k)) the set of k−dimensional cones in Σ (respectively in σ).We refer to the cones ρ ∈ Σ(1) as rays and we set n(ρ) ∈ N to be the first non-zero lattice point along ρ.We denote by M = Hom(N, Z) ∼ = Z n its character lattice and for m ∈ M , we set χ m : T N → C * the corresponding algebraic group homomorphism.For any cone σ ∈ Σ, let σ ∨ be its dual cone, let S σ := σ ∨ ∩ M be the associated semigroup of characters and C[S σ ] the corresponding C−algebra.Then U σ = Spec(C[S σ ]) ⊂ X is a T N −invariant affine subvariety of X.For any two cones τ ≺ σ ∈ Σ, there is a character m ∈ M such that S τ = S σ + Z m and we have an inclusion U τ ֒→ U σ given by the natural morphism of There is a bijection between rays ρ ∈ Σ(1) and T N −invariant Weil divisors D ρ .Furthermore, the T N −invariant Weil divisors generate the class group Cl(X) of X.Indeed, we have the exact sequence ] be a polynomial ring in |Σ(1)| variables.The Cox ring of X is the C−algebra R endowed with a grading, not necessarily standard, given by the class group Cl(X) of X.We set deg(x ρ ) := [D ρ ] ∈ Cl(X), for each ray ρ ∈ Σ(1).We write R = C[x 1 , . . ., x r ] whenever Σ(1) = {ρ 1 , . . ., ρ r } is the (ordered) set of rays of Σ.For a cone σ, we set x i , and B := B is called the irrelevant ideal.In fact, one has B = x σ | σ ∈ Σ max .
Remark 2.1.In general, the Cox ring can be defined for any variety X as the ring R(X) = [D]∈Pic(X) H 0 (X, O(D)).
In the special case when X is a smooth toric variety it coincides with the polynomial ring we defined above.
For any cone σ ∈ Σ, the localization of R at x σ is a Cl(X)−graded algebra R x σ .For any Weil divisor D = ρ∈Σ(1) a ρ D ρ , there is an isomorphism between C[S σ ] and the homogeneous [D]−graded piece (R . We have the following: we say that E is B−torsion free.2.2.Equivariant sheaves and Klyachko filtrations.Let X be a smooth complete toric variety with fan Σ and R = C[x 1 , . . ., x r ] its associated Cl(X)−graded Cox ring.In this subsection, we introduce the notion of a Σ−family to describe equivariant sheaves on X.We refer the reader to [22] and [17] for further details.Definition 2.4.For any t ∈ T N , let µ t : X → X be the morphism given by the action of T N on X.A quasi-coherent sheaf E on X is equivariant if there is a family of isomorphisms {φ t : Notice that any Z r −graded R−module is also Cl(X)−graded.In [2], Batyrev and Cox proved the following result: Proposition 2.5.Let E be a Cl(X)−graded R−module.The quasi-coherent sheaf E is equivariant if and only if E is also Z r −graded.
In [17] and [18], Klyachko observed that to any equivariant torsion-free sheaf we can associate a family of filtered vector spaces, the so-called Klyachko filtration.In what follows we recall how this family can be constructed.Let E be an equivariant sheaf on X corresponding to a Z r −graded module E. For any degree α ∈ Cl(X), the exact sequence (1) endows the homogeneous degree−α piece of E with an M −grading: Now, for any σ ∈ Σ we consider the monomial x σ , and the localized R x σ −module E x σ remains Z r −graded.As before, for any α ∈ Cl(X), (E x σ ) α is M −graded.In particular, taking α = 0 we have: (2) Recall that the semigroup S σ induces a preorder on the character lattice On the other hand, let τ ≺ σ be two cones in Σ and m ∈ M the character such that given by the localization at χ m .Thus, we have a morphism [22,Definition 4.8]).In [22,Theorem 4.9] it is proved that Σ−families characterize equivariant sheaves on X or equivalently, B−saturated R−modules.When E is torsion-free, we have the following result.
Proposition 2.6.Let E be an equivariant torsion-free sheaf of rank s and { Êσ } its associated Σ−family.The following holds: We have the following commutative diagram: Moreover, for any character m ∈ M , we have Proof.See [22,Section 4.4] and [18, Section 1.2 and 1.3].
Remark 2.7.(i) By Proposition 2.6, the Σ−family { Êσ } σ∈Σ of a torsionfree sheaf E of rank s can be seen as a filtered collection of linear subspaces of a fixed ambient vector space E. Geometrically, the vector space E can be identified with the s−dimensional vector space Γ(T N , E) m for any character m ∈ M .
(ii) The description of equivariant torsion-free sheaves given above is based on [22,Section 4].We note that our order of filtrations is reverse of that of Klyachko [17,18].In these references, the filtration is taken as a collection of linear subspaces of E(x 0 ), the fiber of E at a point in the open orbit U {0} = T N ⊂ X (see [22,Remark 4.25]).Definition 2.8.Let E be an equivariant torsion-free sheaf, the filtered collection of vector spaces {E σ m | m ∈ M } σ∈Σ given by its Σ−family is called the collection of Klyachko filtrations of E.
In this note we focus on monomial ideals I in the Cl(X)−graded Cox ring R. Since monomial ideals are naturally Z r −graded they correspond to torsion-free equivariant sheaves of rank 1.Therefore, Proposition 2.6 shows that the Σ−family of a monomial ideal I is structured as a system of vector space filtrations of a 1−dimensional vector space I ∼ = C, which can be identified with I We finish this preliminary section with an example which illustrates Proposition 2.6, and shows how to compute the collection of Klyachko filtrations of a monomial ideal.
Example 2.10.Let R = C[x 0 , x 1 , x 2 ] be the Cox ring of P 2 with fan Σ as in Example 2.2(i).Consider the monomial ideal I = (x 2 2 , x 0 x 2 , x 0 x 1 ), we will compute the Σ−family associated to I. We present I as follows: Next, we localize at x {0} = x 0 x 1 x 2 and we set R {0} := R x {0} the localized ring.For any multidegree (α , the vector space spanned by the monomial (1).To compute I {0} m we take the degree m component of (3).This yields the following exact sequence of vector spaces Thus, and there are isomorphisms φ and restricting the exact sequence (3) to degree m = (d 1 , d 2 ), we have Similarly, we obtain It only remains to compute the components in the Σ−family associated to the two dimensional cones in Σ.Let us consider σ 0 ∈ Σ(2) with rays σ 0 (1) = {ρ 1 , ρ 2 }.We set R σ 0 := R x σ 0 the localization at x σ 0 = x 0 and for any multidegree As before, taking the component of degree m = (d 1 , d 2 ) of (3) we obtain Similarly, we obtain the remaining components of the Σ−family:

Klyachko diagrams of monomial ideals
In this section, we focus our attention on monomial ideals I in the Cox ring R of a smooth complete toric variety X.Using the theory of Klyachko filtrations, we define the Klyachko diagram of I, and we show how it is determined combinatorially by the monomials generating I. Conversely, we give a method to compute a minimal set of generators of a B−saturated monomial ideal I from its Klyachko diagram.Finally, we compute the first local cohomology module H 1 B (I) for any monomial ideal I using its Klyachko diagram.
From now on, we fix a smooth complete toric variety X with fan Σ.We set r = |Σ(1)|, we denote by R = C[x 1 , . . ., x r ] its associated Cl(X)−graded Cox ring and by B its irrelevant ideal.

3.1.
From a monomial ideal to a Klyachko diagram.Let I = (m 1 , . . ., m t ) be a monomial ideal.We write the monomials , and 1 ≤ i ≤ t. and we present I as the image of a Z r −graded map as follows: (4) and there are isomorphisms φ ≥0 , be an ideal generated by a single monomial.Then, for any cone σ = cone(ρ i 1 , . . ., ρ ic ) ∈ Σ, where Proof.The lemma follows from (4), when t = 1 and using that otherwise.
We set C σ 0 := C σ (0,...,0) , and notice the inclusion C σ k ⊂ C σ 0 corresponding to (x k ) σ ⊂ R σ for any k ∈ Z r .Applying Lemma 3.1, repeatedly, we have: ≥0 and 1 ≤ i ≤ t.Then, for any cone σ ∈ Σ, Proof.It follows from ( 4) that I σ m = 0 if and only if R σ m (−k i ) = 0 for all 1 ≤ i ≤ t.By Lemma 3.1 this occurs if and only if m ∈ M \ t i=1 C σ k , and the result follows.

and we obtain
Proof.We write I = (x k 1 , . . ., x k t ) and J = (x l 1 , . . ., x l s ).Then, s I j = min{k 1 j , . . ., k t j }, s J j = min{l 1 j , . . ., l s j } and I + J = (x k 1 , . . ., x k t , x l 1 , . . ., x l s ).It follows that s I+J j = min{s I j , s J j } and then ), and the result follows.
The following example illustrates Corollary 3.6.
2 ) be a monomial ideal.We have s 0 = s 1 = s 2 = s 3 = 0 and • and the shadowed part to We compute ∆ σ 0 I .We order the monomials with respect to ρ 3 : deg 2 }), ordering the two monomials with respect to ρ 2 : 2 ) = 2.We get Similarly, we obtain ∆ Applying the same procedure for the remaining cones, we get We notice that in this example ∆ σ 1 I and ∆ σ 3 I are unbounded.
Since I is Cl(X)−graded and finitely generated, the monomials minimally generating I belong in a finite number of homogeneous pieces.
Proof.For any m ′ ∈ M , the monomial x m ′ +E is divisible by x m+D if and only if m ′ , ρ +b ρ ≥ m, ρ +a ρ for any ρ ∈ Σ(1), and the lemma follows.Now, let G = {x m 1 +k 1 , . . ., x mt+k t } be a finite set of monomials of possibly different degrees.For any E = (b ρ ) ρ∈Σ(1) ∈ Z r such that b ρ ≥ k i ρ for ρ ∈ Σ(1) and 1 ≤ i ≤ t, we define which describes the span of the monomials of G inside Finally, we can describe a finite set of generators of a B−saturated monomial ideal I corresponding to a given Klyachko diagram.Since X is smooth we can assume that Cl(X) ∼ = Z [D ρ i 1 ], . . ., [D ρ i ℓ ] ∼ = Z ℓ .Up to permutation of variables, we may also assume that i 1 = 1, . . ., i ℓ = ℓ, and for any a = (a 1 , . . ., a ℓ ) ∈ Z ℓ we set a := (a 1 , . . ., a ℓ , 0, . . ., 0) ∈ Z r .For any u, v ∈ Z ℓ we say that v u, if u i ≥ v i for 1 ≤ i ≤ ℓ, which defines a partial order.We set G 0 = ∅ and for any u ∈ Z ℓ , we define assuming we have determined G v for any v u.Since I is finitely generated, there are only finitely many degrees u ∈ Z ℓ such that G u = ∅.
If R is the Cox ring of P r−1 , and so R has the standard Z−grading, then by construction that method gives directly a minimal set of monomial generators for I.The following example illustrates the method: x 2 x 3 } and G j = ∅ for j ≥ 3. Hence, the saturated monomial ideal corresponding to this Klyachko diagram is I = (x 1 , x 2 2 , x 2 x 3 ).In more general gradings, we cannot assure that this method gives a minimal set of generators of I, but a finite set of monomials generating I.However, we can extract from it a minimal set of monomials generating I by using suitable monomial divisions.The following example illustrates this situation: 1 y 1 } is a set of generators for a saturated monomial ideal I corresponding to this Klyachko diagram.However, the first monomial divides the three last monomials.Therefore, I is minimally generated by {x 1 , x 3 0 y 1 }.3.3.Non-saturated monomial ideals.The previous subsections have shown that the theory of Klyachko diagrams is well suited to describe saturated monomial ideals, but we cannot retrieve directly information of nonsaturated monomial ideals.In this subsection, we describe the quotient I sat /I ∼ = H 1 B (I) using the Klyachko diagram {(C σ I , ∆ σ I )} σ∈Σ and the generators of I. Proposition 3.15.Let I = (x m 1 +k t , . . ., x mt+k t ) be a monomial ideal with Klyachko  for any α ∈ Cl(X).
Proof.By Lemma 3.10, there is a bijection between a monomial basis of I α (respectively of R α ) and σ∈Σmax (C σ I (α) \ ∆ σ I (α)) (respectively C 0 (α)).In the following we remark how the above formula simplifies when I is an ideal generated by monomials with no common factor.Remark 4.2.Let I ⊂ R be a monomial ideal and x k ∈ R a monomial.We recall that the Hilbert function of J = x k I is Thus, we can assume that the Klyachko diagram of I has s ρ = 0 for any ρ ∈ Σ(1) and, its Hilbert function is  2 , x 0 x 2 , x 0 x 1 ) as in Example 3.5 (i).For any a ∈ Z, we set a = (a, 0, 0) and ∆ σ 0 I (a) = {(0, 0)} ∆ σ 1 I (a) = {(a, 0), (a − 1, 1)} ∆ σ 2 I (a) = ∅.Since s 0 = s 1 = s 2 = 0, by (7) we have the following Hilbert function: In particular, the Hilbert polynomial of R/I is P R/I ≡ 3 constant.
In the following result we characterize the Klyachko diagram of a monomial ideal I with constant Hilbert polynomial.In particular, notice that I is necessarily generated by monomials without common factors.Proof.The left implication follows directly from Proposition 4.1 and Remark 4.2.Conversely, if s ρ > 0 for some ρ ∈ Σ(1), then there is σ ∈ Σ max such that ρ ∈ σ(1) and C 0 (α) \ C σ I (α) increases with α, and P R/I would not be constant.Now, assume that there is some σ ∈ Σ max such that ∆ σ I is not finite.By construction, ∆ σ I contains a set ∆ ′ of the form {m ∈ M | m, ρ i 1 = k 1 , . . ., m, ρ i l = k l , m, ρ i l+1 ≥ k l+1 , . . ., m, ρ ic ≥ k c }.

Declarations
Funding or conflicts of interests.The authors have no competing interests to declare that are relevant to the content of this article.

{0}m
for any character m ∈ M .Remark 2.9.Let {I σ m | m ∈ M } σ∈Σ be the collection of Klyachko filtrations of a monomial ideal I. Let m 0 ∈ M be a character and identify I with I {0} m 0 .For each σ ∈ Σ and m ∈ M , the linear subspace I σ m ⊂ I can be either I σ m ∼ = I or I σ m = 0. Therefore, the collection of Klyachko filtrations of a monomial ideal is characterized by attaching to each cone σ ∈ Σ, the set of characters {m ∈ M | I σ m = 0}.

{0} m : I {0} m
∼ = I.Our first objective is to describe the subspaces I σ m ⊂ I for any character m = (d 1 , . . ., d n ) and any cone σ ∈ Σ.As observed in Remark 2.9, we want to characterize the sets of characters {m ∈ M | I σ m = 0} for each cone σ ∈ Σ.Each of this sets can be seen as the staricase diagram for the inclusion

Definition 3 . 3 .
We call the collection of pairs {(C σ I , ∆ σ I )} σ∈Σ the Klyachko diagram of I. Observe that each of the pairs (C σ I , ∆ σ I ) depicts a staircase diagram of the inclusion I σ ⊂ R σ (see also Example 3.5 below).Precisely, we have the following proposition showing that the Klyachko diagram characterizes the Σ−family of I. Proposition 3.4.Let I = (m 1 , . . ., m t ) ⊂ R be a monomial ideal with m i = x k i for k i ∈ Z r ≥0 and 1 ≤ i ≤ t.Let {(C σ I , ∆ σ I )} σ∈Σ be the Klyachko diagram of I.Then, for any cone σ ∈ Σ,

Figure 2 .
Figure 2. The Klyachko diagram of Figure 1 represented together in a single figure.The shadowed region corresponds to each set C σ i I \ ∆ σ i I for i = 0, 1, 2.
otherwise which coincides with the Σ−family computed in Example 2.10.

Finally, ( 5 )
follows from a comparison with the description of I σ m in Proposition 3.2.Combining Propositions 3.2 and 3.4, the next result shows how to obtain the Klyachko diagram of the sum of two monomial ideals.Corollary 3.6.Let {(C σ I , ∆ σ I )} and {(C σ J , ∆ σ J )} be the Klyachko diagrams of two monomial ideals I and J, respectively.Then, the Klyachko diagram of I + J is given by 

Figure 3 .
Figure 3.The part of the Klyachko diagram associated to the cone σ 0 of I, J and I + J respectively.The dotted part corresponds to C σ 0• and the shadowed part to C σ 0 • \ ∆ σ 0 • .

Example 4 . 3 .
Let R = C[x 0 , x 1 , x 2 ] be the Cox ring of P 2 and I = (x 2

Remark 4 . 5 .
Notice that by Corollary 4.4 we have characterized all monomial ideals I ⊂ R with dim R/I = 1, in terms of the Klyachko diagram.We finish by illustrating Corollary 4.4 with the following example.
be the Cox ring of P 3 (See Example 2.2(i)).