Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 2, 455-467.

Initially Koszul Algebras

Stefan Blum

FB6 Mathematik und Informatik, Universität – GHS – Essen Postfach 103764, 45117 Essen, Germany e-mail: stefan.blum@uni-essen.de

Introduction

In this paper we study initially Koszul algebras. Let R = K[X_{1}, … X_{n}]/I be a homogeneous K-algebra where K is a field and I ⊂ (X_{1}, …, X_{n})^{2} is graded ideal with respect to the standard grading deg(X_{i}) = 1. Such an algebra R is Koszul, if its residue class field has an R-free linear resolution. So far, Koszul algebras have been discussed in several contexts. In [9] Fröberg gives a survey on this subject.

An effective method to show that an algebra is Koszul has been introduced by Conca, Trung and Valla [6]. They have defined Koszul filtrations, that is a family F of ideals generated by linear forms with the following properties: The ideal (0) and the maximal homogeneous ideal 𝔪 of R belong to F, and for every I ∈ F, I ≠ (0), there exists J ∈ F such that J ⊂ I, I/J is cyclic and J: I ∈ F. It is easy to see that an algebra which admits a Koszul filtration is Koszul (see [6]).

We call a K-algebra R initially Koszul (i-Koszul for short) with respect to a sequence x_{1}, …, x_{n} ∈ R_{1}, if the flag F = {(x_{1}, …, x_{i}): i = 0, …, n} is a Koszul filtration of R. Conca, Rossi and Valla have proved that i-Koszulness implies a quadratic Gröbner basis with respect to the reverse lexicographic order on K[X_{1}, …, X_{n}] induced by X_{1} < … < X_{n} (see [5]).

In the first section of the article we give a condition on in(I) which characterizes i-Koszulness of R with respect to X_{1} + I, …, X_{n} + I. Using this criterion we show that i-Koszulness is preserved under tensor products over K. Moreover, if R is i-Koszul and the defining ideal I is generated by monomials of degree 2, then the d-th Veronese subring of R is again i-Koszul.

In Section 3 we study algebras for which generic flags are Koszul filtrations. We will see that this is equivalent to the property that I has a 2-linear resolution. Furthermore we

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

discuss algebras which are i-Koszul with respect to any K-basis of R_{1}. We call such algebras universally initially Koszul (u-i-Koszul for short). In case that K is algebraically closed and char(K) ≠ 2 we classify all u-i-Koszul algebras, showing that I = (0) or I = (X_{1}, …, X_{n})^{2} or I = (g^{2}) for some linear form g.

In the last section we study homogeneous semigroup rings. Let G = {α_{1}, …, α_{k}} be a minimal system of generators of an affine semigroup in ℕ^{n}. We say a semigroup ring R is i-Koszul, if R is i-Koszul with respect to the semigroup generators X^{α_{1}}, …, X^{α_{k}}. R is said to be u-i-Koszul, if R is i-Koszul with respect to all permutations of the semigroup generators.

We consider natural shellability of the divisor poset Σ of R which is closely related to Λ-shellability in [1]. Let λ : G → Λ be a map which totally orders the generators. For any semigroup element α the lexicographic order on Λ^{r} gives a linear order > on the maximal chains of the interval [1, α]. R is said to be naturally shellable, if for each semigroup element α the interval [1, α] is shellable with order >. Using a lemma of Hibi we show that an i-Koszul semigroup ring is natural shellable. We also show that a u-i-Koszul semigroup is already a polynomial ring.

The author is grateful to Prof. Herzog for several helpful comments regarding this paper.

1. Notation and definitions and background

In this paper S = K[X_{1}, …, X_{n}] denotes always the polynomial ring and 𝔪 = (X_{1}, …, X_{n}) the graded maximal ideal of S. We set R = S/I where I ⊂ 𝔪^{2} is a homogeneous ideal. We recall from [6] the following definition:

Definition 1.1. Let R be a homogeneous K-algebra. A family F of ideals of R is called a Koszul filtration of R, if:

(a) Every ideal J ∈ F is generated by linear forms,

(b) The ideal (0) and the maximal homogeneous ideal of R belong to F and

(c) For every J ∈ F, J ≠ 0, there exists L ∈ F such that L ⊂ J, J/L is cyclic and L : J ∈ F.

The following is noted in [6].

Proposition 1.2. Let F be a Koszul filtration of R. Then Tor_{i}^{R}(R/J, K)_{j} = 0 for i ≠ j and for all J ∈ F. In particular, the homogeneous maximal ideal of R has a system of generators x_{1}, …, x_{n} such that all ideals (x_{1}, …, x_{j}) with j = 1, …, n have a linear R-free resolution and R is Koszul.

Definition 1.3. Let x_{1}, …, x_{n} ∈ R_{1}. We call R initially Koszul (i-Koszul for short) with respect to x_{1}, …, x_{n}, if F = {(x_{1}, …, x_{i}): i = 0, …, n} is a Koszul filtration.

In order to simplify notation we say that R = S/I is i-Koszul, if R is initially Koszul with respect to X_{1} + I, …, X_{n} + I. Koszul filtrations as in 1.3 which are generated by a flag of linear subspaces of R, first considered in [4], are called Gröbner flags. The reason for this naming is the following result.

Theorem 1.4. [Conca, Rossi, Valla] Let R = K[X_{1}, …, X_{n}]/I be i-Koszul. Then I has a quadratic Gröbner basis with respect to the reverse lexicographic order induced by X_{1} < X_{2} < … < X_{n}.

By 1.2 any Koszul filtration of R contains a flag. Thus i-Koszulness is equivalent to the existence of a Koszul filtration which is as smallest as possible.

2. Characterization of i-Koszulness

In this section > denotes the reverse lexicographic order induced by X_{1} < X_{2} < … < X_{n}. The following result, which was shown independently in [5], characterizes i-Koszulness in terms of initial ideals.

Theorem 2.1. The following statements are equivalent:

(a) R = K[X_{1}, …, X_{n}]/I is i-Koszul.

(b) R′ = K[X_{1}, …, X_{n}]/in_{>}(I) is i-Koszul.

(c) I has a quadratic Gröbner basis with respect to > and if X_{i}X_{j} ∈ in_{>}(I) for some i < j, then X_{i}X_{k} ∈ in_{>}(I) for all i ≤ k < j.

For the proof of 2.1 we need the following property concerning the chosen term order >.

Lemma 2.2. Let I ⊂ S be a graded ideal and set S̄ = K[X_{2}, …, X_{n}]. Let σ : S → S̄ be the K-algebra homomorphism with X_{1} ↦ 0, X_{i} ↦ X_{i} for i > 1. Suppose that g_{1}, …, g_{t} is a Gröbner basis of I such that X_{1} ∤ in(g_{i}) for i = 1, …, r and X_{1} | in(g_{i}) for i = r + 1, …, t. Then σ(g_{1}), …, σ(g_{r}) is a Gröbner basis of Ī = (σ(f): f ∈ I). In particular, it holds \overline{in(I)} = in(Ī).

Proof. We use Buchberger’s criterion. Since > is the reverse lexicographic order we have for any f ∈ S: If X_{1} | in(f), then X_{1} | f (see [7] 15.4). Thus we get S(σ(g_{i}), σ(g_{j})) = σ(S(g_{i}, g_{j})) for all i, j ∈ {1, …, r}, i ≠ j and the assertion follows immediately. □

We return now to 2.1.

Proof. We prove the equivalence of (a) and (b) by induction on n. The case n = 1 is trivial. Let x_{i} = X_{i} + I and x′_{i} = X_{i} + in(I) for i = 1, …, n. Note that R is i-Koszul, if and only if

(i) R/x_{1}R is i-Koszul and (ii) 0 : x_{1} = (x_{1}, …, x_{k}) for some k. Using in(X_{1} + I) = (X_{1}) + in(I) ([7] 15.12) and 2.2 we see that (i) is equivalent to R′/x′_{1}R′ being i-Koszul. Since in(I : X_{1}) = in(I) : X_{1} (see [7] 15.12) we get 0 : x′_{1} = (x′_{1}, …, x′_{k}) if and only if (ii) holds. This proves the equivalence of (a) and (b). For the equivalence of (b) and (c) we need □

Proposition 2.3. Let R = S/I where I = (m_{1}, …, m_{r}) is generated by monomials of degree 2. Then the following statements are equivalent:

(a) R is i-Koszul.

(b) If X_{i}X_{j} ∈ I for some j > i, then X_{i}X_{k} ∈ I for all i ≤ k < j.

Proof. Let x_{k} = X_{k} + I for k = 1, …, n and J_{i} = (x_{1}, …, x_{i}) for i = 0, …, n.

(a) implies (b). If X_{i}X_{j} ∈ I with i < j, then x_{i}x_{j} = 0 and so x_{j} ∈ J_{i−1} : J_{i}. Since R is i-Koszul it follows J_{i−1} : J_{i} = J_{l} for some l ≥ i − 1. But then for each i ≤ k < j we get x_{i}x_{k} ∈ J_{i−1}. Therefore X_{i}X_{k} − X_{l}X_{s} ∈ I for some l ≤ i − 1 and some s. Since I is a monomial ideal this implies X_{i}X_{k} ∈ I.

(b) implies (a). We have to show that J_{i−1} : J_{i} = (x_{1}, …, x_{k(i)}) for each i = 1, …, n. Let u ∈ J_{i−1} : (x_{i}), u ≠ 0. Since I is a monomial ideal we may assume that u is a monomial. It is clear that J_{i−1} ⊂ J_{i−1} : J_{i}. So we assume u ∉ J_{i−1}. It follows that ux_{i} = 0. There are k ≤ l such that X_{k}X_{l} ∈ I and X_{k}X_{l} | uX_{i}. If i ≠ k and i ≠ l, we have u = 0 which is a contradiction. Since u ∉ J_{i−1} it follows that i = k and u ∈ (x_{l}). Condition (b) implies that (x_{i}, …, x_{l}) ⊂ J_{i−1} : x_{i} which yields the assertion. □

3. Applications and examples

In this chapter we use the criterion of Section 2 to show that certain algebras are i-Koszul. First we need some notation.

Definition 3.1. Let m ∈ S be a monomial. We write max(m) for the largest index i such that X_{i} | m. A set M of monomials is called (combinatorially) stable, if for every m ∈ M and j < max(m) the monomial (X_{j}/X_{max(m)})m ∈ M.

With 2.3 we get immediately

Corollary 3.2. Let I be generated by monomials of degree 2 and G(I) the set of minimal generators. If G(I) is stable, then R = S/I is i-Koszul.

Moreover we observe that i-Koszulness is compatible with tensor products over K.

Proposition 3.3. If R = K[X_{1}, …, X_{n}]/I and R′ = K[Y_{1}, … Y_{m}]/J are i-Koszul algebras, then T = R ⊗_{K} R′ is also i-Koszul.

Proof. By 2.1 there are Gröbner bases f_{1}, …, f_{k} of I and g_{1}, …, g_{l} of J, such that in(I) and in(J) satisfy condition 2.1(c). It is T = K[X_{1}, …, X_{n}, Y_{1}, …, Y_{m}]/Q with Q = IT + JT. We take the reverse lexicographic order on K[X_{1}, …, X_{n}, Y_{1}, …, Y_{m}] induced by X_{1} < … < X_{n} < Y_{1} < … < Y_{n}. It follows immediately from the Buchberger criterion that f_{1}, …, f_{k}, g_{1}, …, g_{l} form a Gröbner basis of Q. Thus condition (b) of 2.3 is satisfied for in(Q) and by 2.1 we get the assertion. □

Theorem 3.4. Let I be generated by monomials. If R = S/I is i-Koszul, then the d-th Veronese subring R^{(d)} is i-Koszul for all d > 0.

Proof. We first consider the case R = S. Let M be the set of all monomials of degree d in S. We order the elements of M such that m_{1} >_{lex} m_{2} >_{lex} … >_{lex} m_{t}. Writing S^{(d)} ≅ K[Y_{1}, …, Y_{t}]/J each monomial m_{l} can be identified with a residue class y_{l} = Y_{l} + J. Thus we define J_{l} := (m_{1}, …, m_{l}) for l = 0, …, t. We have to show that for every l = 1, …, t the ideal J_{l−1} : J_{l} is generated by an initial sequence of the m_{i}’s. We set

M_{l} = {m ∈ M: X_{r}|m for some r ≤ l}

for l = 1, …, t and M_{0} = ∅. The elements of each M_{l} form an initial sequence m_{1}, m_{2}, …, m_{i_{l}}. We claim that

J_{l−1} : (m_{l}) = (M_{max(m_{l})−1})

which yields the assertion. In case l = 1 there is nothing to prove, thus we may assume l > 1. Let s = max(m_{l}) − 1. We write m_{l} = X_{i_{1}} ⋯ X_{i_{d}} with i_{1} ≤ … ≤ i_{d} = s + 1. Let u ∈ J_{l−1} : (m_{l}). We may assume that u is a monomial. Then we have that um_{l} = wm_{r} for some monomial w and r ∈ {1, …, l−1}. We write m_{r} = X_{j_{1}} ⋯ X_{j_{d}} with j_{1} ≤ … ≤ j_{d}. Since m_{r} >_{lex} m_{l}, there exists q ∈ {1, …, d} such that j_{m} = i_{m} for all m < q and j_{q} < i_{q} ≤ s + 1. The equation um_{l} = wm_{r} implies

uX_{i_{q}} ⋯ X_{i_{d}} = wX_{j_{q}} ⋯ X_{j_{d}}

and thus we have X_{j_{q}}|u which yields u ∈ (M_{s}). Conversely, let u ∈ M_{s}. Then there exists r ∈ {1, …, s} such that X_{r}|u. We define w = X_{i_{1}} ⋯ X_{i_{d−1}}X_{r}. It follows w >_{lex} m_{l} and hence w ∈ J_{l−1}. Since

um_{l} = (\frac{u}{X_{r}} X_{i_{d}})w,

it follows that u ∈ J_{l−1} : (m_{l}).

We now consider the general case R = S/I. Let x_{i} = X_{i} + I for i = 1, …, n. Since I is monomial ideal, the set of all monomials which do not belong to I forms a K-basis of R. Thus each monomial u = x_{j_{1}}x_{j_{2}} … x_{j_{r}} ∈ R is either 0 or has a unique presentation u = X_{j_{1}}X_{j_{2}} … X_{j_{r}} + I. Therefore we may identify each monomial with its residue class. We have the following relations:

(∗) For any two non-zero monomials m, m′ ∈ R we have mm′ = 0 if and only if there are i, j ∈ {1, …, n} such that X_{i} | m, X_{j} | m′ and X_{i}X_{j} ∈ I.

R^{(d)} is generated as a K-algebra by the set M of all non-zero monomials of degree d in R. As in the first case we order the monomials of M by m_{1} >_{lex} m_{2} >_{lex} … >_{lex} m_{t} and set J_{i} = (m_{1}, …, m_{i}) for i = 0, …, t. We define

N(m_{l}) = {m ∈ M: there exists i ≤ j with X_{i}|m_{l} , X_{j}|m and X_{i}X_{j} ∈ I}

and assert that

J_{l−1} : (m_{l}) = (M_{max(m_{l})−1}, N(m_{l}))

for l = 1, …, t. Let a ∈ J_{l−1} : (m_{l}), a ≠ 0. We may assume that a is a monomial. There are two cases to consider:

(a) am_{l} = 0. We have a relation as in (∗). If a ∉ (M_{max(m_{l})−1}) then, for each index t with X_{t} | a, it holds t ≥ max(m_{l}). Thus, if X_{i} | m_{l} and X_{j} | a with X_{i}X_{j} ∈ I, it follows i ≤ j which yields a ∈ N(m_{l}).

(b) am_{l} ≠ 0. We have am_{l} = bm_{i} for some monomial b ∈ R^{(d)} and some i < l. There is a K-linear, injective map σ : R = S/I → S with m + I ↦ m for all non-zero monomials m ∈ R. If mm′ ≠ 0 for two monomials m, m′ ∈ R, we get σ(m)σ(m′) = σ(mm′). Let π : S → R = S/I be the natural epimorphism. Then it holds π ∘ σ = id_{R}. Since σ and π respect the standard grading, these maps restrict to R^{(d)} respectively S^{(d)}. We apply σ to the equation above and, since am_{l} ≠ 0, obtain that σ(a)σ(m_{l}) = σ(b)σ(m_{i}) in S^{(d)}. The case R = S yields σ(a) ∈ (M_{max(σ(m_{l}))−1}). Applying π we get a ∈ (M_{max(m_{l})−1}).

The converse inclusion (M_{max(m_{l})−1}, N(m_{l})) ⊂ J_{l−1} : (m_{l}) follows immediately from the case R = S and the relations in (∗).

It remains to show that for all l = 1, …, t the ideal J_{l−1} : (m_{l}) is generated by an initial sequence m_{1}, …, m_{k(l)}. Since the elements of M_{max(m_{l})−1} form already an initial sequence it suffices to prove the following: If m_{s} ∈ N(m_{l}) for some s, then m_{s−1} ∈ M_{max(m_{l})−1} ∪ N(m_{l}). Let m_{l} = X_{i_{1}} … X_{i_{d}} with i_{1} ≤ … ≤ i_{d}. It is i_{d} = max(m_{l}). Since m_{s} ∈ N(m_{l}) there are i ≤ j with X_{i}|m_{l} and X_{j}|m_{s} and X_{i}X_{j} ∈ I. By the chosen order we have m_{s−1} >_{lex} m_{s}. Thus there exists a k with X_{k} | m_{s−1} and k ≤ j. If k < i_{d}, we have m_{s−1} ∈ M_{max(m_{l})−1}. Otherwise we have i ≤ k ≤ j. Since R is i-Koszul we have X_{i}X_{k} ∈ I by 2.3. This yields m_{s−1} ∈ N(m_{l}). □

We now consider

Definition 3.5. (see e.g. [2]) Let L be a finite, distributive lattice, and K[{X_{α}}_{α∈L}] the polynomial ring over K. Consider the ideal I_{L} = (X_{α}X_{β} − X_{α∧β}X_{α∨β}: α, β ∈ L) of K[{X_{α}}_{α∈L}]. The quotient algebra

R_{K}[L] = K[{X_{α}}_{α∈L}]/L

is called the Hibi ring of L over K.

Hibi has shown that I_{L} has a quadratic Gröbner basis for any term order which selects, for any two incomparable elements α, β ∈ L, the monomial X_{α}X_{β} as the initial term of X_{α}X_{β} − X_{α∧β}X_{α∨β} (see [10]). Such a term order > is, for example, the reverse lexicographic term order induced by a total ordering of the variables satisfying X_{α} < X_{β}, if rank(α) > rank(β) (see [2]). We get the following characterization:

Remark 3.6. Let L be a finite distributive lattice and > a term order on S = K[{X_{α}}_{α∈L}] as above. Then the Hibi-Ring R = S/I_{L} is i-Koszul if and only if R is a polynomial ring.

Proof. If I_{L} ≠ (0), we have X_{α}X_{β} ∈ in(I) where α and β are some incomparable elements of L, say X_{α} < X_{β}. Since R is i-Koszul, it follows X_{α}^{2} ∈ in(I) by 2.1. But this would mean that α is incomparable with itself, a contradiction. □

4. u-i-Koszulness

Let R be i-Koszul. In 1.2 we have seen that K has a linear R-free resolution. If we consider R as an S-module, we can study the minimal S-free resolution of R. For the next statement we take Gin(I) with respect to the reverse lexicographic order induced by X_{1} > … > X_{n}. It holds

Proposition 4.1. Let K be an infinite field, char(K) ≠ 2, I ⊂ S a graded ideal and I ≠ (0). The following statements are equivalent:

(a) I has a 2-linear S-resolution.

(b) S/ Gin(I) is i-Koszul.

(c) S/ Gin(I) is Koszul.

Proof. We use some results about Gin(I). It is known that Gin(I) is a Borel-fixed ideal and reg Gin(I) = reg(I) (see [7] 20.21). Since char(K) ≠ 2 by hypothesis, we obtain that Gin(I)_{2} is stable (see [7]15.23b). Let us assume (a). Then we have reg(I) = 2 = reg Gin(I) which implies that Gin(I) is generated in degree 2. Using 3.2 S/ Gin(I) is i-Koszul. This is condition (b) which implies (c) by 1.2. Assuming (c) we have that Gin(I) is generated in degree 2. Since by hypothesis K is an infinite field and Gin(I)_{2} is stable, we can use Prop.10 in [8] which yields that Gin(I) is 2-regular. In case I ⊂ 𝔪^{2} this implies reg(Gin(I)) = 2 = reg(I). Thus I has a 2-linear resolution. □

Proposition 4.1 can be interpreted as follows:

Corollary 4.2. I has a 2-linear resolution if and only if all generic flags are Gröbner flags.

We may now ask for which algebras all flags are Gröbner flags. This leads us to the following

Definition 4.3. A K-algebra R = S/I is called universally initially Koszul (for short u-i-Koszul), if R is i-Koszul with respect to every K-basis x_{1}, …, x_{n} ∈ R_{1}.

In the case that R is u-i-Koszul the property of i-Koszulness is preserved under any change of coordinates in S_{1}. Since this is a strong condition, we can classify all u-i-Koszul algebras in the following case:

Theorem 4.4. Let K be algebraically closed, char(K) ≠ 2 and I ⊂ 𝔪^{2}. Then R = S/I is u-i-Koszul if and only if I = (g^{2}) for some linear form g ∈ S_{1} or I = 𝔪^{2}.

We need some preparation.

Lemma 4.5. Let R be u-i-Koszul and x ∈ R_{1}. Then R/xR is also u-i-Koszul.

Proof. Let R̄ = R/xR and x_{2}, …, x_{n} ∈ R̄_{1} an arbitrary K-basis of R̄_{1}. We have to show that R̄ is i-Koszul with respect to this sequence. Since R is u-i-Koszul, R is i-Koszul with respect to x, x_{2}, …, x_{n}. This yields the assertion. □

Lemma 4.6. Let R be u-i-Koszul, char(K) ≠ 2 and let N ⊂ R_{1} denote the set of all zerodivisors in R_{1}. Then N is a linear subspace of R_{1} and N^{2} = 0.

Proof. Since R is u-i-Koszul we have (x) ⊂ 0 : (x) for all x ∈ N. This implies x^{2} = 0 for all x ∈ N. Thus, for x, y ∈ N we have (x + y)(x − y) = x^{2} − y^{2} = 0 and therefore x + y ∈ N. Since char(K) ≠ 2, it follows N^{2} = 0. □

Lemma 4.7. Let I = (L^{2}) for some linear subspace L of S_{1}. Then R = S/I is u-i-Koszul if and only if dim_{K} L ∈ {0, 1, n}.

Proof. Let R be u-i-Koszul. After a change of coordinates we may assume that L = (X_{1}, …, X_{i}) with i = dim_{K} L and I = (X_{1}, …, X_{i})^{2}. If i ∉ {0, 1, n}, we interchange X_{i} and X_{i+1}. We obtain a new defining ideal J with X_{1}X_{i+1} ∈ J, but X_{1}X_{i} ∉ J which is a contradiction to i-Koszulness of S/J by 2.3. Conversely, let i = dim_{K} L ∈ {0, 1, n}. If i = 0, there is nothing to prove. If i = 1, then I = (g^{2}) for some g ∈ S_{1}. For any transformation we obtain a new defining ideal J = (h^{2}) with h ∈ S_{1}. We observe that in(h^{2}) is a square in the term order of 2.1. The assertion follows from 2.1. If i = n, we have I = (X_{1}, …, X_{n})^{2}. In this case the defining ideal does not change of any transformation and we get the claim by 2.1. □

Lemma 4.8. Let K be algebraically closed, char(K) ≠ 2 and R = S/I. If I ⊂ 𝔪^{2} is a principal ideal, then R is u-i-Koszul if and only if I = (g^{2}) for some g ∈ S_{1}.

Proof. If I = (g^{2}) for some g ∈ S_{1}, then R is u-i-Koszul by 4.7. Let R be u-i-Koszul. Since K is algebraically closed and since char(K) ≠ 2, there exists a K-basis Y_{1}, …, Y_{n} of S_{1} such that the generator of I is of the form Y_{1}^{2} + … + Y_{i}^{2} for some i ≤ n (see [13]). We claim that i = 1 and argue by contradiction. If i > 1, we apply Y_{i−1} ↦ Y_{i−1} + \sqrt{−1}Y_{i} and Y_{j} ↦ Y_{j} for j ≠ i − 1. Then the generator f in the new coordinates has in(f) = −2\sqrt{−1}Z_{r−1}Z_{r} and thus R is not i-u-Koszul by 2.1. Therefore we have i = 1, and f = Y_{1}^{2}. □

Remark 4.9. Let I ⊂ S have a quadratic Gröbner basis and let f_{1}, …, f_{k} be a minimal system of generators of I. Then there exists a minimal Gröbner basis of I which consists of K-linear combinations of f_{1}, …, f_{k}.

Proof of 4.4. In 4.7 and 4.8 we have already observed that R is u-i-Koszul, if I = (g^{2}) or I = 𝔪^{2}. Let R be u-i-Koszul. By 4.6, the set N of all zerodivisors in R_{1} is a linear subspace of R_{1} and N^{2} = 0. Thus, in the case that dim(R) = 0 we have N = R_{1} and so I = (X_{1}, …, X_{n})^{2}. Let now dim(R) > 0. We have to show that I = (g^{2}) for some g ∈ R_{1}. We use induction on d = dim(R). Let d = 1. We have two cases:

(a) N = 0. In this case R is a 1-dimensional Cohen-Macaulay ring with minimal multiplicity and every l ∈ R_{1} is a non-zerodivisor. Suppose I ≠ (0). We show that R must be a domain and deduce a contradiction. Since dim(R) = 1 and I ≠ (0) we have emb dim(R) > 1. Let x_{i} = X_{i} + I for i = 1, …, n. x_{1} is a non-zerodivisor of R. Since dim(R/x_{1}R) = 0 and since R/x_{1}R is u-i-Koszul by 4.5, we get R/x_{1}R = K[X_{2}, …, X_{n}]/(X_{2}, …, X_{n})^{2} as we have already observed above. Since x_{1} is a non-zerodivisor of R, we have X_{1}^{2} ∉ I. By 2.1, S/ in(I) is i-Koszul. The term order of 2.1 implies that X_{1}^{2} ∉ in(I). By 2.3, we get in(I) = (X_{2}, …, X_{n})^{2}. It is a general fact that the set of monomials which do not belong to in(I) forms a K-basis of R. In our case x_{1}^{i}x_{1}, …, x_{1}^{i}x_{n} form a K-basis of R_{i+1} for all i ≥ 0. If a ∈ R_{i}, i ≥ 2, is a homogeneous element we have a ∈ (x_{1})^{i−1}. Suppose ar = 0 for some r ∈ R. We can write a = x_{1}^{i−1}l with some linear form l ∈ R_{1}. It is ar = x_{1}^{i−1}lr = 0. Since x_{1} and l are non-zerodivisors by assumption it follows that r = 0. Thus, every homogeneous element of R is a non-zerodivisor which implies that R is a domain. Therefore R is a polynomial ring in one variable because K is algebraically closed and I is homogeneous. This is a contradiction to emb dim(R) > 1.

(b) N ≠ 0. It is I ≠ (0). We start induction on n = emb dim(R). Let n = 2. By 2.1 and 4.9 I ⊂ K[X_{1}, X_{2}] has a minimal system of generators f_{1}, …, f_{k} which forms a minimal Gröbner basis. Since we are in the case that d = dim(R) = 1 we have I ≠ 𝔪^{2} and thus k ≤ 2. If k = 0, then R is a polynomial ring. For k = 1 we get the assertion by 4.8. If k = 2, we deduce a contradiction. Since R is i-Koszul, we obtain by 2.1 that in(I) = (X_{1}^{2}, X_{1}X_{2}) with respect to the term order of 2.1. It follows that I = (X_{1}^{2}, X_{1}X_{2}) because X_{1}^{2}, X_{1}X_{2} are the smallest two monomials of degree two. Thus, by interchanging X_{1} and X_{2} we get the defining ideal J = (X_{1}X_{2}, X_{2}^{2}). By 2.1 S/J is not i-Koszul which is a contradiction to R being u-i-Koszul. Let n > 2. We choose x ∈ N, x ≠ 0. We may assume x = x_{1} = X_{1} + I. Since x_{1}^{2} = 0 by 4.6, we have that dim(R/x_{1}R) = 1 and emb dim(R/x_{1}R) = n − 1. R/x_{1}R is u-i-Koszul by 4.5. Let N̄ be the set of all zerodivisors of R/x_{1}R. If N̄ ≠ 0, by induction hypothesis on n, if N̄ = 0, by case (a), it follows that R/x_{1}R is a hypersurface ring of the form R/x_{1}R = K[X_{2}, …, X_{n}]/(g^{2}) for some g ∈ K[X_{2}, …, X_{n}]_{1}. Let L ⊂ S_{1} be the linear subspace with (I : X_{1})_{1} = L. Then we have I = (X_{1}L, g^{2} + X_{1}l) for some linear form l ∈ S_{1}. By 4.6, we get X_{1} ∈ L and thus X_{1} ∈ Rad(I). It follows that g ∈ Rad(I) which implies g + I ∈ N. Again by 4.6, we get g^{2} ∈ I and X_{1}g ∈ I. This implies g, l ∈ L and therefore I = (L^{2}). Since d = 1 we have L^{2} = I ≠ 𝔪^{2}. By 4.7 we get the assertion.

We finish now the induction on d. Let d > 1. Then we have N ≠ R_{1}. Thus there exists x ∈ R_{1} \ N, x ≠ 0. We may assume x = x_{1} = X_{1} + I. By 4.5 R/x_{1}R is u-i-Koszul. We have dim(R/x_{1}R) = dim(R) − 1 ≥ 1 and thus by induction hypothesis R/x_{1}R = K[X_{2}, …, X_{n}]/(g^{2}). It follows I = (g^{2} + X_{1}l) for some l ∈ R_{1}. If I ≠ (0), we obtain the assertion by 4.8. □

In the monomial case we have a more precise statement.

Proposition 4.10. Let I ⊂ S a proper monomial ideal. R = S/I is u-i-Koszul if and only

if I = 𝔪^{2} or I is of the form \begin{cases}(X_{i}^{2}) & if char(K) ≠ 2 \\ (X_{i_{1}}^{2}, …, X_{i_{r}}^{2}) & if char(K) = 2 . \end{cases}

Proof. In the case that I = 𝔪^{2} or I is of the form (X_{i}^{2}) for some i the algebra R is u-i-Koszul by 4.7. Now let char(K) = 2 and I = (X_{i_{1}}^{2}, …, X_{i_{r}}^{2}) for some indices i_{1} < … < i_{r}. For any

transformation X_{i} ↦ ∑_{j=1}^{n} a_{ji}X_{j} i = 1, …, n, we obtain a new defining ideal J = (g_{1}, …, g_{r})

with g_{k} = ∑_{j=1}^{n} a_{ji_{k}}^{2}X_{j}^{2} for k = 1, …, r. Then J has a minimal system of generators which

forms a Gröbner basis of J. In the term order of 2.1 in(J) is of the form (X_{j_{1}}^{2}, …, X_{j_{s}}^{2}) for some indices j_{1} < … < j_{s}. By 2.1, S/J is i-Koszul and thus R = S/I is u-i-Koszul. Conversely, let us assume that R is u-i-Koszul. There are two cases:

(a) char(K) ≠ 2. By 4.6 and 4.7, we get I = (X_{i}^{2}) for some i or I = 𝔪^{2}.

(b) char(K) = 2. Let G(I) be the set of the minimal generators of I. We need some facts which follow immediately from 2.3. If

(1) X_{i}X_{j} ∈ G(I) with i < j and X_{i}X_{k} ∉ G(I) for some k > j or if

(2) X_{i}X_{j} ∈ G(I) with i < j and X_{k}^{2} ∉ G(I) for some k > i or if

(3) X_{i}^{2}, X_{i}X_{i+1}, X_{i}X_{n} ∈ G(I), X_{1}^{2}, …, X_{n}^{2} ∈ G(I) and X_{i+1}X_{i+2} ∉ G(I) for some i < n−1 or if

(4) X_{i}^{2}, X_{i}X_{i+1}, X_{i+1}^{2} ∈ G(I) and X_{i−1}X_{i} ∉ G(I) for some 1 < i < n, then R is not u-i-Koszul. We have to show the following: If I ≠ 𝔪^{2} and I is not of the form (X_{i_{1}}^{2}, …, X_{i_{r}}^{2}), then R is not u-i-Koszul. Under this assumption we have X_{i}X_{j} ∈ G(I) for some i < j. By 2.3 and (1), we have X_{i}^{2}, …, X_{i}X_{n} ∈ G(I). By (2), we get X_{i+1}^{2}, …, X_{n}^{2} ∈ G(I). Then (4) implies X_{i−1}X_{i} ∈ G(I). By iteration and using (3), we obtain that I = 𝔪^{2} which is a contradiction. □

As a direct consequence from 4.4 and 4.10 we have

Corollary 4.11. Let char(K) ≠ 2 and K algebraically closed. If R = S/I is u-i-Koszul, then R′ = S/ in(I) is also u-i-Koszul.

The converse of 4.11 is not true. For example, take n = 3 and I = (X_{1}X_{3} − X_{2}^{2}). Since in(I) = (X_{2}^{2}) the algebra R′ is u-i-Koszul by 4.10, but R is not, as follows from 4.4. We get immediately from 4.4:

Corollary 4.12. Let K be algebraically closed, char(K) ≠ 2 and R = S/I a u-i-Koszul domain. Then I = (0).

The statements in 4.4 and 4.12 are not true for more general base fields. Take, for example,

R = ℚ[X_{1}, X_{2}]/(X_{1}^{2} − \frac{1}{2}X_{2}^{2}).

Then X_{1}^{2} − \frac{1}{2}X_{2}^{2} is not a square in ℚ[X_{1}, X_{2}] and R is a u-i-Koszul domain. Moreover,

R = ℤ/2ℤ[X_{1}, …, X_{4}]/(X_{1}^{2} + X_{2}^{2}, X_{3}^{2} + X_{4}^{2})

is u-i-Koszul. Therefore we need char(K) ≠ 2 in 4.4.

In [4] we find the following concept: A homogeneous K-algebra R is called universally Koszul, if the set of all ideals of R which are generated in degree 1 defines a Koszul filtration of R. There is no direct relation to i-Koszulness. Since on the one hand the algebra

K[X_{1}, X_{2}]/(X_{1}X_{2})

is u-Koszul by [4] 1.5., but not i-Koszul by 2.3. On the other hand

K[X_{1}, …, X_{n}]/(X_{1}^{2}, …, X_{n}^{2})

is i-Koszul due to 2.3, but not u-Koszul, if n > 3 and char(K) ≠ 2 (see [4]).

5. i-Koszulness of semigroup rings

In this chapter we want to study semigroup rings. In this case we only consider flags spanned by semigroup generators. We identify a monomial X^{α} with the corresponding exponent α ∈ ℕ^{n}. Thus if R = K[α_{1}, …, α_{k}] is a homogeneous semigroup ring with minimal set of semigroup generators G = {α_{1}, …, α_{k}}, then R is called i-Koszul if R is i-Koszul with respect to α_{1}, …, α_{k}. R is said to be u-i-Koszul , if R is i-Koszul with respect to α_{π(1)}, …, α_{π(k)} for any permutation π ∈ S_{k}. We will see that i-Koszulness implies a certain shellability of the finite intervals in the divisor poset of R. The set Σ of all monomials in R is partially ordered by divisibility. If there is an injective map λ : G → Λ, Λ totally ordered, then all unrefineable finite chains of divisors

C : β_{0} →^{·α_{i_{1}}} β_{1} → ⋯ →^{·α_{ir}} β_{r}

are labeled by λ(C) = (λ(α_{i_{1}}), …, λ(α_{i_{r}})) ∈ Λ^{r}. We have a total order > on Λ^{r} induced by the lexicographic order and the order on Λ.

Definition 5.1. (see [1],[3]) R is called naturally shellable, if for every monomial α ∈ Σ the order complex Δ([1, α]) is shellable with order λ(C_{1}) < ⋯ < λ(C_{r}) where {C_{1}, …, C_{r}} is the set of all unrefineable chains in the interval [1, α].

Let R = K[Y_{1}, …, Y_{k}]/I. Natural shellability can be translated into a condition on in(I) with respect to the reverse lexicographic order induced by Y_{1} < ⋯ < Y_{k}.

Proposition 5.2. (T. Hibi) The following statements are equivalent:

(a) R is naturally shellable.

(b) in(I) is quasi-poset, i.e. if i < k < j and Y_{i}Y_{j} ∈ in(I), then it follows Y_{i}Y_{k} ∈ in(I) or Y_{k}Y_{j} ∈ in(I).

Consequently, by 2.1 the following is evident:

Corollary 5.3. Let R be an i-Koszul semigroup ring. Then R is naturally shellable.

It is known (see [12]) that shellability of divisor posets implies Koszulness. Thus the corollary above gives us an alternative proof for the statement that an i-Koszul semigroup ring is Koszul. We have seen in 3.3 that the property of i-Koszulness is preserved under tensor products. This is not true for Segre products of semigroup rings. For example,

R = K[X_{1}Y_{1}, X_{1}Y_{2}, X_{2}Y_{1}, X_{2}Y_{2}] ≅ K[Z_{1}, Z_{2}, Z_{3}, Z_{4}]/(Z_{1}Z_{4} − Z_{2}Z_{3})  (1)

is not i-Koszul with respect to any permutation of the semigroup generators by 2.1. But it can be shown that R is naturally shellable ([3]). Therefore the converse of 5.3 is not true in general.

We now compare i-Koszulness with other Koszul properties. In [11] strongly Koszul algebras are introduced. In the semigroup case this property is preserved under Segre products (see [11]). Thus the ring R = K[X_{1}, X_{2}] ⋆ K[Y_{1}, Y_{2}] in example (1) is strongly Koszul. In [11] it is shown that strongly Koszul algebra is sequentially Koszul. It is obvious from the definition that:

Remark 5.4. Any i-Koszul algebra R is sequentially Koszul.

As (1) shows the converse is not true in general. Furthermore i-Koszulness does not imply the strongly Koszul property. Take, for example,

T = K[X_{1}^{3}, X_{1}^{2}X_{2}, X_{1}X_{2}^{2}, X_{1}X_{2}X_{3}, X_{2}^{2}X_{3}, X_{2}X_{3}^{2}].

If we order the generators lexicographically descending, we get by computation that T is i-Koszul. However, T is not strongly Koszul by [11] Prop.1.4. because

(X_{1}X_{2}X_{3}) :_{T} (X_{2}^{3}) = (X_{2}^{3}, X_{1}^{3}X_{2}X_{3}^{2}).

Concerning u-i-Koszulness we have the following Proposition 5.5. Let R = K[α_{1}, …, α_{k}] ⊂ S be a semigroup ring. If R is u-i-Koszul, then R is a polynomial ring.

Proof. We may assume that α_{1} >_{lex} … >_{lex} α_{k} where >_{lex} is the total lexicographic order on ℕ^{n}. Let R = K[Y_{1}, …, Y_{k}]/I with α_{i} = Y_{i} + I for i = 1, …, n. By hypothesis, R is i-Koszul with respect to this sequence. We argue by contradiction. If I ≠ (0), we get by 2.1 that I has quadratic Gröbner basis with respect to the reverse lexicographic term order induced by Y_{1} < … < Y_{k}. The toric ideal I is minimally generated by binomials of degree 2. By 4.9, I has a quadratic Gröbner basis G which consists of binomials. The chosen order of the semigroup generators implies that every f ∈ G is of the form f = Y_{i}Y_{j} − Y_{k}Y_{l} with k < i ≤ j < l where in(f) = Y_{i}Y_{j}. We choose the smallest index i such that Y_{i}Y_{j} ∈ in(I) for some i ≤ j. Since R is i-Koszul, we have Y_{i}^{2} ∈ in(I) by 2.1. Thus there exists f ∈ G such that f = Y_{i}^{2} − Y_{i−r}Y_{i+s} for some r, s > 0. Interchanging Y_{i} and Y_{i−r} we get a new defining ideal J and an element g = Y_{i−r}^{2} −Y_{i}Y_{i+s} ∈ J. Taking the same term order on S/J we observe that in(g) = Y_{i}Y_{i+s}. Since S/J is i-Koszul, it follows that Y_{i}^{2} ∈ in(J) by 2.1. Thus there exists a binomial h ∈ J such that h = Y_{i}^{2} − Y_{a}Y_{b} for some a, b ∈ {1, …, k}. But then, there is a relation u = Y_{i−r}^{2} − Y_{c}Y_{d} ∈ G for some c, d ∈ {1, …, k} and the order of the semigroup generators implies c < i − r < d. Thus we have in(u) = Y_{i−r}^{2} ∈ in(I) which is a contradiction to the choice of i. □

References

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[3] Blum, S.: Natürliche Schälbarkeit homogen erzeugter Halbgruppenringe. Diplomarbeit, Universität Essen 1999.

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[11] Herzog, J.; Hibi, T.; Restuccia, G.: Strongly Koszul algebras. To appear in Math. Scand. 1997.

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Received August 9, 1999