QF RINGS AND QF ASSOCIATED GRADED RINGS

HIROYUKI TACHIKAWA

Abstract. The associated graded ring of QF (quasi-Frobenius, generally not commutative) ring R is not necessarily QF. We shall prove that the associated graded ring of R is QF if and only if R is QF and for any primitive idempotent e the upper Loewy series of Re and eR is coincident with the lower Loewy series of Re and eR respectively. In connection with the above result we consider for any pair of positive integers t, n

a ring Λ = K[x_{0}, x_{1}, ⋯, x_{n}]/(x_{i}^{t} − \frac{1}{x_{i}} ∏_{j=0}^{n} x_{j}| i = 0, 1, ⋯, n), because for t ≠ n, the

associated graded ring of Λ is different from Λ but they are both QF (=0-dimensional Gorenstein). So we expect that for t = n, Λ is Gorenstein even if Krull dimension > 0. We pointed out however that if t = n = 2, Λ is not Gorenstein, but Cohen-Macauley. Further if n = t = 3, Λ is neither Cohen-Macauley nor toric, of course not Gorenstein.

1. A characterization of QF associated graded rings

For an Artinian ring R having the Jacobson radical J with J^{n+1} = 0, the series : R ⊃ J ⊃ J^{2} ⊃ ⋯ ⊃ J^{n} ⊃ J^{n+1} = 0 is called the upper Loewy series of _{R}R (resp. R_{R}). If we put A_{i} = J^{i}/J^{i+1}, we can naturally define the multiplication of elements a+J^{i+1} ∈ A_{i} and b + J^{j+1} ∈ A_{j} to be ab + J^{i+j+1} ∈ A_{i+j}. Then by using this multiplication we make the (formal) direct sum A_{0} ⊕A_{1} ⊕⋯⊕A_{n} into a ring R_{G}. Clearly this ring R_{G} is positive Z−graded and A_{1} generates the radical of R_{G}. R_{G} is called the associated graded ring of R. Cf.[3]. R and R_{G} may be not isomorphic to each other. Cf. Example 2.1

By Morita equivalence [8] we can assume without loss of generality that rings are basic. Let e be a primitive idempotent of ring R. Then e + J ∈ A_{0} is a primitive idemotent of R_{G} which we shall denote by e_{G} for short. If we denote the right (resp. left) annihilator of a subset M of R by r(M) (resp. l(M)), then Soc (Re) = r(J)e (resp. Soc (eR) = e l(J)). At first we have

Proposition 1.1. If _{R_{G}}Soc(R_{G}e_{G}) is simple for a primitive idempotent e_{G}, then the RSoc(Re) is simple. And if Soc(R_{G}e_{G}) ≃ R_{G}f_{G}/Rad(R_{G})f_{G} for a primitive idempotent f, then Soc(Re) ≃ Rf/Jf.

Proof. Let J^{ρ}e ≠ 0 and J^{ρ+1}e = 0. Then A_{ρ}e_{G} ≠ 0. Let us denote the set {α ∈ R_{G}|A_{1}α = 0} by r(A_{1}). Since A_{1} generates the radical of R_{G}, by the assumption Soc(R_{G}e_{G}) = r(Rad(R_{G}))e_{G} = r(A_{1})e_{G} is a unique simple R_{G}-submodule of R_{G}e_{G}. Hence r(A_{1})e_{G} ⊆ A_{ρ}e_{G}. On the other hand r(A_{1})e_{G} ⊇ A_{ρ}e_{G} by A_{1}A_{ρ}e_{G} = 0. Hence r(A_{1})e_{G} = A_{ρ}e_{G}.

The paper is in a final form and no version of it will be submitted for publication elsewhere.

Now take u ∈ r(J)e. Then there is a unique positive integer j such that u ∈ J^{j}e\J^{j+1}e. As au = 0 for any a ∈ J we have that (a + J^{2})(u + J^{j+1}) = au + J^{(j+1)+1} = 0 + J^{(j+1)+1} ∈ A_{j+1} for a + J^{2} ∈ A_{1}. Therefore u + J^{j+1} ∈ r(A_{1})e_{G} = A_{ρ}e_{G} and it follows that j = ρ. This implies that r(J)e ⊆ J^{ρ}e. On the other hand r(J)e ⊇ J^{ρ}e by J(J^{ρ}e) = 0. Hence J^{ρ}e = r(J)e. Further J^{ρ}e can be identified with A_{ρ}e_{G} because J^{ρ+1}e = 0.

Now R/J can be identified with R_{G}/Rad(R_{G}). From Rad(R_{G})^{ρ}e_{G} = A_{ρ}e_{G} is simple as a left R_{G}-module it follows that r(J)e is a simple R-module.

The latter statement follows from that f_{G}A_{ρ}e_{G} ≠ 0 if and only if fJ^{ρ}e ≠ 0. This completes the proof.

Following Thrall [12] a ring R is said to be left QF -2 if the socle of Re is simple for every primitive idempotent e. Then we have immediately

Corollary 1.2. R is left QF -2 if R_{G} is left QF -2.

Now we shall prove

Theorem 1.3. If R_{G} is QF, then R is QF.

Proof. By the assumption that R is basic there is a set of primitive idempotents e_{i} such

that 1 = ∑_{i=1}^{n} e_{i} and Re_{i} ≄ Re_{j} for i ≠ j.

Since R_{G} is QF, for all e_{i}, Soc(R_{G}e_{i_{G}}) = r(Rad(R_{G}))e_{i_{G}} = r(A_{1})e_{i_{G}} (resp. Soc(e_{i_{G}}R_{G}) = e_{i_{G}}l(Rad(R_{G})) = e_{i_{G}}l(A_{1})) is a simple left (resp. right) R_{G} -module.

Hence by Proposition 1.1 r(J)e_{i} (resp. e_{i}l(J)) is a simple left (resp. right) R -module. On the other hand it holds that _{R}l(J)e_{i} ≃ _{R}Hom_{R}((e_{i}R/e_{i}J)_{R},_{R}R_{R}) and as is quoted

above _{R}l(J)e_{i} is simple. Similarly e_{i}r(J)_{R} ≃ Hom_{R}((Re_{i}/Je_{i},_{R}R_{R}) is simple.

Therefore by [6, Theorem 2.1] it holds the duality Hom_{R}(−,_{R}R_{R}) between the categories of finitely generated left R-modules and right R-modules. and hence R is QF. Cf. also [5].

In Example 2.1 we shall show that both the converses of Corollary 1.2 and Theorm 1.3 do not hold.

Now it needs to give a characterization of QF ring R for which the associated graded ring R_{G} is QF.

We say that the series : Re = r(J^{ρ+1})e ⊃ r(J^{ρ})e ⊃ r(J^{ρ−1})e ⊃ ⋯ ⊃ r(J)e ⊃ r(R)e = 0 is the lower Loewy series of Re.

In their book [2] Artin-Nesbitt-Thrall proved that subquotient modules J^{k}e/J^{k+1}e and r(J^{ρ+1−k})e/r(J^{ρ−k})e have non-zero isomorphic constituents for every 0 ≦ k ≦ ρ. Then we have the following question:

Which kind of rings do satisfy the coincidence of every non-zero isomorphic constituent of J^{k}e/J^{k+1} with J^{k}e/J^{k+1}e itself ?

We can provide Proposition 1.4 as an answer to the question.

A positive Z-graded ring R = A_{0} ⊕A_{1} ⊕⋯⊕A_{n} is called to be standard if A_{1} generates the radical of R.

Then we have

Proposition 1.4. If R is a standard positive Z-graded QF ring , then the upper Loewy series of Re coincides with the lower Loewy series of Re for any primitive idempotent e.

Proof. For a primitive idempotent e let Re = A_{0}e ⊕ A_{1}e ⊕ A_{2}e ⊕ ⋯ ⊕ A_{ρ}e, ρ ≦ n, be a grading of Re. Then A_{i}A_{j}e ⊆ A_{i+j}e and the rad(Re)= A_{1}e ⊕ A_{2}e ⊕ ⋯ ⊕ A_{ρ}e.

Then from the assumption that R is QF it holds that _{R}l(J)e = r(J)e = Soc(Re) = J^{ρ}e = A_{ρ}e.

Assume that r(J^{s})e = J^{ρ+1−s}e for an integer s ≥ 1 (as pointed out above this assumption is satisfied for s = 1), and suppose that r(J^{s+1})e ≠ J^{ρ−s}e for r(J^{s+1})e ⊇ J^{ρ−s}e.

Then there is 0 ≠ y = ∑_{l≦j<ρ−s}y_{j} ∈ r(J^{s+1})e such that 0 ≠ y_{j} ∈ A_{j}e. From 0 = J^{s+1}y =

J^{s}(Jy) it follows Jy ∈ r(J^{s})e = J^{ρ+1−s}e = ⊕_{ρ+1−s≦k} A_{k}e.

On the other hand Jy = ∑_{j<ρ−s} Jy_{j} ∈ ⊕_{l≦j<ρ−s}A_{j+1}e = ⊕_{l+1≦k<ρ+1−s} A_{k}e.

Hence Jy = 0. Then A_{1}y_{l} = 0 since A_{1} generates J and y_{j} ∈ A_{j}e for l ≦ j < ρ − s. This implies y_{l} ∈ A_{ρ}e and this is a contradiction because l ≠ ρ.

Therefore we conclude that r(J^{s+1})e = J^{ρ−s}e.

Now by induction on s we complete the proof.

Corollary 1.5. If R_{G} is QF then for any primitive idempotent e the upper Loewy series of Re and eR are coincident with the lower Loewy series of Re and eR respectively.

Proof. Let J^{ρ}e ≠ 0 but J^{ρ+1}e = 0. Then by Proposition 1.4 it follows that (Rad(R_{G}))^{ρ+1−k}e_{G} = Soc^{k}(R_{G}e_{G}) = r((Rad(R_{G}))^{k})e_{G} for k = 1, 2, ⋯, ρ. Now we want to prove that J^{ρ+1−k}e = Soc^{k}(Re) = r(J^{k})e for k = 1, 2, ⋯, ρ. Suppose x ∈ r(J^{k})e \ J^{ρ+1−k}e since J^{ρ+1−k}e ⊆ r(J^{k})e. Let j be the maximal integer

such that x ∈ J^{j} \ J^{j+1}e. Then j < ρ + 1 − k. For x + J^{j+1} ∈ A_{j}e_{G} it holds that A_{k}(x + J^{j+1}e) = (J^{k}/J^{k+1})(x + J^{j+1}e = 0 + J^{j+1+k}e = 0. This implies that x + J^{j+1}e ∈ r(A_{k})e_{G} = r(A_{1}^{k})e_{G} = r((Rad(R_{G}))^{k})e_{G} = Rad(R_{G})^{ρ+1−k}e_{G} = (A_{ρ+1−k} ⊕A_{ρ+2−k} ⊕⋯)e_{G}. Thus we have j ≥ ρ + 1 − k. But this contradicts to j < ρ + 1 − k.

We can prove similarly that eJ^{σ+1−k} = Soc^{k}(eR) = e l(J^{k}) for k = 1, 2, ⋯, σ , where eJ^{σ} ≠ 0 but eJ^{σ+1} = 0. This completes the proof.

Proposition 1.6. If R is QF and for any primitive idempotent e the upper Loewy series of Re and eR are coincident with the lower Loewy series of Re and eR respectively, then the associated graded ring R_{G} is QF.

Proof. Let J^{ρ}e ≠ 0 but J^{ρ+1}e = 0. From the coincidence of series of the upper Loewy series and the lower Loewy series of Re it follows that r(J^{i})e = Soc^{i}(_{R}Re) = J^{ρ+1−i}e, i = 1, 2, ⋯, ρ. And especially Soc(Re) = J^{ρ}e is a simple left R-module since R is QF.

For x ∈ J^{k}e \ J^{k+1}e and k ≦ ρ − 1 suppose Jx ∈ J^{k+2}e = Soc^{{ρ+1−(k+2)}}(Re). Then J^{{ρ+2−(k+2)}}x = 0 and x ∈ r(J^{ρ−k})e = J^{k+1}e. But this is a contradiction.

Therefore if k ≦ ρ − 1 and if x + J^{k+1}e ≠ 0 ∈ A_{k}e_{G} = (J^{k}/J^{k+1})e_{G} it holds that 0 ≠ A_{1} (x + J^{k+1}e) ∈ A_{k+1}e_{G}. Therefore Soc(R_{G}e_{G}) = r(Rad(R_{G}))e_{G} ⊆ J^{ρ}e = Rad(R_{G})^{ρ}e_{G}.

As r(Rad(R_{G}))e_{G} ⊇ Rad(R_{G})^{ρ}e_{G} we have Soc(R_{G}e_{G}) = r(Rad(R_{G}))e_{G} = J^{ρ}e, which can be considered as a simple left R_{G}-module because J^{ρ}e is a simple left R-module.

Now it is clear that Hom_{R_{G}}(e_{G}R_{G}/e_{G}Rad(R_{G}),_{R_{G}}R_{G}_{R_{G}}) ≃ r(Rad(R_{G}))e_{G}. This implies that the dual module Hom_{R_{G}}(e_{G}R_{G}/e_{G}Rad(R_{G}),_{R_{G}}R_{G}_{R_{G}}) of a simple right R_{G}-module e_{G}R_{G}/e_{G}Rad(R_{G}) is a simple left R_{G}-module.

Similarly we have that the dual module Hom_{R_{G}}(R_{G}e_{G}/Rad(R_{G})e_{G},_{R_{G}}R_{G}_{R_{G}}) of a simple left R_{G}-module R_{G}e_{G}/Rad(R_{G})e_{G} is a simple right R_{G}-module.

Therefore by [6, Theorem 2.1] it holds the duality Hom_{R_{G}}(−,_{R_{G}}R_{G}_{R_{G}}) between the categories of finitely generated left R_{G}-modules and finitely generated right R_{G}-modules. Hence R_{G} is QF.

Now by Propositions 1.5 and 1.6 we have the following characterization of QF associated graded rings:

Theorem 1.7. The following conditions (i), (ii) and (iii) are equivalent to each other: (i) The associated graded ring R_{G} is QF, (ii) R is QF and for any primitive idempotent e the upper Loewy series of Re and eR

are coincident with the lower Loewy series of Re and eR respectively,

(iii) R is QF and for any primitive idempotent e_{i} and integer 0 ≦ k ≦ ρ_{i} it holds that RJ^{k}e_{i}/J^{k+1}e_{i} ≃ _{R}Hom_{R}(e_{i}J^{ρ−k}/e_{i}J^{ρ−k+1}_{R},_{R}R_{R}) (resp.e_{i}J^{k}/e_{i}J^{k+1} ≃ Hom_{R}(_{R}J^{σ−k}e_{i}/J^{σ−k+1}e_{i},_{R}R_{R}), where J^{ρ_{i}}e_{i} ≠ 0 but J^{ρ_{i}+1}e_{i} = 0 (resp. e_{i}J^{σ_{i}} ≠ 0 but e_{i}J^{σ_{i}+1} = 0).

Let π be a Nakayama permutation of QF ring R on the set of all non isomorphic primitive idempotents e_{i}, i = 1, 2, ⋯, n.

Then it holds that _{R}Re_{π(j)}/Je_{π(j)} ≃ _{R}Hom_{R}(e_{j}R/e_{j}J_{R},_{R}R_{R}).

Corollary 1.8. R_{G} is QF if and only if R is QF and for any primitive idempotent e_{i} it holds that _{R}J^{k}e_{i}/J^{k+1}e_{i} ≃ ⊕_{j}^{n} n_{i,j} × Re_{π(j)}/Je_{π(j)} for a direct sum decomposition : e_{i}J^{ρ−k}/e_{i}J^{ρ−k+1}_{R} ≃ ⊕_{j}^{n} n_{i,j} × e_{j}R/e_{j}J, where n_{i,j} × e_{j}R/e_{j}J means the direct sum of n_{i,j} copies of e_{j}R/e_{j}J.

As indecomposable commutative algebras are local, for them there are no difference between QF -2, QF -3 and QF rings. Hence Theorem 1.7 and Corollary 1.8 are considered to be results for non commutative rings, though Theorem 1.7 seems to be a generalization of Iarrobino’s result [4; Proposition 1.7] for 0-dimensional Gorenstein algebras.

2. examples

Example 2.1. (i) Let R be an algebra over a field K defined by the following quiver.

[figure: >; x; v; u; \begin{matrix}x^{3} = vu, \\ 0 = uv, \\ 0 = xv, \\ 0 = ux. \end{matrix}]

Then the K-bases : R = {e_{1}, x, x^{2}, x^{3}, u; e_{2}, v}, J = Rad(R) = {x, x^{2}, x^{3}, u; v}, J^{2} = {x^{2}, x^{3}},J^{3} = {x^{3}} and J^{4} = {0}. By 0 ≠ x^{3} = vu and 0 = uv, R is not commutative. As r(J)e_{1} = Kx^{3} ≃ Re_{1}/Je_{1} and r(J)e_{2} = Kv ≃ Re_{1}/Je_{1}, R is left QF -2.

It happens however that (v + J^{2})(u + J^{2}) = vu + J^{3} = 0 ∈ G(R) for the contrary vu = x^{3} ≠ 0 ∈ R. Then r(rad(G(R)))e_{1} = K(u + J^{2}) + K(x^{3} + J^{4}) ≃ G(R)e_{2}/Rad(G(R))e_{2} ⊕ G(R)e_{1}/Rad(G(R))e_{1} is not simple. Hence G(R) is not left QF -2.

This shows that the converse of Proposition 1.1 does not hold.

The next example (ii) shows that the converse of Theorem 1.3 does not hold.

(ii)

[figure: >; <; x; y; v; u; \begin{matrix}x^{3} = vu, \\ y^{2} = uv, \\ 0 = xv, \\ 0 = ux, \\ 0 = vy, \\ 0 = yu. \end{matrix}]

Then the K-bases = {e_{1}, x, x^{2}, x^{3}, u, e_{2}, y, v}, where e_{2} and e_{2} are primitive idempotents.

By x^{3} = vu and y^{2} = uv, R is not commutative. J = Rad(R) = {x, x^{2}, x^{3}, u; y, y^{2}, v}, J^{2} = {x^{2}, x^{3}, y^{2}},J^{3} = {x^{3}}, J^{4} = 0. r(J) = {x^{3}, y^{2}} = l(J), r(J)e_{1} = Kx^{3} ≃ Re_{1}/Je_{1} and r(J)e_{2} = Ky^{2} ≃ Re_{2}/Je_{2}. Hence R is QF.

As (v +J^{2})(u+J^{2}) = 0+J^{3} and (y +J^{2})(u+J^{2}) = 0+J^{3}, Soc(R_{G}(e_{1})_{G}) = K(u+J^{2})+ K(x^{3} + J^{3}) ≃ R_{G}(e_{2})_{G}/Rad(R_{G})(e_{2})_{G} ⊕ R_{G}(e_{1})/Rad(R_{G})(e_{1})_{G}). Hence Soc(R_{G}(e_{1})_{G}) is not simple. Therefore R_{G} is not QF.

We know that the upper Loewy series of Re_{1} and Re_{2} are (1, 1+2, 1, 1) and (2, 1+2, 2) respectively. On the other hand lower Loewy series of Re_{1} and Re_{2} are (1, 1, 2 + 1, 1) and (2, 1 + 2, 2) respetively. From Theorem 1.7 it follows also that R_{G} is not QF.

Example 2.2. Let Λ be a quotient ring K[x_{0}, x_{1}, ⋯, x_{n}]/I such that the ideal I are

generated by n + 1 polynomials x_{i}^{t} − \frac{1}{x_{i}} ∏_{j=0}^{n} x_{j}, i = 0, 1, ⋯, n, for the pairs (t, n).

–49–

In case of t ≠ n, for min{n, t} ≦ |t − n|s < max{n, t} there is an idempotent e ≡

∏_{i=0}^{n} x_{i}^{|t−n|s} mod I of Λ and Γ = (1−e)Λ ≃ K[x_{0}, x_{1}, ⋯, x_{n}]/(∏_{i=0}^{n} x_{i}^{|t−n|s}, I) is an Artinian

local algebra. Cf.[12] and Kikumasa-Yoshimura [6].

Let us denote the associated graded algebra Γ_{G} = A_{0} ⊕ A_{1} ⋯ ⊕ A_{m}. Then if t > n, m = (t+1)(n−1) and dim_{K}(A_{k}) = ♯{(d_{0}, d_{1}, ⋯, d_{i}, ⋯ d_{j}, ⋯, d_{n})| ∑_{l=0}^{n} d_{l} = k, 0 ≦ d_{l} ≦ t + 1 and d_{i} = d_{j} = 0 for i ≠ j}. Hence dim_{K}A_{k} = dim_{K}A_{m−k}. It follows by Corollary 1.8 that Γ_{G} (and hence by Theorem 1.3 Γ) is QF.

If t < n, m = (n + 1)(t − 1) and dim_{K}(A_{k}) = ♯{(d_{0}, d_{1}, ⋯, d_{n})| ∑_{l=0}^{n} d_{l} = k, 0 ≦ d_{l} ≦ t − 1}. Hence we have similarly dim_{K}A_{k} = dim_{K}A_{m−k} and Γ_{G} (and hence Γ) is QF.

Now we can extend our consideration for Λ to the case of n = t. Then as \frac{1}{x_{i}} ∏_{j=0}^{n} x_{j} ≡ x_{i}^{n}

mod I for i = 1, 2, ⋯, n, Λ is a positive Z-graded with respect to homogeneous elements

x^{−}_{0}, x^{−}_{1}, ⋯, x^{−}_{n} of degree 1. If we put u = ∏_{i=1}^{n} x^{−}_{i}, then K[u] is a subalgebra of Λ, which is a

polynomial ring over K of a variable u . Further all x^{−}_{i}’s satisfy the equation X^{n+1} − u = 0 ∈ K[u][X]. Hence by Noether’s normalization theorem the Krull dimension of Λ = 1.

Let t = n = 1, then the generators of I are formally \{x_{0}−\frac{1}{x_{0}} x_{0}x_{1} = x_{0}−x_{1}, x_{1}−\frac{1}{x_{1}} x_{0}x_{1} = x_{1}−x_{0}} = {x_{0}−x_{1}} and Λ is a polynomial ring of one variable and is obviously Gorenstein.

Let t = n = 2. Then {f_{2} = x_{0}x_{1} − x_{2}^{2}, f_{0} = x_{1}x_{2} − x_{0}^{2}, f_{1} = x_{2}x_{0} − x_{1}^{2}} defines an intersection of quadratic cones. In [9] Stanley commented that the following Theorm 2.1 was proved first by Macauley.

Theorem 2.1. If a K-algebra Λ is standard positively Z-graded and Gorenstein of Krull dimension d, then for Poincare^{,} series F (Λ, λ) it holds that F (Λ, \frac{1}{λ}) = (−1)^{d}λ^{ρ} F (Λ, λ) (as rational functions of λ) for some integer ρ.

By the Buchberger’s algorithm we obtain the reduced Gröbner bases {f_{0}, f_{1}, f_{2}, f_{3} = S(f_{0}, f_{1}) = −x_{0}^{3} − x_{1}^{3}} of I with respect to the degree- lexicographical order x_{0} < x_{1} < x_{2}.

As the leading terms are Lt(f_{0}) = x_{0}^{0}x_{1}^{1}x_{2}^{1}, Lt(f_{1}) = x_{0}^{1}x_{1}^{0}x_{2}^{1}, Lt(f_{2}) = x_{0}^{0}x_{1}^{0}x_{2}^{2}, Lt(f_{3}) =

x_{0}^{0}x_{1}^{3}x_{2}^{0} it holds α_{1} < 3, α_{2} < 2 for the standard bases x^{−}_{0}^{α_{0}} x^{−}_{1}^{α_{1}} x^{−}_{2}^{α_{2}}∈ Λ = K[x_{0}, x_{1}, x_{2}]/I.

Therefore we know that

{1^{−}, x^{−}_{0}, x^{−}_{0}^{2}, x^{−}_{0}^{3}, ⋯, x^{−}_{1}, x^{−}_{0}x^{−}_{1}, x^{−}_{0}^{2}x^{−}_{1}, x^{−}_{0}^{3}x^{−}_{1}, ⋯, x^{−}_{1}^{2}, x^{−}_{0}x^{−}_{1}^{2}, x^{−}_{0}^{2}x^{−}_{1}^{2}, x^{−}_{0}^{3}x^{−}_{1}^{2}, ⋯, x^{−}_{2}}

are the K-bases of Λ. Cf. [1:Theorem 1.7.4 and Proposition 2.1.6].

A_{0} = K 1^{−}, A_{1} = K x^{−}_{0} +K x^{−}_{1} +K x^{−}_{2}, A_{2} = K x^{−}_{0}^{2} +K x^{−}_{1}x^{−}_{0} +K x^{−}_{1}^{2},

A_{n} = K x^{−}_{0}^{n} +K x^{−}_{1}x^{−}_{0}^{n−1} +K x^{−}_{1}^{2}x^{−}_{0}^{n−2} for n ≥ 3,

are Z^{+} ∪ {0}-grading of Λ and the set of homogeneous generators is {x_{0}^{−}, x_{1}^{−}, x^{−}_{2}} with degree 1.

Thus the Poincare^{,} series F (Λ, λ) = 1 + ∑_{n=1}^{∞} 3λ^{n} = \frac{3}{1 − λ} − 2 = \frac{2λ + 1}{1 − λ}.

Now there is no ρ which satisfies (−1)^{1}λ^{ρ}F (Λ, λ) = (−1)λ^{ρ} \frac{2λ + 1}{(1 − λ)} =

= \frac{λ + 2}{(λ − 1)} = \frac{2\frac{1}{λ} + 1}{1 − \frac{1}{λ}} = F (Λ, \frac{1}{λ}). Therefore by Theorem 2.1 Λ is not Gorenstein.

By the way we notice that in this case Λ is Cohen-Macaulay because Λ = K[x^{−}_{0}] ⊕ K[x^{−}_{0}] x^{−}_{1} ⊕K[x^{−}_{0}] x^{−}_{2} is K[x^{−}_{0}]-free module. Here we notice that

K[x^{−}_{0}] x^{−}_{1}^{2}⊂ K[x^{−}_{0}] x^{−}_{2} by x_{0}x_{2} ≡ x_{1}^{2} mod I.

This arises a new question whether Λ is Cohen-Macaulay.

In order to answer the question let us consider Λ for n = t = 3. In this case the binomials {f_{3} = x_{0}x_{1}x_{2} − x_{3}^{3}, f_{0} = x_{1}x_{2}x_{3} − x_{0}^{3}, f_{1} = x_{2}x_{3}x_{0} − x_{1}^{3}, f_{2} = x_{3}x_{0}x_{1} − x_{2}^{3}} generates I and by using the Buchberger’s algorithm we obtain the following Gröbner bases

Gr = {f_{0}, f_{1}, f_{2}, f_{3}, f_{4} = S(f_{0}, f_{1}) = x_{0}^{4} − x_{1}^{4}, f_{5} = S(f_{1}, f_{2}) = x_{1}^{4} − x_{2}^{4}, f_{6} = S(f_{0}, f_{3}) = x_{0}x_{1}^{2}x_{2}^{2} − x_{0}^{3}x_{3}^{2}, f_{7} = S(f_{1}, f_{3}) = x_{0}^{2}x_{1}x_{2}^{2} − x_{1}^{3}x_{3}^{2}, f_{8} = S(f_{2}, f_{3}) = x_{0}^{2}x_{1}^{2}x_{2} − x_{2}^{3}x_{3}^{2}, f_{9} = S(f_{2}, S(f_{0}, f_{1})) = x_{1}^{3}x_{2}^{3} − x_{0}^{5}x_{3}} of I

and the leading terms {Lt(f_{0}) = x_{1}x_{2}x_{3}, Lt(f_{1}) = x_{2}x_{3}x_{0}, Lt(f_{2}) = x_{3}x_{0}x_{1}, Lt(f_{3}) = x_{3}^{3}, Lt(f_{4}) = x_{1}^{4}, Lt(f_{5}) = x_{2}^{4}, Lt(f_{6}) = x_{0}^{3}x_{3}^{2}, Lt(f_{7}) = x_{1}^{3}x_{3}^{2}, Lt(f_{8}) = x_{2}^{3}x_{3}^{2}, Lt(f_{9}) = x_{0}^{5}x_{3}} with respect to the degree-lexicographical order x_{0} < x_{1} < x_{2} < x_{3}.

Now there is no positive integer n such that x_{0}^{n} ⟶^{Gr}_{+} 0. Therefore K[x^{−}_{0}] is a polynomial

ring in the variable x^{−}_{0}. Further f_{6} = x_{0}x_{1}^{2}x_{2}^{2} − x_{0}^{3}x_{3}^{2} = x_{0}(x_{1}^{2}x_{2}^{2} − x_{0}^{2}x_{3}^{2}) ∈ I and (x_{1}^{2}x_{2}^{2} −

x_{0}^{2}x_{3}^{2}) ∉ I because the terms x_{1}^{2}x_{2}^{2} and x_{0}^{2}x_{3}^{2} are not reduced by any Gröbner base. Hence

x^{−}_{0} (x^{−}_{1}^{2}x^{−}_{2}^{2} − x^{−}_{0}^{2}x^{−}_{3}^{2}) =0^{−}, but (x^{−}_{1}^{2}x^{−}_{2}^{2} − x^{−}_{0}^{2}x^{−}_{3}^{2}) ≠0^{−}.

Hence Λ is not a K[x^{−}_{0}]-free module. This implies Λ is not Cohen-Macaulay.

As all generators of I are binomials, Λ may be a toric variety. Cf. [10]. Toric varieties are defined to be Noetherian integral domains. However as we prove just

now Λ has a zero divisor we cannot expect that Λ is toric.

Proposition 2.2. If n = t = 3, then Λ is neither Cohen-Macaulay nor toric. Of course Λ is not Gorenstein.

References

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[10] R. P. Stanley, Hilbert functions of graded algebra, Advances in Mathematics 28 (1978) 57-83.

[11] B. Sturmfels, Gröbner bases and convex polytopes, University Lecture Series Vol. 8, Amer. Math. Soc., 1996.

[12] H. Tachikawa, QF algebras and Gröbner bases of pure binomials, Preprint 00-014, SFB 343, Bielefeld, 2000.

Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, 305-8571 Japan pytha345@mail2.accsnet.ne.jp