Contemporary Mathematics

Rees algebras of the second syzygy module of the residue field of a regular local ring

Shiro Goto, Futoshi Hayasaka, Kazuhiko Kurano, and Yukio Nakamura

Abstract. Let (A, 𝔪) be a regular local ring and let M ⊆ F = A^{d} be the second syzygy module of the residue field A/𝔪. In this paper, we introduce the concept of generalized Grassmann algebras and investigate the Rees algebra R(M) of M. We explore the properties of generalized Grassmann algebras and prove that the Rees algebra R(M) of the second syzygy module M is a Gorenstein factorial domain.

Introduction

Let (A, 𝔪) be a regular local ring of dimension d. Let k be the residue field of A. Let F_{•} be the minimal free resolution of k and M_{n} the n-th syzygy module of k. The Rees algebra of M_{n} is defined to be the residue ring Sym_{A}(M_{n})/t_{A}(Sym_{A}(M_{n})) of the symmetric algebra Sym_{A}(M_{n}) of M_{n}, where t_{A}(∗) means the A-torsion submodule of the inside module (cf. [10]). We denote by R(M_{n}) the Rees algebra of M_{n}. For n = 1, 2, …, d, it is easily seen that R(M_{n}) is isomorphic to the image of the natural homomorphism from Sym_{A}(M_{n}) to Sym_{A}(F_{n−1}) ([5]). When n = 1, the Rees algebra R(M_{1}) is exactly the classical Rees algebra R(𝔪) of the maximal ideal 𝔪. So as is well known, it is a Cohen-Macaulay normal domain. In this paper, we want to study the Rees algebra R(M_{2}) of the second syzygy module M_{2}. With this notation, our main result is the following.

Theorem 4.1. The Rees algebra of the second syzygy module R(M_{2}) is a Gorenstein factorial domain.

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

1991 Mathematics Subject Classification. Primary 13F50; Secondary 13A50; 13A30.

Key words and phrases. Rees algebras, regular local rings, ASL, invariant rings.

The first, the third and the fourth authors are supported by a Grant-in-Aid for scientific Research Japan.

In order to prove the theorem, we investigate the associated graded ring of the graded maximal ideal of R(M_{2}), which is isomorphic to the k-subalgebra

S = k[A_{1}, …, A_{d}] [det ( \begin{matrix}A_{i} & A_{j} \\ X_{i} & X_{j} \end{matrix} ) | 1 ≤ i < j ≤ d]

of the polynomial ring k[A_{1}, …, A_{d}, X_{1}, … X_{d}] over the residue field k. To analyze the ring S, we introduce the generalized Grassmann algebra and explore its properties. In particular, the generalized Grassmann algebra has the structure of a graded algebra with straightening law (ASL) on a certain poset. That is the key of our theory. Let B be a commutative Noetherian ring and let ℓ, m, n be integers with 0 ≤ ℓ ≤ m ≤ n. Let X = (X_{ij}) be the m × n matrix of indeterminates over B and let B[X] be the polynomial ring B[X_{ij} | 1 ≤ i ≤ m, 1 ≤ j ≤ n]. We denote by X_{ℓ} the ℓ × n submatrix of X consisting of the 1st, …, ℓ-th rows. Let Γ(X) be the set of maximal minors of X. We denote by R_{ℓ}(X) the B-subalgebra of B[X] generated by the entries of X_{ℓ} and the elements of Γ(X). Of course, R_{m}(X) coincides with the polynomial ring B[X]. We note that our ring R_{1}(X) coincides with S when B is the residue field k, m = 2 and n = d. This is a natural generalization of Grassmann algebras. In fact, when B is a field, R_{0}(X) is a homogeneous coordinate ring of Grassmann variety of m-dimensional vector subspaces in B^{n}. Also R_{0}(X) over any ring B was investigated by many authors (for instance see [1]). So we will study the B-algebras R_{ℓ}(X) for any ℓ and show that R_{ℓ}(X) enjoys good properties being similar to R_{0}(X). Our approach is based on the method developed by Bruns and Vetter in [1].

We explain the construction of this paper. In Section 1, we prove that R_{ℓ}(X) is a graded ASL on a certain wonderful poset. Hence it follows that R_{ℓ}(X) is a Cohen-Macaulay ring if and only if so is B. In Section 2, we first determine the dimension of R_{ℓ}(X) and prove that the normality comes from that of the base ring. Furthermore, we also prove that the divisor class group of R_{ℓ}(X) is isomorphic to that of the base ring. Consequently we know that R_{ℓ}(X) is a factorial domain if and only if so is B. In particular, R_{ℓ}(X) is a Gorenstein ring when B is a field. Besides, since R_{ℓ}(X) is a free B-module and the fibre rings are Gorenstein, we conclude that R_{ℓ}(X) is a Gorenstein ring if and only if so is B. In Section 3, we suppose that the base ring B is an infinite field. It is known that R_{0}(X) is the ring of invariants of the special linear group in B[X], which is called the first fundamental theorem of invariant theory; see [3] or [1]. We see that our B-algebra R_{ℓ}(X) has similar properties. We prove that R_{ℓ}(X) is the invariant subring of a certain subgroup of the special linear group. Finally in Section 4, we apply the property of generalized Grassmann algebras and give a proof of Theorem 4.1 .

1. Generalized Grassmann algebras

Let B be a commutative Noetherian ring and X = (X_{ij}) the m × n matrix of indeterminates over B (m ≤ n). Let B[X] = B[X_{ij} | 1 ≤ i ≤ m, 1 ≤ j ≤ n]. Let X_{ℓ} stand for the ℓ × n submatrix of X consisting of the first ℓ rows, where 0 ≤ ℓ ≤ m. Let Δ(X_{ℓ}) be the set of all minors of X_{ℓ} and Γ(X) the set of maximal minors of X. We denote by R_{ℓ}(X) the B-subalgebra of B[X] generated by the entries of X_{ℓ} and the elements of Γ(X).

We first recall the definition of a graded algebra with straightening law (ASL). For the fundamental theory of a graded ASL we refer the reader to [1], [2] and [4]. Let A be a B-algebra and H a finite poset (a partially ordered set) in A. A product of the form α_{1}α_{2} ⋯ α_{p} where α_{1} ≤ α_{2} ≤ ⋯ ≤ α_{p} is called a standard monomial. Then A is a graded ASL on H over B if the following three conditions are satisfied:

(ASL-0) A = ⊕_{i≥0}A_{i} is a graded B-algebra such that B = A_{0}, H generates A as a B-algebra and H consists of homogeneous elements of positive degree.

(ASL-1) The set of standard monomials are linearly independent over B. (ASL-2) For all incomparable elements α, β ∈ H, the product αβ has a rep-

resentation αβ = ∑ b_{μ}μ, b_{μ} ∈ B, b_{μ} ≠ 0, where μ is a standard monomial and satisfying the following condition: every μ contains a factor ν ∈ H such that ν ≤ α, ν ≤ β. (It is allowed that αβ = 0, the sum ∑ b_{μ}μ being empty.) This relation is called the straightening relation.

Let {a_{1}, …, a_{t}} (resp. {b_{1}, …, b_{t}}) be a non-empty subset of {1, …, m} (resp. {1, …, n}). We denote by

[a_{1}, …, a_{t}|b_{1}, …, b_{t}]

the determinant of the submatrix (X_{a_{i}b_{j}})_{{1≤i≤t, \atop 1≤j≤t}} of X. To simplify the notation, we put

[a_{1}, …, a_{m}] = [1, …, m|a_{1}, …, a_{m}],

for the maximal submatrix (X_{ia_{j}})_{{1≤i≤m, \atop 1≤j≤m}} of X. We always assume that these indices are given in ascending order a_{i} < a_{i+1}, b_{i} < b_{i+1}. We define the partial order on the set Δ(X) as follows:

If α = [a_{1}, …, a_{u}|b_{1}, …, b_{u}] and β = [c_{1}, …, c_{v}|d_{1}, …, d_{v}], then

α ≤ β ⟺ u ≥ v and a_{i} ≤ c_{i}, b_{i} ≤ d_{i}, for i = 1, …, v.

We regard any subset of Δ(X) as an ordered set in this way. Let deg X_{ij} = 1 for all i, j. Then each element of Δ(X) is a homogeneous element of positive degree and R_{ℓ}(X) has the structure of a graded algebra with R_{ℓ}(X)_{0} = B. It is known that the algebra R_{ℓ}(X) is a graded ASL when ℓ = 0 or ℓ = m over B; see [1, Section 4].

Theorem 1.1. (1) (Hodge) R_{0}(X) is a graded ASL on the poset Γ(X) over B.

(2) (Doubilet-Rota-Stein) R_{m}(X) = B[X] is a graded ASL on the poset Δ(X) over B.

Our theorem tells that the same holds true for any 0 ≤ ℓ ≤ m.

Theorem 1.2. R_{ℓ}(X) is a graded ASL on the poset Δ(X_{ℓ}) ∪ Γ(X) over B.

From now on, we put H_{ℓ} = Δ(X_{ℓ}) ∪ Γ(X) for short. Before the proof, we recall the following:

Lemma 1.3 (Plücker relations). Let A be a ring and 0 < m ≤ n integers. For every m × n matrix with entries in A and all indices f_{1}, …, f_{p}, g_{1}, …, g_{q}, h_{1}, …, h_{r} ∈ {1, …, n} such that q > m, s > 0, m = p + s = r + q − s, one has

∑_{{i_{1}<···<is \atop i_{s+1}<···<iq} ; \{i_{1},...,iq \}=\{1,...,q\}} σ(i_{1}, …, i_{q})[f_{1}, …, f_{p}, g_{i_{1}}, …, g_{i_{s}}][g_{i_{s+1}}, …, g_{i_{q}}, h_{1}, …, h_{r}] = 0.

Proof of Theorem (1.2). As stated above, R_{ℓ}(X) is a graded B-algebra such that R_{ℓ}(X)_{0} = B, the poset H_{ℓ} consists of homogeneous elements of positive degree and H_{ℓ} generates R_{ℓ}(X) over B. Furthermore, since H_{ℓ} is a subposet of Δ(X) and R_{ℓ}(X) is a subring of B[X], the standard monomials in R_{ℓ}(X) are linearly independent by Theorem 1.1 (2). Thus, it is enough to show that the algebra R_{ℓ}(X) satisfies condition (ASL-2).

We build the matrix X̃ of indeterminates over B from X as follows:

X̃ = ( \begin{matrix}& X_{1,n+1} & ⋯ & X_{1,m+n} \\ X & ⋮ & ... & ⋮ \\ & X_{m,n+1} & ⋯ & X_{m,m+n} \end{matrix} ) .

Let φ̃ be the B-algebra map from B[X̃] to B[X] substituting for each entry of X̃ the corresponding entry of the matrix

( \begin{matrix}& 0 & ⋯ & ⋯ & 0 & 1 \\ & ⋮ & & … & … & 0 \\ X & ⋮ & … & … & … & ⋮ \\ & 0 & … & … & & ⋮ \\ & 1 & 0 & ⋯ & ⋯ & 0 \end{matrix} ) .

Let φ : R_{0}(X̃) → B[X̃] → B[X] be the composition of the inclusion and φ̃. If δ = [b_{1}, …, b_{m}] ∈ Γ(X̃) \ {[n + 1, …, n + m]}, then

φ(δ) = ±[a_{1}, …, a_{t}|b_{1}, …, b_{t}]

where t = max{i | b_{i} ≤ n} and {a_{1}, …, a_{t}} = {1, …, m} \ {n + m + 1 − b_{m}, …, n + m + 1 − b_{t+1}}. Note that φ gives a bijective correspondence between Γ(X̃) \ {[n + 1, …, n + m]} and Δ(X). Moreover, this correspondence preserves the order (see [1, (4.9) Lemma]).

To check condition (ASL-2), we need the following lemma.

Lemma 1.4. Let α = [a_{1}, …, a_{t}|b_{1}, …, b_{t}] ∈ Δ(X), β = [c_{1}, …, c_{m}] ∈ Γ(X). Then the product αβ has a representation

αβ = ∑ b_{μ_{1}μ_{2}}μ_{1}μ_{2}, b_{μ_{1}μ_{2}} ∈ B, b_{μ_{1}μ_{2}} ≠ 0

such that μ_{1} ∈ Γ(X), μ_{2} ∈ Γ ((X_{a_{i}j})_{{1≤i≤t, \atop 1≤j≤n}} ), and μ_{1} ≤ μ_{2}, μ_{1} ≤ α, μ_{1} ≤ β.

Proof. We may assume that α and β are incomparable. Let α̃ = [b_{1}, …, b_{t}, b_{t+1}, …, b_{m}] be an element of Γ(X̃) such that φ(α̃) = ±α. It is clear that φ(β) = β. Since φ preserves the order, α̃ and β are incomparable. Hence there is an integer p such that

c_{1} ≤ b_{1}, …, c_{p} ≤ b_{p}, c_{p+1} > b_{p+1}, 0 ≤ p ≤ t − 1.

We now apply the Plücker relation 1.3 in the case:

{f_{1}, …, f_{p}} = {c_{1}, …, c_{p}},
{g_{1}, …, g_{q}} = {b_{1}, …, b_{p+1}, c_{p+1}, …, c_{m}},
{h_{1}, …, h_{r}} = {b_{p+2}, …, b_{m}}.

Any non-zero term [d_{1}, …, d_{m}][e_{1}, …, e_{m}] being different from βα̃, in the Plücker relation corresponding to these data, have the following properties after arranging the indices in ascending order ([1, (4.5) Lemma]):

[d_{1}, …, d_{m}] ≤ β, d_{1} ≤ e_{1}, …, d_{p+1} ≤ e_{p+1}.

By induction on p the product βα̃ has a representation

βα̃ = ∑ b_{δ_{1}δ_{2}}δ_{1}δ_{2}, b_{δ_{1}δ_{2}} ∈ B, b_{δ_{1}δ_{2}} ≠ 0

such that δ_{1} ∈ Γ(X), δ_{2} ∈ Γ(X̃), and δ_{1} ≤ δ_{2}, δ_{1} ≤ β. We note that δ_{2} has the form [ν_{1}, …, ν_{t}, b_{t+1}, …, b_{m}] such that ν_{t} ≤ n. Similarly, we straighten it in the order α̃β. Then we have the straightening relation α̃β = ∑ c_{γ_{1}γ_{2}}γ_{1}γ_{2} where c_{γ_{1}γ_{2}} ∈ B, γ_{1} ≤ γ_{2} and γ_{1} ≤ α̃. By Theorem 1.1 (1), the standard monomials of the poset Γ(X̃) are linearly independent. Therefore both representations of βα̃ = α̃β coincide. Hence we get δ_{1} ≤ α̃. Since φ(α̃β) = ±αβ and φ preserves the order, the lemma follows.

□

Now we return to the proof of Theorem 1.2. Let α and β be elements of poset H_{ℓ} which are incomparable. We have to show that αβ has the straightening relation.

When both α and β are in Γ(X) (resp. in Δ(X_{ℓ})), it follows from Theorem 1.1. Otherwise, it follows from Lemma 1.4.

□

It is easy to check that the poset H_{ℓ} satisfies the condition of the distributive lattice in Δ(X). In particular, H_{ℓ} is a wonderful poset and we get the following.

Corollary 1.5. The ring B satisfies Serre’s condition (S_{n}) if and only if the ring R_{ℓ}(X) satisfies (S_{n}). In particular, B is a Cohen-Macaulay ring if and only if R_{ℓ}(X) is a Cohen-Macaulay ring.

2. Normality, the class group and Gorensteinness

In this section we argue the normality, the class group and the Gorenstein property of R_{ℓ}(X). We first observe the dimension of R_{ℓ}(X).

Lemma 2.1. Let B = K be a field and let p be an integer such that 1 ≤ p ≤ n. Let

Ψ_{1} = {X_{1j} | 1 ≤ j ≤ n} ∪ {X_{ip} | 2 ≤ i ≤ ℓ}
Ψ_{2} = {X′_{ij} | 2 ≤ i ≤ m, 1 ≤ j ≤ n, j ≠ p}

where X′_{ij} = X_{ij} − X_{1j}X_{ip}X_{1p}^{−1}, and let X′ = (X′_{ij}) be the (m − 1) × (n − 1) matrix. If ℓ > 0, then the set Ψ = Ψ_{1} ∪ Ψ_{2} is algebraically independent over K and

R_{ℓ}(X)[X_{1p}^{−1}] = (R_{ℓ−1}(X′)) [Ψ_{1}][X_{1p}^{−1}].

In other words, R_{ℓ}(X)[X_{1p}^{−1}] is isomorphic to (R_{ℓ−1}(Y)) [T_{1}, …, T_{n+ℓ−1}][T_{1}^{−1}] where Y is an (m − 1) × (n − 1) matrix of indeterminates over K and T_{1}, …, T_{n+ℓ−1} are indeterminates over R_{ℓ−1}(Y).

Proof. Let p be an integer such that 1 ≤ p ≤ n. Then dim K[Ψ, {X_{ip}}_{ℓ<i≤m}] = mn, since K[X][X_{1p}^{−1}] = K[Ψ, {X_{ip}}_{ℓ<i≤m}][X_{1p}^{−1}]. The cardinality of the set Ψ ∪ {X_{ip}}_{ℓ<i≤m} is also mn, hence these are algebraically independent over K. For the second statement, we have

R_{ℓ}(X)[X_{1p}^{−1}] = K[X_{ℓ}, Γ(X)][X_{1p}^{−1}]
= K[X′_{ℓ−1}, Γ(X′), Ψ_{1}][X_{1p}^{−1}]
= (R_{ℓ−1}(X′)) [Ψ_{1}][X_{1p}^{−1}].
□

Proposition 2.2. If ℓ < m, then

dim R_{ℓ}(X) = dim B + m(n − m + ℓ) + 1.

Proof. For a Noetherian ring B, dim R_{ℓ}(X) = dim B+rank H_{ℓ} because R_{ℓ}(X) is a graded ASL on H_{ℓ}. So we may assume that B is a field. We prove it by induction on ℓ. The case ℓ = 0 is true (e.g. see [1, (5.12) Corollary]). Suppose ℓ > 0. By Lemma 2.1 and the induction hypothesis, we have

dim R_{ℓ}(X) = dim R_{ℓ}(X)[X_{11}^{−1}]
= dim (R_{ℓ−1}(Y)) [T_{1}, …, T_{n+ℓ−1}][T_{1}^{−1}]
= (m − 1)(n − 1 − (m − 1) + ℓ − 1) + 1 + (n + ℓ − 1)
= m(n − m + ℓ) + 1.
□

We need some more notation. Let A be a commutative ring and M = (x_{ij}) be an m × n matrix over A with 0 < m ≤ n. Let δ = [a_{1}, …, a_{m}] be a maximal minor of M and {b_{1}, …, b_{n−m}} = {1, …, n} \ {a_{1}, …, a_{m}}. For each ℓ ≤ min{m, n−m}, take two subsets {i_{1}, …, i_{ℓ}} and {j_{1}, …, j_{ℓ}} such that {i_{1}, …, i_{ℓ}} ⊆ {a_{1}, …, a_{m}} with i_{1} < ⋯ < i_{ℓ} and {j_{1}, …, j_{ℓ}} ⊆ {b_{1}, …, b_{n−m}} with j_{1} < ⋯ < j_{ℓ}. Let c_{1}, …, c_{m} be the sequence of integers such that 1 ≤ c_{1} < … c_{m} ≤ n and {c_{1}, …, c_{m}} = ({a_{1}, …, a_{m}} \ {i_{1}, …, i_{ℓ}}) ∪ {j_{1}, …, j_{ℓ}}. We denote by δ_{j_{1}···j_{ℓ}}^{i_{1}···i_{ℓ}} the maximal minor [c_{1}, …, c_{m}].

Lemma 2.3. Let A_{1} ⊆ A_{2} be commutative rings and 1 ≤ m ≤ n integers. Let M = (x_{ij}) be an m × n matrix over A_{2} and δ = [a_{1}, …, a_{m}] a maximal minor of M. Then we have

A_{1} [Γ(M), δ^{−1}] = A_{1} [δ, δ^{−1}, \{δ_{j}^{i}\}_{{i∈\{a_{1},...,am\} \atop j∈\{b_{1},...,b_{n−m}\}}} ] .

Proof. We may assume that m ≥ 2. Note that any element in Γ(M) is the form of δ_{j_{1}···j_{ℓ}}^{i_{1}···i_{ℓ}}. It is enough to show that δ_{j_{1}···j_{ℓ}}^{i_{1}···i_{ℓ}} ∈ A_{1} [δ, δ^{−1}, \{δ_{j}^{i}\}_{{i∈\{a_{1},...,am\} \atop j∈\{b_{1},...,b_{n−m}\}}} ] for any 1 ≤ ℓ ≤ min{m, n − m}. We shall prove it by induction on ℓ. When ℓ = 1, it is true by definition. Suppose ℓ ≥ 2. Let {a′_{1}, …, a′_{m−ℓ}} = {a_{1}, …, a_{m}} \ {i_{1}, …, i_{ℓ}}. We now apply the Plücker relation 1.3 in the case:

{f_{1}, …, f_{p}} = {j_{ℓ}},
{g_{1}, …, g_{q}} = {a′_{1}, …, a′_{m−ℓ}, i_{1}, …, i_{ℓ}, j_{1}, …, j_{ℓ−1}},
{h_{1}, …, h_{r}} = {a′_{1}, …, a′_{m−ℓ}}.

Any non-zero term being different from δ_{j_{1}···j_{ℓ}}^{i_{1}···i_{ℓ}}δ, in the Plücker relation corresponding to these data, has the form ξ_{1}ξ_{2} such that ξ_{1} and ξ_{2} have only ℓ−1 indices in \{j_{1}, …, j_{ℓ}\} at most. Thus, such a term is in A_{1} [δ, δ^{−1}, \{δ_{j}^{i}\}_{{i∈\{a_{1},...,am\} \atop j∈\{b_{1},...,b_{n−m}\}}} ] by

induction hypothesis. Hence δ_{j_{1}···j_{ℓ}}^{i_{1}···i_{ℓ}} ∈ A_{1} [δ, δ^{−1}, \{δ_{j}^{i}\}_{{i∈\{a_{1},...,am\} \atop j∈\{b_{1},...,b_{n−m}\}}} ], because δ^{−1} is contained in the ring. □

Lemma 2.4. Let δ = [a_{1}, …, a_{m}] be an element of Γ(X) and set {b_{1}, …, b_{n−m}} = {1, …, n} \ {a_{1}, …, a_{m}}. We put

Φ = \{\{X_{ia_{j}}\}_{{1≤i≤ℓ \atop 1≤j≤m}} , δ, \{δ_{j}^{i}\}_{{i∈\{a_{1},...,am\} \atop j∈\{b_{1},...,b_{n−m}\}}} \} .

If ℓ < m, then

R_{ℓ}(X)[δ^{−1}] = B[Φ][δ^{−1}]

in the total quotient ring of R_{ℓ}(X), and moreover the set Φ is algebraically independent over B. Therefore R_{ℓ}(X)[δ^{−1}] is isomorphic to B[T_{1}, …, T_{d}][T_{1}^{−1}] where d = ^{#}Φ = m(n − m + ℓ) + 1 and T_{1}, …, T_{d} are indeterminates over B.

Proof. By Lemma 2.3, we have

R_{ℓ}(X)[δ^{−1}] = B[X_{ℓ}][Γ(X), δ, δ^{−1}] = B[X_{ℓ}] [δ, δ^{−1}, \{δ_{j}^{i}\}_{{i∈\{a_{1},...,am\} \atop j∈\{b_{1},...,b_{n−m}\}}} ] .

Furthermore we can check that X_{ib_{j}} ∈ B[Φ][δ^{−1}] for all 1 ≤ i ≤ ℓ and 1 ≤ j ≤ n−m, because

0 = det ( \begin{matrix}X_{ia_{1}} & ⋯ & X_{ia_{m}} & X_{ib_{j}} \\ X_{1a_{1}} & ⋯ & X_{1a_{m}} & X_{1b_{j}} \\ ⋮ & ... & ⋮ & ⋮ \\ X_{ma_{1}} & ⋯ & X_{ma_{m}} & X_{mb_{j}} \end{matrix} ) = (∑_{k=1}^{m} ±X_{ia_{k}}δ_{b_{j}}^{a_{k}}) + X_{ib_{j}}δ

and δ is an invertible element. Hence we get R_{ℓ}(X)[δ^{−1}] = B[Φ][δ^{−1}]. In order to prove the algebraic independence of Φ, we first consider the case that B (= K) is a field. Then R_{ℓ}(X) is a finitely generated K-algebra of dimension d, whence it follows that dim K[Φ] = d. Thus Φ is algebraically independent. The general case follows by the same argument in [1, the proof of (6.1) Lemma]. In fact, when B = Z, Φ is algebraically independent over Z (since it is true over Q), besides one can check that R_{ℓ}(X)/Z[Φ] is Z-flat. Therefore, we have an isomorphism between R_{ℓ}(X)[δ^{−1}] and Z[Φ][δ^{−1}] ⊗_{Z} B for any B. Hence Φ is algebraically independent over B.

□

Next we argue the normality of R_{ℓ}(X) and its divisor class group. For the theory of the divisor class group we refer the reader to [6]. Let Cl(A) stand for the divisor class group of a normal domain A.

Theorem 2.5. If B is a normal domain, then R_{ℓ}(X) is a normal domain and its divisor class group Cl(R_{ℓ}(X)) is isomorphic to Cl(B).

Proof. We may assume ℓ < m. Let δ = [1, …, m]. Note that δ is the minimal element in Δ(X) and also in H_{ℓ}. Now we claim that δB[X] ∩ R_{ℓ}(X) = δR_{ℓ}(X). Take an element δα ∈ δB[X] ∩ R_{ℓ}(X), where α ∈ B[X]. We write α = ∑ b_{μ}μ, where b_{μ} ∈ B and μ is a standard monomial in B[X]. Then δα = ∑ b_{μ}(δμ) and this is in R_{ℓ}(X). Each δμ is also a standard monomial, whence it should be in R_{ℓ}(X). Hence μ ∈ R_{ℓ}(X) and α ∈ R_{ℓ}(X). We have completed the proof of the claim. Because δ is a prime element in B[X], it follows that δ is a prime element in R_{ℓ}(X). R_{ℓ}(X)[δ^{−1}] is a normal domain by Lemma 2.4 and the residue class ring R_{ℓ}(X)/δR_{ℓ}(X) is an integral domain. Hence R_{ℓ}(X) is a normal domain ([1, (16.24) Lemma]).

Next we see the class group of R_{ℓ}(X). By Nagata’s theorem [6, 7.2], we have the following exact sequence:

0 → U → Cl(R_{ℓ}(X)) → Cl(R_{ℓ}(X)[δ^{−1}]) → 0,

where U is a subgroup generated by the classes of the minimal prime ideals of δ. But now δ is a prime element of R_{ℓ}(X), we get U = (0) and Cl(R_{ℓ}(X)) ≅ Cl(R_{ℓ}(X)[δ^{−1}]). Furthermore, by Lemma 2.4 again, R_{ℓ}(X)[δ^{−1}] is a localization of a polynomial ring over B. Therefore we have Cl(R_{ℓ}(X)) ≅ Cl(B).

□

Since a normal domain A is factorial if and only if its divisor class group Cl(A) is zero, we get the following.

Corollary 2.6. R_{ℓ}(X) is a factorial domain if and only if B is a factorial domain.

In the case that B is a field, the ring R_{ℓ}(X) is a Cohen-Macaulay factorial domain by Corollary 1.5 and Corollary 2.6. Hence it follows that R_{ℓ}(X) is a Gorenstein ring from Murthy’s theorem. More generally we get the following theorem.

Theorem 2.7. R_{ℓ}(X) is a Gorenstein ring if and only if B is a Gorenstein ring.

Proof. Since R_{ℓ}(X) is a faithfully flat extension of B, B is a Gorenstein ring if R_{ℓ}(X) is a Gorenstein ring. Suppose B is a Gorenstein ring. Take a prime ideal Q of R_{ℓ}(X) and let P = Q ∩ B. Then R_{ℓ}(X)_{Q} is a Gorenstein ring if and only if the fibre ring (B_{P} /P B_{P}) ⊗ R_{ℓ}(X)_{Q} is Gorenstein. But this is a localization of (B_{P} /P B_{P}) ⊗ R_{ℓ}(X) and it is a graded ASL on the same poset H_{ℓ} over the field B_{P} /P B_{P}. Thus the fibre ring is a Gorenstein ring. Hence R_{ℓ}(X)_{Q} is a Gorenstein ring.

□

Remark 2.8. When B is a field and m = 2, R_{1}(X) coincides with the Rees algebra of second cycle of the Koszul complex with respect to a sequence of n-variables of a polynomial ring over B. The Gorenstein property of R_{1}(X) is independently proved in case B is a field and m = 2 by Herzog, Tang and Zarzuela [7]. Weyman [11] also recently proved that the Rees algebras of all cycles of the Koszul complex with respect to a sequence of variables of a polynomial ring over a field of characteristic zero are Cohen-Macaulay and have only rational singularities.

3. R_{ℓ}(X) as an invariant ring

Before the proof of Theorem 4.1, we look at our generalized Grassmann algebra R_{ℓ}(X) from another point of view. In this section we assume that B is an infinite field and put B = K. Let 0 ≤ ℓ ≤ m ≤ n be integers. Let GL(m, K) be the general linear group over K. Let SL(m, K) = {σ ∈ GL(m, K) | det σ = 1} be the special linear group over K. We consider the subgroup of SL(m, K):

H_{m}^{ℓ} = \{ ( \begin{matrix}E_{ℓ} & 0 \\ C & D \end{matrix} ) C ∈ M_{(m−ℓ)×ℓ}(K), D ∈ SL(m − ℓ, K) \}

where E_{ℓ} is the ℓ×ℓ identity matrix and M_{(m−ℓ)×ℓ}(K) denotes the set of (m−ℓ)×ℓ matrices over K. The group H_{m}^{ℓ} acts on the polynomial ring K[X] as follows: for σ ∈ H_{m}^{ℓ} and f ∈ K[X], we define σ(f)(X) = f(σX). Let

K[X]^{H{ℓ \atop m}} = \{f ∈ K[X] | σ(f) = f , ∀σ ∈ H_{m}^{ℓ}\}

be the ring of invariants under this action of H_{m}^{ℓ} on K[X]. We want to describe the invariant subring K[X]^{H{ℓ \atop m}}. When ℓ = 0, it is the first fundamental theorem of invariant theory (e.g. see [1], [3]).

Theorem 3.1 (The first fundamental theorem). Let K be an infinite field. Then R_{0}(X) is the ring of invariants K[X]^{SL(m,K)}.

Our result is the following.

Theorem 3.2. Let K be an infinite field. Then R_{ℓ}(X) coincides with the ring of invariants K[X]^{H{ℓ \atop m}}.

Proof. We prove it by induction on ℓ. When ℓ = 0, the assertion follows from the first fundamental theorem. Suppose ℓ ≥ 1. Of course we may assume that ℓ < m. We fix an integer p such that 1 ≤ p ≤ n and use the notation of Lemma 2.1. For σ ∈ H_{m}^{ℓ}, we let σ̃ denote the element of H_{m−1}^{ℓ−1} obtained by deleting the 1st column and the 1st row. By Lemma 2.1, the set Ψ = Ψ_{1} ∪ Ψ_{2} is algebraically independent over K. Let H_{m−1}^{ℓ−1} act on the ring K[X′] = K[Ψ_{2}] in the same manner. Then we first claim that:

Claim 1. If g ∈ K[X′] and σ ∈ H_{m}^{ℓ}, then σ(g) = σ̃(g).

Let σ = (σ_{ij}). It is enough to show that σ(X′_{ij}) = σ̃(X′_{ij}) for each element X′_{ij} ∈ Ψ_{2}.

σ(X′_{ij}) = σ(X_{ij}) − X_{1j}X_{1p}^{−1}σ(X_{ip})
= ∑_{k=1}^{m} σ_{ik}X_{kj} − X_{1j}X_{1p}^{−1} ∑_{k=1}^{m} σ_{ik}X_{kp}
= ∑_{k=2}^{m} σ_{ik}X_{kj} − X_{1j}X_{1p}^{−1} ∑_{k=2}^{m} σ_{ik}X_{kp}
= ∑_{k=2}^{m} σ_{ik}X′_{kj}
= σ̃(X′_{ij}).

Therefore Claim 1 follows. The inclusion R_{ℓ}(X) ⊆ K[X]^{H{ℓ \atop m}} is obvious. We have to show the opposite inclusion. Take any element f ∈ K[X]^{H{ℓ \atop m}}. Of course, the element f ∈ K[X][X_{1p}^{−1}] = K[Ψ, {X_{ip}}_{ℓ+1≤i≤m}][X_{1p}^{−1}]. Here we claim that:

Claim 2. f ∈ K[Ψ][X_{1p}^{−1}].

Let S = K[Ψ, X_{1p}^{−1}][{X_{ip}}_{ℓ+1≤i≤m−1}]. X_{mp} is algebraically independent over S. Since f ∈ S[X_{mp}], we can write f = f(X_{mp}) = ∑_{k=0}^{u} f_{k}X_{mp}^{k} where f_{k} ∈ S. For c ∈ K, let

σ_{c} = ( \begin{matrix}1 & 0 & ⋯ & 0 \\ 0 \\ ⋮ & & E_{m−1} \\ 0 \\ c \end{matrix} )

be the element of H_{m}^{ℓ}. Then we have f = σ_{c}(f) = ∑_{k=0}^{u} f_{k}(X_{mp} + cX_{1p})^{k}. Hence f(cX_{1p}) = f_{0} for all c ∈ K. Therefore we get that f − f_{0} = 0 because K is an infinite field. So f ∈ S.

Repeating this procedure, we get f ∈ K[Ψ][X_{1p}^{−1}]. Thus X_{1p}^{t}f ∈ K[Ψ] for some t ≥ 0. We write

X_{1p}^{t}f = ∑_{a} f_{a}\underline{X}^{a},

where a = (a_{11}, …, a_{1n}, a_{2p}, …, a_{ℓp}), \underline{X}^{a} = X_{11}^{a_{11}} ⋯ X_{1n}^{a_{1n}}X_{2p}^{a_{2p}} ⋯ X_{ℓp}^{a_{ℓp}} and f_{a} belongs to K[Ψ_{2}]. For σ ∈ H_{m}^{ℓ},

X_{1p}^{t}f = σ(X_{1p}^{t}f) = ∑_{a} σ(f_{a})\underline{X}^{a} = ∑_{a} σ̃(f_{a})\underline{X}^{a}.

Hence σ̃(f_{a}) = f_{a} for every a. Therefore f_{a} ∈ K[X′]^{H{ℓ−1 \atop m−1}}. Since K[X′]^{H{ℓ−1 \atop m−1}} = R_{ℓ−1}(X′) by induction hypothesis, f ∈ (R_{ℓ−1}(X′)) [Ψ_{1}][X_{1p}^{−1}] = R_{ℓ}(X)[X_{1p}^{−1}] by Lemma 2.1. In particular f ∈ R_{ℓ}(X)[X_{11}^{−1}] ∩ R_{ℓ}(X)[X_{12}^{−1}]. We can write f = X_{11}^{−u}g = X_{12}^{−v}h for some g, h ∈ R_{ℓ}(X). Hence we have that f ∈ R_{ℓ}(X) because the ring R_{ℓ}(X) is a factorial domain. This completes the proof. □

Remark 3.3. The groups H_{m}^{ℓ} are not linearly reductive even in characteristic zero, so Boutot’s theorem cannot be used to conclude the Cohen-Macaulay property.

4. the Rees algebra of the second syzygy module

Let (A, 𝔪) be a regular local ring and d the dimension of A. Let k be the residue field of A. Let F_{•} be the minimal free resolution of k and M_{n} the n-th syzygy module of k. The Rees algebra R(M_{n}) of M_{n} is given as the image of the natural homomorphism from Sym_{A}(M_{n}) to Sym_{A}(F_{n−1}).

In this section we prove the following:

Theorem 4.1. The Rees algebra of the second syzygy module R(M_{2}) is a Gorenstein factorial domain.

Proof of Theorem 4.1. Let a_{1}, …, a_{d} be a regular system of parameters in A. Let S = A[x_{1}, …, x_{d}] be the polynomial ring over A (here x_{1}, …, x_{d} are indeterminates). Because R(M_{2}) is the image of the natural map from Sym_{A}(M_{2}) to S. One can regard R(M_{2}) as the subalgebra

A [det ( \begin{matrix}a_{i} & a_{j} \\ x_{i} & x_{j} \end{matrix} ) | 1 ≤ i < j ≤ d]

of S. We put R = R(M_{2}). And we set deg(x_{i}) = 1. Let 𝔐 be the graded maximal ideal of R. Let

X = ( \begin{matrix}A_{1} & … & A_{d} \\ X_{1} & … & X_{d} \end{matrix} )

the generic 2 × d matrix and R_{1}(X) stands for our generalized Grassmann algebra of X over the field k. We denote by gr_{𝔐}(R) the associated graded ring of R with respect to 𝔐. Then we have the following isomorphism.

Claim 1. R_{1}(X) ≅ gr_{𝔐}(R).

To show Claim 1, we first determine the defining ideal of R_{1}(X). Let

Y = ( \begin{matrix}0 & T_{12} & ⋯ & T_{1d} & A_{1} \\ −T_{12} & 0 & ⋯ & T_{2d} & A_{2} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ −T_{1d} & −T_{2d} & ⋯ & 0 & A_{d} \\ −A_{1} & −A_{2} & ⋯ & −A_{d} & 0 \end{matrix} )

be the (d + 1) × (d + 1) skew symmetric matrix of indeterminates over k and let Pf_{2t}(Y) be the ideal generated by all 2t-order pfaffians of Y. And let T = k[{T_{ij}}_{1≤i<j≤d}, {A_{i}}_{1≤i≤d}] be the polynomial ring with d(d + 1)/2-indeterminates over k. Then it is known that Pf_{2t}(Y) is a prime ideal in T with height(Pf_{2t}(Y)) = (d+2−2t)(d+3−2t)/2 ([8], [9]). Let φ : T → R_{1}(X) be the k-algebra map such that T_{ij} ↦ A_{i}X_{j} − A_{j}X_{i} and A_{i} ↦ A_{i}. Then R_{1}(X) is isomorphic to the residue ring T/ Pf_{4}(Y). In fact, one can check that Pf_{4}(Y) ⊆ Ker(φ). Since R_{1}(X) is a domain, we have that Ker(φ) is a prime ideal. Since φ is surjective and dim R_{1}(X) = 2d−1, it follows that height(Ker(φ)) = dim T − dim R_{1}(X) = (d − 2)(d − 1)/2. Therefore Ker(φ) = Pf_{4}(Y), hence the isomorphism T/ Pf_{4}(Y) ≅ R_{1}(X) follows. Let ϕ : T → gr_{𝔐}(R) be the k-algebra map such that T_{ij} ↦ \overline{a_{i}x_{j} − a_{j}x_{i}} and A_{i} ↦ \overline{a_{i}}. Here \overline{∗} denotes the images in 𝔐/𝔐^{2}. Similarly, one can check that Pf_{4}(Y) ⊆ Ker(ϕ). By [10, Proposition 2.2], dim R = dim A + rank M_{2} = 2d − 1, hence we have that dim gr_{𝔐}(R) = 2d − 1. Therefore, by the same reason, we get the isomorphism R_{1}(X) ≅ gr_{𝔐}(R). Hence it follows that gr_{𝔐}(R) is a Gorenstein factorial domain by Theorem 2.7 and Corollary 2.6. So the Rees ring R is also a Gorenstein ring.

To show that the Rees algebra R is a factorial domain, it is enough to check the following.

Claim 2. a_{1} is a prime element of R and the ring R[\frac{1}{a_{1}} ] is a factorial domain.

First, there is the following isomorphism as graded rings,

gr_{𝔐/a_{1}R}(R/a_{1}R) ≅ gr_{𝔐}(R)/a_{1}^{∗}gr_{𝔐}(R)

where a_{1}^{∗} is the initial form of a_{1}. Thus gr_{𝔐/a_{1}R}(R/a_{1}R) is an integral domain because a_{1}^{∗} is an irreducible and the ring gr_{𝔐}(R) is a factorial domain. In particular, the residue ring R/a_{1}R is an integral domain. Hence a_{1} is a prime element of R. One can check that

R [\frac{1}{a_{1}} ] = A [x_{2} − \frac{a_{2}}{a_{1}} x_{1}, …, x_{d} − \frac{a_{d}}{a_{1}} x_{d}] [\frac{1}{a_{1}} ] ,

and the elements \{x_{i} − \frac{a_{i}}{a_{1}} x_{i}\}_{2≤i≤d} are algebraically independent over A. Hence we have the ring R[\frac{1}{a_{1}} ] is a factorial domain. This completes the proof. □

References

[1] W. Bruns and U. Vetter, Determinantal Rings, Lecture Notes in Math. 1327, Springer-Verlag Berlin Heidelberg, 1988.

[2] C. De Concini, D. Eisenbud and D. Procesi, Hodge algebras, Asterisque 91 (1982).

[3] C. De Concini and D. Procesi, A characteristic free approach to invariant theory, Adv. in Math. 21 (1976), 330–354.

[4] D. Eisenbud, Introduction to algebras with straightening laws, Ring Theory and Algebra � edited by B. R. McDonald, Lect. Notes in Pure and Appl. Math. 55, M. Dekker, 1980, 243–268

[5] D. Eisenbud, C. Huneke and B. Ulrich, What is the Rees algebra of a module?, Proc. Amer. Math. Soc. 131 (2003), no. 3, 701–708

[6] R.Fossum, The Divisor Class Group of a Krull Domain, Springer-Verlag Berlin Heidelberg New York 1973

[7] J. Herzog, Z. Tang and S. Zarzuela, Symmetric and Rees algebras of Koszul cycles and their Gröbner bases, preprint

[8] T. Józefiak and P. Pragacz, Ideals generated by pfaffians, J. Alg. 61 (1979), 189–198

[9] H. Kleppe, Deformation of schemes defined by vanishing of pfaffians, J. Alg. 53 (1978), 84–92 [10] A. Simis, B. Ulrich and W. V. Vasconcelos, Rees algebras of modules, Proc. London Math.

Soc. (3) 87 (2003), 610–646 [11] J. Weyman, Application of the geometric technique of calculating syzygies to Rees algebras,

preprint

Shiro Goto Department of Mathematics, School of Science and Technology, Meiji University, 214-8571 JAPAN e-mail : goto@math.meiji.ac.jp

Futoshi Hayasaka Department of Mathematics, School of Science and Technology, Meiji University, 214-8571 JAPAN e-mail : ee68048@math.meiji.ac.jp

Kazuhiko Kurano Department of Mathematics, School of Science and Technology, Meiji University, 214-8571 JAPAN e-mail : kurano@math.meiji.ac.jp

Yukio Nakamura Department of Mathematics, School of Science and Technology, Meiji University, 214-8571 JAPAN e-mail : ynakamu@math.meiji.ac.jp