1 Introduction

This paper concerns a result in commutative algebra but is intended mainly for experts in complex analysis and CR-geometry. The background required for understanding the algebraic content is rather minimal and is given in Section 2.

We consider Artinian local commutative associative algebras over a field k. Such an algebra A is Gorenstein if and only if the socle Soc(A) of A is a one-dimensional vector space over k. Gorenstein algebras frequently occur in various areas of mathematics and its applications to physics (see, e.g., [1],[12]). In the case when the field k has characteristic zero, in articles [6],[11], a surprising criterion for isomorphism of Artinian Gorenstein algebras was found. The criterion is given in terms of a certain algebraic hypersurface S π in the maximal ideal of A associated to a linear projection π on with range Soc(A), where we assume that dimk A>1. The hypersurface S π passes through the origin and is the graph of a polynomial map P π : kerπ→Soc(A)≃k. It is shown in [6],[11] that two Artinian Gorenstein algebras A and are isomorphic if and only if any two hypersurfaces S π and arising from A and , respectively, are affinely equivalent, that is, there exists a bijective affine map such that .

Currently, there are two different proofs of the above criterion. The one given in [11] is purely algebraic, whereas the one proposed in [6] reduces the case of an arbitrary field to that of . A proof of the criterion in the latter case is contained in our earlier article [7] and, quite surprisingly, is based on a complex-analytic argument. It turns out that one can give an independent complex-analytic proof of the criterion for as well. This proof has not been previously published and is derived from ideas developed in paper [5], which never appeared in print (see Remark 4.1). However, the argument utilized in the case is rather important as it inspired our proof for the case in [7] and therefore laid the foundation of the general result for an arbitrary field. Thus, the purpose of the present article is to provide a complex-analytic proof of the criterion for and to compare it with that for given in [7]. These proofs emphasize an intriguing connection between complex analysis and commutative algebra. In each of the two cases, the idea is to consider certain tube submanifolds in complex space associated to the algebras in question and utilize their CR-automorphisms. In fact, as stated in Remark 5.1, there is a deep relationship between Artinian Gorenstein algebras for and tube hypersurfaces locally CR-equivalent to Levi non-degenerate hyperquadrics. We believe that this relationship has much to offer if fully understood.

The paper is organized as follows. Section 2 contains algebraic preliminaries and the precise statement of the criterion in Theorem 2.1. The proof of the necessity implication of Theorem 2.1 is given in Section 3. This part of the proof has no relation to complex analysis and is only included for the completeness of our exposition. Further, Sections 4 and 5 contain the complex-analytic proofs of the sufficiency implications for and , respectively. Finally, in Section 6, we demonstrate how powerful our criterion can be in applications. Namely, we apply it to a one-parameter family A t of 15-dimensional Gorenstein algebras. While directly finding all pairwise isomorphic algebras in the family A t seems to be quite hard, this problem is easily solved with the help of Theorem 2.1

2 Preliminaries

Let A be a commutative associative algebra over a field k. We assume that A is unital and denote by 1 its multiplicative identity element. Further, we assume that A is local, that is, (i) A has a unique maximal ideal (which we denote by ), and (ii) the natural injective homomorphism is in fact an isomorphism. In this case, one has the vector space decomposition , where 〈·〉 denotes linear span. Furthermore, A is isomorphic to the unital extension of its maximal ideal , which is the direct sum endowed with an operation of multiplication in the obvious way. For example, the algebra of germs of holomorphic functions at the origin in is a complex local algebra whose maximal ideal consists of all germs vanishing at 0. Clearly, every element of is the sum of the germ of a constant function and a germ vanishing at the origin.

Next, we suppose that dimk A>1 and that A is Artinian, i.e., dimk A<∞. In addition, we let A to be Gorenstein, which means that the socle of A, defined as , is a one-dimensional vector space over k (see, e.g., [9]). For example, if I is the ideal in generated by the germs of , then is a complex Artinian Gorenstein algebra with and spanned by the element of represented by the germ of z 1 z 2.

We now assume that the field k has characteristic zero and consider the exponential map

where x 0:=1. By Nakayama’s lemma (see, e.g., Theorem 2.2 on p. 8 in [13]), is a nilpotent algebra, and therefore the above sum is in fact finite, with the highest-order term corresponding to m=ν, where ν≥1 is the socle degree of A, i.e., the largest among all integers μ for which . Observe that . The map exp is bijective with the inverse given by the polynomial transformation

(2.1)

Fix a linear projection π on A with range Soc(A) and kernel containing 1 (we call such projections admissible). Set and let S π be the hypersurface in given as the graph of the polynomial map of degree ν defined as follows:

(2.2)

(note that for dimk A=2, one has P π =0). Observe that the Soc(A)-valued quadratic part of P π is non-degenerate on since the Soc(A)-valued bilinear form

(2.3)

is non-degenerate on A (see, e.g., p. 11 in [8]). Numerous examples of hypersurfaces S π explicitly computed for particular algebras can be found in [2],[6],[7] (see also Section 6 below).

We will now state the criterion for isomorphism of Artinian Gorenstein algebras obtained in [6],[11].

Theorem 2.1.

Let A and be Gorenstein algebras of finite vector space dimension greater than 1 over a field of characteristic zero and π and admissible projections on A and , respectively. Then A and are isomorphic if and only if the hypersurfaces S π and are affinely equivalent.

Remark 2.2.

For every hypersurface S π , we let be the graph over of the polynomial map -P π (see (2.2)). Observe that

(2.4)

Clearly, S π and are affinely equivalent if and only if and are affinely equivalent. Therefore, in order to prove Theorem 2.1, it is sufficient to obtain its statement with and in place of S π and , respectively. The hypersurfaces and are easier to work with, and we utilize them instead of S π and in our proofs below.

3 Proof of the necessity implication

First of all, we explain how the necessity implication of Theorem 2.1 is derived. As stated in the introduction, this part of the proof has no relation to complex analysis and is only included in the paper for the completeness of our exposition. The proof below works for any field of characteristic zero. The idea is to show that if π 1 and π 2 are admissible projections on A, then for some . Clearly, the necessity implication is a consequence of this fact (cf. [7]).

For every , let M y be the multiplication operator from A to defined by a↦y a and set . The correspondence defines a linear map from into the space of linear maps from to Soc(A). Since for every admissible projection π the form b π defined in (2.3) is non-degenerate on A and since , it follows that is an isomorphism.

Next, let λ:=π 2-π 1 and observe that λ(1)=0, λ(Soc(A))=0. Clearly, lies in , and therefore there exists such that . We then have everywhere on A, hence

for x 0:= log(1+y 0), which implies as claimed. □

4 Proof of the sufficiency implication for

By assumption, , and we denote this common dimension by N. If N=2, then A and are clearly isomorphic, and thus, from now on, we suppose that N>2.

Let be the complexification of the real algebra A. Then , and is a complex Artinian Gorenstein algebra with maximal ideal . Next, we denote by the complex-linear extension of π to and by the conjugation on defining the real form A, for all , with x,y∈A. Then is a -valued Hermitian form on that coincides with b π on A (see (2.3)). Since the bilinear form b π is non-degenerate on A, the Hermitian form h is non-degenerate on .

Consider the following real Levi non-degenerate hyperquadric in the complex projective space :

where [z] denotes the point of represented by . We think of as the affine part of and of

as the affine part of the hyperquadric . One can choose complex coordinates w 1,…,w N-1 in so that, upon identification of with , the affine quadric is given by the equation

(4.1)

where w:=(w 1,…,w N-2) and H is a non-degenerate Hermitian form on .

Next, consider the following real tube hypersurface in :

(4.2)

Let be the exponential map associated to . It is straightforward to check that the biholomorphic transformation from to given by

maps T onto .

Analogously, for the algebra , we obtain a Hermitian -valued form on , a real Levi non-degenerate hyperquadric in , the corresponding affine hyperquadric in , and a tube hypersurface in .

Now, let be a bijective affine map that establishes equivalence between and . We extend it to a complex affine map . Clearly, transforms T into . Consider the biholomorphism from to defined as follows:

(4.3)

where and is the exponential map associated to . Observe that Φ maps onto .

We will now show that, upon identification of with and with , the map Φ is affine. Indeed, since and are biholomorphically equivalent, the signatures of their Levi forms coincide. Therefore, one can choose complex coordinates in so that, upon identification of with , the affine quadric is given by the equation

where (cf. (4.1)). Thus, when written in the coordinates (w,w N-1), , the map Φ becomes an automorphism of preserving quadric (4.1). It is well-known that every such transformation has the form

where satisfies H(U w,U w)≡σ H(w,w), , , λ>0, σ=±1, and σ may be equal to -1 only if the numbers of positive and negative eigenvalues of H are equal. In particular, Φ is an affine map as claimed.

By formulas (2.1) and (4.3), for we have

(4.4)

where x 0:=f(0), g:=f-x 0 is the linear part of f, and is the exponential map associated to . Since Φ is affine, formula (4.4) implies g(x)2=g(x 2) for all , which is equivalent to the statement that is an algebra isomorphism. Therefore, and are isomorphic, hence A and are also isomorphic as required. □

Remark 4.1.

The method utilized in the above proof can be extracted, in principle, from ideas contained in paper [5] (which should be read in conjunction with [4]), but is by no means explicitly articulated there.

5 Proof of the sufficiency implication for

The argument that follows is contained in [7] and is included in this paper for comparison with the proof in the case given in Section 4.

By assumption, , and we denote this common dimension by N. If N=2, then A and are isomorphic, and thus, from now on, we suppose that N>2.

Consider the maximal ideal of A. We will now forget the complex structure on and treat it as a real algebra. Let be the (real) unital extension of (see Section 2) and 1 the multiplicative identity element in . Observe that is not Gorenstein since . We now consider the complexification of . Then , and is a complex Artinian local algebra with maximal ideal . The complex algebra is not Gorenstein since .

Next, let be the extension of to defined by the condition and denote by the complex-linear extension of to . Further, denote by the conjugation on defining the real form , for all , with . Then is a -valued Hermitian form on that coincides on with the -valued bilinear form defined analogously to (2.3).

Consider the following subset of the complex projective space :

where [z] denotes the point of represented by . We think of as the affine part of and of

as the affine part of . Observe that is a real-analytic Levi non-degenerate CR-submanifold of of real codimension 2. In fact, one can choose complex coordinates w 1,…,w 2N-2 in so that, upon identification of with , the affine quadric is given by the equations

(5.1)

This can be seen by choosing coordinates in (regarded as a complex algebra) in which the restriction of the Soc(A)-valued bilinear form b π to is given by the identity matrix.

Next, consider the following real tube codimension 2 submanifold in :

Let be the exponential map associated to . It is straightforward to check that the biholomorphic transformation from to given by

maps T onto .

Analogously, for the algebra , we obtain algebras and , a Hermitian -valued form on , a real Levi non-degenerate codimension 2 affine quadric in , the corresponding coordinates in , and a tube hypersurface in .

Now, let be a bijective affine map that establishes equivalence between and . We treat f as a real affine map and extend it to a complex affine map . Notice that transforms T into . Consider the biholomorphism from to defined as follows:

(5.2)

where and is the exponential map associated to . Observe that Φ maps onto .

We claim that, upon identification of with and with , the map Φ is affine. Indeed, when written in the coordinates w 1,…,w 2N-2, , the map Φ becomes an automorphism of preserving quadric (5.1). The fact that Φ is affine now follows from a description of CR-automorphism of this quadric (see the elliptic case on pp. 37–38 in [3]).

By formulas (2.1) and (5.2), for we have

(5.3)

where x 0:=f(0), g:=f-x 0 is the linear part of f, and is the exponential map associated to . Since Φ is affine, formula (5.3) implies g(x)2=g(x 2) for all , i.e., is an algebra isomorphism. Therefore, and are isomorphic, hence A and are also isomorphic as required. □

Remark 5.1.

The proofs of Theorem 2.1 for the cases presented in this paper are based on considering real tube submanifolds in complex space CR-equivalent to Levi non-degenerate affine quadrics. For , we utilized hypersurfaces, whereas for , codimension 2 submanifolds were required. The former are called spherical tube hypersurfaces (see [10]), and there is in fact an intriguing relationship between them and real and complex Artinian Gorenstein algebras. This relationship was outlined in [5] (see also Section 9.2 in [10] for a brief survey). It turns out that

  • to every real Artinian Gorenstein algebra of dimension greater than 2 and to every complex Artinian Gorenstein algebra one can associate a (closed) spherical tube hypersurface, where in the real case one obtains exactly hypersurfaces with bases (2.4) as in (4.2);

  • any two such hypersurfaces are affinely equivalent if and only if the corresponding algebras are isomorphic (one way to obtain the necessity implication in the real case is to proceed as in Section 4);

  • in a certain sense, all spherical tube hypersurfaces can be obtained by combining hypersurfaces of these two types.

We note that no higher-codimensional analogues of spherical tube hypersurfaces were considered in [5]. However, as the proof of Theorem 2.1 for suggests, analogues of this kind are related to Artinian Gorenstein algebras as well. We believe that the curious connection between complex analysis and commutative algebra manifested through tube submanifolds CR-equivalent to affine quadrics deserves further investigation.

6 Example of application of Theorem 2.1

Theorem 2.1 is particularly useful when at least one of the hypersurfaces S π and is affinely homogeneous (recall that a subset S of a vector space V is called affinely homogeneous if for every pair of points p,q∈S there exists a bijective affine map g of V such that g(S)=S and g(p)=q). In this case, the hypersurfaces S π and are affinely equivalent if and only if they are linearly equivalent. Indeed, if, for instance, S π is affinely homogeneous and is an affine equivalence between S π and , then f∘g is a linear equivalence between S π and , where g is an affine automorphism of S π such that g(0)=f -1(0). Clearly, in this case, is affinely homogeneous as well.

The proof of Theorem 2.1 in [11] shows that every linear equivalence f between S π and has the block-diagonal form with respect to the decompositions and , where , that is, there exist linear isomorphisms and such that f(x+y)=f 1(x)+f 2(y), with , y∈Soc(A). Therefore, for the corresponding polynomial maps P π and (see (2.2)), we have

(6.1)

where and are the homogeneous components of degree m of P π and , respectively.

Thus, Theorem 2.1 yields the following corollary (cf. Theorem 2.11 in [7]).

Corollary 6.1

Let A and be Gorenstein algebras of finite vector space dimension greater than 1 over a field of characteristic zero and π and admissible projections on A and , respectively.

  1. (i)

    If A and are isomorphic and at least one of S π and is affinely homogeneous, then for some linear isomorphisms and identity (6.1) holds. In this case, both S π and are affinely homogeneous.

  2. (ii)

    If for some linear isomorphisms and identity (6.1) holds, then the hypersurfaces S π and are linearly equivalent, and therefore the algebras A and are isomorphic.

In [11] (see also Corollary 4.10 in [6]), we found a criterion for the affine homogeneity of some (hence every) hypersurface S π arising from an Artinian Gorenstein algebra A. Namely, S π is affinely homogeneous if and only if the action of the automorphism group of the algebra on the set of all hyperplanes in complementary to Soc(A) is transitive. Furthermore, we showed (see also Corollary 4.11 in [6]) that this condition is satisfied if A is non-negatively graded in the sense that it can be represented as a direct sum

where A j are linear subspaces of A, with A 0=k (in this case and Soc(A)=A d for d:= max{j:A j≠0}). It then follows that part (i) of Corollary 6.1 applies in the situation when one (hence the other) of the algebras A and is non-negatively graded (see also [7] for the case ). Note, however, that the existence of a non-negative grading on A is not a necessary condition for the affine homogeneity of S π (see, e.g., Remark 2.6 in [7]). Also, as shown in Section 8.2 in [6], the hypersurface S π need not be affinely homogeneous in general.

To demonstrate how our method works, we will now apply Corollary 6.1 to a one-parameter family of non-negatively graded Artinian Gorenstein algebras. As before, let k be a field of characteristic zero. For t∈k, t≠±2, define

It is straightforward to verify that every A t is a Gorenstein algebra of dimension 15. We will prove the following proposition.

Proposition 6.2

A r and A s are isomorphic if and only if r=±s.

Proof.

The sufficiency implication is trivial (just replace y by -y). For the converse implication, consider the following monomials in k[x,y]:

Let e 0=1,e 1,…,e 14, respectively, be the vectors in A t arising from these monomials (to simplify the notation, we do not indicate the dependence of e j on t). They form a basis of A t . Define

where, as before, 〈·〉 denotes linear span. It is straightforward to check that the subspaces form a non-negative grading on A t .

Next, denote by the maximal ideal of A t and let π t be the projection on A t with range and kernel 〈e 0,…,e 13〉. Denote by w 1,…,w 14 the coordinates in with respect to the basis e 1,…,e 14. In these coordinates the corresponding polynomial map is written as

Suppose that for some r≠s the algebras A r and A s are isomorphic. By part (i) of Corollary 6.1, there exist C∈GL(13,k) and c∈k ∗ such that

(6.2)

where w:=(w 1,…,w 13). Since 0 is the only value of t for which P t has degree 7, we have r,s≠0. Comparing the terms of order 7 in identity (6.2), we obtain that the second row in the matrix C has the form (0,μ,0,…,0), and

(6.3)

Next, comparing the terms of order 6 in (6.2), we see that the first row in the matrix C has the form (σ,ρ,0,…,0), and

(6.4)

Further, comparing the terms of order 5 in (6.2) that do not involve , we obtain

(6.5)

Now, (6.3), (6.4), and (6.5) yield r 2=s 2 as required. □