A Criterion for Isomorphism of Artinian Gorenstein Algebras

Let $A$ be an Artinian Gorenstein algebra over an infinite field $k$ with either $\hbox{char}(k)=0$ or $\hbox{char}(k)>\nu$, where $\nu$ is the socle degree of $A$. To every such algebra and a linear projection $\pi$ on its maximal ideal ${\mathfrak m}$ with range equal to the socle $\hbox {Soc}(A)$ of $A$, one can associate a certain algebraic hypersurface $S_{\pi}\subset{\mathfrak m}$, which is the graph of a polynomial map $P_{\pi}:\hbox{ker}\,\pi\to \hbox{Soc}(A)\simeq k$. Recently, the author and his collaborators have obtained the following surprising criterion: two Artinian Gorenstein algebras $A$, $\tilde A$ are isomorphic if and only if any two hypersurfaces $S_{\pi}$ and $S_{\tilde\pi}$ arising from $A$ and $\tilde A$, respectively, are affinely equivalent. The proof is indirect and relies on a geometric argument. In the present paper we give a short algebraic proof of this statement. We also discuss a connection, established elsewhere, between the polynomials $P_{\pi}$ and Macaulay inverse systems.


Introduction
We consider Artinian local commutative associative algebras over a field k. Recall that such an algebra A is Gorenstein if and only if the socle Soc(A) of A is a 1-dimensional vector space over k (see, e.g. [Hu]). Gorenstein algebras frequently occur in various areas of mathematics and its applications to physics (see, e.g. [B], [L]). For k = C, in [FIKK] we found a surprising criterion for two Artinian Gorenstein algebras to be isomorphic. The criterion was given in terms of a certain algebraic hypersurface S π in the maximal ideal m of A associated to a linear projection π on m with range Soc(A), where we assume that dim k A > 1. The hypersurface S π passes through the origin and is the graph of a polynomial map P π : ker π → Soc(A) ≃ C. In [FIKK] we showed that for k = C two Artinian Gorenstein algebras A, A are isomorphic if and only if any two hypersurfaces S π and S π arising from A and A, respectively, are affinely equivalent, that is, there exists a bijective affine map A : m → m such that A(S π ) = S π .
It should be noted that the above criterion differs from the well-known criterion in terms of Macaulay inverse systems, where to an Artinian Gorenstein algebra A one also associates certain polynomials (see, e.g. Proposition 2.2 in [ER] and references therein). Indeed, as the discussion in Section 5 shows, in general none of the polynomials P π is an inverse system for A. The value of the method of [FIKK] lies in the fact that affine equivalence for the hypersurfaces S π and S π is sometimes easier to verify than the kind of to our work.
Utilizing the above relationship between nil-polynomials and Macaulay inverse systems, at the end of Section 5 we revisit Example 4.2 and explain how isomorphism between A t and A 0 can be established by using inverse systems. This turns out to be significantly harder to do than by applying the method based on nil-polynomials. In general, to use inverse systems one needs to compute them first, and Theorem 5.1 provides an explicit way for producing them from nil-polynomials. This fact alone shows that nilpolynomials are a new rather useful tool for the study of Artinian Gorenstein algebras.
Acknowledgements. We are grateful to N. Kruzhilin for stimulating discussions and to the referee for the thorough reading of the paper and helpful suggestions. This work is supported by the Australian Research Council.

Preliminaries
Let A be an Artinian Gorenstein algebra over a field k with identity element 1, maximal ideal m and socle Soc(A) := {x ∈ A : x m = 0}. Suppose that dim k A > 1 (i.e. m = 0) and denote by ν the socle degree of A, that is, the largest among all integers µ for which m µ = 0. Observe that ν ≥ 1 and Soc(A) = m ν .
Assume now that either char(k) = 0 or char(k) > ν and define the exponential map exp : m → 1 + m by the formula where x 0 := 1. This map is bijective with the inverse given by Next, fix a linear projection π on A with range Soc(A) and kernel containing 1 (we call such projections admissible). Set K := ker π ∩ m and let S π be the graph of the polynomial map P π : K → Soc(A) of degree ν defined as follows: (note that for dim k A = 2 one has P π = 0). Everywhere below we identify any point (x, P π (x)) ∈ S π with the point x + P π (x) ∈ K ⊕ Soc(A) = m and therefore think of S π as a hypersurface in m. Observe that the Soc(A)-valued quadratic part of P π is non-degenerate on K since the Soc(A)-valued bilinear is non-degenerate on A (see, e.g. p. 11 in [He]). Examples of hypersurfaces S π explicitly computed for particular algebras can be found in [FIKK], [FK], [EI] (see also Section 4 below).
We will now state the criterion for isomorphism of Gorenstein algebras that was obtained in [FIKK], [FK] for algebras over fields of zero characteristic.
THEOREM 2.1 Let A, A be Gorenstein algebras of finite vector space dimension greater than 1 and socle degree ν over an infinite field k with either char(k) = 0 or char(k) > ν. Let, further, π, π be admissible projections on A, A, respectively. Then A, A are isomorphic if and only if the hypersurfaces S π , S π are affinely equivalent. Moreover, if A : m → m is a linear isomorphism such that A(S π ) = S π , then A is an algebra isomorphism.

Proof of Theorem 2.1
For every hypersurface S π , we let S π be the graph over K of the polynomial map −P π (see (2.1)). Observe that S π = {x ∈ m : π(exp(x)) = 0}. (3.1) Clearly, S π and S π are affinely equivalent if and only if S π and S π are affinely equivalent, and below we will obtain the first statement of the theorem with S π and S π in place of S π and S π , respectively. The necessity implication is proved as in Proposition 2.2 of [FIKK]. The argument is short, and for the completeness of our exposition we reproduce it below. The idea is to show that if π 1 , π 2 are admissible projections on A, then S π 1 = S π 2 + x 0 for some x 0 ∈ m. Clearly, the necessity implication is a consequence of this fact.
For every y ∈ m, let M y be the multiplication operator from A to m defined by a → ya and set K 1 := ker π 1 ∩ m. The correspondence y → π 1 • M y | K 1 defines a linear map L from K 1 into the space L(K 1 , Soc(A)) of linear maps from K 1 to Soc(A). Since for every admissible projection π the form b π defined in (2.2) is non-degenerate on A and since dim k L(K 1 , Soc(A)) = dim k K 1 , it follows that L is an isomorphism.
We will now obtain the sufficiency implication. Let A : m → m be an affine equivalence with A(S π ) = S π , and z 0 := A(0). Consider the linear map L(x) := A(x) − z 0 , with x ∈ m. We will show that L : m → m is an algebra isomorphism, which will imply that A and A are isomorphic.
Clearly, L maps S π onto S π − z 0 . Consider the admissible projection on A given by the formula π ′ (a) := π( exp(z 0 )a), where exp is the exponential map associated to A. Formula (3.1) then implies S π − z 0 = S π ′ , hence L maps S π onto S π ′ .
Recall that S π is the graph of the polynomial map −P π : K → Soc(A) (see (2.1)). Set n := dim k m − 1 = dim k m − 1 and choose coordinates α = (α 1 , . . . , α n ) in K and a coordinate α n+1 in Soc(A). In these coordinates the hypersurface S π is written as where g ij and h ijℓ are symmetric in all indices, (g ij ) is non-degenerate, and the dots denote the higher-order terms. In Proposition 2.10 in [FIKK] (which works over any field of characteristic either zero or greater than ν) we showed that the above equation of S π is in Blaschke normal form, that is, one has n ij=1 g ij h ijℓ = 0 for all ℓ, where (g ij ) := (g ij ) −1 . We will now need the following lemma, which is a variant of the second statement of Proposition 1 in [EE].
Lemma 3.1 Let V , W be vector spaces over an infinite field k, with dim k V = dim k W = N + 1, N ≥ 0. Choose coordinates β = (β 1 , . . . , β N ), β N +1 in V and coordinates γ = (γ 1 , . . . , γ N ), γ N +1 in W . Let S ⊂ V , T ⊂ W be hypersurfaces given, respectively, by the equations where P, Q are polynomials without constant and linear terms. Assume further that equations (3.2) are in Blaschke normal form. Then every bijective linear transformation of V onto W that maps S into T has the form where C ∈ GL(N, k), c ∈ k * and β, γ are viewed as column-vectors.
Proof: Let L : V → W be a bijective linear transformation. Write L in the most general form for some c 1 , . . . , c N , c ∈ k, d ∈ k N , and N × N-matrix C with entries in k.
Since k is an infinite field, identity (3.3) implies that the coefficients at the same monomials on the left and on the right are equal. Then, since P and Q do not contain linear terms, it follows that c 1 = . . . = c N = 0 and therefore C ∈ GL(N, k), c ∈ k * . Further, comparing the second-and thirdorder terms in (3.3), it is straightforward to see that the equations of both S and T cannot be in Blaschke normal form unless d = 0.
By Lemma 3.1, writing the hypersurface S π ′ in some coordinates α = ( α 1 , . . . , α n ), α n+1 in m chosen as above, we see that the map L has the form for some C ∈ GL(n, k), c ∈ k * . In coordinate-free formulation, (3.4) means that with respect to the decompositions m = K ⊕ Soc(A) and m = K ⊕ Soc( A), where K := ker π ′ ∩ m, the map L has the blockdiagonal form, that is, there exist linear isomorphisms L 1 : K → K and . Therefore, for the corresponding polynomial maps P π and P π ′ (see (2.1)) we have Clearly, identity (3.5) yields π ′ are the homogeneous components of degree m of P π , P π ′ , respectively, and ν is the socle degree of each of A and A (observe that the socle degrees of A and A are equal since by (3.5) one has deg P π = deg P π ′ ).
Let π m be the symmetric Soc(A)-valued m-form on K defined as follows: We will focus on the forms π 2 and π 3 (in fact, it is shown in Proposition 2.8 in [FIKK] that every π m with m > 3 is completely determined by π 2 , π 3 ). As in [FIKK], we define a commutative product (x, y) → x * y on K by requiring the identity π 2 (x * y, z) = π 3 (x, y, z) (3.9) to hold for all x, y, z ∈ K. Owing to the non-degeneracy of the form b π defined in (2.2), the form π 2 is non-degenerate and therefore for any x, y ∈ K the element x * y ∈ K is uniquely determined by (3.9). We need the following lemma.
Lemma 3.2 For any two elements x + u, y + v of m, with x, y ∈ K and u, v ∈ Soc(A), one has (3.10) Proof: We write the product xy with respect to the decomposition m = K ⊕ Soc(A) as xy = (xy) 1 + (xy) 2 , where (xy) 1 ∈ K and (xy) 2 ∈ Soc(A). It is then clear that (xy) 2 = π 2 (x, y). Therefore, to prove (3.10) we need to show that x * y = (xy) 1 . This identity is obtained by a simple calculation similar to the one that occurs in the proof of Proposition 2.8 of [FIKK] (cf. Proposition 5.13 in [FK]). Indeed, from (3.7), (3.9) for any z ∈ K one has Since π 2 is non-degenerate, the above identity implies x * y = (xy) 1 as required.
(3.11) Denote by * the product on K defined by π 2 , π 3 as in (3.9). Now, for all x, y ∈ K and u, v ∈ Soc(A), Lemma 3.2 and identity (3.11) with m = 2 yield ( 3.12) Next, for any z ∈ K, from (3.9) and identity (3.11) with m = 2, 3 one obtains Finally, we will obtain the last statement of the theorem. Let A : m → m be a linear isomorphism such that A(S π ) = S π . Consider the linear automorphisms F of m and F of m defined by Then the composition A 0 := F • A • F is a linear transformation from m to m that maps S π onto S π . By the above argument, it follows that A 0 is an algebra isomorphism. On the other hand, one has A 0 = A. Thus, A is an algebra isomorphism, and the proof of the theorem is complete.
4 Example of application of Theorem 2.1 Theorem 2.1 is particularly useful when at least one of the hypersurfaces S π , S π 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 A of V such that A(S) = S and A(p) = q). In this case the hypersurfaces S π , S π are affinely equivalent if and only if they are linearly equivalent. Indeed, if, for instance, S π is affinely homogeneous and A : m → m is an affine equivalence between S π , S π , then A • A ′ is a linear equivalence between S π , S π , where A ′ is an affine automorphism of S π such that A ′ (0) = A −1 (0). Clearly, in this case S π is affinely homogeneous as well.
The proof of Theorem 2.1 shows that every linear equivalence L between S π , S π has the block-diagonal form with respect to the decompositions m = K ⊕ Soc(A) and m = K ⊕ Soc( A), where K := ker π ∩ m, that is, there exist linear isomorphisms L 1 : K → K and L 2 : Soc(A) → Soc( A) such that L(x + u) = L 1 (x) + L 2 (u), with x ∈ K, u ∈ Soc(A). Therefore, analogously to (3.6), for the corresponding polynomial maps P π and P π (see (2.1)) we have where, as before, P [m] π , P [m] π are the homogeneous components of degree m of P π , P π , respectively. Thus, Theorem 2.1 yields the following corollary (cf. Theorem 2.11 in [FIKK]) .
Corollary 4.1 Let A, A be Gorenstein algebras of finite vector space dimension greater than 1 and socle degree ν over an infinite field k with either char(k) = 0 or char(k) > ν. Let, further, π, π be admissible projections on A, A, respectively. (i) If A and A are isomorphic and at least one of S π , S π is affinely homogeneous, then for some linear isomorphisms L 1 : K → K and L 2 : Soc(A) → Soc( A) identity (4.1) holds. In this case both S π and S π are affinely homogeneous.
(ii) If for some linear isomorphisms L 1 : K → K and L 2 : Soc(A) → Soc( A) identity (4.1) holds, then the hypersurfaces S π , S π are linearly equivalent and therefore the algebras A and A are isomorphic.
As shown in Section 8.2 in [FK], the hypersurface S π need not be affinely homogeneous in general. In [Is] (see also [FK]) 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 nilpotent algebra m on the set of all hyperplanes in m complementary to Soc(A) is transitive. Furthermore, we showed 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 m = ⊕ j>0 A j and Soc(A) = A d for d := max{j : A j = 0}). We stress that the grading {A j } in the above statement is not required to be standard, i.e. A j may not coincide with (A 1 ) j . Note, however, that the existence of a non-negative grading on A is not a necessary condition for the affine homogeneity of S π . For example, for any Artinian Gorenstein algebra of socle degree not exceeding 4 the hypersurfaces S π are affinely homogeneous (see Proposition 6.5 in [FK]), whereas not every such algebra admits a non-negative grading. The case ν = 3 is relatively easy; in fact, all algebras with this property were completely described in Theorem 4.1 of [ER]. The case ν = 4 is much harder (see Section 7.2 in [FK]), and the affine homogeneity of S π makes Corollary 4.1 an important tool in this case. We are confident that the approach discussed in this paper will help make significant advances on the classification problem at least for ν = 4. Note that, as shown in Proposition 7.5 of [FK], the classification result of Theorem 4.1 of [ER] can indeed be obtained by utilizing this method.
We will now give an example of how one can hope to apply our technique to algebras of socle degree 4.
Example 4.2 Let A 0 be the Artinian Gorenstein algebra introduced at the end of Section 8.1 in [FK], namely, y, z]] is the algebra of formal power series in x, y, z and J(f ) is the Jacobian ideal of f , i.e. the ideal generated by the first-order partial derivatives of f . As explained in [FK], this algebra does not admit any nonnegative grading. Furthermore, the Hilbert function of A 0 is {1, 3, 3, 1, 1}, hence the associated graded algebra of A 0 is not Gorenstein (cf. Proposition 9 in [W]). Also, one has ν = 4 and dim k A 0 = 9.
Let e t,0 , . . . , e t,7 be the basis in the maximal ideal m t of A t whose elements are represented, respectively, by monomials (4.3). Let π t be the admissible projection on A t defined by ker π t ∩ m t = e t,1 , . . . , e t,7 =: K t (observe that Soc(A t ) = e t,0 ). Then, denoting by y 1 , . . . , y 7 the coordinates in K t with respect to the basis e t,1 , . . . , e t,7 and identifying Soc(A t ) with k by means of e t,0 , we have P t := P π t = y 1 y 3 + 1 2 y 2 2 + 6y 2 y 4 − 8 3 y 4 y 7 − 8 3 y 5 y 6 + 15t 2 y 1 y 4 − 5t 2 y 2 4 + 1 2 y 2 1 y 2 + 3y 2 1 y 4 − 2y 1 y 2 4 + 4 9 y 3 4 − 4 3 y 4 y 2 5 + 1 24 y 4 1 . (4.5) Thus, we see that in the coordinates chosen as above the polynomials P 0 and P t coincide in homogeneous components of degrees 3 and 4 but their quadratic terms differ. We now identify Soc(A 0 ) and Soc(A t ) by means of the vectors e 0,0 and e t,0 . Then, by part (ii) of Corollary 4.1, to prove that A t is isomorphic to A 0 , it suffices to find a linear isomorphism L : 3, 4, (4.6) where, as before, P are the homogeneous components of degree m of P 0 , P t , respectively. Define L by x j = y j , for j = 3, 7, x 7 = 15t 16 y 4 + y 7 . * Here and below the relevant polynomials P π were found by using a Singular-based computer program, which had been kindly made available to us by W. Kaup.
Clearly, this linear transformation satisfies (4.6), which shows that A t is indeed isomorphic to A 0 for every t as claimed.
By the last statement of Theorem 2.1, the map L together with the identification of Soc(A 0 ) and Soc(A t ) is in fact an algebra isomorphism between m t and m 0 . More precisely, the map defined on the basis elements as e t,j → e 0,j , for j = 4, e t,4 → 15t 2 e 0,3 + e 0,4 + 15t 16 e 0,7 is an isomorphism from m t onto m 0 . The corresponding isomorphism between A t and A 0 is induced by the automorphism of k [[x, y, z]] given by the following change of variables: x → x, y → y + 15t 16 It would be interesting to see whether formula (4.7) could be obtained directly using ideal generators. One can produce an alternative proof of isomorphism of A t and A 0 by making use of Macaulay inverse systems. We will explore this possibility in the next section and compare it with the proof given above.

Nil-polynomials and Macaulay inverse systems
With the exception of a further discussion of Example 4.2 given below, the material of this section is contained in [AI]. Since this material is highly relevant to the present paper's theme and the proofs involved are not long, we reproduce it here for the reader's benefit. As before, let A be an Artinian Gorenstein algebra over a field k, with maximal ideal m and socle degree ν, where we assume that dim k A > 1 and either char(k) = 0 or char(k) > ν. Next, let Notice that in (5.1) the element (x 1 e 1 + . . . + x M e M ) j ∈ A may have a nontrivial projection to Soc(A) parallel to ker ω even for j < ν, in which case ω(x 1 e 1 + . . . + x M e M ) j = 0. In formulas (5.9) and (5.10) below we compute polynomials of this kind for the algebras A 0 and A t from Example 4.2. Further, the elements e 1 , . . . , e M generate A as an algebra, hence A is isomorphic to k[x 1 , . . . , x M ]/I, where I is the ideal of all relations among e 1 , . . . , e M , i.e. polynomials f ∈ k[x 1 , . . . , x M ] with f (e 1 , . . . , e M ) = 0. Observe that I contains the monomials x ν+1 1 , . . . , x ν+1 M . From now on we assume that char(k) = 0 (cf. Remark 5.2 below). For It is well-known that, since the quotient k[x 1 , . . . , x M ]/I is Gorenstein, there is a polynomial g ∈ k[x 1 , . . . , x M ] of degree ν such that I = Ann(g), where is the annihilator of g. The freedom in choosing g with Ann(g) = I is fully understood, namely, if g 1 , g 2 ∈ k[x 1 , . . . , x M ] satisfy Ann(g 1 ) = Ann(g 2 ) = I, with h(0) = 0. Any polynomial g with I = Ann(g) is called a Macaulay inverse system for the Artinian Gorenstein quotient k[x 1 , . . . , x M ]/I. Conversely, for any non-zero element g ∈ k[x 1 , . . . , x M ] the quotient k[x 1 , . . . , x M ]/ Ann(g) is a (local) Artinian Gorenstein algebra of socle degree deg g, hence any polynomial is an inverse system of some Artinian Gorenstein quotient. It is well-known that inverse systems can be used for solving the isomorphism problem for quotients of this kind as explained, for example, in Proposition 2.2 in [ER] (a precise statement is given at the end of this section). For details on inverse systems we refer the reader to [M], [Em], [Ia] (a brief survey given in [ER] is also helpful).
In applications it is desirable to have an explicit formula for computing an inverse system for any Artinian Gorenstein quotient. As the following theorem shows, formula (5.1) achieves this purpose.  Further, since the bilinear form (a, b) → ω(ab) is non-degenerate on A, identity (5.5) implies f (e 1 , . . . , e M ) = 0. Therefore f ∈ I, which shows that I = Ann(Q ω,B ) as required.
We will now make a number of useful remarks.
Remark 5.2 For simplicity, we have chosen to discuss inverse systems only under the assumption char(k) = 0. For fields of positive characteristics one needs to pass to divided power rings (see, e.g. Lemma 1.2 in [Ia]). If char(k) > ν, one can naturally think of Q ω,B as an element of the corresponding divided power ring, and Theorem 5.1 remains correct.
Remark 5.3 Fix a hyperplane Π in m complementary to Soc(A). A kvalued polynomial P on Π is called a nil-polynomial for A if there exists a linear form ρ : A → k such that ker ρ = Π, 1 and P = ρ • exp | Π . Note that deg P = ν. Upon identification of Soc(A) with k, the class of nil-polynomials coincides with that of Soc(A)-valued polynomial maps P π introduced in (2.1). If dim k A > 2, then ν ≥ 2 and Π contains an Mdimensional subspace that forms a complement to m 2 in m. Fix any such subspace V , choose a basis B = {e 1 , . . . , e M } in V and let x 1 , . . . , x M be the coordinates in V with respect to this basis. Then the polynomial Q ρ,B given by formula (5.1) is exactly the restriction of the nil-polynomial P to V written in the coordinates x 1 , . . . , x M . Thus, one way to explicitly obtain an inverse system for an Artinian Gorenstein quotient is to restrict a soclevalued map P π to a complement to m 2 in m lying in ker π ∩m and identify the socle with the field k. This observation provides a link between the classical approach to Artinian Gorenstein algebras by means of inverse systems and our criterion in Theorem 2.1.
Remark 5.4 Suppose that A is a standard graded algebra, i.e. A can be represented in the form (4.2) with A j = (A 1 ) j for j ≥ 1. Set In this case, for any choice of a basis B = {e 1 , . . . , e M } in V , the ideal I is homogeneous, i.e. generated by homogeneous relations. For an arbitrary nilpolynomial P on Π its restriction Q ρ,B to V coincides with the homogeneous component of degree ν of P : Thus, Theorem 5.1 yields a simple proof of the well-known fact that a standard graded Artinian Gorenstein algebra, when written as a quotient by a homogeneous ideal, admits a homogeneous inverse system and all homogeneous inverse systems for such algebras are mutually proportional (see [Em] and pp. 79-80 in [M]). where ω is a linear form on k[x 1 , . . . , x m ]/I with kernel complementary to the socle and ν is the socle degree of k[x 1 , . . . , x m ]/I. Then, arguing as in the proof of Theorem 5.1, we see that R is an inverse system for k[x 1 , . . . , x m ]/I. Thus, (5.6) is an explicit formula providing an inverse system for any Artinian Gorenstein quotient, and any other inverse system is obtained as h ⋆ R, where h ∈ k[x 1 , . . . , x m ] does not vanish at the origin. Notice that for m > M, no inverse system as in (5.6) comes from restricting a nil-polynomial to a subspace of m complementary to m 2 .
Thus, in order to decide whether two Artinian Gorenstein algebras are isomorphic, one can use either the classical approach, which utilizes inverse systems, or our method, which relies on nil-polynomials, and the two techniques are related as explained in Theorem 5.1 and Remark 5.3. In a particular situation one of the approaches may work better than the other. For instance, as we saw in the preceding section, the technique based on nil-polynomials is very appropriate for establishing that the algebras A t in Example 4.2 are all isomorphic to A 0 . We will now obtain a different proof of this statement by the method based on inverse systems.