Maximal families of Gorenstein algebras

Jan O. Kleppe

Abstract

The purpose of this paper is to study maximal irreducible families of (e.g. Artinian) Gorenstein quotients of a polynomial ring R. If GradAlg^{H}(R) is the scheme parametrizing graded quotients of R with Hilbert function H, we prove there is a close relationship between the generically smooth irreducible components of GradAlg^{H}(R) whose general member is a Gorenstein codimension (c + 1) quotient, and the generically smooth irreducible components of GradAlg^{H′}(R) whose general member B is a codimension c Cohen-Macaulay algebra. Moreover the dimension of the “Gorenstein” components are computed in terms of the dimension of the corresponding “Cohen-Macaulay” component and of other invariants of B. Furthermore we show that linkage is quite effective in computing all invariants of the dimension formula as well as verifying the assumptions which appears in a generalization of the main Theorem. AMS Subject Classification. 14C05, 13D10, 13D03, 13D07, 13C40. Keywords. Parametrization, Gorenstein algebra, Artinian algebra, liaison, licci, Cohen-Macaulay, canonical module, normal module, Hilbert scheme.

1 Introduction

The main goal of this paper is to contribute to the classification of (e.g. Artinian) Gorenstein quotients of a polynomial ring R. If GradAlg^{H} = GradAlg^{H}(R) is the scheme parametrizing graded quotients A of R of depth A ≥ min(1, dim A) and with Hilbert function H, we let CM_{c}^{H} (resp. Gor_{c}^{H}) be the open subscheme parametrizing Cohen-Macaulay (resp. Gorenstein) quotients of codimension c in R. In the Artinian case Gor_{c}^{H} and the scheme PGor(H) defined by certain catalecticant minors ([13]) are essentially the same. We shortly call an irreducible component of GradAlg^{H} which properly intersects CM_{c}^{H} (resp. Gor_{c}^{H}) a ”Cohen-Macaulay (resp. “Gorenstein) codimension c” component. The main Theorem of this paper establishes a strong connection between the generically smooth “Gorenstein codimension (c + 1)” components of GradAlg^{H}(R) and the generically smooth “Cohen-Macaulay codimension c” components of GradAlg^{H′}(R) for some H′. Indeed the connection between these components allows us to deduce the generically smoothness and the dimension of the “Gorenstein codimension (c+1)” component from the corresponding property and number of the “Cohen-Macaulay codimension c” component. Heuristically speaking, there are at least as many generically smooth “Gorenstein codimension (c + 1)” components of GradAlg^{H} as there are generically smooth “Cohen-Macaulay codimension c components of GradAlg^{H′}(R).

We all known the nice classification of CM_{2}^{H}, given by the Hilbert-Burch theorem, and of Gor_{3}^{H}, proved by Buchsbaum and Eisenbud, in terms of the maximal minors (resp. the Pfaffians) of a suitable matrix. In this case the schemes CM_{2}^{H} and Gor_{3}^{H} are smooth and irreducible. There is, in our opinion, little hope of finding one such nice classification of Gor_{4}^{H} in codimension 4 because this scheme consists of several irreducible components ([3] and [14]). We think a more promising classification strategy should consist of classifying the irreducible components of Gor_{4}^{H} together with describing the members of each family concretely. In this context the main Theorem shows that the corresponding classification of CM_{3}^{H′} is sentral for the classification of Gor_{4}^{H}, and it provides a method for answering Iarrobino et al’s question ([14]) about whether the non-irreducibility of Gor_{4}^{H} is a rare phenomenon, or not.

More precisely if we have A well defined (and hence B generically Gorenstein, see the beginning of section 2) by the sequence

0 → K_{B}(−s) → B → A → 0,  (1)

and K_{B}^{∗} is the B-dual of the canonical module of B ≃ R/I_{B}, we show

Theorem 1. Let B ≃ R/I_{B} be licci (resp. Proj(B) licci and B Cohen-Macaulay and unobstructed) and let A be a codimension 1 quotient of B, well defined by a twist of the canonical module K_{B}. Then A is unobstructed (resp. unobstructed if s >> 0) as a graded R-algebra, A is Gorenstein and,

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + dim(K_{B}^{∗})_{s} − 1 − δ(B)_{−s}
(resp. dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + dim(K_{B}^{∗})_{s} − 1)

where δ(B)_{−s} = dim(_{−s}Hom_{B}(I_{B}/I_{B}^{2}, K_{B})) − dim(_{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B})).

In the licci case of the Theorem, we have in fact the unobstructedness of A in the sense H^{2}(R, A, A) = 0, which holds in the non-graded case as well (Theorem 21). Moreover we prove a more general result (Proposition 10) having Theorem 1 as a Corollary. That result requires some skill in verifying the vanishing of the homology groups involved. In the final section 3 we show how we can use linkage to compute these homology groups as well as finding the other dimension invariants of Theorem 1. Note that if B = R/I and R/J are geometrically linked, then A = R/(I +J) is given by (1), so Theorem 1 covers this important case. The unobstructedness of A in the last mentioned case is known ([28]).

To achieve our results we study more generally deformations of a quotient A of R which happens to be a quotient of some quotient B of R. Firstly we recall and further develop (Theorem 4) some partially known results on the unobstructedness and the “family-dimension” of such a quotient A of R with fixed Hilbert function H_{A}. Here we neither assume B to be Cohen-Macaulay, nor A to be Gorenstein. The Theorem is central for the main application of this paper because we manage to treat the two main cases of interest: when the generic member of an irreducible family R → A factorizes via some deformation of B → A, and when it does not.

In this work the author is inspired by joint works with Chris Peterson where we in various ways construct Gorenstein quotients of R. We show for instance that if M is a maximal Cohen-Macaulay B-module and ∧^{r}M̃ ≃ K̃_{B}(s) as sheaves (in a large enough open set) of Proj(B), then any regular section of M̃^{∗}(s) defines a Gorenstein quotient R ↠ A of codimension r + c (cf. [20] for details). The idea of this paper is to look to the construction and vary (via deformation theory) every object and morphism in the construction as much as possible, to see how large the corresponding stratum in GradAlg^{H} will be. This paper looks at the r = 1 case because it is so far only in this case the author is able to prove his results also in the Artinian Gorenstein case. Note that C. Peterson and J. Migliore already have observed that not every codimension 3 quotent is obtained by (1) ([24]). On the other hand A. Geramita and J. Migliore ([7]) use geometric linkage to show that every non-empty Gor_{3}^{H} contains at least one point given by (1). All this fits nicely with Theorem 1 in which the number dim(_{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B})) essentially measures the codimension of the strata. To apply Theorem 1 we need to have at least one point of Gor_{c}^{H} given by (1), to apply it. In a forthcoming paper the author generalizes the results of this paper considerable by considering deformations of regular sections of a maximal Cohen-Macaulay module M of any rank, but, unfortunately, where we must restrict to positively dimensional Gorenstein algebras. We thank Olav A. Laudal, Roy Skjelnes and Johannes Kleppe for clarifying discussions.

For the convinciency of the reader we include a section of preliminaries where we give some space to the theory of deformations. Indeed it is not always easy to find a nice reference for the results we use because the litterature is often to general (with general cohomology groups of algebras which are difficult to handle) or to special (to makes the mentioned groups vanish or managable for being computed).

1.1 Preliminaries

Let B be an n-dimensional graded quotient of a polynomial k-algebra R (k is algebraically closed) in n + c variables (of degree 1), and let M and N be finitely generated graded Bmodules. Let depth_{J} M denote the length of a maximal M-sequence in a homogeneous ideal J and let depth M = depth_{𝔪} M where 𝔪 the irrelevant maximal ideal. Let H_{J}^{i}(−) be the right derivative of the functor, Γ_{J}(−), of sections with support in Spec(B/J). Recall that depth_{J} M = inf{depth M_{℘} | ℘ ⊇ J, ℘ graded} and that depth_{J} M ≥ r iff H_{J}^{i}(M) = 0 for i < r, cf. [8] or [10].

If Y = Proj(B) and Z is closed in Y and U = Y − Z, we let H_{∗}^{0}(U, M̃) = ⊕_{v} H^{0}(U, M̃(v)). Then we have an exact sequence

0 → H_{I(Z)}^{0}(M) → M → H_{∗}^{0}(U, M̃) → H_{I(Z)}^{1}(M) → 0  (2)

and isomorphisms H_{I(Z)}^{i}(M) ≃ H_{∗}^{i−1}(U, M̃) for i ≥ 2. In the case depth_{I(Z)} N ≥ 2, the graded Ext_{B}^{1}(M, N) injects into the corresponding global Ext_{O_{U}}^{1} -group of sheaves. Indeed we have in this case an exact sequence ([8], exp. VI)

0Ext_{B}^{1}(M, N) ↪ Ext_{O_{U}}^{1} (M̃|_{U}, Ñ|_{U}) → _{0}Hom_{B}(M, H_{I(Z)}^{2}(N)) → _{0}Ext_{B}^{2}(M, N) → Ext_{O_{U}}^{2} (M̃|_{U}, Ñ|_{U})

(3)

where the form of the middle term comes from a spectral sequence also treated in [8].

A Cohen-Macaulay (resp. maximal Cohen-Macaulay) B-module satisfies depth M = dim M (resp. depth M = dim B) by definition, or equivalently, H_{𝔪}^{i}(M) = 0 for i < dim M (resp. i < dim B). If B is Cohen-Macaulay and and K_{B} = Ext_{R}^{c}(B, R(−n − c)) is the canonical module of B, we know that v-graded piece of H_{𝔪}^{i}(M) is by Gorenstein duality

vH_{𝔪}^{i}(M) ≃ _{−v}Ext_{B}^{n−i}(M, K_{B})^{∨} .

Recall that two graded quotients, R/J and R/J′, are said to be (complete intersection) linked if there exists a homogeneous complete intersection ideal L such that J = L : J′ and J′ = L : J (with L ⊆ J ∩ J′). The relationship of being linked generates the equivalence relation, “linkage”. B = R/I_{B} is said to be licci (and hence Cohen-Macaulay) if it is in the linkage class of a complete intersection (cf. [22] for a survey).

Even though we in this paper tries to avoid using algebra (co)homology groups, e.g. H_{2}(R, B, B) and H^{2}(R, B, B) (we will instead use the corresponding Tor- of Ext-groups), we occasionally need to consider them. For this we recall that the former group is given by an exact sequence

0 → H_{2}(R, B, B) → H_{1} → G_{1} ⊗_{R} B → I_{B}/I_{B}^{2} → 0.  (4)

in which G_{1} is R-free, G_{1} ↠ I_{B} is surjective and minimal, and H_{1} = H_{1}(I_{B}) is the 1. Koszul homology of I_{B} [27]. An ideal I_{B} of R is called syzygietic if H_{2}(R, R/I_{B}, R/I_{B}) = 0. If I_{B} is syzygietic, then (I_{B})_{℘} is syzygietic for any prime ideal ℘ of R. Using (4) one shows that if R → B is generically a complete intersection and B is licci, then H_{2}(R, B, B) = 0 because H_{1} is a maximal Cohen-Macaulay module in the licci case [12]. Thus, generically complete intersection licci ideals are syzygietic. For the graded group H^{2}(R, B, B) we remark that there is an exact sequence

0 → _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, N) → _{0}H^{2}(R, B, N) → _{0}Hom_{B}(H_{2}(R, B, B), N) →  (5)

(coming from a well known spectral sequence because I_{B}/I_{B}^{2} ≃ H_{1}(R, B, B), cf. [1] or [21]).

A Cohen-Macaulay quotient B = R/I_{B} of codimension c = dim R − dim B in R has a minimal R-free resolution of the following form [6]

0 → G_{c} → ... → G_{1} → R → B → 0  ,  G_{j} = ⊕_{i=1}^{r_{j}}R(−n_{j,i})  (6)

Let G_{i}^{∗} be the R-dual of G_{i}. Since K_{B} = Ext_{R}^{c}(B, R(−n − c)), the R-dual complex of (6) yields a minimal R-free resolution of the twisted canonical module K_{B}(n + c). In particular if c = 3, it induces a complex

0 → N_{B} → G_{1}^{∗} ⊗_{R} B ⟶^{ψ_{1}} G_{2}^{∗} ⊗_{R} B ⟶^{ψ_{2}} G_{3}^{∗} ⊗_{R} B → K_{B}(n + c) → 0  (7)

which is exact except in the middle, cf. [20], p.795. Indeed ker ψ_{2}/ im ψ_{1} ≃ Tor_{1}^{R}(B, K_{B})(n + c) ≃ I/I^{2} ⊗ K_{B}(n + c), and this module is a maximal Cohen-Macaulay module if B is licci [4]. If c = 2, then (7) makes an exact sequence provided we have removed G_{3}^{∗} ⊗_{R} B.

Recall that a closed subscheme Y of ℙ = Proj R is called unobstructed if the Hilbert scheme, Hilb^{p} := Hilb^{p}(ℙ), is smooth at (Y ⊂ ℙ). Let GradAlg^{H} = GradAlg^{H}(ℙ) be the strata of Hilb^{p} given by deforming (Y ⊂ ℙ) with constant Hilbert function H_{Y} = H (i.e. its functor deforms both Y ⊂ ℙ and its homogeneous coordinate ring, B, flatly), cf. [17]. GradAlg^{H} allows a natural scheme structure whose tangent (resp. “obstruction”) space at (Y ⊂ ℙ) is _{0}Hom_{B}(I_{B}/I_{B}^{2}, B) ≃ _{0}Hom_{R}(I_{B}, B) (resp. _{0}H^{2}(R, B, B)), i.e. it is given by deforming B as a graded R-algebra [15]. In the case H(v) does not vanish for large v (i.e. B is non-Artinian), we may look upon GradAlg^{H} as parametrizing graded R-quotients, R → B, satisfying depth_{𝔪} B ≥ 1 and with Hilbert function H_{B} = H. If B is Artinian, GradAlg^{H} still represents a functor parametrizing graded R-quotients with Hilbert function H_{B} = H (see Proposition 6), and its open subscheme of Gorenstein quotients coincides, at least topologically and infinitesimally, with PGor(H) (the corresponding scheme of forms with “catalectical structure”). B is called unobstructed as a graded R-algebra iff GradAlg^{H} is smooth at (R → B). This definition of unobstructedness coincides with the one given in Remark 2 by [21] or [25], p.151. By [15], Remark 3.7,

GradAlg^{H} ≃ Hilb^{p} at (Y ⊂ ℙ)  (8)

provided _{0}Hom_{R}(I_{B}, H_{𝔪}^{1}(B)) = 0. This happenes for quotients B of depth_{𝔪} B ≥ 2.

Similarly we let GradAlg(H_{B}, H_{A}) be the representing object of the functor deforming flags (surjections) B → A of graded quotients of R of positive depth (for non-Artinian quotients) and with Hilbert functions H_{B} and H_{A} of B and A respectively. The tangent space T_{B→A} of GradAlg(H_{B}, H_{A}) at (B → A) and the tangent maps T_{pr_{i}} of the projection morphisms pr_{1} : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{A}} and pr_{2} : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{B}} are given by the cartesian square in the following diagram of exact sequences

0Hom_{R}(I_{B}, I_{A/B})
↓
T_{B→A}  →^{T_{pr_{2}}}  _{0}Hom_{R}(I_{B}, B)
↓  □  ↓
0 → _{0}Hom_{R}(I_{A/B}, A) → _{0}Hom_{R}(I_{A}, A) → _{0}Hom_{R}(I_{B}, A)

(9)

Here B → A ≃ R/I_{A} is a surjection of graded R-algebras with kernel I_{A/B}. Since the lower sequence in the diagram above may continue by _{0}H^{2}(B, A, A) to make a long exact sequence of algebra cohomology groups, and since _{0}H^{2}(B, A, A) injects in general into 0Ext_{B}^{1}(I_{A/B}, A) ([9], exp. VI), it follows that _{0}Hom_{R}(I_{A}, A) → _{0}Hom_{R}(I_{B}, A) is surjective provided _{0}Ext_{B}^{1}(I_{A/B}, A) = 0. (One may also prove this surjectivity more elementary by dualizing the exact sequence 0 → I_{B} ⊗_{R} A ≃ I_{B}/I_{A}I_{B} → I_{A} ⊗_{R} B → I_{A/B} → 0.)

Remark 2. Let R → B be a graded quotient. Let (T, 𝔪_{T}) → (S, 𝔪_{S}) be a small artin surjection (i.e. of local Artinian k-algebras with residue fields k whose kernel 𝔞 satisfies 𝔞·𝔪_{T} = 0). A graded deformation B_{S} of B to S is a graded S-flat quotient of R⊗_{k}S satisfying B_{S} ⊗_{S} k ≃ B. A graded deformation B_{T} of B_{S} to T is T -flat and satisfies B_{T} ⊗_{T} S ≃ B_{S}.

A flag (i.e. a surjection) of graded quotients of R, B → A, is said to be unobstructed if for every small artin T ↠ S and for every graded deformation B_{S} → A_{S} of B → A to S, there exists a graded deformation B_{T} → A_{T} (of T-flat quotients) reducing to B_{S} → A_{S} via (−) ⊗_{T} S. Similarly a quotient A of R is unobstructed if every graded deformation A_{S} deforms further to T

The obstruction of deforming A_{S} (resp. B_{S} → A_{S}) to T (resp. to a given graded deformation B_{T} of B_{S}) sits in _{0}H^{2}(R, A, A)⊗_{k} 𝔞 (resp. in _{0}H^{2}(B, A, A)⊗_{k} 𝔞) and _{0}H^{1}(R, A, A)⊗_{k} 𝔞 (resp. _{0}H^{1}(B, A, A) ⊗_{k} 𝔞) corresponds to the set of deformations (e.g. the tangent space). Note that, by definition, the vanishing of the obstruction is equivalent to the existence of corresponding desired deformation. Hence the vanishing of the obstruction groups (“spaces”) lead to the unobstructedness of the corresponding objects.

One may consider non-graded deformations of e.g. a quotient B of R in which case the whole algebra cohomology group Hom_{R}(I_{B}, B) ≃ H^{1}(R, B, B) (resp. H^{2}(R, B, B)) serves as tangent (resp. “obstruction”) space of the deformations of B as an R-algebra. For graded deformations it is the degree zero piece of the cohomology groups which counts. For a generically a complete intersection, R ↠ B, (or weaker, for a B which satisfies_{0}Hom_{B}(H_{2}(R, B, B), B) = 0, its obstructions sit in _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) ⊗ 𝔞 by (5).

It is rather straightforward to get (i) in the Proposition below from Remark 2 and the injection _{0}H^{2}(B, A, A) ↪ _{0}Ext_{B}^{1}(I_{A/B}, A). The latter statement is deeper and relies on a closer study of the obstructions involved, see [16], p.135 for a related proof.

Proposition 3. Let B be unobstructed as a graded R-algebra.

(i) If _{0}Ext_{B}^{1}(I_{A/B}, A) = 0 (or if _{0}H^{2}(B, A, A)), then B → A is unobstructed, i.e. GradAlg(H_{B}, H_{A}) is smooth at (B → A).

(ii) If _{0}Hom_{R}(I_{B}, B) → _{0}Hom_{R}(I_{B}, A) is surjective, then the first projection pr_{1} : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{A}} is smooth at (B → A).

Using Proposition 3, we get easily the “resp.” part of the following Theorem which is not new ([16], sect.1 or [19], ch.9), nor is special cases of the “non-resp” version ([19], ch.7). Note that the “resp.” part and in particular the parenthetical condition in Theorem 4(iii) has as a consequence that the generic member of a family of R ↠ A comes from a member of the family of R ↠ B ↠ A. Since it is in general unlikely that a general member factorizes via a deformation of B even though a special member does, we need a result to take care of this possibility. The “non-resp.” part of the following theorem does that (in which we use small letters for the k-dimension), and we only include a proof of this variation.

Theorem 4. Let R be a finitely generated graded k-algebra and let B ≃ R/I_{B} ↠ A ≃ B/I_{A/B} be a graded morphism of quotients of R. If

(i) _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 and (I_{B})_{℘} is syzygietic for any graded ℘ of Ass(A) ∪ Ass(B) (resp. B is unobstructed as a graded R-algebra),

(ii) _{0}Ext_{B}^{1}(I_{A/B}, A) = 0, and

(iii) _{0}Ext_{B}^{2}(I_{B}/I_{B}^{2}, I_{A/B}) ↪ _{0}Ext_{B}^{2}(I_{B}/I_{B}^{2}, B) is injective (resp. _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) ↪ _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) ) is injective),

then A is unobstructed as a graded R-algebra. Moreover, if depth_{𝔪} A ≥ min(1, dim A), then

dim_{(A)} GradAlg^{H_{A}} = _{0}hom_{R}(I_{B}, B)+_{0}hom_{B}(I_{A/B}, A)−_{0}hom_{R}(I_{B}, I_{A/B})+_{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}).

(resp. dim_{(A)} GradAlg^{H_{A}} = _{0}hom_{R}(I_{B}, B) + _{0}hom_{B}(I_{A/B}, A) − _{0}hom_{R}(I_{B}, I_{A/B}) .)

Furthermore if k is of characteristic zero and if W (B → A) and W (A) are the irreducible components of GradAlg(H_{A}, H_{B}) and GradAlg^{H_{A}} respectively to which (B → A) and (A) belong, then

dim W (A) − dim pr_{1}(W (B → A)) = _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}).

Proof. Using the assumptions of (i) and (iii) and the long exact sequence of_{0}Ext_{B}^{i}(I_{B}/I_{B}^{2}, −), we get _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, A) = 0. By the definition of a syzygetic ideal we have H_{2}(R, B, B)_{℘} = 0 for any ℘ of Ass(A). It follows that_{0}Hom_{B}(H_{2}(R, B, B), A) = 0 and hence that_{0}H^{2}(R, B, A) = 0 by (5). Since we have in general an injection _{0}H^{2}(B, A, A) ↪ _{0}Ext_{B}^{1}(I_{A/B}, A) ([9], exp.

VI), we get _{0}H^{2}(B, A, A) = 0 by (ii) above. It follows that _{0}H^{2}(R, A, A) = 0 by the long exact sequence of algebra cohomology [21],

→ _{0}H^{2}(B, A, A) → _{0}H^{2}(R, A, A) → _{0}H^{2}(R, B, A) →  (10)

This proves the unobstructedness of A and dim_{(A)} GradAlg^{H_{A}} = _{0}hom_{R}(I_{A}, A). Now, looking to (9), noting that vanishing of _{0}Ext_{B}^{1}(I_{A/B}, A) translates to a certain surjectivity in the cartesian square of (9) and that the assumption (i) above implies that the cokernel of the vertical Hom-map in the cartesian square of (9) is precisely _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}), it is straightforward to get the dimension formula.

To see that _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) measures the codimension, we claim that B ↠ A is unobstructed and hence that dim_{(B→A)} GradAlg(H_{A}, H_{B}) = dim T_{B→A}. Indeed the assumption (i) and (5) show that B is unobstructed as a graded R-algebra because we have H_{2}(R, B, B)_{℘} = 0 for any ℘ of Ass(B). Using (ii) above and Proposition 3(i), we get the claim. Now we remark that pr_{1} : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{A}} is a morphism between smooth schemes (i.e. smooth at (B → A) and A respectively) and with fiber dimension _{0}hom_{B}(I_{B}/I_{B}^{2}, I_{A/B}), from which we may conclude by using the Theorem of generic smoothness to pr_{1} at (B → A), cf. [19], first part of the proof of Proposition 9.14 for more details (from a similar proof).

Remark 5. a) Note that the “non-resp. part” of Theorem 4 leads to _{0}H^{2}(R, A, A) = 0 (and in fact to _{v}H^{2}(R, A, A) = 0 by twisting all three Ext_{B}-assumptions by v), cf. proof.

b) It is possible to weaken assumption (ii) in Theorem 4 to “ _{0}H^{2}(B, A, A) = 0” and still conclude as in Theorem 4. Similarly one may also weaken “(I_{B})_{℘} is syzygietic for any ℘ of Ass(A)∪Ass(B)” to “(I_{B})_{℘} is syzygietic for any ℘ of Ass(A) and _{0}Hom_{B}(H_{2}(R, B, B), A) = 0” and still get the same conclusions. Moreover by possibly forgetting the conclusion that the codimension is equal to _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) in Theorem 4, we may replace the “non-resp. assumptions” of both (i) and (iii) by “ _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, A) = 0 and _{0}Hom_{B}(H_{2}(R, B, B), A) = 0” and still conclude that A is unobstructed, but now that dim_{(A)} GradAlg^{H_{A}} is given by

0hom_{R}(I_{B}, B)+ _{0}hom_{B}(I_{A/B}, A)− _{0}hom_{R}(I_{B}, I_{A/B})+ _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B})− _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, B).

There is no real change in the proof. This variations turn out to have an application notably in the case where A is Artinian.

We finish this section by looking more closely to deformations of an Artinian algebra A. Firstly we recall the representability of the functor (the sheaf) GRADALG^{H}(−), locally given by letting GRADALG^{H}(Spec(S)) (S a noetherian k-algebra) be the set of graded S-flat quotients R_{S} → A_{S} such that H_{A_{s}} = H and depth_{𝔪} A_{s} ≥ min(1, dim A_{s}) for any s ∈ Spec(S). Here R_{S} = R ⊗_{k} S and A_{s} is the “fiber” of Spec(A_{S}) → Spec(S) at s ∈ Spec(S). To simplify and clarify [17] (being more careful with the topologies involved), we partially include proofs.

Proposition 6. The functor GRADALG^{H}(−) is representable. Its representing object, GradAlg^{H}, is a k-scheme of finite type.

Proof. Here we sketch a proof which is slightly different from that in [17] in the Artinian case (i.e. when H(v) = 0 for v >> 0). Indeed the flattening stratification described in [25] and Grothendieck’s representability Theorems of the Hilbert functor are the results we need to conclude. In the non-Artinian case, H(v) is a polynomial p(v) for v >> 0 and GradAlg^{H} is the stratum of Hilb^{p}(ℙ^{n+c−1}) of constant postulation (cf. [17], Theorem 1.1 for details). If H(v) = 0 for v >> 0 and t = ∑_{v} H(v), then there exists a scheme Hilb := Hilb^{t}(𝔸^{n+c}) of finite type which parametrizes quotients R → A of length t. The restriction (pullback) R_{S} → A_{S} of the universal quotient to Spec(S) → Hilb defines the subscheme GradAlg^{H} of Hilb by requiring ker((R_{S})_{v} → A_{S}) to be flat over S, of rank dim R_{v} − H(v) for every 0 ≤ v ≤ t at each fiber of Spec(S). Since such a subscheme GradAlg^{H} of finite type exists by [25], Lect. 8 or [23], VI, Proposition 1.1, we are done.

Remark 7. We may in the case H(v) = 0 for v >> 0 use flattening stratification to show the existence of the scheme Z(H) of not neccessarily graded Artinian quotients A of R with Hilbert function H (by requiring gr(A_{S}) to be S-flat, of Hilbert function H at each fiber of Spec(S)), to get Z(H) as a locally closed subscheme (stratum) of Hilb^{t}(𝔸^{n+c}). Its local deformation functor at a quotient A ≃ R/I_{A} (with \sqrt{I_{A}} = 𝔪) of R, defined on the category of local Artinian k-algebras with residue field k, consists of quotients A_{S} of R_{S} = S[X_{1}...., X_{n+c}] (or equivalently, of S[[X_{1}...., X_{n+c}]]) such that A_{S} and gr(A_{S}) are deformations of A and gr(A) respectively. If R → A is graded and satisfies _{v}H^{2}(R, A, A) = 0 for v ≥ 0, then [17], Theorem 1.10 and Remark 1.11 shows that Z(H) is smooth at (R → A) and dim_{(A)} Z(H) = ∑_{v≥0} _{v}hom_{R}(I_{A}, A).

Inside GradAlg^{H} there is an open set (and hence a scheme Gor_{c}^{H} with induced scheme structure) consisting of Gorenstein quotients R → A with Hilbert function H. In the Artinian case there is another known scheme parametrizing graded Gorenstein quotients with Hilbert function H, namely the determinantal loci PGor(H) of forms of degree s in ℙ^{N}, N = \binom{s+n+c−1}{n+c−1} − 1, given by requiring the “catalecticant matrices” to have ranks given by the Hilbert function, see Iarrobino-Kanev’s book [13], Sect.1.1 for an introduction, and denote the restriction of the generic s-form of ℙ^{N} to PGor(H) by F_{PGor(H)} ([13], Definition 1.3). The underlying set of closed poins of Gor_{c}^{H} and PGor(H) are the same by apolarity (the Macaulay correspondence), and one may see from the proof of our next result that they are in fact isomorphic as topological spaces. Moreover since Iarrobino and Kanev have proved that the tangent space of PGor(H) at A is isomorphic to _{0}Hom_{A}(I_{A}/I_{A}^{2}, A) ([13], Theorem 3.9) and it is well known that the tangent space of GradAlg^{H} is the same group [15], we get

Theorem 8. Let R → A be a graded Gorenstein Artinian quotient with Hilbert function H. Then dim_{(A)} GradAlg^{H} = dim_{(A)} PGor(H). Hence PGor(H) is smooth at (R → A) if and only if GradAlg^{H} is smooth at (R → A). In particular PGor(H) is smooth at (R → A) provided _{0}H^{2}(R, A, A) = 0, i.e. provided _{s}H_{2}(R, A, A) = 0 where s is the socle degree of A.

Proof. The vanishing of the two last mentioned groups is equivalent by duality. Hence we only need to prove dim_{(A)} GradAlg^{H} = dim_{(A)} PGor(H) because we know their tangent spaces are isomorphic. Let V ⊂ PGor(H) be a closed irreducible subset (e.g. of the form V = Spec(T)), and let V have the reduced scheme structure. By the definition of PGor(H), the restriction F_{V} of the “universal” s-form F_{PGor(H)} to V defines via apolarity a family of graded Artinian Gorenstein quotients over V (e.g. the family A_{T} ≃ R_{T} ∘ F_{V} over V = Spec(T) where R_{T} is interpreted as the T -algebra of contractions “partial derivations without coefficients”) with constant Hilbert function H. Since V is integral, it follows that the family (e.g. the morphism Spec(A_{T}) → Spec(T)) is flat. Hence we have a morphism π : V → GradAlg^{H} by the universal property of GradAlg^{H}. π(V) is irreducible and closed in Gor_{c}^{H} (it is closed because an “invers” (Gor_{c}^{H})_{red} → PGor(H) on closed points exists by [13], Ch. 9). So chains of closed irreducible subsets in PGor(H) and Gor_{c}^{H} correspond, and we are done.

Problem 9. It will be interesting to prove that PGor(H) and Gor_{c}^{H} are isomorpic as schemes, cf. [17], Remark 1.9. The arguments in the proof above, which leads to the isomorphism of PGor(H) and Gor_{c}^{H} as topological spaces, are implicitely contained in [17], Remark 1.9. In that Remark we also use the mentioned arguments to see the existence of a natural π : PGor(H) → Gor_{c}^{H}. Here an index “red” is missing, i.e. we can from the arguments only be sure of the existence of a natural morphism π : (PGor(H))_{red} → Gor_{c}^{H}.

2 Families of Gorenstein quotients of codimension 1 on Cohen-Macaulay algebras

The main purpose of this paper is to study families of graded Gorenstein quotients A obtained by taking a regular section of the anticanonical sheaf K̃_{B}^{∗}(s)|_{U}, i.e. quotients A given by

0 → K_{B}(−s) → B → A → 0  (11)

for some integer s. Our main results apply to Artinian Gorenstein algebras as well as to Gorenstein quotients of higher dimension. In what follows B is Cohen-Macaulay. In order to make the description of A in (11) meaningfull, we must suppose B ≃ R/I_{B} is generically Gorenstein (cf. [19], Lemma 5.2 and 5.4). Then we shortly say “A is a codimension 1 quotient of B, well defined by a twist of the canonical module”. In this case there is an open set U = Y −Z of Y = Proj B such that dim B−dim B/I(Z) ≥ 1 with K̃_{B}|_{U} locally free (let I(Z) = 𝔪 if U = Y), and a regular section σ ∈ H^{0}(U, K̃_{B}^{∗}(s)) whose dual section σ^{∗} ∈ H_{∗}^{0}(Y, K̃_{B}^{∗∗}(−s)) → H_{∗}^{0}(Y, B̃) factorizes via K_{B}(−s) → B, thus defining A by A := coker(K_{B}(−s) → B). Note that if U above is a little larger, namely such that dim B − dim B/I(Z) ≥ 2 (i.e. B is Gorenstein in codimension ≤ 1), then

A := coker(K_{B}(−s) = H_{∗}^{0}(U, K̃_{B}^{∗∗}(−s)) →^{σ^{∗}} B = H_{∗}^{0}(U, B̃)).

The following result determines dim_{(A)} GradAlg^{H_{A}} in terms of invariants of B, such as N_{B} := Hom_{B}(I_{B}/I_{B}^{2}, B) and K_{B}^{∗} := Hom_{B}(K_{B}, B) ≃ Hom_{B}(S_{2}(K_{B}), K_{B}) (S_{2}(K_{B}) is the second symmetric power of K_{B}), and leads to the main theorem of this section.

Proposition 10. Let B be a generically Gorenstein, graded Cohen-Macaulay quotient of a polynomial ring R and let A be a codimension 1 quotient of B, defined by (11) for some s. If

(i) _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 and (I_{B})_{℘} is syzygietic for any graded prime ℘ ⊇ I_{B} of R satisfying dim B_{℘} ≤ 1 (resp. B is unobstructed as a graded algebra),

(ii) _{s}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0, and

(iii) _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) = 0 (resp. _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = 0), then A is unobstructed as a graded R-algebra, A is Gorenstein and

dim_{(A)} GradAlg^{H_{A}} = dim(N_{B})_{0} + dim(K_{B}^{∗})_{s} − 1 − δ(B)_{−s}

(resp. dim_{(A)} GradAlg^{H_{A}} = dim(N_{B})_{0} + dim(K_{B}^{∗})_{s} − 1 − _{−s}hom_{R}(I_{B}, K_{B})) where δ(B)_{−s} = _{−s}hom_{R}(I_{B}, K_{B}) − _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}). Moreover if the characteristic of k is zero, then

−sext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = dim W (A) − dim pr_{1}(W (B → A))

where W (B → A) and W (A) are the irreducible components of GradAlg(H_{A}, H_{B}) and GradAlg^{H_{A}} respectively to which (B → A) and (A) belong.

Remark 11. (a) With the notations, 0 → ⊕_{i}R(−n_{c,i}) → ... → ⊕_{i}R(−n_{1,i}) → I_{B} → 0, of

(6), recalling dim R = n + c and dim B = n, we claim that

δ(B)_{−s} = 0 provided s > max n_{c,i} + max n_{2,i} − n − c

(in fact _{−s}Ext_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}) = 0 for i = 0 and 1 for such s). Indeed dualizing (6) we get a surjection ⊕_{i}R(n_{c,i}) ↠ K_{B}(n + c), and it follows that (K_{B})_{−v} = 0 for v > max n_{c,i} − n − c. We conclude by applying _{0}Hom_{B}(−, K_{B}) to the right-exact sequence ⊕_{i}B(−n_{2,i}) → ⊕_{i}B(−n_{1,i}) → I_{B}/I_{B}^{2} → 0 and the definiton Ext_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}).

(b) If dim B = 1, we have Ext_{B}^{i}(−, K_{B}) = 0 for i ≥ 2 by duality. In this case the assumption _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) = 0 of Proposition 10 is always satisfied. Moreover looking to Remark 5 and using that A ≃ K_{A}(−s), we may weaken “I_{B} is syzygietic in codimension ≤ 1” to “I_{B} is generically syzygietic and _{s}H_{2}(R, B, B) = 0” and still conclude as in Proposition 10. By possibly forgetting the final conclusion on −sext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) we may replace the whole “non-resp. assumption” of (i) by the assumption “(H_{2}(R, B, B) ⊗_{B} A)_{s} = 0” and conclude that A is unobstructed and that

dim_{(A)} GradAlg^{H_{A}} = dim(N_{B})_{0} + dim(K_{B}^{∗})_{s} − 1 − δ(B)_{−s} − _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, B)

(c) Slightly reformulating Proposition 10 (by replacing the vanishing of the three _{0}Ext_{B}^{i} group of the ordinary assumptions by the corresponding vanishing of Ext_{B}^{i}) one may use the proof below to show that A is unobstructed as an R-algebra in the sense H^{2}(R, A, A) = 0, cf. Remark 5. A further slight reformulation (replacing Proj by Spec and skip the twists) allows us to prove a corresponding result for non-graded objects as well. Since all three Ext_{B} groups vanish if B is licci ([4], [12], [27] Thm. 4.2.6, or Corollary 27 of this paper), we get the unobstructedness of the Gorenstein quotient A if B is licci. In the interesting codimension c = 3 case, the Cohen-Macaulayness of S_{2}(K_{B}) and the Koszul homology group, H_{1} = H_{1}(I_{B}), are equivalent [27], Thm. 4.2.9. This Cohen-Macaulayness holds in particular if the Cohen-Macaulay type of B is 2 [27], Cor. 4.2.11.

(d) If I_{B} is generically syzygietic, then Hom_{B}(H_{2}(R, B, B), K_{B}) = 0. Hence applying Hom_{B}(−, K_{B}) to (4), we see that _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) ↪_{−s} Ext_{B}^{1}(H_{1}, K_{B}) is injective and that

δ(B)_{−s} = dim(G_{1}^{∗} ⊗_{R} K_{B})_{−s} − _{−s}hom_{B}(H_{1}, K_{B})

allowing us to restate the dimension formula in terms of H_{1} and G_{1}. If B is licci, then _{−s}Hom_{B}(H_{1}, K_{B}) ≃ (H_{r−1})_{−t−s} by [20], Prop. 18, where r and t is given by ∧^{r}H̃_{1}|_{U} ≃ K̃_{B}(t)|_{U}.

Proof. Note that A ≃ K_{A}(−s) and hence that A is Gorenstein by applying the mapping cone construction to (11), (or see [19], Lemma 5.2 and 5.4). We claim that the Proposition above follows from Theorem 4 and Remark 5 by letting I_{A/B} = K_{B}(−s). Indeed all assumptions of Theorem 4 are straightforward to get from the corresponding assumptions of the Proposition above, except possibly (ii). Suppose temporarily that B is Gorenstein in codimension ≤ 1. Then _{0}Ext_{B}^{1}(K_{B}(−s), B) ≃ _{0}Ext_{B}^{1}(K_{B} ⊗ K_{B}(−s), K_{B}) ≃ _{0}Ext_{B}^{1}(S_{2}(K_{B})(−s), K_{B}) = 0 by some spectral sequence of [11]. Moreover, since K_{B} is a maximal Cohen-Macaulay Bmodule, one knows _{0}Ext_{B}^{i}(K_{B}, K_{B}) = 0 for i > 0 and we get _{0}Ext_{B}^{1}(K_{B}(−s), A) = 0 by applying _{0}Hom(K_{B}(−s), −) to (11), i.e. we get (ii) of Theorem 4. In the general case where B is generically Gorenstein we get the conclusion of Theorem 4 by combining Lemma 12 and Remark 5(b), recalling K_{B}(−s) ≃ I_{A/B}. Finally to get the dimension formula we note that _{0}Hom(K_{B}(−s), K_{B}(−s)) is 1 dimensional. Hence by again applying _{0}Hom(K_{B}(−s), −) to (11), we get the dimension formula from the corresponding formula of Theorem 4.

Lemma 12. If B is Cohen-Macaulay and A ≃ B/I_{A/B} is Gorenstein of socle degree s and if dim B − dim A = c, then _{0}H^{2}(B, A, A) and _{−s}Ext_{B}^{c}(S_{2}(I_{A/B}), K_{B}) vanish simultaneously.

Proof. Using (5) we see the vanishing of _{0}H^{2}(B, A, A) is equivalent to _{0}Ext_{A}^{1}(I_{A/B}/I_{A/B}^{2}, A) = 0 and the injectivity of the natural map _{0}Hom_{A}(H_{2}(B, A, A), A) ↪ _{0}Ext_{A}^{2}(I_{A/B}/I_{A/B}^{2}, A). Moreover since A ≃ K_{A}(−s) we have by Gorenstein duality (applied both to A and B) that

0Ext_{A}^{i}(I_{A/B}/I_{A/B}^{2}, A) ≃ _{s}H_{𝔪}^{dim A−i}(I_{A/B}/I_{A/B}^{2})^{∨} ≃ _{−s}Ext_{B}^{i+c−1}(I_{A/B}^{2}, K_{B}).

for i ≥ 1 because the middle term is isomorphic to _{s}H_{𝔪}^{dim A+1−i}(I_{A/B}^{2}) by the long exact sequence of local cohomology applied to 0 → I_{A/B}^{2} → I_{A/B} → I_{A/B}/I_{A/B}^{2} → 0. Correspondingly 0Hom_{A}(H_{2}(B, A, A), A) ≃ _{−s}Ext_{B}^{c}(H_{2}(B, A, A), K_{B}). Hence _{0}H^{2}(B, A, A) is equivalent to −sExt_{B}^{c}(I_{A/B}^{2}, K_{B}) = 0 and the injectivity of of the natural map _{−s}Ext_{B}^{c}(H_{2}(B, A, A), K_{B}) ↪ −sExt_{B}^{c+1}(I_{A/B}^{2}, K_{B}). Recalling the exact sequence

0 → H_{2}(B, A, A) → S_{2}(I_{A/B}) → I_{A/B}^{2} → 0  (12)

(cf. [1], p.106 or [27], Section 2.1) and that _{−s}Ext_{B}^{c−1}(H_{2}(B, A, A), K_{B}) = 0 since H_{2}(B, A, A) is supported at the codimension c quotient A of B, we get the conclusion by the long exact sequence of _{−s}Ext_{B}^{i}(−, K_{B}) applied to (12).

Theorem 13. Let B ≃ R/I_{B} be licci (resp. Proj(B) licci and B Cohen-Macaulay and unobstructed) and let A be a codimension 1 quotient of B, well defined (hence B generically Gorenstein) by a twist of the canonical module K_{B}. Then A is unobstructed (resp. unobstructed if s >> 0) as a graded R-algebra, A is Gorenstein and,

dim_{(A)} GradAlg^{H_{A}} = dim(N_{B})_{0} + dim(K_{B}^{∗})_{s} − 1 − δ(B)_{−s}
(resp. dim_{(A)} GradAlg^{H_{A}} = dim(N_{B})_{0} + dim(K_{B}^{∗})_{s} − 1)

where δ(B)_{−s} = _{−s}hom_{R}(I_{B}, K_{B}) − _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}).

Remark 14. We may think of Theorem 13 (e.g. for fixed large enough s) as an application from the set of generically smooth irreducible components of GradAlg^{H_{B}} containing locally licci, Cohen-Macaulay codimension c quotients B of R, to the set of genercally smooth irreducible components of GradAlg^{H_{A}} containing Gorenstein codimension c + 1 quotients A of R. We expect this map to be injective, because the dimension formula seems to show that the “Cohen-Macaulay” components of GradAlg^{H_{B}} above which have different dimension, yields “Gorenstein” components of GradAlg^{H_{A}} of different dimension (their dim(N_{B})_{0} are different while their dim(K_{B}^{∗})_{s} usually (always?) coincide, cf. preceding examples).

Proof. Since B is licci and generically Gorenstein, it is generically syzygietic by e.g. [19], Proposition 6.17 which tells H^{2}(R, B, B)_{℘} = 0 which in turn is equivalent to H_{2}(R, B, B)_{℘} = 0 in the case dim B_{℘} = 0 by Gorenstein-duality and the spectral sequence relating algebrahomology to algebra-cohomology. Since all Ext_{B}^{i} groups of the ordinary assumptions of Proposition 10 vanish by Remark 11(c), we conclude by Proposition 10.

If Y = Proj(B) is licci and B unobstructed, the sheaf { \atop S_{2}(K_{B})} is Cohen-Macaulay by Remark 11(c). Hence Ext_{O_{Y}}^{1} ({ \atop S_{2}(K_{B})}, K̃_{B}) = 0 and it follows that _{s}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) vanishes for large s because of the exact sequence

→ _{s}Hom_{B}(S_{2}(K_{B}), H_{𝔪}^{1}(K_{B})) → _{s}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) → H^{1}(Y, Hom_{B}({ \atop S_{2}(K_{B})}, K̃_{B})(s))

(which is analogously to (3)). Since the condition (iii) of Proposition 10 is obviously true for s >> 0, we conclude by the resp. assumptions of the Proposition.

To apply Proposition 10 and Theorem 13, we need to compute dim(K_{B}^{∗})_{s} and verify its assumptions. In several cases it is easy to compute dim(K_{B}^{∗})_{s} at least for large s. For instance if B is 2-dimensional and C = Proj(B) is smooth, then K̃_{B}^{∗} is just the tangent sheaf (sheaf of derivations), θ_{C}, of C, and we can get the information we need from vanishing of cohomology and Riemann-Roch’s theorem. In this case the most difficult assumption to verify is perhaps the unobstructedness of B (or _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0), for which we point out a couple of important cases where the unobstructedness is known;

1. _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 provided C is linked (in possibly several steps) to a generically Gorenstein curve C′ = Proj(B′) satisfying _{0}Ext_{B′}^{1}(I_{B′}/I_{B′}^{2}, B′) = 0 (cf. next section).

2. B is unobstructed if C = Proj(B) sits on a smooth rational surface in ℙ^{4} and the degree of C is ≥ 24 ([19], Proposition 9.13, see Theorem 9.4 for a generalization).

3. C is determinantal curve with some conditions on the degree matrix ([19], Proposition 10.7 and Corollary 10.11).

By 1) we get_{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 if B is licci (e.g. if C′ is a complete intersection), or if C′ is a local complete intersection curve and H^{1}(N_{C′}) = 0 (because _{0}Ext_{B′}^{1}(I_{B′}/I_{B′}^{2}, B′) ↪ H^{1}(N_{C′}) is injective by (3)). Indeed for a local complete intersection curve C = Proj(B) we have H^{1}(N_{C}(v)) ≃ Ext_{O_{C}}^{1}({ \atop I_{B}/I_{B}^{2}}, B̃(v)) and by (3) an exact sequence

vExt_{B}^{1}(I_{B}/I_{B}^{2}, B) ↪ H^{1}(N_{C}(v)) → _{v}Hom_{B}(I_{B}/I_{B}^{2}, H_{𝔪}^{2}(B)) → _{v}Ext_{B}^{2}(I_{B}/I_{B}^{2}, B) → 0 . (13)

We illustrate by taking a relatively simple example, remarking that far more complex examples can be given by the same methods (by increasing s).

Example 15. Let B be the homogeneous coordinate ring of a smooth rational arithmetically Cohen-Macaulay curve of degree d = 4 in ℙ^{4} = Proj(R), with minimal resolution

0 → R(−4)^{3} → R(−3)^{8} → R(−2)^{6} → R → B → 0,

and let A be defined by a regular section of K_{B}^{∗}(s) for s ≥ 0. C is the determinantal curve given by the maximal minors of a 2 by 4 matrix with linear entries. Even though B is not licci, Theorem 13 applies for s >> 0. Indeed, looking to Proposition 10 and Remark 11(a), we claim that the conclusions of Theorem 13 holds for s ≥ 3 (and in fact for s ≥ 0, see the next section). Indeed we only need to verify its assumptions. Since C is smooth and θ_{C} := K̃_{B}^{∗} is of degree deg θ_{C} = − deg K̃_{B} = 2 − 2g (g = the genus of C), we get H^{1}(θ_{C}(v)) = 0 and h^{0}(θ_{C}(v)) = dv + 3 − 3g = 4v + 3 for v ≥ 0 ([25], Lect.11). Since we have an exact sequence

0 → θ_{C} → θ_{ℙ}|_{C} → N_{C} → 0

where θ_{ℙ}|_{C} = coker(O_{C} → O_{C}(1)^{⊕5}) is the tangent sheaf of ℙ = ℙ^{4} restricted to C, we get

0Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) ↪ H^{1}(N_{C}) = 0.

Moreover _{v}Ext_{B}^{1}(S_{2}(K_{B}), K_{B})^{∨} ≃ _{−v}H_{𝔪}^{1}(K_{B}) and since H_{∗}^{0}(Y, { \atop S_{2}(K_{B})}) → H_{𝔪}^{1}(K_{B}) is surjective and H^{0}(Y, { \atop S_{2}(K_{B})}(−v))^{∨} ≃ H^{1}(θ_{C}(v)) we get

vExt_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 for v ≥ 0.

Finally by Remark 11(a), we know _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = 0 for s > 2 and the claim is proved for s ≥ 3. In conclusion the quotient A is an unobstructed Gorenstein algebra for s ≥ 3 and

dim_{(A)} GradAlg^{H} = h^{0}(N_{C}) + h^{0}(θ_{C}(s)) − 1 = 5d + 1 − g + ds + 3 − 3g − 1 = 4s + 23

Since it is straightforward to see that the Hilbert function H of A is given by dim(A)_{v} = 4v+1 for 0 ≤ v ≤ s and dim(A)_{v} = 4s+2 for v > s, it follows that dim_{(A)} GradAlg^{H} = 21+H(s+1).

Now we will look to examples of Artinian Gorenstein quotients, and in particular concentrate on Artinian reductions (by modding out with a general linear form L) of the class of examples which we can treat according to the methods illustrated in Example 15. More generally let B ≃ R/I_{B} be an n-dimensional Cohen-Macaulay algebra, let A ≃ B/K_{B}(−s), and let R_{L} := R/(L) → B_{L} = B/(L) → A_{L} = A/(L) be their quotients modulo (L). Note that an R_{L}-free minimal resolution of B_{L} has the same graded Betti numbers as the R-free minimal resolution of B ([22], p.28), i.e. their numbers n_{j,i} given in (6) coincide. Dualizing (6) we see the minimal resolutions of K_{B}(n + c) and K_{B_{L}}(n + c − 1) also coincide, i.e. K_{B} ⊗ B_{L} ≃ K_{B_{L}}(−1). Looking to (11) it follows that A_{L} is given by

0 → K_{B_{L}}(−s − 1) → B_{L} → A_{L} → 0,  (14)

(note −s−1 instead of −s!) and A_{L} is an Artinian Gorenstein quotient, well defined by a twist of the canonical module. Hence we may apply Proposition 10 and Theorem 13. To compute dim(K_{B_{L}}^{∗})_{s+1} and verify its assumptions, we see, thanks to the arguments of Remark 11(a), that δ(B_{L})_{−s−1} = 0 and _{−s−1}Ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, K_{B_{L}}) = 0 (where I_{B_{L}} = ker(R_{L} → B_{L})), provided s + 1 > max n_{c,i} + max n_{2,i} − (n + c − 1) which is equivalent to the inequality of that Remark, used to show δ(B)_{−s} = 0. To check the other assumptions, we claim that Lemma 16. (i) If _{v−1}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0, then dim(K_{B_{L}}^{∗})_{v+1} = dim(K_{B}^{∗})_{v}−dim(K_{B}^{∗})_{v−1}.

(ii) If _{v}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = _{v−1}Ext_{B}^{2}(S_{2}(K_{B}), K_{B}) = 0, then

v+1Ext_{B_{L}}^{1}(S_{2}(K_{B_{L}}), K_{B_{L}}) = 0.

(iii) If B is an unobstructed graded R-algebra and the map _{−1}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) ↪ 0Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) induced by L is injective, then B_{L} is an unobstructed graded R_{L}-algebra and

dim(N_{B_{L}})_{0} = dim(N_{B})_{0} − dim(N_{B})_{−1}.

(iv) If _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = _{−1}Ext_{B}^{2}(I_{B}/I_{B}^{2}, B) = 0, then _{0}Ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, B_{L}) = 0.

Proof. (i) and (ii). Apply Hom_{B}(S_{2}(K_{B}), −) to the exact sequence 0 → K_{B} → K_{B}(1) → K_{B_{L}} → 0 and use that S_{2}(K_{B}) ⊗ B_{L} ≃ S_{2}(K_{B_{L}})(−2) and that K_{B}^{∗} = Hom_{B}(S_{2}(K_{B}), K_{B}).

(iii) The unobstructedness is just a very particular case of Theorem 4 letting I_{A/B} ≃ B(−1). To compute N_{B_{L}}, just apply _{0}Hom_{B}(I_{B}/I_{B}^{2}, −) to 0 → B(−1) → B → B_{L} → 0. Continuing this sequence into a long exact sequence of Ext-groups, we prove (iv) as well because _{0}Ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, B_{L}) ≃ _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B_{L}).

Remark 17. (a) In the case Proj(B) is a curve, we have by duality_{v}Ext_{B}^{2}(S_{2}(K_{B}), K_{B})^{∨} ≃ −vH_{𝔪}^{0}(S_{2}(K_{B})) ↪ S_{2}(K_{B})_{−v}. Looking to Remark 11(a), we see there is a surjection S_{2}(⊗R(n_{c,i})) ↠ S_{2}(K_{B}(n + c)) showing that

vExt_{B}^{2}(S_{2}(K_{B}), K_{B}) = 0 provided v > 2 max n_{c,i} − 2(n + c)

(b) If the curve C = Proj(B) is a local complete intersection, then (13) shows that −1Ext_{B}^{2}(I_{B}/I_{B}^{2}, B) = 0 provided H^{1}(O_{C}(n_{1,i} − 1)) ≃ H^{0}(K̃_{B}(−n_{1,i} + 1))^{∨} = 0, i.e.

−1Ext_{B}^{2}(I_{B}/I_{B}^{2}, B) = 0 provided min n_{1,i} > max n_{c,i} − n − c + 1.

Example 18. We consider the Artinian reduction A_{L} of the Gorenstein algebra of Example 15. Since B is 2-dimensional we get the vanishing of _{0} Ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, B_{L}), and hence the unobstructedness of B_{L}, from Lemma 16(iii) and Remark 17(b). Moreover since we have seen that Remark 11(a) also applies to the Artinian reduction, we get _{−s−1}Ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, K_{B_{L}}) = 0 for s > 2. Finally Lemma 16(ii) and Remark 17(a) shows that _{v+1}Ext_{B_{L}}^{1}(S_{2}(K_{B_{L}}), K_{B_{L}}) = 0 for v ≥ 0. Hence all “resp.-assumtions” of Proposition 10 concerning B_{L} are satisfied. It follows that A_{L} is an unobstructed Artinian Gorenstein algebra for s ≥ 3, i.e. PGor(H_{L}) is smooth at A_{L} by Theorem 8 and

dim_{(A_{L})} PGor(H_{L}) = dim(N_{B_{L}})_{0} + dim(K_{B_{L}}^{∗})_{s+1} − 1 = 3d + d − 1 = 15

Finally we claim that this conclusion (about A_{L}) holds for any Artinian Gorenstein quotient A (well defined by a twist of the canonical module of B) with numerically the same minimal resolution (i.e. the same graded Betti numbers) as A_{L}. Indeed to use Remark 11(a) we only need the numbers of the minimal resolution, and since we may prove Remark 17 for a 1-dimensional B by replacing Ext_{B}^{2}(−, −) by the corresponding Ext_{B}^{1}(−, −) (in (b) we use 0Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) ≃ _{0}Ext_{B}^{1}(I_{B} ⊗_{R}K_{B}, K_{B}) ≃ _{0}H_{𝔪}^{0}(I_{B} ⊗_{R}K_{B})^{∨} to see that this group vanishes provided min n_{1,i} > max n_{c,i} − n − c + 1), we get the claim.

We end this section by using Theorem 13 to reprove the smoothness and give a dimension formula of GradAlg^{H} for (e.g. Artinian) Gorenstein codimension 3 quotients equivalent to those in [17], [18] or [5].

Example 19. Recall that if A ≃ R/I_{A} is Gorenstein of codimension 3 with minimal resolution

0 → R(−f) ≃ F_{3} → F_{2} → F_{1} → I_{A} → 0.  (15)

one knows that A is unobstructed and that N_{A} ≃ ∧^{2}I_{A}(f) (char k ≠ 2). It follows that

dim_{(A)} GradAlg^{H_{A}} = dim(∧^{2}F_{1})_{f} − dim(F_{1} ⊗ F_{2})_{f} + dim(S_{2}(F_{2}))_{f}  (16)

cf. [18], Theorem 2.6 and its Remark. In (16) it is not really neccessary to suppose (15) to be minimal because we easily see that (16) is invariant under adding a common free factor to both F_{1} and F_{2}.

If B ≃ R/I_{B} is Cohen-Macaulay of codimension 2 (and hence licci), and A is given as in Theorem 13, then we claim the dimension formula of dim_{(A)} GradAlg^{H_{A}} in Theorem 13 coincides with (16). Note that since every irreducible component of the Hilbert scheme of Gorenstein codimension 3 quotients of R contains a quotient constructed via (11) by [7], this example reproves the smoothness of GradAlg^{H} and PGor(H), as well as (16). The argument holds in the case char k = 2 as well.

Firstly we compute dim(K_{B}^{∗})_{s}. Indeed the dual of (7) (for c = 2) yields the exact sequence

0 → K_{B}^{∗}(−n − c) → G_{2} ⊗ B → G_{1} ⊗ B → I_{B}/I_{B}^{2} → 0  (17)

in which H_{1} is the kernel of G_{1} ⊗ B → I_{B}/I_{B}^{2} by (4). By [2] one knows that

0 → ∧^{i+1}G_{2} → ∧^{i+1}G_{1} → ∧^{i}G_{2} → H_{i} → 0  (18)

is exact for any i. Let f = s + n + c. Using the last mentioned sequence for i = 1, (17) and the R-free resolution of G_{2} ⊗ B which we easily deduce from (6), we get

dim(K_{B}^{∗})_{s} = dim(∧^{2}G_{1})_{f} − dim(G_{1} ⊗ G_{2})_{f} + dim(S_{2}(G_{2}))_{f}.  (19)

Next we will compute δ = δ(B)_{−s} = dim(G_{1}^{∗} ⊗_{R} K_{B})_{−s} − _{−s}hom_{B}(H_{1}, K_{B}), cf. Remark 11(d). If r = rank H_{1}, we have by (18) a resolution of H_{r−1} leading to

0 → G_{2}^{∗} → (∧^{2}G_{1})^{∗} → (∧^{2}G_{2})^{∗} → H_{r−1} ⊗(∧^{r+1}G_{2})^{∗} ≃ Hom(H_{1}, K_{B}(n + c)) → 0 (20)

We have an R-free resolution of K_{B} by dualizing (6) and hence a resolution of G_{1}^{∗} ⊗ K_{B}. Combining with (20), we get

−δ = dim(G_{1} ⊗ G_{2})_{−f}^{∗} − dim(S_{2}(G_{1})^{∗})_{−f} − dim(∧^{2}G_{2})_{−f}^{∗} + dim(G_{1})_{−f}^{∗} − dim(G_{2})_{−f}^{∗}. (21)

Note that dim(N_{B})_{0} is well known, e.g. one may easily deduce it from the exact sequence

0 → G_{1}^{∗} ⊗ G_{2} → ((G_{1}^{∗} ⊗ G_{1}) ⊕ (G_{2}^{∗} ⊗ G_{2}))/k → G_{2}^{∗} ⊗ G_{1} → N_{B} → 0  (22)

of [20], Remark 16 (k is a 1-dimensional subspace). Putting these formulas patiently together, using Theorem 13, we get exactly (16) because we find the following resolution of I_{A},

0 → R(−f) → G_{1}^{∗}(−f) ⊕ G_{2} → G_{2}^{∗}(−f) ⊕ G_{1} → I_{A} → 0.  (23)

by applying the mapping cone construction onto (11).

Inside the scheme Z(H) of Remark 7 we have the open subscheme ZGor(H) consisting of not neccessarily graded Gorenstein quotients R → A with Hilbert function H. ZGor(H) has been studiet in different contexts (see [13], p.126 for references). Since we by Remark 11(c) easily see that _{v}H^{2}(R, A, A) = 0 for v ≥ 0 provided B is licci, we can use Remark 7 to prove the smoothness and find the dimension of the scheme ZGor(H) at A. Indeed _{v}hom_{R}(I_{A}, A) is found using (9) as in the proof of Theorem 4. And if we apply _{v}Hom(K_{B}(−s), −) to (11), we get _{v}hom(K_{B}(−s), A) = dim(K_{B}^{∗})_{s+v} − dim B_{v}. Since A_{v} = 0 for v > s, we have

Proposition 20. Let B ≃ R/I_{B} be licci and generically Gorenstein, and let A be a codimension 1 quotient of B, defined by (11) for some s. Then ZGor(H) is smooth at (R → A) and

dim_{(A)} Z(H) = ∑_{v≥0}^{s} (dim(N_{B})_{v} − δ(B)_{−s+v}) + ∑_{v≥0}^{s} (dim(K_{B}^{∗})_{s+v} − dim B_{v}).

Invoking Remark 11(c), we take the opportunity to state the main result of this section in the non graded case. The proof is omited because it is the same as in the graded case.

Theorem 21. Let B ≃ R/I_{B} be Cohen-Macaulay, generically Gorenstein and generically syzygietic of dimension ≥ 1 and suppose B satisfies

Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 , Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 and Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) = 0.

Consider K_{B} as an ideal of B and let A := B/K_{B}. Then A is Gorenstein and strongly unobstructed in the sense H^{2}(R, A, A) = 0.

In particular if B ≃ R/I_{B} be licci and generically Gorenstein, then A is Gorenstein and strongly unobstructed in the sense H^{2}(R, A, A) = 0. Indeed licci and generically Gorenstein implies Cohen-Macaulay and generically syzygietic as well as the vanishing of the three Extgroups above.

3 Computing the dimension via liaison

It is not really difficult to compute δ(B)_{−s} = _{−s}hom_{R}(I_{B}, K_{B}) − _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) and dim(K_{B}^{∗})_{s}. For instance it turns out that the difference dim(K_{B}^{∗})_{s} −δ(B)_{−s} (which appears in Theorem 13) is quite easy to compute because, after adding a simple term to it, it becomes a liaison invariant among syzygetic ideals I_{B}. Moreover δ(B)_{−s} is computable in terms of dim(K_{B′}^{∗})_{s} of the linked algebra B′. To allow any complete intersection linkage of Cohen-Macaulay quotients and still get the liaison invariance described above, we need to define δ(B) also for Cohen-Macaulay quotients R → B which satisfy H_{2}(R, B, B) ≠ 0 generically. To do this we consider the exact sequence [27]

0 → H_{2}(R, B, B) → S_{2}I_{B} → I_{B} → I_{B}/I_{B}^{2} → 0.  (24)

Let ψ : H_{𝔪}^{n}(H_{2}(R, B, B)) → H_{𝔪}^{n}(S_{2}I_{B}), dim B = n, be the induced map.

Definition 22. Let B = R/I_{B} be Cohen-Macaulay of dimension n, not neccessarily (generically) syzygietic. Then

δ^{∗}(B)_{v} = _{v}hom_{R}(I_{B}, K_{B}) − _{v}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) − dim(im ψ)_{−v}

Remark 23. If B is generically syzygietic, then _{v}Hom_{B}(H_{2}(R, B, B), K_{B}) = 0, and we get −vH_{𝔪}^{n}(H_{2}(R, B, B)) = 0 by duality and hence δ^{∗}(B)_{v} = δ(B)_{v}, i.e. the “old and the new” definition of δ coincides.

Using (24) and_{v}Ext_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}) ≃ _{−v}H_{𝔪}^{n−i}(I_{B}/I_{B}^{2})^{∨} and letting h_{𝔪}^{j}(−)_{v} = dim(_{v}H_{𝔪}^{j}(−)), one shows

δ^{∗}(B)_{v} = h_{𝔪}^{n+1}(S_{2}I_{B})_{−v} − h_{𝔪}^{n}(S_{2}I_{B})_{−v} − h_{𝔪}^{n+1}(I_{B})_{−v}  (25)

Indeed splitting (24) into two short exact sequences we see that h_{𝔪}^{n−1}(I_{B}/I_{B}^{2})_{−v} = h_{𝔪}^{n}(I_{B}^{2})_{−v} and h_{𝔪}^{n}(I_{B}/I_{B}^{2})_{−v} = h_{𝔪}^{n+1}(I_{B}^{2})_{−v} − h_{𝔪}^{n+1}(I_{B})_{−v} and moreover h_{𝔪}^{n+1}(I_{B}^{2})_{−v} = h_{𝔪}^{n+1}(S_{2}I_{B})_{−v} and h_{𝔪}^{n}(I_{B}^{2})_{−v} = h_{𝔪}^{n}(S_{2}I_{B})_{−v} − dim(im ψ)_{−v} and we get (25) from Definition 22.

Proposition 24. Let R be a finitely generated polynomial k-algebra with canonical module R(e) (so e = −n − c), and let B and B′ be graded Cohen-Macaulay quotients, algebraically linked by a complete intersection R ↠ D of dimension n ≥ 1, and with minimal resolution

0 → R(−a) → ... → ⊕_{i=1}^{c}R(−c_{i}) → I_{D} → 0.

Let β(B, D)_{v} := dim B_{v} + ∑_{i=1}^{c} dim(I_{B/D})_{v+c_{i}}. Then I_{B/D} ≃ K_{B′}(−e − a) and

(i) dim(N_{B})_{0} + ∑_{i=1}^{c} dim(I_{B/D})_{c_{i}} = dim(N_{B′})_{0} + ∑_{i=1}^{c} dim(I_{B′/D})_{c_{i}}

(ii) If B and B′ are generically Gorenstein (resp. Gorenstein in codimension ≤ 1), then vExt_{B}^{i}(I_{B}/I_{B}^{2}, B) and _{v}Ext_{B′}^{i}(I_{B′}/I_{B′}^{2}, B′) are isomorphic for i = 1 (resp. i = 2).

(iii) δ^{∗}(B)_{v−2e−2a} = β(B′, D)_{v−e−a} − dim(K_{B′}^{∗})_{v} for any v. Moreover if β(B′, D)_{v−e−a} = 0 and I_{B} is generically syzygietic, then _{v−2e−2a}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = dim(K_{B′}^{∗})_{v}.

(iv) dim(K_{B}^{∗})_{v} − δ^{∗}(B)_{−v} − β(B, D)_{v−e−a} = dim(K_{B′}^{∗})_{−v+2e+2a} − δ^{∗}(B′)_{v−2e−2a} − β(B′, D)_{−v+e+a}.

(v) If I_{B} is generically syzygietic (resp. syzygietic in codimension ≤ 1), then vExt_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) ↪ _{v+2e+2a}Ext_{B′}^{1}(S_{2}(K_{B′}), K_{B′}) is injective (resp. an isomorphism).

(vi) The conclusion of (ii) holds if B is linked to B′ in possibly several steps. Here the algebras of the intermediate steps need not neccessarily be generically Gorenstein.

Proof. (i) is proved in [19], Proposition 9.17. (ii) Also this part, i.e. the liaison invariance of I_{B}/I_{B}^{2} ⊗ K_{B}, is known ([4], or [19], ch.6). Here the Gorenstein assumption is required for having _{v}Ext_{B}^{i}(I_{B}/I_{B}^{2}, B) ≃ _{v}Ext_{B}^{i}(I_{B}/I_{B}^{2} ⊗ K_{B}, K_{B}). Note that the liaison invariance of I_{B}/I_{B}^{2} ⊗ K_{B} takes care of (vi) as well.

To prove (iii), recall that K_{D} ≃ D(e+a) and hence, I_{B/D} ≃ Hom_{D}(B, D) ≃ K_{B′}(−e−a), i.e. we get exact sequences

0 → I_{D} · I_{B} → S_{2}I_{B} → S_{2}(K_{B′})(−2e − 2a) → 0  (26)
0 → I_{D} · I_{B} → I_{D} → I_{D} ⊗_{R} R/I_{B} ≃ I_{D}/I_{D}^{2} ⊗_{D} B ≃ ⊕_{i=1}^{c}B(−c_{i}) → 0,  (27)

cf. [27], Lemma 4.2.7 to see (26). Since β(B′, D)_{v} := dim B′_{v} + ∑_{i=1}^{c} dim(I_{B′/D})_{v+c_{i}} and I_{B′/D} ≃ K_{B}(−e−a), then β(B′, D)_{v} = h_{𝔪}^{n}(K_{B′})_{−v} +∑_{i} h_{𝔪}^{n}(B)_{e−v+a−c_{i}} by Gorenstein duality.

Using (25) and (26), we get dim(K_{B′}^{∗})_{v} = h_{𝔪}^{n}(S_{2}K_{B′})_{−v} = −δ^{∗}(B)_{−2e+v−2a}−h_{𝔪}^{n+1}(I_{B})_{2e−v+2a}+ h_{𝔪}^{n+1}(I_{D}I_{B})_{2e−v+2a} while (27) and h_{𝔪}^{n}(K_{B′})_{−v+e+a} = h_{𝔪}^{n+1}(I_{D})_{2e−v+2a} − h_{𝔪}^{n+1}(I_{B})_{2e−v+2a} shows −h_{𝔪}^{n+1}(I_{B})_{2e−v+2a} + h_{𝔪}^{n+1}(I_{D}I_{B})_{2e−v+2a} = β(B′, D)_{v−e−a}. Hence we have proved

dim(K_{B′}^{∗})_{v} = −δ^{∗}(B)_{−2e+v−2a} + β(B′, D)_{v−e−a}  (28)

Moreover if β(B′, D)_{v−e−a} = 0, then dim B′_{v−e−a} = 0, i.e. v − e − a < 0 and hence H_{𝔪}^{n+1}(I_{D})_{2e−v+2a} = 0 by duality (e.g. if dim R > n+1, then h_{𝔪}^{n+1}(I_{D})_{2e−v+2a} = h_{𝔪}^{n}(D)_{2e−v+2a} = h_{𝔪}^{n}(K_{D})_{−v+e+a} = 0 by duality). Since h_{𝔪}^{n}(K_{B′})_{−v+e+a} = 0, it follows that h_{𝔪}^{n+1}(I_{D}I_{B})_{2e−v+2a} = β(B′, D)_{v−e−a} = 0 which easily leads to the final conclusion of (iii).

Note that (28) also shows dim(K_{B}^{∗})_{v} = −δ^{∗}(B′)_{v−2e−2a} + β(B, D)_{v−e−a}. Moreover if we replace v by −v + 2e + 2a in (28), we get (iv) by combining these formulas.

To prove (v), we use Gorensteinduality. Indeed

vExt_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) ≃ _{−v}H_{𝔪}^{n−2}(I_{B}/I_{B}^{2})^{∨} ≃ _{−v}H_{𝔪}^{n−1}(I_{B}^{2})^{∨} ↪ _{−v}H_{𝔪}^{n−1}(S_{2}I_{B})^{∨},

where the final injection follows from Gorenstein duality and _{v}Hom_{B}(H_{2}(R, B, B), K_{B}) = 0 because I_{B} is generically syzygietic. Note that the injection is an isomorphism if I_{B} is syzygietic in codimension ≤ 1. Now combining (26) and (27) we get

−vH_{𝔪}^{n−1}(S_{2}I_{B})^{∨} ≃ _{−v}H_{𝔪}^{n−1}(S_{2}(K_{B′})(−2e − 2a))^{∨} ≃ _{v+2e+2a}Ext_{B′}^{1}(S_{2}(K_{B′}), K_{B′}) (29)

and we conclude easily.

Example 25. We now complete Example 15 for the remaining cases (0 ≤ s ≤ 2) by verifying the assumptions of Proposition 10 and computing its invariants, using liaison. Indeed if we link B to a B′ via a complete intersection of three hypersurfaces, all of degree 2, the linked curve C′ = Proj(B′) has degree d′ = 8 − 4 = 4 and genus g′ = 0, i.e. we see that C and C′ are essentially the same curve. Hence _{v}Ext_{B′}^{1}(S_{2}(K_{B′}), K_{B′}) = 0 for v ≥ 0 since B satisfies this vanishing. By Proposition 24 it follows that _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) ↪ −s+2Ext_{B′}^{1}(S_{2}(K_{B′}), K_{B′}) = 0 for s ≤ 2, which precisely covers the cases of Example 15 which is not already analyzed. So, for 0 ≤ s ≤ 2, the non-resp. version of Proposition 10 applies! In Example 15 we saw that dim(K_{B}^{∗})_{s} = 4s + 3 for any s ≥ 0. It remains to compute δ(B)_{−s}. Since δ(B)_{v−2} = β(B′, D)_{v−1} − dim(K_{B′}^{∗})_{v} and β(B′, D)_{v} = dim B′_{v} + 3 dim(I_{B′/D})_{v+2} by Proposition 24, we get (δ(B)_{0}, δ(B)_{−1}, δ(B)_{−2}) to be equal to (15, 3, −3), and as previously

dim_{(A)} GradAlg^{H} = h^{0}(N_{C}) + h^{0}(θ_{C}(s)) − 1 − δ(B)_{−s} = 4s + 23 − δ(B)_{−s}

Note that it is easy, for s = 2, to see that _{−s}Hom(I_{B}/I_{B}^{2}, K_{B}) = 0 and hence _{−2}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = −δ(B)_{−2} = 3 by looking to the minimal resolution of I_{B} (or use Proposition 24(iii)). By the interpretation of _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) of Proposition 10 we get that the codimension of the strata of Gorenstein quotients given by (11) is 3 in this case.

Example 26. Now we consider the Artinian reduction A_{L} of the Example 18 of socle degree s + 1, to treat the remaining cases of interest (0 ≤ s ≤ 2). Since s ≤ 2 and I_{B} has no linear generator, _{s+1}H_{2}(R, B, B) ↪ (S_{2}I_{B})_{s+1} = 0. Hence in using Proposition 10, Remark 17(b) shows that no syzygietic assumptions on I_{B} is neccessary to assume and that Proposition 10(iii) always holds. Looking to Example 18 we see that all assumptions of the “non-resp.”version of Proposition 10 are satisfied. Now examining H^{i}(θ_{C}(v)) of Example 15 for v = −1 a little further and invoking (S_{2}K_{B})_{1} = 0 we get _{−1}ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 1 in addition to _{v}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 for v ≥ 0. By Lemma 16(i) we get dim(K_{B}^{∗})_{v} = 4 for v ≥ 1. Moreover if we apply Hom_{B}(I_{B}/I_{B}^{2}, −) to the exact sequence 0 → K_{B} → K_{B}(1) → K_{B_{L}} → 0, we get

δ(B_{L})_{v−1} = δ(B)_{v} − δ(B)_{v−1} + _{v}ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) − _{v−1}ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}).

Since _{−3}ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) = _{−1}ext_{B′}^{1}(S_{2}(K_{B′}), K_{B′}) = 1 and _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) = 0 for s ≤ 2, we get that (δ(B_{L})_{−1}, δ(B_{L})_{−2}, δ(B_{L})_{−3}) is equal to (12, 6, −4) from the values of δ(B)_{v} in Example 25. It follows that A_{L} is an unobstructed Artinian Gorenstein algebra for 0 ≤ s ≤ 2, i.e. PGor(H_{L}) is smooth at A_{L} by Theorem 8 and

dim_{(A_{L})} PGor(H_{L}) = dim(N_{B_{L}})_{0} + dim(K_{B_{L}}^{∗})_{s+1} − 1 − δ(B_{L})_{−s−1} = 15 − δ(B_{L})_{−s−1} .

Moreover, for s = 2, we see that −δ(B_{L})_{−3} = _{−3}ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, K_{B_{L}}) = 4 is the codimension of the strata of Gorenstein quotients given by (14).

Finally we remark that the conclusions (about A_{L}) also hold for any Artinian Gorenstein quotient A (well defined by a twist of the canonical module of B) with the same graded Betti numbers as A_{L} (cf. Example 18). In this case we can compute the numbers δ(B)_{v} of a 1-dimensional B as we did in Example 25 by essentailly the same linkage argument.

Mainly using Proposition 24 twice (for (i) below, we utilize (29)), we get

Corollary 27. If B and B′ are Cohen-Macaulay quotients of R, algebraically linked by a complete intersection D of type (c_{1}, c_{2}, .., c_{c}), and B′ and B′′ are algebraically linked by a complete intersection D′ of type (c′_{1}, c′_{2}, .., c′_{c}), and if a = ∑ c_{i}, a′ = ∑ c′_{i} and dim B > 0, then

(i) _{v+2a′}Ext_{B′′}^{1}(S_{2}(K_{B′′}), K_{B′′}) ≃ _{v+2a}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) for any v.

(ii) _{v−2a′}Ext_{B′′}^{2}(I_{B′′}/I_{B′′}^{2}, K_{B′′}) ≃ _{v−2a}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) provided I_{B} and I_{B′′} are syzygietic in codimension ≤ 1.

(iii) δ^{∗}(B′′)_{v−2e−2a′} = δ^{∗}(B)_{v−2e−2a} − β(B′, D)_{v−e−a} + β(B′, D′)_{v−e−a′}.

(iv) dim(K_{B′′}^{∗})_{v+2e+2a′} = dim(K_{B}^{∗})_{v+2e+2a} −β(B′, D)_{v+e+a} +β(B′′, D′)_{v+e+a′} or equivalently, dim(K_{B′′}^{∗})_{v+2e+2a′} = dim(K_{B}^{∗})_{v+2e+2a}+dim B′′_{v+e+a′} −dim B_{v+e+a}+∑_{i=1}^{c}(dim(K_{B′})_{v+c′_{i}} − dim(K_{B′})_{v+c_{i}}).

We include one example of biliaison starting with the smooth rational curve of Example 15. Of course starting with a licci curve, the verification of the assumptions of Proposition 10 is immediate (Theorem 13). Some part of the argument below is, however, still needed to find dim(K_{B}^{∗})_{s} − δ^{∗}(B)_{−s} and to see how small we can let the number s in Teorem 13 be.

Example 28. If we link the homogeneous coordinate ring B of the smooth rational quartic curve of Example 15 with minimal resolution

0 → R(−4)^{3} → R(−3)^{8} → R(−2)^{6} → R → B → 0,

to a B′ using a complete intersection of type (2, 2, 3) and further link B′ to a B′′ via a complete intersection of type (2, 3, 4) (using the same hypersurface of degree 2 and 3 as previously) one may (notably due to I_{B′/D} ≃ K_{B}(−2) and the mapping cone construction, used twice, or see [26]) see that the minimal resolution of B′′ is

0 → R(−7) ⊕ R(−6)^{3} → R(−5)^{10} → R(−4)^{5} ⊕ R(−3) ⊕ R(−2) → R → B′′ → 0.

If A is defined by a regular section of K_{B′′}^{∗}(s), we can easily find the minimal resolution of A and its Hilbert function by using (11). For instance dim(A)_{v} = 16v − 19 for 3 ≤ v ≤ s − 3 and dim(A)_{v} = 16s − 38 for v > s. Let us suppose s > 3 since otherwise Proj(A) is degenerate. To use Proposition 10, we may argue as in Example 15 to see that δ(B′′)_{−s} = 0 for s > 7 by Remark 11(a), and that H^{1}(θ_{C′′}(v)) = 0 and hence dim(K_{B′′}^{∗})_{v} = h^{0}(θ_{C′′}(v)) = 16v − 57 for v ≥ 5 because C′′ := Proj(B′′) is smooth curve of degree d′′ = 16 and genus g′′ = 20. Corollary 27 leads, however, to more precise information. Indeed it is rather straightforward to combine the Corollary and the computations of Example 15 and Example 25 to see that the conclusions of Theorem 13 holds for s ≥ 4, that dim(K_{B′′}^{∗})_{4} = 8, and that (δ(B′′)_{−4}, δ(B′′)_{−5}, δ(B′′)_{−6}) is equal to (7, −6, −6) and δ(B′′)_{−s} = 0 for s > 6. Finally patiently using Proposition 24(i) several times, one shows dim(N_{B′′})_{0} = 62 (It is, however, faster to show _{0}Ext_{B′′}^{i}(I_{B′′}/I_{B′′}^{2}, B′′) = 0 for i = 1 and 2 by Proposition 24(ii), using the computations of Example 15 and (13), and then use (13) once more to see h^{1}(N_{C′′}) = 1 and hence dim(N_{B′′})_{0} = 5d′′ + 1 − g′′ + h^{1}(N_{C′′}) = 62). Putting this together we have by Proposition 10 that A is an unobstructed Gorenstein algebra for s ≥ 4 and

dim_{(A)} GradAlg^{H} = dim(N_{B′′})_{0} + dim(K_{B′′}^{∗})_{s} − 1 − δ(B′′)_{−s} = 16s + 4 for s ≥ 7,

dim_{(A)} GradAlg^{H} = 16s + 10 for 5 ≤ s ≤ 6 and dim_{(A)} GradAlg^{H} = 62 for s = 4. By the interpretation of _{−s}ext_{B′′}^{1}(I_{B′′}/I_{B′′}^{2}, K_{B′′}) of Proposition 10, we get by e.g. Proposition 24(iii) that the codimension of the strata of Gorenstein quotients given by (11) is 6 for 5 ≤ s ≤ 6.

We may treat the Artinian reduction of A ≃ B′′/K_{B′′}(−s) in Example 28 satisfactorily by remarking that _{s+1}H_{2}(R, B′′, B′′) = 0 for 4 ≤ s ≤ 6 (since one may show (S_{2}I_{B′′})_{s+1} ≃ (I_{B′′}^{2})_{s+1} by using that the generators of I_{B′′} of degree 2 and 3 form a regular sequence), and that B′′_{L} is unobstructed by Lemma 16(iii) for any s > 3. For the latter, we use the liaison invariance of _{v}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) and the fact that H^{1}(N_{C}(v)) = 0 for v ≥ −1 for a smooth rational curve C. Instead of working out these details, we will finish this subsection by considering two general classes of examples where B is licci. We concentrate on licci codimension 3 quotients, leading to unobstructed Gorenstein codimension 4 quotients by Theorem 13. In one of the cases the use of Proposition 24 is quite effective.

Example 29. Let B be a Gorenstein quotient of R of codimension 3 given by

0 → R(−f) ≃ F_{3} → F_{2} → F_{1} → I_{B} → 0.  (30)

(as A in Example 19). In this case e = −n − 3 and K_{B} ≃ B(−f + n + 3) and since B is licci ([27]) and therefore satisfies _{v}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 for any v by Proposition 24, we get δ(B)_{−s} = dim(N_{B})_{−(f+s−n−3)}. If A is defined by a section of K_{B}^{∗}(s) as in Theorem 13, then A is Gorenstein and unobstructed and

dim_{(A)} GradAlg^{H_{A}} = dim(N_{B})_{0} + dim(B)_{f+s−n−3} − 1 − dim(N_{B})_{−f−s+n+3}

where dim(N_{B})_{v} is given by [18], Theorem 2.6, i.e. by modifying (16) to

dim(N_{B})_{v} = dim(∧^{2}F_{1})_{f+v} − dim(F_{1} ⊗ F_{2})_{f+v} + dim(S_{2}(F_{2}))_{f+v} + dim(F_{1})_{v} − dim(F_{2})_{v}

Example 30. Let B be an almost complete intersection of R of codimension 3 with a minimal resolution

0 → G_{3} → G_{2} → G_{1} = ⊕_{i=1}^{3}R(−c_{i}) ⊕ R(−h) → I_{B} → 0  (31)

and let B′ be the Gorenstein quotient obtained by linking B to B′ via a complete intersection whose minimal generators (of degree c_{i}) corresponds to three of those of I_{B}. A standard mapping cone construction and duality yields

0 → R(h − a) → G_{2}^{∗}(−a) → G_{3}^{∗}(−a) ⊕ (⊕_{i=1}^{3}R(−c_{i})) → I_{B′} → 0  (32)

and we see that K_{B′} ≃ B′(a − h − n − 3) where a = ∑ c_{i}. If A is defined by the zero locus of a section of K_{B}^{∗}(s) as in Theorem 13, then A is Gorenstein and unobstructed and invoking Proposition 24 and dim(K_{B′}^{∗})_{−s+2a−2n−6} − δ(B′)_{s−2a+2n+6} = dim(B′)_{(−s+h+a−n−3)} − dim(N_{B′})_{(s−h−a+n+3)} we get

dim_{(A)} GradAlg^{H_{A}} = dim(N_{B′})_{0} + dim(B′)_{(−s+h+a−n−3)} − dim(N_{B′})_{(s−h−a+n+3)} − 1 + γ

where γ := ∑_{i=1}^{3}(dim(I_{B′/D})_{c_{i}}−dim(I_{B/D})_{c_{i}}+dim(I_{B/D})_{s−a+n+3+c_{i}}−dim(I_{B′/D})_{−s+a−n−3+c_{i}})+ dim B_{s−a+n+3} − dim B′_{−s−n−3+a}. Using I_{B/D} ≃ K_{B′}(n + 3 − a) ≃ B′(−h), we easily express γ in terms of the Hilbert function of B′ (or of B) and D. Moreover dim(N_{B′})_{v} is given by the final formula of Example 29.

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OSLO UNIVERSITY COLLEGE, FACULTY OF ENGINEERING, CORT ADELERSGT. 30, N-0254 OSLO, NORWAY. E-mail address: JanOddvar.Kleppe@iu.hio.no