FAMILIES OF ARTINIAN AND LOW DIMENSIONAL DETERMINANTAL RINGS.

JAN O. KLEPPE

Abstract. Let GradAlg(H) be the scheme parameterizing graded quotients of R = k[x_{0}, …, x_{n}] with Hilbert function H (it is a subscheme of the Hilbert scheme of ℙ^{n} if we restrict to quotients of positive dimension, see definition below). A graded quotient A = R/I of codimension c is called standard determinantal if the ideal I can be generated by the t × t minors of a homogeneous t × (t + c − 1) matrix (f_{ij}). Given integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}, we denote by W_{s}(\underline{b}; \underline{a}) ⊂ GradAlg(H) the stratum of determinantal rings where f_{ij} ∈ R are homogeneous of degrees a_{j} − b_{i}.

In this paper we extend previous results on the dimension and codimension of W_{s}(\underline{b}; \underline{a}) in GradAlg(H) to artinian determinantal rings, and we show that GradAlg(H) is generically smooth along W_{s}(\underline{b}; \underline{a}) under some assumptions. For zero and one dimensional determinantal schemes we generalize earlier results on these questions. As a consequence we get that the general element of a component W of the Hilbert scheme of ℙ^{n} is glicci provided W contains a standard determinantal scheme satisfying some conditions. We also show how certain ghost terms disappear under deformation while other ghost terms remain and are present in the minimal resolution of a general element of GradAlg(H).

Keywords. Determinantal rings, Deformations, Hilbert schemes, Artinian rings. AMS Subject Classification. 13C40, 14C05, 13D10 (Primary), 14M12, 13E10, 13D02, 13D07, 13A02, 13H10 (Secondary).

1. Introduction

The main goal of this paper is to generalize previous results on maximal families of determinantal schemes, notable to cover maximal families of artinian determinantal kalgebras. Recall that a k-algebra A ≃ R/I, R = k[x_{0}, …, x_{n}], of codimension c is called determinantal if the ideal I can be generated by the r × r minors of a homogeneous p × q matrix (f_{ij}) with c = (p−r+1)(q−r+1). A is called standard (resp. good) determinantal. if r = min(p, q) (resp. r = min(p, q) and A is a generic complete intersection of R).

Let GradAlg(H) be the “Hilbert scheme of constant Hilbert function”, i.e. the scheme parameterizing graded quotients A of R of depth A ≥ min(1, dim A) and with Hilbert function H. Given integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}, we denote by W (\underline{b}; \underline{a}) (resp. W_{s}(\underline{b}; \underline{a})) the stratum in GradAlg(H) consisting of good (resp. standard) determinantal k-algebras where f_{ij} are homogeneous polynomials of degree a_{j} − b_{i}. Then W_{s}(\underline{b}; \underline{a}) is irreducible, W_{s}(\underline{b}; \underline{a}) ≠ ∅ if a_{i−1} > b_{i} for 1 ≤ i ≤ t, and the closures \overline{W_{s}(\underline{b}; \underline{a})} and \overline{W (\underline{b}; \underline{a})} are equal if n ≥ c (Proposition 2.1).

In this paper we focus on the following problems.

(1) Determine when \overline{W_{s}(\underline{b}; \underline{a})} is an irreducible component of GradAlg(H).

(2) Find the codimension of W_{s}(\underline{b}; \underline{a}) in GradAlg(H) if its closure is not a component.

2 JAN O. KLEPPE

(3) Determine when GradAlg(H) is generically smooth along W_{s}(\underline{b}; \underline{a}).

These questions have been considered in several papers, and in [31] we solve all these problems completely provided n−c > 1 and a_{0} > b_{t}. In this case the closure of W_{s}(\underline{b}; \underline{a}) is a generically smooth irreducible component of the usual Hilbert scheme Hilb(ℙ^{n}), as well as of GradAlg(H) (see Theorem 4.6), i.e. [34, Conjecture 4.2] holds, and dim W_{s}(\underline{b}; \underline{a}) is determined (equal to λ in Theorem 4.1, whence [34, Conjecture 4.1] holds for n − c > 0), see also [13] for a somewhat different approach to these problems. So we only need to consider the case n + 1 − c ∈ {0, 1, 2}, and in [29] we solve problems (1)-(3) for determinantal schemes with n + 1 − c ∈ {1, 2} under certain assumptions. In this paper we concentrate on artinian determinantal k-algebras (n+1−c = 0), and we prove the main Theorems 4.1 and 4.6 under conditions that allow c to be n+1. In addition we prove a new result (Theorem 4.8, extending Theorem 4.6) which applies when dim W_{s}(\underline{b}; \underline{a}) ≠ λ. This theorem implies that the general element of an irreducible component W of the Hilbert scheme of ℙ^{n} is glicci (in the Gorenstein liaison class of a complete intersection) provided W contains a standard determinantal scheme satisfying some conditions (Corollary 4.14). For an introduction to glicciness, see [32], and see [11] and its references for further developments. Finally, in Sec. 6, we generalize and complete several results of [29] for families of determinantal schemes of dimension 0 or 1, some of which occurred in [32, 33], and we slightly extend a result of [34].

We have used two different strategies to attack the problems (1) to (3). Indeed in [32, 33, 29, 34] we successively deleted columns of the t × (t + c − 1) matrix A associated to a determinantal scheme X := Proj(A) to get a nest (“flag”) of closed subschemes X = X_{c} ⊂ X_{c−1} ⊂ ... ⊂ X_{2} ⊂ ℙ^{n} and we proved our results inductively by considering the smoothness of the Hilbert flag scheme of pairs and its natural projections to Hilbert schemes. On the other hand, in [31] we compared deformations of A with deformation of the cokernel M of the map ⊕_{j=0}^{t+c−2}R(−a_{j}) → ⊕_{i=1}^{t}R(−b_{i}) induced by A. This latter approach turned out to be quite successful, and it indeed solved problem (1)-(3) for n − c > 1. It is this approach that we generalize to the artinian case, only introducing an extra assumption (_{0}Hom_{A}(M, M) ≃ k) in the theorems. In fact we show that the main results of [31] hold, whence partially solving problems (1) to (3) also for n = c − 1.

For c = 2 all assumptions of the theorems are fulfilled. We even replace R by any Cohen-Macaulay quotient \overline{R} of R and solve the problems above for determinantal quotients of codimension c = 2 of \overline{R} (Theorem 4.15). For c > 2 we need to verify that certain Ext^{i}- groups vanish to apply our results when n + 1 − c ∈ {0, 1, 2}. In this paper we accomplish this by using Macaulay 2 ([15]). We give many examples, supported by Macaulay 2 computations, to illustrate the theorems in the artinian case.

In Sec. 5 we consider ghost terms, i.e. common free summands in consecutive terms of the minimal R-free resolution of A, and we show that some of them disappear, others remain unchanged under suitable generizations (deformations to a more general algebra) in GradAlg(H). If A is general in GradAlg(H) and the assumptions of Theorem 4.8 are fulfilled (e.g. dim A ≥ 3 and a_{0} ≥ b_{t}), one may easily describe all ghost terms in its minimal free resolution while it seems hard to get improved results when dim A ≤ 2 (Proposition 5.1, Remark 5.3), although examples indicate a possible connection (Remark 4.12).

In the proofs we use the exactness of the Buchsbaum-Rim complex ([6, 10]) and the 5term exact sequence associated to the spectral sequence E_{2}^{p,q} := Ext_{A}^{p}(Tor_{q}^{R}(A, M), M) ⇒ Ext_{R}^{p+q}(M, M) which, in view of Fitting’s lemma, plays a surprisingly important role in comparing the deformation functors of M and A.

Notation: Throughout ℙ := ℙ^{n} is the projective n-space over an algebraically closed field k (except in Theorem 4.15 where we allow k ≠ \overline{k}), R = k[x_{0}, x_{1}, …, x_{n}] is a polynomial ring and 𝔪 = (x_{0}, …, x_{n}). If X ⊂ Y are closed subschemes of ℙ^{n}, we denote by I_{X/Y} (resp. N_{X/Y}) the ideal (resp. normal) sheaf of X in Y, and we omit /Y if Y = ℙ^{n}. Let I_{X} = H_{∗}^{0}(I_{X}) ⊂ R be the saturated homogeneous ideal of X ⊂ ℙ^{n}. When we write X = Proj(A) we always let A := R/I_{X} and K_{A} = Ext_{R}^{c}(A, R)(−n − 1) be the canonical module of A or X where c = codim_{ℙ} X := n − dim X. Note that by the codimension, codim_{Y} X, of an irreducible X in a not necessarily equidimensional scheme Y we mean dim O_{Y,x} − dim X, where x is a general k-point of X. We let vExt(F, G) be the degree-v part of the graded module Ext(F, G). Moreover we denote by hom(F, G) = dim_{k} Hom(F, G) the dimension of the group of morphisms between coherent O_{X}-modules and we use small letters for the k-vector space dimension of similar groups.

2. Preliminaries

2.1. Hilbert schemes and Hilbert function strata. We denote the Hilbert scheme with the Hilbert polynomial p ∈ ℚ[s] by Hilb^{p}(ℙ^{n}), cf. [16] for existence and [39] for the local theory. Similarly GradAlg^{H}(R), or Hilb^{H}(ℙ^{n}) when dim A > 0, is the representing object of the functor that parameterizes flat families of graded quotients A of R of depth_{𝔪} A ≥ min(1, dim A) and with Hilbert function H, H(i) = dim A_{i} ([26, 27, 17]). We allow calling it “the postulation Hilbert scheme” ([28, §1.1]) even though it may be different from the parameter space studied by Gotzmann and Iarrobino ([14, 21]) who study the “same” scheme with the reduced scheme structure. We let (A), or (X) where X = Proj(A), denote the point of GradAlg(H) := GradAlg^{H}(R) that corresponds to A. Note that if depth_{𝔪} A ≥ 1 and _{0}Hom_{R}(I_{X}, H_{𝔪}^{1}(A)) = 0, then

(2.1)  GradAlg(H) ≃ Hilb^{p}(ℙ^{n}) at (X) ,

and hence we have an isomorphism _{0}Hom(I_{X}, A) ≃ H^{0}(N_{X}) of their tangent spaces (cf.

[12] for the case depth_{𝔪} A ≥ 2, and (9) of [27] for the general case). If X is generically a complete intersection, then _{0}Ext_{A}^{1}(I_{X}/I_{X}^{2}, A) is an obstruction space of GradAlg(H) at (A), and hence of Hilb^{p}(ℙ^{n}) at (X) if (2.1) holds. Indeed for any quotient A = R/I, we may define H_{2}(R, A, A) := ker(S_{2}(I) → I^{2}) where S_{2}(I) is the 2^{nd} symmetric power of I, cf. [1], p.106. Then GradAlg(H) is smooth at (A) if

(2.2)  _{0}Ext_{A}^{1}(I/I^{2}, A) = 0 and _{0}Hom_{A}(H_{2}(R, A, A), A) = 0 ,

and observe that the latter group vanishes if A is generically a complete intersection of R [27, §1.1]. By definition X (resp. A) is called unobstructed if Hilb^{p}(ℙ^{n}) (resp. GradAlg(H)) is smooth at (X) (resp. (A)).

We say that X is general in some irreducible subset W ⊂ Hilb^{p}(ℙ^{n}) if (X) belongs to a sufficiently small open subset U of W such that any (X) in U has all the openness properties that we want to require. We define ”A general in GradAlg(H)” similarly.

4 JAN O. KLEPPE

2.2. Determinantal rings and schemes. We mainly maintain the notions and notations from [29, 31], but we need to extend some results to artinian determinantal k-algebras. For a more general background of determinantal rings and schemes, see [4, 10, 3, 44].

Indeed let

(2.3)  φ : F = ⨁_{i=1}^{t} R(b_{i}) ⟶ G := ⨁_{j=0}^{t+c−2} R(a_{j})

be a graded morphism of free R-modules, let A = (f_{ij})_{i=1,...t}^{j=0,...,t+c−2}, deg f_{ij} = a_{j} − b_{i}, be a matrix which represents the dual φ^{∗} := Hom_{R}(φ, R) and let I(A) := I_{t}(A) be the ideal of R generated by the maximal minors of A. We always suppose t ≥ 2 (unless explicitly allowing t = 1) and

(2.4)  c ≥ 2, b_{1} ≤ ... ≤ b_{t} and a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2}.

A codimension c quotient A = R/I (resp. subscheme X ⊂ ℙ^{n} if n ≥ c) is called standard determinantal if I := I(A) (resp. I_{X} := I(A)) for some homogeneous t×(t+c−1) matrix A as above. They are good determinantal if additionally, the ideal I_{t−1}(A) of submaximal minors has at least codimension c + 1 in R, i.e. R/I(A) is a generic complete intersection of R. If t = 1, standard, as well as good determinantal are complete intersections of R.

Given integers b_{i} and a_{j} satisfying (2.4) we let W_{s}(\underline{b}; \underline{a}; R) (resp. W (\underline{b}; \underline{a}; R)) be the stratum in GradAlg(H) := GradAlg^{H}(R) consisting of standard (resp. good) determinantal quotients, cf. (2.1), and we usually omit R in this notation. Note that we do not require A to be minimal (i.e. f_{ij} = 0 when b_{i} = a_{j}) for R/I_{t}(A) to belong to W_{s}(\underline{b}; \underline{a}) := W_{s}(\underline{b}; \underline{a}; R). In examples, however, we mainly consider determinantal rings with positive degree matrix (i.e. for every i, j, b_{i} < a_{j}, and recall that a_{j} − b_{i} = 1 is called the linear case).

Proposition 2.1. The closure of W_{s}(\underline{b}; \underline{a}) in GradAlg(H) is irreducible, and

W_{s}(\underline{b}; \underline{a}) ≠ ∅ ⇔ a_{i−1} ≥ b_{i} for all i and a_{i−1} > b_{i} for some i .

Moreover if n − c ≥ 0, then \overline{W_{s}(\underline{b}; \underline{a})} = \overline{W (\underline{b}; \underline{a})}. And if n − c = −1, then every A = R/I of W (\underline{b}; \underline{a}) satisfies I_{t−1}(A) = R, i.e. is a complete intersection (c.i.) of R.

Proof. For W_{s}(\underline{b}; \underline{a}) to be non-empty we refer to (2.2) of [34] whose arguments carry over to the artinian case. The text after [34, (2.2)] shows also that \overline{W_{s}(\underline{b}; \underline{a})} = \overline{W (\underline{b}; \underline{a})} and that this locus is irreducible in the non-artinian case (n ≥ c). To see that W_{s}(\underline{b}; \underline{a}) is irreducible also when W_{s}(\underline{b}; \underline{a}) parametrizes artinian k-algebras we consider the affine scheme 𝕍 = Hom_{R}(G^{∗}, F^{∗}) whose rational points correspond to t × (t + c − 1) matrices. Since the vanishing of Ext_{R}^{i}(R/I_{t}(A), R) is an open property, the subset U of 𝕍 of matrices such that I_{t}(A) has maximal codimension in R is open and irreducible. Then, since there is an obvious morphism from U onto W_{s}(\underline{b}; \underline{a}), it follows that \overline{W_{s}(\underline{b}; \underline{a})} is irreducible.

Finally, for A artinian and good determinantal, I_{t−1}(A) is not contained in 𝔪 because this would imply dim R/I_{t−1}(A) = dim R/I_{t}(A) = 0. Thus I_{t−1}(A) = R, and it follows that A is a c.i. of R by Corollary 5.2. □

Remark 2.2. In earlier papers on determinantal rings we often assumed that A was good determinantal. By [33, Lem. 3.2] good determinantal rings satisfy A ≃ Hom_{A}(M, M) where M := coker φ^{∗}. This and the fact that they are generic complete intersections of R, was often used in the proofs of the deformation results in [32, 33, 29, 31]. To generalize earlier results to the artinian case, we see from Proposition 2.1 that we can not assume A good determinantal any more. Also A ≃ Hom_{A}(M, M) has to be dropped because this seems to never hold for A artinian. It turns, however, out that only assuming 0Hom_{A}(M, M) ≃ k and A standard determinantal often allows us to generalize the proofs.

In the following let A = R/I_{t}(A) be standard determinantal, i.e. (A) ∈ W_{s}(\underline{b}; \underline{a}) and let X = Proj(A) if dim A > 0. Then R-free resolutions of A and M := M_{A} := coker φ^{∗} are given by the Eagon-Northcott and Buchsbaum-Rim complexes respectively, see [6, 9, 10]. These resolutions are minimal if A is minimal. By e.g. [31, (2.3), (2.4)] the resolutions are

\begin{matrix}0 ⟶ ∧^{t+c−1}G^{∗} ⊗ S_{c−1}(F) ⊗ ∧^{t}F ⟶ ⋯ → ∧^{t+i}G^{∗} ⊗ S_{i}(F) ⊗ ∧^{t}F \\ → … ⟶ ∧^{t}G^{∗} ⊗ S_{0}(F) ⊗ ∧^{t}F ⟶ R ⟶ A ⟶ 0 , and \end{matrix} ; (2.5)

\begin{matrix}0 → ∧^{t+c−1}G^{∗} ⊗ S_{c−2}(F) ⊗ ∧^{t}F → ⋯ → ∧^{t+i+1}G^{∗} ⊗ S_{i}(F) ⊗ ∧^{t}F \\ → … → ∧^{t+1}G^{∗} ⊗ S_{0}(F) ⊗ ∧^{t}F → G^{∗} ⟶^{φ^{∗}} F^{∗} → M → 0 . \end{matrix} ; (2.6)

The matrix A_{i} obtained by deleting the last c − i columns defines a morphism

(2.7)  φ_{i} : F = ⨁_{i=1}^{t} R(b_{i}) ⟶ G_{i} := ⨁_{j=0}^{t+i−2} R(a_{j}) .

and we suppose that all entries of the deleted columns belong to 𝔪. Then the k-algebra D_{i} ≅ R/I_{D_{i}} given by the maximal minors of A_{i} is standard determinantal by [2] and φ_{i} is injective. Letting B_{i} = coker φ_{i} and M_{i} = coker φ_{i}^{∗}, we have an exact sequence

(2.8)  0 → B_{i}^{∗} → G_{i}^{∗} ⟶^{φ_{i}^{∗}} F^{∗} ⟶ M_{i} ≅ Ext_{R}^{1}(B_{i}, R) ⟶ 0 .

Then M_{i} is a D_{i}-module, R ↠ D_{i} ↠ D_{i+1}... ↠ D_{c} = A, and there is an exact sequence

(2.9)  0 ⟶ D_{i} ⟶ M_{i}(a_{t+i−1}) ⟶ M_{i+1}(a_{t+i−1}) ⟶ 0

in which D_{i} ⟶ M_{i}(a_{t+i−1}) is a regular section that defines D_{i+1} (also when i + 1 = c and D_{c} is artinian [36, Lem. 3.6]). Hence the sequence

(2.10)  0 ⟶ I_{D_{i+1}/D_{i}} ≃ Hom_{D_{i}}(M_{i}(a_{t+i−1}), D_{i}) ⟶ D_{i} ⟶ D_{i+1} ⟶ 0 .

is exact. By (2.6) M_{i} is a maximal Cohen-Macaulay D_{i}-module and so is I_{D_{i+1}/D_{i}} by (2.10). Moreover by e.g. (2.6), K_{D_{i}}(n + 1) ≅ S_{i−1}M_{i}(ℓ_{i}) where ℓ_{i} := ∑_{j=0}^{t+i−2} a_{j} − ∑_{q=1}^{t} b_{q}.

Lemma 2.3. With the above notation (and M = M_{c}), there are exact sequences

0 → Hom_{R}(M_{i}, M) → F ⊗_{R} M → G_{i} ⊗_{R} M → B_{i} ⊗_{R} M → 0 , and
0 → Hom_{R}(B_{i}, F) → Hom_{R}(B_{i}, G_{i}) → Hom_{R}(B_{i}, B_{i}) → Hom_{R}(M_{i}, M_{i}) → 0 .

Proof. (cf. [33, Lem. 3.1 and 3.10]). We apply Hom_{R}(B_{i}, −) to

(2.11)  0 ⟶ F ⟶ G_{i} ⟶ B_{i} ⟶ 0

and using M_{i} ≃ Ext_{R}^{1}(B_{i}, R) we deduce the exact sequence

0 → Hom(B_{i}, F) → Hom_{R}(B_{i}, G_{i}) → Hom_{R}(B_{i}, B_{i}) → F ⊗_{R} M_{i} → G_{i} ⊗_{R} M_{i} .

Hence we get the lemma by applying Hom(−, M_{i}) to (2.8) and (−) ⊗_{R} M to (2.11). □

Proposition 2.4. Set K_{i} := _{0}hom(B_{i−1}, R(a_{t+i−2})) and ℓ_{i} := ∑_{j=0}^{t+i−2} a_{j} − ∑_{q=1}^{t} b_{q}. Letting h_{i−3} := 2a_{t+i−2} − ℓ_{i} + n for 3 ≤ i ≤ c and \binom{a}{n} = 0 if a < n we have

K_{i+3} = ∑_{{r+s=i \atop r,s≥0}} ∑_{{0≤i_{1}<...<ir≤t+i \atop 1≤j_{1}≤...≤js≤t}}(−1)^{i−r}\binom{h_{i} + a_{i_{1}} + ⋯ + a_{i_{r}} + b_{j_{1}} + ⋯ + b_{j_{s}}}{n} for 0 ≤ i ≤ c − 3 ,

e.g. K_{3} = \binom{h_{0}}{n} and K_{4} = ∑_{j=0}^{t+1} \binom{h_{1}+a_{j}}{n} − ∑_{i=1}^{t} \binom{h_{1}+b_{i}}{n}. Moreover if _{0}hom(M, M) = 1 and the entries of the last c − 2 columns belong to 𝔪 (e.g. A is minimal), then

aut(B_{c}) := _{0}hom(B_{c}, B_{c}) = 1 + K_{3} + K_{4} + ⋯ + K_{c} .

Proof. Using (2.6) we get a resolution of M_{i−1} which due to (2.8) leads to the following resolution:

0 → E_{i−3} → ⋯ → E_{j} → ⋯ → E_{0} → B_{i−1}^{∗} → 0

where E_{j} := ∧^{t+j+1}G^{∗} ⊗ S_{j}(F) ⊗ ∧^{t}F. Taking dimensions of the degree a_{t+i−2} part of the free modules in this resolution and using K_{i} = dim B_{i−1}^{∗}(a_{t+i−2})_{0}, we get that K_{i} coincides with the sum of binomials appearing in the proposition (see [33, p.2882] for details).

Now dualizing the exact sequence 0 → R(a_{t+c−2}) → B_{c} → B_{c−1} → 0, we get

0 → Hom(B_{c−1}, R) → Hom_{R}(B_{c}, R) → R(−a_{t+c−2}) → M_{c−1} → M_{c} → 0

by (2.8). Combining with (2.9) we get the exact sequence

0 → Hom(B_{c−1}, R) → Hom_{R}(B_{c}, R) → I_{D_{c−1}}(−a_{t+c−2}) → 0

which implies the vanishing of the lower down-arrows in the following commutative diagram

0Hom(B_{c−1}, F) ⟶ _{0}Hom(B_{c−1}, G_{c})
↓  ↓
0Hom(B_{c}, F) ⟶ _{0}Hom(B_{c}, G_{c})
↓  ↓
0Hom(R(a_{t+c−2}), F) ⟶ _{0}Hom(R(a_{t+c−2}), G_{c}) .

It follows that the upper down-arrows are bijections. Since Hom(B_{c−1}, −) is exact on 0 → R(a_{t+c−2}) → G_{c} → G_{c−1} → 0 (because Ext^{1}(B_{c−1}, −) ≅ M_{c−1} ⊗ (−)) we get

(2.12) _{0}hom(B_{c}, G_{c}) − _{0}hom(B_{c}, F) = _{0}hom(B_{c−1}, G_{c−1}) − _{0}hom(B_{c−1}, F) + K_{c} .

By Lemma 2.3 and the assumption _{0}hom(M_{c}, M_{c}) = 1, we have

aut(B_{c}) = 1 + _{0}hom(B_{c}, G_{c}) − _{0}hom(B_{c}, F) ,

and since there is a corresponding expression for aut(B_{c−1}) we get

(2.13)  aut(B_{c}) = 1 + K_{c} + aut(B_{c−1}) − _{0}hom(M_{c−1}, M_{c−1})

by (2.12). D_{c−1} is, however, good determinantal because I_{t}(A_{c−1}) ⊂ I_{t}(A_{c}) ⊂ I_{t−1}(A_{c−1}) and codim_{D_{c−1}} D_{c} = 1. Indeed all D_{j}, 2 ≤ j ≤ c − 1 are good determinantal for the same reason and we get _{0}hom(M_{j}, M_{j}) = 1 for 2 ≤ j ≤ c − 1 by Remark 2.2. This simplifies (2.13) which holds also if we replace c by j + 1 in (2.13). Then we conclude the proof by induction using that aut(B_{2}) = _{0}hom(I_{D_{2}}, I_{D_{2}}) = 1. □

Finally recall that if J = I_{t−1}(A) and dim A > 0 then X = Proj(A) is a local complete intersection (l.c.i.) in X−V (J). In the following we always take Z ⊃ V (J) and U = X−Z, i.e. so that X ↪ ℙ^{n} is an l.c.i. in U. Since the 1^{st} Fitting ideal of M is equal to J, we get that M̃ and Ĩ/I^{2} are locally free on X − V (J), cf. [50, Lem. 1.8] and [3, Lem. 1.4.8]. Note that depth_{J} A ≥ codim_{X} Sing(X) which we use in the following.

Remark 2.5. Let X = X_{c}, Y = X_{c−1} and let α be a positive integer. If X is general in W (\underline{b}; \underline{a}) ≠ ∅ and a_{i−min(α,t)} − b_{i} ≥ 0 for min(α, t) ≤ i ≤ t, then

(2.14)  codim_{X_{j}} Sing(X_{j}) ≥ min{2α − 1, j + 2} for 2 ≤ j ≤ c ,

cf. the arguments in [33, Rem. 2.7] which use [7, 51]. In particular letting α = 3 we get that X ↪ ℙ^{n} (resp. Y ↪ ℙ^{n} and X ↪ Y) are l.c.i.s outside a subset Z ⊂ X of codimension at least min(5, c + 2) (resp. min(4, c)) in X. Indeed we may take Z = V (I_{t−1}(A_{c−1})) for the statement involving X and Y because (2.10) implies that I_{X/Y} is locally free on Y − Z, noting that Z ⊂ V (I_{t}(A)) = X. Moreover if a_{i−2} − b_{i} ≥ 0 for 2 ≤ i ≤ t, then X ↪ ℙ^{n} (resp. Y ↪ ℙ^{n} and X ↪ Y) are l.c.i.s outside a subset Z ⊂ X of codimension at least min(3, c + 2) (resp. min(2, c)). Notice that we interpret I(Z) as 𝔪 if Z = ∅.

2.3. The dimension of the determinantal locus. In [31] we proved that the dimension of a non-empty W (\underline{b}; \underline{a}) in the case a_{i−2} − b_{i} ≥ 0 for i ≥ 2 and n − c ≥ 1 is given by

(2.15)  dim W_{s}(\underline{b}; \underline{a}) = dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + K_{4} + ... + K_{c} ,

where K_{i} is defined in Proposition 2.4 and λ_{c} is defined by

(2.16) λ_{c} := ∑_{i,j} \binom{a_{j} − b_{i} + n}{n}+∑_{i,j} \binom{b_{i} − a_{j} + n}{n}−∑_{i,j} \binom{a_{i} − a_{j} + n}{n}−∑_{i,j} \binom{b_{i} − b_{j} + n}{n}+1

where the indices belonging to a_{j} (resp. b_{i}) range over 0 ≤ j ≤ t + c − 2 (resp. 1 ≤ i ≤ t).

Remark 2.6. Note that we assume (2.4) and delete columns from the right-hand side for (2.15) to hold. From the proof of Proposition 2.4 we see that we also need all algebras D_{i} in R ↠ D_{2} ↠ D_{3}... ↠ D_{c} = A to be standard determinantal. If D_{i} is standard determinantal and we delete a column whose entries belong to 𝔪, then D_{i−1} is standard determinantal by [2], see Proposition 5.1 for deleting columns containing units. In particular (2.15) holds if the entries of the last c − 2 columns of A belong to 𝔪 (e.g. if A is minimal).

Using, however, the Hilbert function, H_{M}(−), of M and assuming _{0}hom(M, M) = 1, we have by [31, Rem. 3.9], for A possibly non-minimal, that

(2.17)  dim W_{s}(\underline{b}; \underline{a}) = dim W (\underline{b}; \underline{a}) = ∑_{j=0}^{t+c−2} H_{M}(a_{j}) − ∑_{i=1}^{t} H_{M}(b_{i}) + 1 .

For zero-dimensional determinantal schemes (n − c = 0) we have the following.

Remark 2.7. (i) Assume a_{0} > b_{t}. Then (2.15) holds provided 3 ≤ c ≤ 5 (resp. c > 5) and a_{t+c−2} > a_{t−2} (resp. a_{t+3} > a_{t−2}) by [34, Thm. 3.2], see Sec. 6 for a generalization.

(ii) If A is a linear 2 × (c + 1) matrix, we showed in [29, Ex. 3.3] that dim W (\underline{b}; \underline{a}) is strictly smaller that the right-hand side of (2.15) for every c > 2. To our knowledge this is the only known case where the equality of (2.15) fails to hold when n = c.

8 JAN O. KLEPPE

2.4. The determinantal locus as components of Hilbert schemes. It was shown in [31] that \overline{W (\underline{b}; \underline{a})}, for W (\underline{b}; \underline{a}) ≠ ∅, is an irreducible component of Hilb^{p}(ℙ^{n}) provided n − c ≥ 2 and a_{i−min(3,t)} − b_{i} ≥ 0 for min(3, t) ≤ i ≤ t. If n − c ≤ 1 then \overline{W (\underline{b}; \underline{a})} may fail to be an irreducible component. Indeed \overline{W (0, 0; 1, 1, ..., 1, 2)} is not an irreducible component of GradAlg(H) for all c ≥ 3 both when n = c and n = c + 1, see [29, Ex. 4.1]. In both cases the h-vector of a general A is (1, c, c), e.g. the Hilbert function is given by (dim A_{i})_{i=0}^{∞} = (1, c + 1, 2c + 1, 2c + 1, ...) if n = c.

The reason why \overline{W (\underline{b}; \underline{a})} is not a component is that there exist determinantal rings allowing “deformations that do not come from deforming the matrix A”. Let us recall this notion. We briefly say “T a local ring” (resp. “T artinian”) for a local k-algebra (T, 𝔪_{T}) essentially of finite type over k = T/𝔪_{T} (resp. of finite type over k = T/𝔪_{T} such that 𝔪_{T}^{r} = 0 for some r ∈ ℤ). The local deformation functor Def_{A/R} is defined for each artinian T as the set of graded (T -flat) deformations A_{T} of A to T (i.e. A_{T} ⊗_{T} k = A). Moreover “T → S is called a small artinian surjection” if there is a morphism (T, 𝔪_{T}) → (S, 𝔪_{S}) of local artinian k-algebras whose kernel 𝔞 satisfies 𝔞 · 𝔪_{T} = 0.

If T is a local ring, we let A_{T} = (f_{ij,T}) be a matrix of homogeneous polynomials of the graded algebra R_{T} := R ⊗_{k} T, satisfying f_{ij,T} ⊗_{T} k = f_{ij} and deg f_{ij,T} = a_{j} − b_{i} for all i, j. Here all elements of T are considered to be of degree zero. Once having A_{T}, it induces a morphism

(2.18)  φ_{T} : F_{T} := ⊕_{i=1}^{t}R_{T} (b_{i}) → G_{T} := ⊕_{j=0}^{t+c−2}R_{T} (a_{j}) .

Lemma 2.8. If A = R/I_{t}(A) is standard determinantal, then A_{T} := R_{T} /I_{t}(A_{T}) and M_{T} := coker φ_{T}^{∗} are graded deformations of A and M respectively for every choice of A_{T}, T a local ring. Moreover every graded deformation of M is of the form M_{T} for some A_{T}.

Proof. See [29, Lem. 4.2] for a proof. Indeed A_{T} = R_{T} /I_{t}(A_{T}) and M_{T} are T -flat because all maps in their Eagon-Northcott and Buchsbaum-Rim complexes are defined in terms of A_{T}, and these complexes become exact after tensoring by k over T. Note also that the final statement in the lemma follows immediately from M = coker φ^{∗}. □

The final statement may be formulated as “every graded deformation of M to a local ring T comes from deforming A”. This may not hold for A.

Definition 2.9. Let A = R/I_{t}(A). We say “every deformation of A (or X if dim A > 1, see (2.1)) comes from deforming A” if for every local ring T and every graded deformation R_{T} → A_{T} of R → A to T, then A_{T} is of the form A_{T} = R_{T} /I_{t}(A_{T}) for some A_{T} as above.

Lemma 2.10. Let A = R/I_{t}(A) be a standard determinantal ring, (A) ∈ W_{s}(\underline{b}; \underline{a}). If every deformation of A comes from deforming A, then A is unobstructed (i.e. Def_{A/R} is smooth). Moreover \overline{W_{s}(\underline{b}; \underline{a})} is an irreducible component of GradAlg(H).

Proof. See [29, Lem. 4.4], only replacing Hilb(ℙ^{n}) by GradAlg(H) in its proof. □

Remark 2.11. By these lemmas we get T -flat determinantal schemes by just parameterizing the polynomials of A over a local ring T, see Rem. 4.5 of [29] and Laksov’s papers [38, 37] for somewhat similar results for more general determinantal schemes.

3. deformations of modules and determinantal rings

The main goal of this section is to generalize to A artinian the close relationship between the local deformation functor, Def_{M/R}, of the graded R-module M = coker φ^{∗} and the corresponding local functor, Def_{A/R}, of graded deformations of the standard determinantal ring A = R/I_{t}(A). Note that I := I_{t}(A) = ann(M) by [5]. In [31] these functors were shown to be isomorphic (resp. the first a natural subfunctor of the latter) provided dim X ≥ 2 (resp. dim X = 1) and X = Proj(A) general and good determinantal. The comparison between these deformation functors relied on understanding the spectral sequence E_{2}^{p,q} := Ext_{A}^{p}(Tor_{q}^{R}(A, M), M) ⇒ Ext_{R}^{p+q}(M, M) and its induced 5-term exact sequence

(3.1)  0 → Ext_{A}^{1}(M, M) → Ext_{R}^{1}(M, M) ⟶^{δ} E_{2}^{0,1} → Ext_{A}^{2}(M, M) → Ext_{R}^{2}(M, M) → .

Indeed Tor_{1}^{R}(A, M) ≃ I ⊗_{R} M implies E_{2}^{0,1} ≃ Hom_{R}(I, Hom_{R}(M, M)) in general, and since Hom_{A}(M, M) ≃ A provided A is good determinantal (i.e. depth_{I_{t−1}(A)A} A ≥ 1) by [33, Lem. 3.2], it follows in this case that the morphism δ of (3.1) induces a natural map

(3.2)  _{0}Ext_{R}^{1}(M, M) ⟶ (E_{2}^{0,1})_{0} ≃ _{0}Hom_{R}(I, A)

between the tangent spaces of Def_{M/R} and Def_{A/R}. For A standard determinantal, there is still a natural map e_{M}(T) : Def_{M/R}(T) → Def_{A/R}(T), T artinian, obtained by taking a matrix A_{T} whose corresponding morphism has M_{T} as cokernel and letting A_{T} := R_{T} /I_{t}(A_{T}) (see Lemma 2.8). Since matrices inducing the same M_{T} define the same ideal of maximal minors (Fitting’s lemma, [10, Cor. 20.4]), this morphism is well-defined.

Remark 3.1. The main difficulty in generalizing the comparison above to an artinian A is that A → Hom_{A}(M, M) is no longer an isomorphism (for A not a c.i.) which implies that the tangent map of Def_{M/R} → Def_{A/R} may not be the map in (3.2), i.e. (E_{2}^{0,1})_{0} ≃ 0Hom_{R}(I, A) may fail. The morphism Def_{M/R} → Def_{A/R} is, however, well-defined also in the artinian case, and since I_{t}(A) = ann(M) implies that A → Hom_{A}(M, M) is injective and the functors are pro-representable provided _{0}Hom_{A}(M, M) ≃ k, we shall see that we are able to generalize the comparison.

Definition 3.2. Let A = R/I_{t}(A) be a standard determinantal ring and let \underline{ℓ} be the category of artinian k-algebras (cf. the text before (2.18)). Then the local deformation functor Def_{A∈W_{s}(\underline{b};\underline{a})}, defined on \underline{ℓ}, is the subfunctor of Def_{A/R} given by:

Def_{A∈W_{s}(\underline{b};\underline{a})}(T) = \{A_{T} ∈ Def_{A/R}(T)|A_{T} = R_{T} /I_{t}(A_{T}) for some matrix A_{T} lifting A to T \} .

With this definition it is obvious that the natural map e_{M} : Def_{M/R} → Def_{A/R} defined above for any T ∈ ob(\underline{ℓ}) factors via Def_{A∈W_{s}(\underline{b};\underline{a})} ↪ Def_{A/R}. Moreover we have

Lemma 3.3. If D := k[ϵ]/(ϵ^{2}) then the degree zero part, δ_{0}(M), of the connecting homomorphism δ of (3.1) factors through e_{M}(D) : Def_{M/R}(D) → Def_{A/R}(D), i.e. δ_{0}(M) is the composition

0Ext_{R}^{1}(M, M) ⟶^{e_{M} (D)} _{0}Hom_{R}(I, A) ⟶^{i_{A,M}} _{0}Hom_{R}(I, Hom_{A}(M, M)) ≃ (E_{2}^{0,1})_{0}

where the map in the middle is induced by A ↪ Hom_{A}(M, M), 1 ∈ A ↦ id_{M}.

10 JAN O. KLEPPE

Proof. We know that e_{M}(D) : _{0}Ext_{R}^{1}(M, M) → _{0}Hom_{R}(I, A) is well-defined. To describe it, take any \overline{η} ∈ _{0}Ext_{R}^{1}(M, M) and let η′ ∈ _{0}Hom(G^{∗}, M) represent \overline{η} and η ∈ _{0}Hom(G^{∗}, F^{∗}) map to η′, cf. (3.3). A maximal minor f_{1} corresponds to choosing t columns of A which we for simplicity suppose is obtained by deleting the last c−1 columns of A. With notations as in Sec. 2, f_{1} is the determinant of φ_{1}^{∗} : G_{1}^{∗} = ⊕_{j=0}^{t−1}R(−a_{j}) → F^{∗} = ⊕_{i=1}^{t}R(−b_{i}). By definition e_{M}(D)(\overline{η}) is given by the ideal generated by maximal minors of φ^{∗} + ϵη. Let I_{1} be the principal ideal generated by f_{1}. Since it is easy to compute det(φ_{1}^{∗} + ϵη_{1}) where η_{1} is the composition, G_{1}^{∗} ↪ G^{∗} ⟶^{η} F^{∗}, of the natural inclusion G_{1}^{∗} ↪ G^{∗} by η, we get that the image of e_{M}(D)(\overline{η}) in _{0}Hom_{R}(I_{1}, A) ≃ A_{(deg f_{1})} via the natural map I_{1} ↪ I is tr(φ_{1}^{a} · η_{1}) ⊗_{R} 1 where 1 ∈ A is the unity, φ_{1}^{a} is the adjoint of φ_{1}^{∗} and tr the trace map.

Let M_{1} = coker φ_{1}^{∗} and let π : M_{1} → M be the canonical surjection. To see that δ_{0}(M) = i_{A,M} · e_{M}(D) it suffices to check that the image of δ_{0}(M)(\overline{η}) via the composition

0Hom_{R}(I, Hom(M, M)) ↠ _{0}Hom_{R}(I_{1}, Hom(M, M)) ↪ _{0}Hom_{R}(I_{1}, Hom(M_{1}, M))

is tr(φ_{1}^{a}·η_{1})⊗_{R}π because the maximal minors provide a surjection ⊕R(−n_{i}) ↠ I and hence an injection _{0}Hom_{R}(I, Hom(M, M)) → _{0}Hom_{R}(⊕R(−n_{i}), Hom(M, M)) (and think of f_{1} as an arbitrary minor). Then we can conclude the proof by the functoriality of connecting homomorphisms and Ile’s result for δ_{0}(M_{1}) in [24]. Indeed we have a commutative diagram

0Ext_{R}^{1}(M, M) ⟶^{δ_{0}(M)}  _{0}Hom_{R}(I, Hom_{R}(M, M))
↓  ↓
0Ext_{R}^{1}(M_{1}, M)) ⟶ _{0}Hom_{R}(I_{1}, Hom_{R}(M_{1}, M))
↑  ↑
0Ext_{R}^{1}(M_{1}, M_{1}) ⟶^{δ_{0}(M_{1})} _{0}Hom_{R}(I_{1}, Hom_{R}(M_{1}, M_{1}))

of three connecting homomorphisms. If A_{1} = R/I_{1} then δ_{0}(M_{1})(\overline{η_{1}}) = tr(φ_{1}^{a} · η_{1}) ⊗_{R} id_{M_{1}} by [24, Prop. 2], whence maps to tr(φ_{1}^{a} · η_{1}) ⊗_{R} π via the right up-arrow. Here \overline{η_{1}} ∈ 0Ext_{R}^{1}(M_{1}, M_{1}) is represented by the composition G_{1}^{∗} ⟶^{η_{1}} F^{∗} → M_{1}. Since the elements \overline{η} ∈ _{0}Ext_{R}^{1}(M, M) and \overline{η_{1}} ∈ _{0}Ext_{R}^{1}(M_{1}, M_{1}) map to the same element in _{0}Ext_{R}^{1}(M_{1}, M) (the one represented by the composition G_{1}^{∗} ⟶^{η_{1}} F^{∗} → M) the proof is complete. □

First we will find the dimension of the pro-representing object of Def_{M/R}, i.e. we need to generalize [31, Thm. 3.2] by weakening its conditions so that it applies to an artinian A. Indeed we have (and see Proposition 2.4 and Remark 2.6 for computing the dimension).

Theorem 3.4. Let A = R/I_{t}(A) be standard determinantal and let M = coker φ^{∗}. Then M is unobstructed, i.e. Def_{M/R} is formally smooth. Moreover if _{0}Hom_{A}(M, M) ≃ k, then Def_{M/R} is pro-representable, and the pro-representing object H(M/R) of Def_{M/R} satisfies

dim H(M/R) = dim _{0}Ext_{R}^{1}(M, M) = λ_{c} + K_{3} + K_{4} + ... + K_{c} .

Moreover we have dim _{0}Ext_{R}^{1}(M, M) = ∑_{j=0}^{t+c−2} H_{M}(a_{j}) − ∑_{i=1}^{t} H_{M}(b_{i}) + 1 .

Proof. For the unobstructedness of M, see [31, Thm. 3.1] or Ile’s PhD thesis [22, ch. 6]. To see the dimension formula we claim that there is an exact sequence

(3.3) 0 → _{0}Hom_{R}(M, M) → _{0}Hom_{R}(F^{∗}, M) → _{0}Hom_{R}(G^{∗}, M) → _{0}Ext_{R}^{1}(M, M) → 0.

Indeed the map d_{1} : ∧^{t+1}G^{∗} ⊗ S_{0}(F) ⊗ ∧^{t}F → G^{∗} appearing in the Buchsbaum-Rim complex (2.6), takes an element of ∧^{t+1}G^{∗} ⊗ S_{0}(F) ⊗ ∧^{t}F to a linear combination of maximal minors with coefficients in G^{∗} because I_{t}(A) = im(∧^{t}G^{∗} ⊗ S_{0}(F) ⊗ ∧^{t}F → R) is generated by maximal minors, whence _{0}Hom_{R}(d_{1}, M) = 0 because I = ann(M). So if we apply _{0}Hom_{R}(−, M) to (2.6), we get (3.3) by the definition of _{0}Ext_{R}^{i}(M, M).

Taking dimensions of the groups in (3.3) and using (2.7) and _{0}hom(M, M) = 1 we get the latter dimension formula for dim _{0}Ext_{R}^{1}(M, M). Moreover if we apply the exact functors _{0}Hom_{R}(F^{∗}, −) and _{0}Hom_{R}(G^{∗}, −) onto (2.8) with i = c we get

0hom_{R}(G^{∗}, M) − _{0}hom_{R}(F^{∗}, M) = λ_{c} − 1 + _{0}hom_{R}(G^{∗}, B^{∗}) − _{0}hom_{R}(F^{∗}, B^{∗})

by using the definition (2.16) of λ_{c}. Hence we get the dimension formula (i.e. the rightmost equality) of Theorem 3.4 provided we can prove

0hom_{R}(B, G) − _{0}hom_{R}(B, F) = K_{3} + ... + K_{c}.

This follows from Proposition 2.4 and the second exact sequence of Lemma 2.3.

Finally the condition _{0}Hom_{A}(M, M) ≃ k allows us to lift automorphisms, i.e. Def_{M/R} is pro-representable, cf. [18, Thm. 19.2], whence dim H(M/R) = dim _{0}Ext_{R}^{1}(M, M) by the smoothness of Def_{M/R}. □

Let D := k[ϵ]/(ϵ^{2}) be the dual numbers and denote the dimension in Theorem 3.4 by

(3.4)  λ := dim _{0}Ext_{R}^{1}(M, M) = λ_{c} + K_{3} + K_{4} + ... + K_{c} .

Recalling that W_{s}(\underline{b}; \underline{a}) is a certain quotient of an open irreducible set in the affine scheme 𝕍 = Hom_{R}(G^{∗}, F^{∗}) parameterizing determinantal k-algebras (Proposition 2.1), we get

Theorem 3.5. Let A = R/I, I = I_{t}(A), be a standard determinantal k-algebra and let M = coker φ^{∗}.

(i) If _{0}Hom_{A}(M, M) ≃ k and _{0}Ext_{A}^{1}(M, M) = 0 then

Def_{A∈W_{s}(\underline{b};\underline{a})} ≃ Def_{M/R} .

Hence Def_{A∈W_{s}(\underline{b};\underline{a})} is a formally smooth pro-representable functor and the pro-representing object has dimension

dim W_{s}(\underline{b}; \underline{a}) = λ_{c} + K_{3} + K_{4} + ... + K_{c} ,

cf. Remark 2.6. Moreover the tangent space of Def_{A∈W_{s}(\underline{b};\underline{a})} is the subvector space of 0Hom_{R}(I, A) that corresponds to graded deformation R_{D} → A_{D} of R → A to D of the form A_{D} = R_{D}/I_{t}(A_{D}) for some matrix A_{D} which lifts A to D.

(ii) If in addition _{0}Ext_{A}^{2}(M, M) = 0, then Def_{M/R} ≃ Def_{A∈W_{s}(\underline{b};\underline{a})} ≃ Def_{A/R} and Def_{A/R} is formally smooth. Moreover every deformation of A comes from deforming A (cf. Definition 2.9).

Proof. We already know that Def_{M/R}(T) → Def_{A∈W_{s}(\underline{b};\underline{a})}(T) is well-defined (Fitting’s lemma) and obviously surjective for any (T, 𝔪_{T}) in \underline{ℓ}. To see the injectivity, let M_{i} for i = 1, 2 represent two elements of Def_{M/R}(T) mapping to the same element A_{T} ∈ Def_{A∈W_{s}(\underline{b};\underline{a})}(T). Since M_{i}, i = 1, 2 are cokernels of two morphisms given by matrices (A_{T})_{1} and (A_{T})_{2} lifting A to T, we have A_{T} = R_{T} /I_{t}((A_{T})_{1}) = R_{T} /I_{t}((A_{T})_{2}), cf. Definition 3.2. To show the injectivity, we may by induction on r ≥ 1 suppose

12 JAN O. KLEPPE

𝔪_{T}^{r+1} = 0 and Def_{M/R}(T/𝔪_{T}^{r}) → Def_{A∈W (\underline{b};\underline{a})}(T/𝔪_{T}^{r}) injective. Noticing that the condition_{0}Hom_{A}(M, M) ≃ k allows us to lift automorphisms of M_{i}⊗_{T} T/𝔪_{T}^{r} to automorphisms of M_{i}, it follows from _{0}Ext_{A}^{1}(M, M) = 0 that the graded deformations M_{1} and M_{2} represent the same element of Def_{M/R}(T), and the injectivity is proved.

Moreover using Def_{A∈W_{s}(\underline{b};\underline{a})} ≃ Def_{M/R} and Theorem 3.4 we get that Def_{A∈W_{s}(\underline{b};\underline{a})} is prorepresentable and formally smooth, whence its pro-representing object H satisfies H ≃ H(M/R) and since H and its “universal family” A_{H} are algebraizable by the proof of [31, Thm. 5.2] (or use the 2^{nd} paragraph of the proof of Theorem 4.8 of this paper to see that we get a morphism O_{GradAlg(H),(A)} ↠ S, H ≃ Ŝ with “universal family” A_{S} := R_{S}/I_{t}(A_{S}), which extends to open subsets V ⊂ U, of W_{s}(\underline{b}; \underline{a}) and GradAlg(H) respectively with (A) ∈ V and O_{V,(A)} = S), we get dim H = dim W_{s}(\underline{b}; \underline{a}) by Theorem 3.4. The description of its tangent space follows from (3.1) and Definition 3.2 since _{0}Hom_{R}(I, A) is the tangent space of Def_{A/R}.

If_{0}Ext_{A}^{2}(M, M) = 0 then_{0}Ext_{R}^{1}(M, M) ≃ (E_{2}^{0,1})_{0} ≃ _{0}Hom_{R}(I, Hom_{A}(M, M)) by (3.1). Hence the first part of the proof and A ↪ Hom_{A}(M, M) imply that all the injections in

(3.5)  Def_{M/R}(D) ↪ Def_{A/R}(D) ≃ _{0}Hom_{R}(I, A) ↪ _{0}Hom_{R}(I, Hom_{A}(M, M))

are isomorphisms of finite dimensional vector spaces. It follows that Def_{M/R} → Def_{A/R} is an isomorphism since it is bijective on tangent spaces and Def_{M/R} is formally smooth. Then we conclude the proof by arguing as in the proof of [31, Thm. 5.2]. □

Remark 3.6. The theorems of this section admit substantial generalizations with respect to R being a polynomial ring. Indeed we may let R be any graded quotient of a polynomial ring. The main reason for this is that the spectral sequence of this section, cf. (3.1), Fitting’s lemma and the exactness of the Buchsbaum-Rim complex are valid with almost no assumption on R (but we need to replace the binomials defining λ_{c} and K_{i} with their Hilbert functions; the final formula of Theorems 3.4 is, however, valid as stated).

4. the locus of determinantal k-algebras

In this section we generalize [31, Theorems 5.5 and 5.8] concerning dimension and smoothness of GradAlg(H) along W_{s}(\underline{b}; \underline{a}), to cover the artinian case. Indeed using that Theorem 3.5 extends [31, Thm. 5.2] by only assuming _{0}Hom_{A}(M, M) ≃ k and A standard determinantal instead of good determinantal, the generalizations in Theorems 4.1 and 4.6 from good to standard determinantal are rather immediate. In this section we also prove a new result (Theorem 4.8) to cover cases where dim W_{s}(\underline{b}; \underline{a}) ≠ λ.

In the first theorem we let, as in [31],

ext^{2}(M, M) := dim ker( _{0}Ext_{A}^{2}(M, M) → _{0}Ext_{R}^{2}(M, M) ) , cf. (3.1) .

Clearly ext^{2}(M, M) ≤ _{0}ext_{A}^{2}(M, M), and note that we below may replace GradAlg(H) with Hilb^{p}(ℙ^{n}) if n−c ≥ 1, cf. the text accompanying (2.1) for explanations and notations.

Theorem 4.1. Let A = R/I, I = I_{t}(A) be a standard determinantal k-algebra, i.e. (A) ∈ W_{s}(\underline{b}; \underline{a}), let M = coker φ^{∗} and suppose _{0}Hom_{A}(M, M) ≃ k and _{0}Ext_{A}^{1}(M, M) = 0. Then

dim W_{s}(\underline{b}; \underline{a}) = λ := λ_{c} + K_{3} + K_{4} + ... + K_{c} ,

cf. Remark 2.6. Moreover, for the codimension of W_{s}(\underline{b}; \underline{a}) in GradAlg(H) in a neighborhood of (A) we have

dim_{(A)} GradAlg(H) − dim W_{s}(\underline{b}; \underline{a}) ≤ dim _{0}Hom(I, A) − λ ≤ ext^{2}(M, M) ,

where the first inequality is an equality if and only if GradAlg(H) is smooth at (A). In particular these conclusions hold if dim A ≥ 3 + dim R/I_{t−1}(A), \underline{or} if dim A ≥ 2 and dim R/I_{t−1}(B) = 0 where B is obtained from A by deleting some column of A (e.g. if dim A ≥ 2, a_{i−2} ≥ b_{i} for 2 ≤ i ≤ t and A is general).

Proof. The 1^{st} conclusion follows from Theorem 3.5(i). For the statements on the codimension one knows that dim_{(A)} GradAlg(H) ≤ dim _{0}Hom_{R}(I, A) with equality if and only if A is unobstructed ([25, Thm. 1.5]). Using (3.1), (3.4) and the injection A ↪ _{0}Hom_{A}(M, M), it follows that _{0}Hom_{R}(I, A) − λ ≤ dim (E_{2}^{0,1})_{0} − λ = ext^{2}(M, M), and we get the next conclusion of Theorem 4.1. Now since depth_{I_{t−1}(A)A} A = dim A − dim R/I_{t−1}(A) and A good determinantal is equivalent to depth_{I_{t−1}(A)A} A ≥ 1, the final sentence follows directly from [31, Theorem 5.5], and the parenthesis from Remark 2.5. □

There are many artinian determinantal rings satisfying the conditions_{0}Hom_{A}(M, M) ≃ k and _{0}Ext_{A}^{1}(M, M) = 0 of Theorem 4.1. After having computed many examples using Macaulay 2 the general picture when a_{0} > b_{t} seems to be that the only artinian determinantal rings A = R/I_{t}(A) that do not satisfy these conditions are those with matrix A that is linear except in one column v ∈ R_{m}^{⊕t} where the degree is m ≥ 2, so A is of the form [B, v] where B is linear. The following example avoids this case, but B is otherwise quite close to being linear (the case where B is linear will be considered in Example 4.11). Since the codimension-2 case is straightforward in this context, the first non-trivial case is c = 3. We include a codimension-4 example to see that it can be treated similarly.

Example 4.2. (determinantal artinian quotients of R, using Theorem 4.1)

(i) Let R = k[x_{0}, x_{1}, x_{2}] and let A = [B, v] be a general 2 × 4 matrix with linear (resp. quadratic) entries in the first and second (resp. third) column and where the entries of v are polynomials of the same degree m, i.e. the degree matrix of A is ( \begin{matrix}1 & 1 & 2 & m \\ 1 & 1 & 2 & m \end{matrix}). The vanishing of all 2 × 2 minors defines an artinian ring with h-vector (1, 3, 5, 5, .., 5, 5, 3), where the number of 5s is m − 1. For m ≥ 2 one verifies that the first conditions of Theorem 4.1, i.e. that all conditions of Theorem 3.5(i) hold, and it follows that \overline{W_{s}(\underline{b}; \underline{a})} is an irreducible subset of GradAlg(H) of dimension λ_{3} +K_{3} which is 14 for m ≥ 3 (or one may use (2.17) to find the dimension). For m ≥ 5 one shows that _{0}hom_{A}(I/I^{2}, A) = 16, and we get that the codimension of \overline{W_{s}(\underline{b}; \underline{a})} in GradAlg(H) is at most 2 by Theorem 4.1. The dimensions of the _{0}Ext^{i}-groups above are computed by using Macaulay 2 (over the finite fields ℤ_{101}, ℤ_{701} and ℤ_{3001}), and strictly speaking only for m ≤ 10, from which we clearly see the general pattern also for m > 10. Note that the dimensions we found of the 0Ext^{i}-groups are independent of the characteristic of the fields and since field extensions are flat, they remain unchanged over any field containing one of these three fields. So the conclusions above hold at least when k is the algebraic closure of ℤ_{p}, p ∈ {101, 701, 3001}.

(ii) Let R = k[x_{0}, x_{1}, x_{2}, x_{3}] and let A = [B, v] be a general 2 × 5 matrix with degree matrix ( \begin{matrix}1 & 1 & 1 & 2 & m \\ 1 & 1 & 1 & 2 & m \end{matrix}). The vanishing of all 2 × 2 minors defines an artinian ring with hvector (1, 4, 7, 7, .., 7, 4), where the number of 7s is m − 1. For m ≥ 2 one verifies that

14 JAN O. KLEPPE

all conditions of Theorem 3.5(i) hold, and we get that \overline{W_{s}(\underline{b}; \underline{a})} is an irreducible subset of GradAlg(H) of dimension λ_{4} + K_{3} + K_{4}, which is equal to 25 if m ≥ 3 (or one may use (2.17) to find the dimension). For m ≥ 5, _{0}hom_{A}(I/I^{2}, A) = 33, whence the codimension of \overline{W_{s}(\underline{b}; \underline{a})} in GradAlg(H) is at most 8 by Theorem 4.1. The computations of the _{0}Ext^{i}- groups are verified by Macaulay 2 for m ≤ 10 over the fields ℤ_{101}, ℤ_{701} and ℤ_{3001}, but their dimensions are the same for any field containing one of these fields and for m > 10.

Remark 4.3. In all examples of this section we have verified the dimensions of those 0Ext_{A}^{i}(M, M) and _{0}Ext_{A}^{i}(I/I^{2}, A), i ≥ 0 we need to apply Theorems 4.1, 4.6 and 4.8 by using Macaulay 2 over each of the ground fields ℤ_{p} corresponding to p = 101, 701 and p = 3001. But we have also computed these dimensions for many other values of p > 10, and it seems that they are independent of the characteristic of the field.

Sometimes Theorem 4.1 allows us to find the exact codimension of \overline{W_{s}(\underline{b}; \underline{a})} in GradAlg(H).

Example 4.4. (determinantal artinian quotients of R = k[x_{0}, x_{1}, x_{2}], using Theorem 4.1)

(i) Let A be a general 2 × 4 matrix with quadratic (resp. linear) entries in the first (resp. second) row, i.e. with degree matrix ( \begin{matrix}2 & 2 & 2 & 2 \\ 1 & 1 & 1 & 1 \end{matrix} ). The vanishing of all 2 × 2 minors defines an artinian ring with h-vector (1, 3, 6, 4, 1). Macaulay 2 computations (cf. Remark 4.3) show that all conditions of Theorem 3.5(i) hold, and it follows that \overline{W_{s}(\underline{b}; \underline{a})} is an irreducible subset of GradAlg(H) of dimension λ_{3} + K_{3} = 16. Moreover we get that A is unobstructed by (2.2). Indeed both _{0}Ext_{A}^{1}(I/I^{2}, A) = 0 (verified by Macaulay 2) and 0Hom_{A}(H_{2}(R, A, A), A) = 0; the latter because _{v}H_{2}(R, A, A) ↪ S_{2}(I)_{v} = 0 for v < 6 and the socle degree of A is 4. Since _{0}hom_{A}(I/I^{2}, A) = 20 by Macaulay 2, we get by Theorem 4.1 that codim_{GradAlg(H)} \overline{W (\underline{b}; \underline{a})} = 4.

(ii) Again R = k[x_{0}, x_{1}, x_{2}], but now A is a general 3 × 5 matrix with quadratic (resp. linear) entries in the first (resp. second and third) row. The vanishing of all 3 × 3 minors defines an artinian ring with h-vector (1, 3, 6, 10, 5, 1). Using Macaulay 2 one verifies that all conditions of Theorem 3.5(i) hold, and we get that \overline{W_{s}(\underline{b}; \underline{a})} is an irreducible subset of GradAlg(H) of dimension λ_{3} + K_{3} = 25. Since _{0}Ext_{A}^{1}(I/I^{2}, A) = 0 by Macaulay 2 and _{v}H_{2}(R, A, A) ↪ S_{2}(I)_{v} = 0 for v < 8, we get _{0}Hom_{A}(H_{2}(R, A, A), A) = 0, whence A is unobstructed by (2.2). Then Theorem 4.1 and _{0}hom_{A}(I/I^{2}, A) = 40 (by Macaulay 2) imply that codim_{GradAlg(H)} \overline{W (\underline{b}; \underline{a})} = 15.

Remark 4.5. Another possible way of finding the dimension of \overline{W_{s}(\underline{b}; \underline{a})} and its codimension in GradAlg(H), as well as determining whether GradAlg(H_{A}) is smooth at (A) ∈ \overline{W_{s}(\underline{b}; \underline{a})} ⊂ GradAlg(H_{A}), is to consider some closely related and well-understood family of GradAlg(H_{B}) containing a point (B) for which there is a surjection B → A of k-algebras whose Hilbert functions satisfy H_{B}(v) = H_{A}(v) for every v ≤ s and s sufficiently large. Such a comparison is done in [29] when dim A ≥ 1, using B = R/I_{t}(B), A = R/I_{t}(A) and A = [B, v], v a column, cf. Example 4.2. In [28] we explore this approach more generally, e.g. see [28, Thm. 29] for a result where A is artinian in B → A.

In the case B = R/I_{t}(B) where A = [B, v] and v a column, we generalize in Sec. 6 of this paper several results from [29], see in particular Theorem 6.6 where dim A = 1. It is possible to extend this theorem to A artinian by increasing the set of assumptions in Theorem 6.6, but since this theorem is already quite complicated as stated, we think Theorem 4.1 is much simpler. It is also an option to replace B = R/I_{t}(B) by an artinian Gorenstein quotient A_{g} of B such that B → A factors via A_{g} → A and, for c = 3, use either [26, Thm. 2.3] or [8, Sec. 5] to compute e.g. dim_{(A_{g})} GradAlg(H_{A_{g}}). Applying, however, [27, Thm. 1] to B → A_{g} which states that GradAlg(H_{A_{g}}) and GradAlg(H_{B}) are much related, especially when c = 3 by [27, Example 28], this latter approach is quite similar (concerning assumptions to be fulfilled) to the approach using R/I_{t}(B) → A.

We have the following generalization of [31, Theorem 5.8] in which the good determinantal assumption (depth_{I_{t−1}(A)A} A ≥ 1) is weakened, allowing A to be artinian. Note that we below may replace GradAlg(H) with Hilb^{p}(ℙ^{n}) if dim A ≥ 2, cf. (2.1).

Theorem 4.6. Let A = R/I, I = I_{t}(A) be a standard determinantal k-algebra, i.e. (A) ∈ W_{s}(\underline{b}; \underline{a}), let M = coker φ^{∗} and suppose _{0}Hom_{A}(M, M) ≃ k and _{0}Ext_{A}^{i}(M, M) = 0 for i = 1 and 2. Then GradAlg(H) is smooth at (A) and

dim_{(A)} GradAlg(H) = λ_{c} + K_{3} + K_{4} + ... + K_{c} ,

(cf. Remark 2.6). Moreover \overline{W_{s}(\underline{b}; \underline{a})} ⊂ GradAlg(H) is an irreducible component, and every deformation of A comes from deforming A. In particular these conclusions hold if dim A ≥ 4 + dim R/I_{t−1}(A), \underline{or} if dim A ≥ 3 and dim R/I_{t−1}(B) = 0 where B is obtained from A by deleting some column of A (e.g. if dim A ≥ 3, a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t and A is general).

Proof. The first two sentences of conclusions follow from Theorem 3.5 and Lemma 2.10. Moreover since depth_{I_{t−1}(A)A} A = dim A − dim R/I_{t−1}(A), the final sentence follows directly from [31, Theorem 5.8], and the parenthesis from Remark 2.5. □

The condition _{0}Ext_{A}^{2}(M, M) = 0 of Theorem 4.6 seems harder to satisfy for artinian rings, but it holds quite often if A is not too close to the linear case. Here is an example:

Example 4.7. (determinantal artinian quotients of R = k[x_{0}, x_{1}, x_{2}], using Theorem 4.6)

(i) Let A = [B, v] be a general 2 × 4 matrix with linear (resp. cubic) entries in the first and second (resp. third) column and where the entries of v are polynomials of the same degree m, m ≥ 3. The degree matrix of A is ( \begin{matrix}1 & 1 & 3 & m \\ 1 & 1 & 3 & m \end{matrix}). The vanishing of all 2 × 2 minors defines an artinian ring with h-vector (1, 3, 5, 7, 7, .., 7, 5, 3), where the number of 7s is m − 2. For m = 3 and m ≥ 5 one verifies that all conditions of Theorem 3.5(ii), i.e. the first conditions of Theorem 4.6 hold, and it follows that \overline{W_{s}(\underline{b}; \underline{a})} is a generically smooth irreducible component of GradAlg(H) of dimension λ_{3} + K_{3} = 18 (resp. 17) for m ≥ 5 (resp. m = 3). The computations for the _{0}Ext^{i}-groups above are verified by Macaulay 2 for m ≤ 10 (cf. Remark 4.3), but their dimensions hold also for m > 10. One also verifies that _{0}ext_{A}^{1}(I/I^{2}, A) = 2 for 6 ≤ m ≤ 10, so both the generic smoothness (cf. (2.2)) along \overline{W_{s}(\underline{b}; \underline{a})} and the conclusion that \overline{W_{s}(\underline{b}; \underline{a})} is an irreducible component of GradAlg(H) would be hard to see without using Theorem 4.6.

(ii) Let A be a general 2 × 4 matrix with degree matrix ( \begin{matrix}3 & 3 & 3 & 3 \\ 1 & 1 & 1 & 1 \end{matrix} ) . The vanishing of all 2 × 2 minors defines an artinian ring with h-vector (1, 3, 6, 10, 9, 7, 3, 1). Using Macaulay 2 one verifies that all conditions of Theorem 3.5(ii) hold, (i.e. also _{0}Ext_{A}^{2}(M, M) = 0), and we get that \overline{W_{s}(\underline{b}; \underline{a})} is a generically smooth irreducible component of GradAlg(H) of dimension λ_{3} + K_{3} = 29. In this example _{0}Ext_{A}^{1}(I/I^{2}, A) ≠ 0 and _{0}hom_{A}(I/I^{2}, A) = 29, so it is not straightforward to see that A is unobstructed, but it is, due to Theorem 4.6 which also contains additional information.

As indicated it seems that the only artinian determinantal rings A = R/I_{t}(A) in the case a_{0} > b_{t} that do not satisfy the conditions of Theorem 3.5(i) are those with a matrix A that is linear except in one column v ∈ R_{m}^{⊕t} where the degree is m ≥ 2. Moreover if m ≥ 3 we may even have

0Hom_{A}(M, M) ≃ k , _{0}Ext_{A}^{1}(M, M) ≠ 0 and _{0}Ext_{A}^{2}(M, M) = 0 ,

and the deformation functors of these rings are fully determined by our next Theorem 4.8. For artinian linear determinantal rings (i.e. m = 1 above) all conditions of Theorem 3.5(ii) seem to hold and even more, they are rigid in several senses: _{0}Hom_{R}(I, A) = 0, as well as 0Ext_{R}^{1}(M, M) = 0, whence the linear case may be not so interesting. Recalling (3.4), i.e. λ := dim _{0}Ext_{R}^{1}(M, M) = λ_{c} + K_{3} + K_{4} + ... + K_{c} and Remark 2.6, we get

Theorem 4.8. Let A = R/I, I = I_{t}(A) be a standard determinantal k-algebra, let M = coker φ^{∗} and suppose _{0}Hom_{A}(M, M) ≃ k. If the map _{0}Ext_{A}^{2}(M, M) → _{0}Ext_{R}^{2}(M, M) of (3.1) is injective, or weaker, if the tangent map _{0}Ext_{R}^{1}(M, M) → _{0}Hom_{R}(I, A) of the natural morphism of functors e_{M} : Def_{M/R} → Def_{A/R} is surjective, then e_{M} is smooth. Moreover Def_{M/A} is smooth and its pro-representing object H(M/A) satisfies

dim H(M/A) = dim _{0}Ext_{A}^{1}(M, M) = λ − dim W_{s}(\underline{b}; \underline{a}) .

Furthermore Def_{A∈W_{s}(\underline{b};\underline{a})} = Def_{A/R}, Def_{A/R} is smooth, \overline{W_{s}(\underline{b}; \underline{a})} ⊂ GradAlg(H) is an irreducible component of dimension λ−dim _{0}Ext_{A}^{1}(M, M) and every deformation of A comes from deforming A. In particular these conclusions hold if dim A ≥ 3+min{dim R/I_{t−1}(A)+ 1, dim R/I_{t−1}(B)} where B is obtained from A by deleting some column of A (e.g. if dim A ≥ 3, a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t and A is general).

Remark 4.9. We may consider Def_{M/A} as a fiber functor of e_{M} : Def_{M/R} → Def_{A/R}, using trivial deformations of A. For a description of the obstruction maps of Def_{M/A} for arbitrary M, see [40, 23]. Note that the assumption _{0}Hom_{A}(M, M) ≃ k implies the pro-representability of both Def_{M/A} and Def_{M/R}, cf. end of proof of Theorem 3.4.

Proof. Firstly we observe that the composition of the natural maps:

0Ext_{R}^{1}(M, M) ⟶^{e_{M} (D)} _{0}Hom_{R}(I, A) ↪ _{0}Hom_{R}(I, Hom_{A}(M, M)) ≃ (E_{2}^{0,1})_{0}

described in Lemma 3.3 is just the degree zero part of the map Ext_{R}^{1}(M, M) → E_{2}^{0,1} appearing in (3.1). This composition is surjective if and only if the _{0}Ext_{R}^{2}-map of Theorem 4.8 is injective by (3.1). Since A → Hom_{A}(M, M) is injective, it follows that the injectivity of this _{0}Ext_{R}^{2}-map is equivalent to the surjectivity of e_{M}(D) : _{0}Ext_{R}^{1}(M, M) → 0Hom_{R}(I, A) and _{0}Hom_{R}(I, A) ≃ _{0}Hom_{R}(I, Hom_{A}(M, M)) being bijective.

Let T → S be a small artinian surjection with kernel 𝔞. To prove the (formal) smoothness of e_{M} : Def_{M/R} → Def_{A/R}, we must by definition show that the induced map

Def_{M/R}(T) → Def_{M/R}(S) ×_{Def_{A/R}(S)} Def_{A/R}(T)

is surjective. Let M_{S} be an arbitrary fixed deformation of M to S inducing R_{S}/I_{S} ∈ Def_{A/R}(S), let R_{T} /I_{T} be a deformation of R_{S}/I_{S} to T and let e_{M_{S}} be the functor that takes a deformation M_{T} = coker(φ^{∗}) of M_{S} onto the deformation of R_{S}/I_{S} defined by the maximal minors of a matrix representing φ. Note that since M is unobstructed, there exists a deformation M′_{T} of M_{S} to T inducing a deformation R_{T} /I′_{T} of R_{S}/I_{S}. The difference “R_{T} /I_{T} − R_{T} /I′_{T} ” sits in _{0}Hom_{R}(I, A) ⊗_{k} 𝔞 and since e_{M_{S}}(T) ⊗_{T} id_{𝔞} is up to isomorphism equal to the surjective map e_{M}(D) ⊗_{k} 𝔞 as one may see by the arguments described in the first paragraph of the proof of Lemma 3.3, e_{M_{S}}(T) ⊗_{T} id_{𝔞} is surjective. Hence there exists an element in _{0}Ext_{R}^{1}(M, M) ⊗_{k} 𝔞 that we can “add” to M′_{T} to get a deformation M_{T} of M_{S} that induces R_{T} /I_{T}, i.e. e_{M} is smooth. It follows that Def_{A/R} is smooth. We also get that Def_{A∈W_{s}(\underline{b};\underline{a})}(T) ↪ Def_{A/R}(T) is surjective for any T ∈ ob(\underline{ℓ}), and injective by Definition 3.2, whence an equality, and since Def_{M/A} is a fiber functor of Def_{M/R} → Def_{A/R}, it follows that Def_{M/A} is smooth. Hence dim H(M/A) = dim _{0}Ext_{A}^{1}(M, M), and we get the displayed dimension formulas of Theorem 4.8 from (3.1) and the surjectivity of 0Ext_{R}^{1}(M, M) → _{0}Hom_{R}(I, A) provided dim W_{s}(\underline{b}; \underline{a}) = _{0}hom_{R}(I, A).

If we compare Def_{A∈W_{s}(\underline{b};\underline{a})} = Def_{A/R} with Definition 3.2 we get that every deformation A_{T} of A to an artinian T comes from deforming A. To see that we may replace “an artinian T ” by “T a local ring” in this statement (cf. Definition 2.9) we pick d := _{0}hom_{R}(I, A) elements of _{0}Ext_{R}^{1}(M, M) that map to linearly independent elements of _{0}Hom_{R}(I, A) via the tangent map _{0}Ext_{R}^{1}(M, M) ↠ _{0}Hom_{R}(I, A) of e_{M}, and we let η_{1}, ..., η_{d} ∈ _{0}Hom(G^{∗}, F^{∗}) with presentation matrices A_{1}, ..., A_{d} correspond to the d elements we picked in _{0}Ext_{R}^{1}(M, M), cf. the beginning of the proof of Lemma 3.3 for a similar set-up. Let A_{S} := A + s_{1}A_{1} + ... + s_{d}A_{d} and S := k[s_{1}, ..., s_{d}]_{(s_{1},...,s_{d})} where k[s_{1}, ..., s_{d}] is a polynomial ring, and let O := O_{GradAlg(H),(A)} be the local ring of GradAlg(H) at the k-point (A) ∈ GradAlg(H), k = \overline{k}. Then the algebraic family A_{S} := R_{S}/I_{t}(A_{S}) is S-flat by Lemma 2.8 and e.g. using the explicit description of the pro-representing object H(A/R) of Def_{A/R} appearing in the proof of Thm. 4.2.4 of [39] to see that any deformation of A_{S} ⊗_{S} S/(s_{1}, ..., s_{d})^{2} to H(A/R) may serve as a versal lifting, we get that H(A/R) equals S with“universal object” A_{S} (up to completion, but notice that the versal lifting is defined by polynomials, not power series, in s_{1}, ..., s_{d}). Hence, by the universal property of the representable functor that corresponds to GradAlg(H), there is a morphism O → S whose completions are isomorphic because both pro-represents Def_{A/R} on ℓ ([48, (2,1) and (2.8)]). Thus O → S is injective, and since O/𝔪_{O}^{n} ≃ S/𝔪_{S}^{n} for every n, it follows that O → S is an isomorphism, having A_{O} := A_{S} ⊗_{S} O as the pullback (considered as scheme) of the universal object of GradAlg(H) to Spec(O). This shows that “every deformation of A comes from deforming A“ because the universal property implies that all deformations are given by pullback to Spec(T) (i.e. by taking tensor product of O → A_{O} via O → T). Then Lemma 2.10 implies that \overline{W_{s}(\underline{b}; \underline{a})} ⊂ GradAlg(H) is a generically smooth irreducible component of dimension d, and in particular we have dim W_{s}(\underline{b}; \underline{a}) = _{0}hom_{R}(I, A).

The argument for the statement in the final sentence is as in the proof of Theorem 4.6 provided dim A ≥ 4 + dim R/I_{t−1}(A). If, however, dim A ≥ 3 + dim R/I_{t−1}(B) then depth_{I_{t−1}(B)} R/I_{t}(B) ≥ 4 and we get _{0}Ext_{A}^{2}(M, M) = 0 by [31, Thm. 4.5] and we are done. □

As a corollary to the 3^{rd} paragraph of the proof, we get that if every graded deformation of A to any local artinian k-algebra comes from deforming A, then every deformation of A comes from deforming A (see Definition 2.9). But the proof implies more:

18 JAN O. KLEPPE

Corollary 4.10. Let A = R/I_{t}(A) be a standard determinantal k-algebra. If every graded deformation A_{D} of A to the dual numbers D := k[ϵ]/(ϵ^{2}) comes from deforming A, or equivalently, if the tangent map of the natural morphism e_{M} : Def_{M/R} → Def_{A/R},

e_{M}(D) : Def_{M/R}(D) = _{0}Ext_{R}^{1}(M, M) → Def_{A/R}(D) = _{0}Hom_{R}(I, A) ,

is surjective, then every deformation of A comes from deforming A.

Proof. Since every graded deformation of M comes from deforming A by Lemma 2.8, it is clear that “every graded deformation of A to D comes from deforming A” is equivalent to the surjectivity of the tangent map e_{M}(D). Then we get what we want from the proof appearing in the 3^{rd} paragraph of Theorem 4.8 since that part of the proof only requires the surjectivity of e_{M}(D) and not the condition _{0}Hom_{A}(M, M) ≃ k. □

Let us consider two particular cases of artinian rings:

Example 4.11. (determinantal artinian quotients of R, using Theorem 4.8)

(i) Let R = k[x_{0}, x_{1}, x_{2}] and let A = [B, v] be a general 2 × 4 matrix where B is linear and the entries of v are polynomials of the same degree m ≥ 2. The vanishing of all 2 × 2 minors defines an artinian ring with h-vector (1, 3, 3, .., 3) where the number of 3s is m. Using Macaulay 2 for 3 ≤ m ≤ 10 one verifies (cf. Remark 4.3) that all conditions of Theorem 4.8 hold and that _{0}ext_{A}^{1}(M, M) = 2, and it follows that \overline{W_{s}(\underline{b}; \underline{a})} is a generically smooth irreducible component of GradAlg(H) of dimension λ_{3} + K_{3} − _{0}ext_{A}^{1}(M, M) = 6 (true also for m > 10). If m = 2, Theorem 4.8 does not apply because _{0}Ext_{A}^{2}(M, M) ≠ 0. In fact one may verify (Macaulay 2) that all inclusions of (3.5) are strict, i.e. nonisomorphisms.

(ii) Let R = k[x_{0}, x_{1}, x_{2}, x_{3}] and let A = [B, v] be a general 2 × 5 matrix where B is linear and the entries of v are polynomials of the same degree m ≥ 2. The vanishing of all 2×2 minors defines an artinian ring with h-vector (1, 4, 4.., , 4), where the number of 4s is m. Using Macaulay 2 for 3 ≤ m ≤ 10 one verifies that all conditions of Theorem 4.8 hold and that _{0}ext_{A}^{1}(M, M) = 4. By Proposition 2.4 and (2.16), K_{3} = 0, K_{4} = 4 and λ_{4} = 12, and it follows that \overline{W_{s}(\underline{b}; \underline{a})} is a generically smooth irreducible component of GradAlg(H) of dimension λ_{4} + K_{3} + K_{4} − _{0}ext_{A}^{1}(M, M) = 12 (true also for m > 10).

Remark 4.12. Comparing Theorem 4.8 with Theorem 4.6 we see that the main condition for \overline{W_{s}(\underline{b}; \underline{a})} to be an irreducible component of GradAlg(H_{A}) is that _{0}Ext_{A}^{2}(M, M) (or ext^{2}(M, M), cf. Theorem 4.1) vanishes; the condition _{0}Ext_{A}^{1}(M, M) = 0 of Theorem 4.6 is in this regard less important. An interesting observation to make to all examples of this section is that ghost terms in the minimal resolution of A appear precisely when 0Ext_{A}^{2}(M, M) ≠ 0. Indeed for all A in Examples 4.2 and 4.4 we have _{0}Ext_{A}^{2}(M, M) ≠ 0 (Macaulay 2) as well as ghost terms in the minimal resolution of A while we in Examples 4.7 and 4.11 where \overline{W_{s}(\underline{b}; \underline{a})} is a component due to _{0}Ext_{A}^{2}(M, M) = 0, have no ghost terms in the minimal resolution of A. So one may wonder if there is a connection between 0Ext_{A}^{2}(M, M) ≠ 0 and the existence of ghost terms for small c in the artinian case (and maybe also when dim A ≤ 1, cf. Example 4.13, but not when dim A ≥ 2, cf. Example 5.5). Indeed we devote the next section to a study of ghost terms in the Eagon-Northcott resolution (2.5).

For determinantal zero-schemes we showed in [29] that the dimension formula (2.15) fails when (and only when?) A is a linear matrix consisting of two rows, cf. Remark 2.7(ii). In our next example we use Theorem 4.8 to treat this case.

Example 4.13. (determinantal quotients of R = k[x_{0}, x_{1}, ⋯, x_{c}] of dimension one)

Let A be a general 2 × (c + 1) matrix of linear entries. The vanishing all 2 × 2 minors defines a reduced scheme X = Proj(A) of c + 1 general points in ℙ^{c}. Since we may suppose A is good determinantal, Hom_{A}(M, M) ≃ A by [33, Lem. 3.2], and we have 0ext_{R}^{1}(M, M) = c(c+1)+c−2 by Theorem 3.4. In the range 3 ≤ c ≤ 10 we use Macaulay 2 to see _{0}ext_{A}^{1}(M, M) = c−2 and _{0}Ext_{A}^{2}(M, M) = 0. For c ≤ 10 we get from Theorem 4.8 that \overline{W (0, 0; 1, 1, ..., 1)} is a generically smooth irreducible component of GradAlg(H) of dimension c(c + 1) and that every deformation of A comes from deforming A.

It is possible to describe \overline{W (0, 0; 1, 1, ..., 1)} for all c ≥ 2 without using Macaulay 2. Indeed it is well known that there exists a generically smooth component of the usual Hilbert scheme Hilb^{p}(ℙ^{c}), of dimension c(c + 1), containing an open subset parametrizing c + 1 reduced points in ℙ^{c}. Since the Hilbert function of X is (dim A_{i}) = (1, c + 1, c + 1, ...), we get H^{1}(I_{X}(v)) ≃ _{v}H_{𝔪}^{1}(A) = 0 for v ≥ 1 and hence _{0}Hom_{R}(I_{X}, H_{𝔪}^{1}(A)) = 0. GradAlg(H) is therefore isomorphic to Hilb^{p}(ℙ^{c}) at (X) by (2.1), and it follows that \overline{W (0, 0; 1, 1, ..., 1)} is a generically smooth irreducible component of GradAlg(H) of dimension c(c + 1).

Theorem 4.8 admits a nice consequence concerning glicciness. Indeed since glicciness is not necessarily an open property, the following result may be useful.

Corollary 4.14. Let X = Proj(A), I_{X} = I_{t}(A), be a standard determinantal scheme and suppose that _{0}Hom_{R}(I_{X}, H_{𝔪}^{1}(A)) = 0 and that the map _{0}Ext_{A}^{2}(M, M) → _{0}Ext_{R}^{2}(M, M) of (3.1) is injective. Then the Hilbert scheme Hilb^{p}(ℙ^{n}) is smooth at (X) and (X) belongs to a unique irreducible component of Hilb^{p}(ℙ^{n}) whose general element X̃ ⊂ ℙ^{n} is glicci. In particular this conclusion holds if dim X ≥ 2+min{dim R/I_{t−1}(A)+1, dim R/I_{t−1}(B)} where B is obtained from A by deleting some column of A.

Proof. Indeed X̃ is standard determinantal by Theorem 4.8 and (2.1) and since standard determinantal schemes are glicci by [32, Thm. 3.6], we are done. □

Finally we generalize [31, Thm. 5.16] which is about codimension-2 determinantal quotients of an ACM scheme, to cover the artinian case. Below \overline{R} is a Cohen-Macaulay quotient of a polynomial ring (i.e. Proj(\overline{R}) ⊂ ℙ_{k}^{N} is ACM) and A is a standard determinantal quotient of \overline{R}, but notice that it is not necessarily a determinantal quotient of k[x_{0}, ⋯, x_{N}]. Moreover letting \underline{b}, \underline{a} be as in (2.4) and (A) ∈ W_{s}(\underline{b}; \underline{a}) observe that, for dim A ≥ 2, we may replace GradAlg^{H}(\overline{R}) by Hilb^{p}(Proj(\overline{R})) since (2.1) extends to hold in this generality by [25, Thm. 3.6, Rem. 3.7]. For the irreducibility and dimension of \overline{W_{s}(\underline{b}; \underline{a})} in the case \overline{R} = k[x_{0}, x_{1}], see also [19, Thms. 2.9, 2.12, 3.13].

Theorem 4.15. Let P = Proj(\overline{R}) ⊂ ℙ_{k}^{N} be an ACM scheme where k is any field and let X = Proj(A) ⊂ P, A = R/I_{t}(A), be any standard determinantal scheme of codimension 2 in P. Then GradAlg^{H}(\overline{R}) is smooth at (A) and dim_{(A)} GradAlg^{H}(\overline{R}) = λ(\overline{R})_{2} where

λ(\overline{R})_{2} := ∑_{i,j} dim \overline{R}_{(a_{j}−b_{i})} + ∑_{i,j} dim \overline{R}_{(b_{i}−a_{j})} − ∑_{i,j} dim \overline{R}_{(a_{i}−a_{j})} − ∑_{i,j} dim \overline{R}_{(b_{i}−b_{j})} + 1.

20 JAN O. KLEPPE

Moreover every deformation of A comes from deforming A. In particular if k = \overline{k}, then GradAlg^{H}(\overline{R}) is smooth along W_{s}(\underline{b}; \underline{a}) and the closure \overline{W_{s}(\underline{b}; \underline{a})} in GradAlg^{H}(\overline{R}) is an irreducible component of dimension λ(\overline{R})_{2}.

Proof. Since M is a twist of the canonical module of A when c = 2, it is well known that 0Ext_{A}^{i}(M, M) = 0 for i > 0 and Hom_{A}(M, M) ≃ A. We get Def_{M/\overline{R}} ≃ Def_{A/\overline{R}} and every statement of the theorem from Theorem 3.5, Lemma 2.10 and Remark 3.6, except the displayed formula. Using, however, the last conclusion of Theorem 3.4, we easily prove the displayed formula, cf. (2.6). Finally note that we have included the assumption k = \overline{k} in the statements of W_{s}(\underline{b}; \underline{a}) for the only reason that W_{s}(\underline{b}; \underline{a}) is by definition a certain locus (of k-points) of GradAlg^{H}(\overline{R}). □

So there are no singular points of GradAlg^{H}(\overline{R}) at (A) ∈ W_{s}(\underline{b}; \underline{a}) when c = 2 while singular points of GradAlg^{H}(\overline{R}) for c > 2 are quite common (see [34, Rem. 3.6] and [49]).

5. ghost terms

Let N be a graded R-module with a minimal R-free resolution 0 → P_{•} ↠ N. By a ghost term of P_{•} we mean a direct free summand that appears in consecutive terms of P_{•}. As in the minimal resolution conjecture ([41, 46], and see [43, 42, 45] and its references for other contributions), one expects that a general element R/I of GradAlg(H) contains “as few ghost terms as possible”, while ghost terms for special elements of GradAlg(H) are more common. Indeed the semicontinuity of graded Betti numbers imply that these numbers decrease under generization (deformation to a more general element), but there may still be ghost terms left in the minimal free resolution of the general element of GradAlg(H). In this section we shall see that some ghost terms of a determinantal ring R/I easily disappear under generizations while other ghost terms do not. Letting \underline{a}î = a_{0}, ...a_{i−1}, a_{i+1}..., a_{t+c−2} and \underline{b}î = b_{1}, ...b_{i−1}, b_{i+1}..., b_{t} we have the following result (allowing t = 2, in which case W_{s}(\underline{b}î; \underline{a}ĵ) consists of complete intersections of R).

Proposition 5.1. Suppose a_{j} = b_{i} for some j ∈ {0, t + c − 2} and i ∈ {1, t}. Then we have an inclusion W_{s}(\underline{b}î; \underline{a}ĵ) ⊂ W_{s}(\underline{b}; \underline{a}) of open irreducible subsets of GradAlg(H). Moreover if W_{s}(\underline{b}; \underline{a}) \ W_{s}(\underline{b}î; \underline{a}ĵ) ≠ ∅, then every R/I of W_{s}(\underline{b}; \underline{a}) \ W_{s}(\underline{b}î; \underline{a}ĵ) admits a generization (R/I_{g}) ∈ W_{s}(\underline{b}î; \underline{a}ĵ) removing exactly all ghost terms in the Eagon-Northcott resolution (2.5) coming from a_{j} = b_{i}; in particular a resolution of R/I_{g} of W_{s}(\underline{b}î; \underline{a}ĵ) is given by its Eagon-Northcott resolution.

Proof. As observed in the proof of Lemma 2.1, the vanishing of Ext_{R}^{i}(R/I_{t}(A), R) is an open property. This holds also for the elements R/I_{t}(A) of GradAlg(H), i.e. the subset of GradAlg(H) such that I_{t}(A) has maximal codimension in R (this subset is W_{s}(\underline{b}; \underline{a})) is open. Since it is irreducible by Lemma 2.1, we have proved the open irreducible property stated in Proposition 5.1, and it remains to see the inclusion W_{s}(\underline{b}î; \underline{a}ĵ) ⊂ W_{s}(\underline{b}; \underline{a}) and the existence of generizations.

We claim that the elements R/I_{t}(A) of W_{s}(\underline{b}; \underline{a}) whose matrix A contains a unit of the field k at the (i, j)-entry, belong to W_{s}(\underline{b}î; \underline{a}ĵ). Indeed A is a presentation matrix of M = coker φ^{∗} and by rearranging the direct summands of the source and target of the morphism φ^{∗}, we may assume (i, j) = (1, 0). By elementary row operations we transform A to a matrix with only zeros (and one 1) in the first column, i.e. there is an invertible t × t matrix C such that the cokernel of the map induced by C · A is isomorphic to M. By Fitting’s lemma, I_{t}(A) = I_{t}(C · A). Moreover the (t − 1) × (t + c − 2) matrix A′ obtained by deleting the first row and column of C · A satisfies I_{t−1}(A′) = I_{t}(C · A) and defines a determinantal ring of W_{s}(\underline{b}î; \underline{a}ĵ), and we have proved the inclusion and the whole claim (but taking any R/I_{t}(A′) of W_{s}(\underline{b}î; \underline{a}ĵ) and putting A = ( \begin{matrix}1 & 0 \\ 0 & A′ \end{matrix}) we get (R/I_{t}(A)) ∈ W_{s}(\underline{b}; \underline{a}) which is an easier argument for the inclusion).

Finally to see the existence of the generization of R/I of W_{s}(\underline{b}; \underline{a}) \ W_{s}(\underline{b}î; \underline{a}ĵ), I = I_{t}(A), we may assume that A has a 0 at the (i, j)-entry by the proven claim. Then we apply Lemma 2.8 to T = k[u]_{℘}, ℘ = (u), letting u = 0 correspond to A and A_{T} to a matrix obtained by replacing the 0 of the (i, j)-entry by u and all other (i′, j′)-entries by polynomials of R ⊗ k[u] of degree a_{j′} − b_{i′} with coefficients in k[u] such that the choice u = 1 makes the corresponding matrix general enough. More precisely, with notations as above, i.e. with (i, j) = (1, 0) etc., we may choose the coefficients of the polynomials of the 1. column such that the choice u = 1 make the entries equal to 0 for all (i, 0) with i > 1 and the entries of A′ general. In some open subset U ∋ {0, 1} of Spec(k[u]) the ideal I_{t}(A_{u}) of the matrices given by A_{u} for u ∈ U \ {0} has maximal codimension in R and we are done. □ Corollary 5.2. Let I = I_{t}(A), R/I ∈ W_{s}(\underline{b}; \underline{a}), and suppose that the ideal of submaximal minors I_{t−1}(A) = R. Then R/I is a complete intersection of R. Proof. Suppose I_{t−1}(A) = R. Since the submaximal minors are homogeneous and generate I_{t−1}(A), one of the submaximal minors must be a unit, whence at least one of the entries of A is a unit. Using the notations and the proven claim in the proof above, we have I_{t}(A) = I_{t}(C · A) = I_{t−1}(A′), and we get I_{t−1}(A) = I_{t−1}(C · A) = I_{t−2}(A′) for t ≥ 3 by the same arguments. Since I_{1}(A′) is a c.i. of R, we conclude by induction on t. □ Remark 5.3. The ghost terms treated in Proposition 5.1 are removable under generization, while all other ghost terms appearing in the Eagon-Northcott resolution of R/I_{t}(A) may sometimes probably be removed, but not always. Indeed the latter can not be removed under generization provided every graded deformation of R/I_{t}(A) comes from deforming A because, in this case, \overline{W_{s}(\underline{b}; \underline{a})} is an irreducible component of GradAlg(H), whence the minimal resolution of its general element is given by some Eagon-Northcott resolution of R/I_{t}(A′) with A′ minimal, cf. Example 5.5 below. Note that Theorem 4.6 and Theorem 4.8 give conditions under which every graded deformation of R/I_{t}(A) comes from deforming A. In particular if dim R/I_{t}(A) ≥ 3 and a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t, only ghost terms as in Proposition 5.1 are removable under generization! Example 5.4. (removing all ghost terms using Proposition 5.1)

(i) Let R = k[x_{0}, x_{1}], let B be a general 2×3 matrix with linear (resp. quadratic) entries in the first (resp. second) row and let A = ( \begin{matrix}u & v \\ w & B \end{matrix}) where u ∈ k, v a row of general linear forms and w a column whose transpose is (0, x_{0}). The vanishing of all 3×3 minors defines an artinian ring with h-vector (1, 2, 3, 1) for every u ∈ k. If I (resp. I_{g}) is the ideal given by all 3 × 3 minors of A with u = 0 (resp. u = 1), then (R/I) ∈ W_{s}(−2, −1, −1; −1, 0, 0, 0) and (R/I_{g}) ∈ W_{s}(−2, −1; 0, 0, 0) and using Macaulay 2 we find minimal resolutions

0 → R(−5) ⊕ R(−4)^{2} → R(−4) ⊕ R(−3)^{3} → I → 0 ,

22 JAN O. KLEPPE

0 ⟶ R(−5) ⊕ R(−4) ⟶ R(−3)^{3} ⟶ I_{g} ⟶ 0.

Here R/I_{g} is a generization of R/I in GradAlg(H).

(ii) Let R = k[x_{0}, x_{1}, x_{2}], let B be a general 2 × 4 matrix with linear (resp. cubic) entries in the first (resp. second) row and let A = ( \begin{matrix}u & v \\ w & B \end{matrix}) where u ∈ k, v a row of general linear forms and w the column whose transpose is (0, x_{0}^{2}). Using Macaulay 2 we get that the vanishing of all 3 × 3 minors defines an artinian ring with h-vector (1, 3, 6, 10, 9, 7, 3). If I (resp. I_{g}) is the ideal given by the 3 × 3 minors of A with u = 0 (resp. u = 1), then (R/I) ∈ W_{s}(−3, −1, −1; −1, 0, 0, 0, 0) and (R/I_{g}) ∈ W_{s}(−3, −1; 0, 0, 0, 0) and we have minimal resolutions

0 → R(−10) ⊕ R(−8) ⊕ R(−6) → R(−7)^{4} ⊕ R(−5)^{4} → R(−4)^{6} → I_{g} → 0 ,

0 → R(−10)⊕R(−8)^{2}⊕R(−6)^{3} → R(−8)⊕R(−7)^{4}⊕R(−6)^{2}⊕R(−5)^{8} → R(−5)^{4}⊕R(−4)^{6} ↠ I.

Example 5.5. (removing some ghost terms using Proposition 5.1)

Let R = k[x_{0}, x_{1}, x_{2}] and let A = ( \begin{matrix}u & v \\ w & B \end{matrix}) be as in Example 5.4 (ii) only replacing cubic by quadratic and w^{T} by (0, x_{0}). Using Macaulay 2 we get that the vanishing of all 3×3 minors defines an artinian ring with h-vector (1, 3, 6, 4, 1). If I (resp. I_{g}) is the ideal generated by the 3 × 3 minors of A for u = 0 (resp. u = 1), then (R/I) ∈ W_{s}(−2, −1, −1; −1, 0, 0, 0, 0) and (R/I_{g}) ∈ W_{s}(−2, −1; 0, 0, 0, 0) and we have minimal resolutions

(5.1)  0 → R(−7) ⊕ R(−6) ⊕ R(−5) → R(−5)^{4} ⊕ R(−4)^{4} → R(−3)^{6} → I_{g} → 0 ,

(5.2) 0 → R(−7) ⊕ R(−6)^{2} ⊕ R(−5)^{3} → R(−6) ⊕ R(−5)^{6} ⊕ R(−4)^{8} → R(−4)^{4} ⊕ R(−3)^{6} → I → 0.

Observe that there is still a ghost term in the resolution of I_{g}. In the artinian case, however, we have seen in Example 4.4 that \overline{W_{s}(−2, −1; 0, 0, 0, 0)} is not an irreducible component of GradAlg(H). So there is a possibility for removing this ghost under a generization to a non-determinantal artinian ring! Indeed M. Boij pointed out in a note he sent me that a general artinian algebra with Hilbert function (1, 3, 6, 4, 1) can be seen as a type two algebra A given by the inverse system (f, g) where deg f = 4 and deg g = 3. Following Boij’s note, this algebra has a minimal resolution as in (5.1) with the ghost term R(−5) removed and A is probably a generization of R/I_{g}.

On the other hand, we may very well consider the determinantal algebras R/I and R/I_{g} described above in the polynomial ring R = k[x_{0}, x_{1}, ⋯, x_{n}] for n ≥ 5. This leads to determinantal rings of dimension greater or equal to 3 with minimal resolutions exactly as in (5.1) and (5.2). Thanks to Theorem 4.6, see Remark 5.3, \overline{W_{s}(−2, −1; 0, 0, 0, 0)} is in this case an irreducible component of GradAlg(H) (as well as of Hilb^{p}(ℙ^{n}), up to closure), whence R/I_{g} has no generization to non-determinantal rings and the ghost term R(−5) of (5.1) remains a ghost term for the general element of the component.

Finally suppose c = 2. Then repeated use of Proposition 5.1 will remove all ghost terms in the minimal resolution of I since the transpose of A is the Hilbert-Burch matrix. This is rather well known (cf. [12, Thm. 2(iii)]); indeed even more precise results are true, see [20] which finds the codimension of all “ghost terms strata” when dim A is small and A is Cohen-Macaulay. Note that any codimension-2 Cohen-Macaulay quotient R/I is determinantal, so Proposition 5.1 applies to such R/I. If A = R/I is the homogeneous coordinate ring of a codimension-2 scheme X in ℙ^{n}, then A is Cohen-Macaulay for dim A = 1 and there are no ghost terms in the minimal resolution of I := I_{X} for A general. If X is a locally Cohen-Macaulay (lCM) curve in ℙ^{3} and A is not necessarily Cohen-Macaulay, then we can generalize the removal of ghost terms above for codimension-2 quotients to the following.

Proposition 5.6. [30, Thm. 2.8] Let X ⊂ ℙ^{3} be any lCM curve and let 0 → L_{4} ⟶^{σ} L_{3} → ⋯ → M → 0 be a minimal R-free resolution of the Rao module M := H_{∗}^{1}(I_{X}). By [47] there is a minimal R-free resolution of the following form

0 → L_{4} ⟶^{σ⊕0} L_{3} ⊕ P_{2} → P_{1} → I_{X} → 0

(i.e. the composition L_{4} ⟶^{σ⊕0} L_{3} ⊕ P_{2} → P_{2} is zero). If P_{1} and P_{2} have a common free summand; P_{2} = P′_{2} ⊕ R(−i), P_{1} = P′_{1} ⊕ R(−i), then there is a generization X′ of X in GradAlg(H) with minimal resolution

0 → L_{4} ⟶^{σ⊕0} L_{3} ⊕ P′_{2} → P′_{1} → I_{X′} → 0 .

Note that the statement “a generization X′ of X in Hilb^{p}(ℙ^{3}) with constant postulation” in [30, Thm. 2.8] really means “a generization X′ of X in GradAlg(H)”. This result applies to a codimension-2 Cohen-Macaulay quotient R/I_{X} by letting M = 0, which implies L_{3} = L_{4} = 0, coinciding with Proposition 5.1 in this case.

6. Upgrading of previous results

In this section we generalize main theorems from [29, 34] by using Theorems 4.1 and 4.6 for dim A ≥ 1, or more precisely we use that the two conjectures in [34] mentioned in the Introduction now are theorems (for n − c > 0). Indeed the previous proofs relied on the part of the conjectures which was proved (in [33]) at the time these papers were written, but we can, using mainly the “same” proofs as in [29, 34], get stronger results (the same conclusions under weaker assumptions). To better understand the proofs presented here, it may be a good idea to consider the corresponding proofs in [29] simultaneously.

We start, however, with [34]. The following theorem was proved in [34, Thm. 3.2] under some assumptions (mainly “a_{0} > b_{t}, a_{t+3} > a_{t−2} and n − c ≥ 0”) and it was in [31, Cor. 5.7] shown to be true in general for n − c ≥ 1 (under elsewhere weaker assumptions: W (\underline{b}; \underline{a}) ≠ ∅ and a_{i−2} ≥ b_{i} for i ≥ 2). For n = c, [34, Thm. 3.2] is still the best result we have with assumptions only on a_{j} and b_{i}, and we can immediately generalize it to

Theorem 6.1. Let X ⊂ ℙ^{n} be a general determinantal scheme of codimension c and suppose a_{0} > b_{t} and a_{t+c−2} > a_{t−2}. Then (cf. Remark 2.6)

dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + K_{4} + ⋯ + K_{c} .

Proof. This follows from [34, Prop. 3.1], see Proposition 6.3(i) below, and [31, Cor. 5.7]. □

Now, to generalize the main results of [29], we first restate [29, Prop. 3.4]. With notations as in Sec. 2 (i.e. we delete the last column of A to get A_{c−1} and the map φ_{c−1} induced by the transpose of A_{c−1}), let B := A_{c−1}, B := R/I_{t}(B) and N := coker φ_{c−1}^{∗}. In this section we let A := R/I_{t}(A) be good determinantal, i.e. (A) ∈ W (\underline{b}; \underline{a}) where \underline{a} = a_{0}, a_{1}, ..., a_{t+c−2}, and as previously we assume that all entries of the deleted column(s) belong to 𝔪. It follows that (B) ∈ W (\underline{b}; \underline{a′}) with \underline{a′} = a_{0}, a_{1}, ..., a_{t+c−3} by [2] and the inclusions I_{t}(B) ⊂ I_{t}(A) ⊂ I_{t−1}(B). We set X = Proj(A) and Y = Proj(B).

Proposition 6.2. Let c ≥ 3, let (X) ∈ W (\underline{b}; \underline{a}) be general and suppose dim W (\underline{b}; \underline{a′}) ≥ λ_{c−1} + K_{3} + K_{4} + ... + K_{c−1} (and ≥ λ_{2} if c = 3). If

(6.1)  _{0}hom_{R}(I_{Y} , I_{X/Y}) ≤ ∑_{j=0}^{t+c−3} \binom{a_{j} − a_{t+c−2} + n}{n} ,

then dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + K_{4} + ... + K_{c}. We also get equality in (6.1), as well as

dim W (\underline{b}; \underline{a}) = dim W (\underline{b}; \underline{a′}) + dim_{k} N(a_{t+c−2})_{0} − 1 − _{0}hom_{R}(I_{Y} , I_{X/Y}).

See [29, Prop. 3.4] for a proof, and notice that since dim A ≥ 1 and X is general in W (\underline{b}; \underline{a}), the assumption depth_{I(Z)} B ≥ 2 of [29, Prop. 3.4] holds letting Z := V (I_{t−1}(B)).

Then [34, Prop. 3.1] is mainly (i) in the following.

Proposition 6.3. (i) If a_{0} > b_{t} and a_{t+c−2} > a_{t−2}, then (6.1) holds for X general.

(ii) If _{0}Ext_{B}^{i}(N, B(−a_{t+c−2})) = 0 for i = 1 and 2, then (6.1) holds. In particular (6.1) holds provided depth_{I_{t−1}(B)} B ≥ 3 (and Ext_{B}^{1}(N, B) = 0 provided depth_{I_{t−1}(B)} B ≥ 2).

Proof. (ii) The arguments we apply in the proof are very similar to those needed in (3.1), (3.2) and (3.3) to get related statements, except that we now use the spectral sequence E_{2}^{p,q} := Ext_{B}^{p}(Tor_{q}^{R}(B, N), B) ⇒ Ext_{R}^{p+q}(N, B) and its corresponding 5-term exact sequence. Indeed due to assumptions and (2.10) we get isomorphisms

vExt_{R}^{1}(N, B) ≃ (E_{2}^{0,1})_{v} ≃ _{v}Hom_{R}(I_{Y} , Hom_{B}(N, B)) ≃ _{v}Hom_{R}(I_{Y} , I_{X/Y} (a_{t+c−2}))

for v = −a_{t+c−2},. Moreover as in (3.3), cf. (2.7), we get for this v the exact sequence

(6.2) 0 → _{v}Hom_{R}(N, B) → _{v}Hom_{R}(F^{∗}, B) → _{v}Hom_{R}(G_{c−1}^{∗}, B) → _{v}Ext_{R}^{1}(N, B) → 0

where _{v}Hom_{R}(F^{∗}, B) ≃ ⊕_{i=1}^{t}B_{(b_{i}+v)} = 0 and _{v}hom_{R}(G_{c−1}^{∗}, B) ≃ dim(⊕_{j=0}^{t+c−3}B_{(a_{j}+v)}) = ∑_{j=0}^{t+c−3} \binom{a_{j}−a_{t+c−2}+n}{n}. This implies (6.1).

The above arguments are used in [35, Rem. 3.4] and the proof of [35, Prop. 3.5] from which we get (6.1) under the assumption depth_{I_{t−1}(B)} B ≥ 4. To see that we can weaken the depth assumption to depth_{I_{t−1}(B)} B ≥ 3, we really need to refine the argument. Following, however, the proof of [31, Thm. 4.5] exactly as in the paragraph after (4.3) in [31], we get Ext_{B}^{i}(N, B) = 0 for i = 1 and 2 (resp. for i = 1) under the assumption depth_{I_{t−1}(B)} B ≥ 3 (resp. depth_{I_{t−1}(B)} B ≥ 2), and we are done. □

Remark 6.4. Using Propositions 6.2 and 6.3(ii) one may reprove the dimension formula dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + K_{4} + ⋯ + K_{c} of Theorem 4.1 in the case n − c ≥ 1, a_{i−2} ≥ b_{i} for i ≥ 2 and W (\underline{b}; \underline{a}) ≠ ∅ by using the recursive strategy of successively deleting columns of A from the right-hand side, see Introduction and Remark 2.6. In [35] we pointed out that [35, Cor. 3.19] shows that the recursive strategy also applies to reprove the generic smoothness of W (\underline{b}; \underline{a}) of Theorem 4.6 in the case n−c ≥ 2 and a_{0} ≥ b_{t} (due to Remark 2.5, the latter assumption may here be weakened to a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t). This means that we have two rather different proofs for the two conjectures of [34].

With the above propositions we start by considering determinantal curves. In this and later results we denote by τ_{X/Y} the following morphism induced by I_{X/Y} ↪ B:

τ_{X/Y} : _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) → _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B).

A key result of [29] (Prop. 4.6) shows that if ker τ_{X/Y} = 0, depth_{I_{t−1}(B)} B ≥ 3 and X good determinantal, then the property “every deformation of a determinantal scheme comes from deforming the matrix” is transferred from Y to X. Recalling that dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + ... + K_{c} is now a theorem when n − c = 1, the following result generalizes [29, Prop. 4.15] to arbitrary c ≥ 3. (For c = 2 we have a more complete result with stronger conclusions in Theorem 4.15).

Proposition 6.5. Let X = Proj(A) ⊂ ℙ^{n}, A = R/I_{t}(A), be general in W (\underline{b}; \underline{a}) and suppose a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t, c ≥ 3 and dim X = n − c = 1. If Y = Proj(B) is defined by the vanishing of the maximal minors of the matrix obtained by deleting the last column of A, then the following statements hold:

(i) If τ_{X/Y} : _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) →_{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B) is injective, then X is unobstructed and \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{p}(ℙ^{n}). Moreover every deformation of X comes from deforming A.

(ii) If _{0}Ext_{A}^{1}(I_{X}/I_{X}^{2}, A) = 0, then X is unobstructed, and

codim_{Hilb^{p}(ℙ^{n})} \overline{W (\underline{b}; \underline{a})} = dim ker τ_{X/Y} − _{0}ext_{B}^{1}(I_{X/Y} , A) .

(iii) We always have

(6.3)  codim_{Hilb^{p}(ℙ^{n})} \overline{W (\underline{b}; \underline{a})} ≤ dim ker τ_{X/Y} .

Moreover if _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0, then we have equality in (6.3) if and only if X is unobstructed.

Proof. Thanks to Proposition 6.3(ii) the proof is the “same” as for [29, Prop. 4.15]. Indeed for (i) we apply Theorem 4.6 onto W (\underline{b}; \underline{a′}) ∋ (Y) instead of the corresponding result of [29] which required c − 1 ≤ 4 and we get (i) from [29, Thm. 4.6] and Lemma 2.10. For (ii) and (iii) we need the final dimension formula of Proposition 6.2 to use the proof of [29]. Using Proposition 6.3(ii) we get the mentioned dimension formula and we are done. □

We consider now zero dimensional determinantal schemes (n − c = 0). In this case we also need to consider the morphism

(6.4)  ρ^{1} := ρ_{X/Y}^{1} : _{0}Ext_{B}^{1}(I_{X/Y} , I_{X/Y}) → _{0}Ext_{B}^{1}(I_{X/Y} , B)

induced by I_{X/Y} ↪ B. In addition to [29, Prop. 4.6] which states that ker τ_{X/Y} = 0, ker ρ^{1} = 0 and depth_{I_{t−1}(B)} B = 2 transfer the property “every deformation of a determinantal ring comes from deforming the matrix” from B (or Y) to A, we need a variation of [29, Prop. 4.6] which don’t require ker ρ^{1} = 0 (see [29, Prop. 4.13] for the details). Below we generalize [29, Thm. 4.19] by replacing the assumption 4 ≤ c ≤ 6 in [29, Thm. 4.19] by c ≥ 4. Note that we denote GradAlg(H) by Hilb^{H}(ℙ^{n}) since dim A = 1, cf. the text before (2.1) and see Remark 2.6 for computing dim W (\underline{b}; \underline{a}). We refer to [29] for examples.

26 JAN O. KLEPPE

Theorem 6.6. Let X = Proj(A) ⊂ ℙ^{n}, A = R/I_{t}(A), be general in W (\underline{b}; \underline{a}) and let Y = Proj(B) and V = Proj(C) be defined by the vanishing of the maximal minors of B and C respectively where B (resp. C) is obtained by deleting the last column of A (resp. B). Suppose dim X = n − c = 0, a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t and suppose that (6.1) (cf. Proposition 6.3) holds. Moreover suppose

either c = 3 or [ c ≥ 4 and ker τ_{Y/V} = 0 ].

Then dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + ... + K_{c} and the following statements are true:

(i) If both ρ^{1} : _{0}Ext_{B}^{1}(I_{X/Y} , I_{X/Y}) → _{0}Ext_{B}^{1}(I_{X/Y} , B) and τ_{X/Y} : _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) → 0Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B) are injective, then A is unobstructed and \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{H}(ℙ^{n}). Moreover every deformation of A comes from deforming A.

(ii) If _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0 and ker τ_{X/Y} = 0, then \overline{W (\underline{b}; \underline{a})} belongs to a unique generically smooth irreducible component Q of Hilb^{H}(ℙ^{n}) and the codimension of \overline{W (\underline{b}; \underline{a})} in Hilb^{H}(ℙ^{n}) is dim ker ρ^{1}. Indeed A is unobstructed and

dim Q = λ_{c} + K_{3} + ... + K_{c} + dim ker ρ^{1}.

(iii) If _{0}Ext_{A}^{1}(I_{X}/I_{X}^{2}, A) = 0, then A is unobstructed and

codim_{Hilb^{H}(ℙ^{n})} \overline{W (\underline{b}; \underline{a})} = dim ker ρ^{1} + dim ker τ_{X/Y} − _{0}ext_{B}^{1}(I_{X/Y} , A).

(iv) We always have codim_{Hilb^{H}(ℙ^{n})} \overline{W (\underline{b}; \underline{a})} ≤ dim ker ρ^{1} + dim ker τ_{X/Y}. Suppose _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0. Then we have

codim_{Hilb^{H}(ℙ^{n})} \overline{W (\underline{b}; \underline{a})} = dim ker ρ^{1} + dim ker τ_{X/Y}

if and only if A is unobstructed.

Proof. Using Proposition 6.2 we get dim W (\underline{b}; \underline{a}) = λ_{c} +K_{3} +...+K_{c} because Theorem 4.1 applies to (B) ∈ W (\underline{b}; \underline{a′}) to find dim W (\underline{b}; \underline{a′}) where \underline{a′} = a_{0}, a_{1}, ..., a_{t+c−3}. Then, due to [29, Prop. 4.6 and Prop. 4.13] the proof of the theorem is “the same” as the proof of [29, Thm. 4.19] since we already have generalized [29, Prop. 4.15] to Proposition 6.5 of this paper to cover determinantal curves Y of arbitrary codimension c ≥ 3. Note also that we have to delete a column of B (thus defining V) when we apply Proposition 6.5 to Y. □

Remark 6.7. Looking at the proofs we see that we don’t need to suppose (6.1) to get (the part of the) conclusions of (i) and (ii) that don’t involve dimension and codimension formulas. Moreover note the overlap in (ii) and (iv) of the theorem.

Using again the variation [29, Prop. 4.13] we generalize [29, Prop. 4.24] to the following.

Proposition 6.8. With notations as in the first sentence of Theorem 6.6, suppose c ≥ 4, dim X = n − c = 0 and X general, a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t and suppose that (6.1) holds. Then dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + ... + K_{c} (cf. Remark 2.6), and the following statements are true:

(i) If _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0, _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) = 0 and _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B) = 0 then A is unobstructed. Moreover W (\underline{b}; \underline{a}) is contained in a unique generically smooth irreducible component of Hilb^{H}(ℙ^{c}) of codimension dim ker ρ_{X/Y}^{1} + dim ker τ_{Y/V} − _{0}ext_{C}^{1}(I_{Y/V} , B).

(ii) We always have codim_{Hilb^{H}(ℙ^{c})} \overline{W (\underline{b}; \underline{a})} ≤ dim ker ρ_{X/Y}^{1} +dim ker τ_{X/Y} +dim ker τ_{Y/V}. Suppose _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0 and _{0}Ext_{C}^{1}(I_{Y/V} , B) = 0. Then we have

codim_{Hilb^{H}(ℙ^{c})} \overline{W (\underline{b}; \underline{a})} = dim ker ρ_{X/Y}^{1} + dim ker τ_{X/Y} + dim ker τ_{Y/V}

if and only if A is unobstructed (e.g. _{0}Ext_{A}^{1}(I_{X}/I_{X}^{2}, A) = 0).

Proof. Again we have dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + ... + K_{c} by Proposition 6.2. Since Proposition 6.5 applies for every c ≥ 3 we can use the proof of [29, Prop. 4.24] to get Proposition 6.8. Indeed we use Proposition 6.5(ii) and [29, Prop. 4.13] to get (i) above while we use the proof of Proposition 6.5(iii) to get (ii) (see [29] for details). □

Acknowledgement. I thank prof. R.M. Miró-Roig at Barcelona for interesting discussions on this topic, prof. A. Iarrobino at Northeastern University, Boston and prof. M. Boij at KTH, Stockholm for comments and helping me approach the artinian case. Indeed this study of families of artinian determinantal rings started some time ago during a visit by A. Iarrobino and M. Boij at Oslo University College where we concretely studied examples of such families. I also thank the referee for some valuable comments.

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Faculty of Technology, Art and Design, Oslo and Akershus University College of Applied Sciences, P.O. Box 4, St. Olavs Plass, NO-0130 Oslo, Norway

 E-mail address: JanOddvar.Kleppe@hioa.no