Linkage classes of grade 3 perfect ideals
Lars Winther Christensen, Oana Veliche, and Jerzy Weyman

TL;DR
This paper explores the linkage properties of grade 3 perfect ideals in regular local rings, showing they are linked to either complete intersections or Golod ideals, and uses this to analyze Cohen-Macaulay ring structures.
Contribution
It proves that grade 3 perfect ideals are linked to complete intersections or Golod ideals, extending linkage theory beyond grade 2 cases.
Findings
Grade 3 perfect ideals are linked to complete intersections or Golod ideals.
Linkage reveals finer structures of Cohen-Macaulay rings of codimension 3.
Homological classification aids in understanding ideal linkages.
Abstract
While every grade 2 perfect ideal in a regular local ring is linked to a complete intersection ideal, it is known not to be the case for ideals of grade 3. We soften the blow by proving that every grade 3 perfect ideal in a regular local ring is linked to a complete intersection or a Golod ideal. Our proof is indebted to a homological classification of Cohen-Macaulay local rings of codimension 3. That debt is swiftly repaid, as we use linkage to reveal some of the finer structures of this classification.
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Linkage classes of grade 3 perfect ideals
Lars Winther Christensen
Texas Tech University, Lubbock, TX 79409, U.S.A.
[email protected] http://www.math.ttu.edu/~lchriste ,
Oana Veliche
Northeastern University, Boston, MA 02115, U.S.A.
[email protected] https://web.northeastern.edu/oveliche and
Jerzy Weyman
University of Connecticut, Storrs, CT 06269, U.S.A.
[email protected] http://www.math.uconn.edu/~weyman
(Date: 4 June 2019)
Abstract.
While every grade perfect ideal in a regular local ring is linked to a complete intersection ideal, it is known not to be the case for ideals of grade . We soften the blow by proving that every grade perfect ideal in a regular local ring is linked to a complete intersection or a Golod ideal. Our proof is indebted to a homological classification of Cohen–Macaulay local rings of codimension . That debt is swiftly repaid, as we use linkage to reveal some of the finer structures of this classification.
Key words and phrases:
Complete intersection, Golod, linkage, local ring, Tor algebra.
2010 Mathematics Subject Classification:
Primary 13C40. Secondary 13D02; 13H10.
This work is part of a body of research that started during the authors’ visit to MSRI in Spring 2013 and continued during a months-long visit by L.W.C. to Northeastern University; the hospitality of both institutions is acknowledged with gratitude. L.W.C. was partly supported by NSA grant H98230-14-0140 and Simons Foundation collaboration grant 428308, and J.W. was partly supported by NSF DMS grants 1400740 and 1802067.
1. Introduction
Let be a local ring with maximal ideal and residue field . The difference between the embedding dimension and depth of ,
[TABLE]
i.e. the number , is called the embedding codepth of . Rings with are regular, rings with are hypersurfaces, and for rings with there are two possibilities: complete intersection or Golod. For the field of possibilities widens: Such rings can be Gorenstein and not complete intersection, or they may not even belong to any of the classes mentioned thus far. There is, nevertheless, a classification of local rings of embedding codepth . It is based on multiplicative structures in homology, and the details are discussed below.
In the 1980s Weyman [14] and Avramov, Kustin, and Miller [4] established a classification scheme which—though it is discrete in the sense that it does not involve moduli—is subtle enough to facilitate a proof of the rationality of Poincaré series of codepth local rings, an open question at the time. For this purpose it was not relevant to know if local rings of every class in the scheme actually exist, and it later turned out that they do not. In 2012 Avramov [3] returned to the classification and tightened it; that is, he limited the range of possible classes. This was necessary to use the classification to answer the codepth 3 case of a question ascribed to Huneke about growth in the minimal injective resolution of a local ring. In the same paper [3, Question 3.8], Avramov formally raised the question about realizability: Which classes of codepth 3 local rings do actually occur? This paper deals with aspects of the realizability question that can be studied by linkage theory and, in turn, provide new insight on linkage of grade perfect ideals.
To discuss our contributions towards an answer to the realizability question, we need to describe the classification in further detail. The -adic completion of is by Cohen’s Structure Theorem a quotient of a regular local ring. To be precise, there is a regular local ring with maximal ideal and an ideal with . The Auslander-Buchsbaum Formula thus yields
[TABLE]
For , Buchsbaum and Eisenbud [7] show that the minimal free resolution
[TABLE]
of over has a structure of a commutative differential graded algebra. The multiplicative structure on induces a graded-commutative algebra structure on ; as a -algebra, is isomorphic to the Koszul homology algebra , in particular it is independent of the presentation of and the choice of . The multiplicative structure on is the basis for the classification. In [4] the possible multiplication tables are explicitly described, up to isomorphism; they are labeled , , , , and , where the parameters , , and are non-negative integers bounded by functions of the invariants , called the first derivation of , and , called the type of . Thus, for fixed and there are only finitely many possible structures, and [3, Question 3.8] asks which ones actually occur. It is known that a complete answer will have to take into account the Cohen–Macaulay defect of the ring; in this paper we only consider Cohen–Macaulay rings, and our main result pertains to the two-parameter family:
1.1 Theorem.
Let be a Cohen–Macaulay local ring of codimension 3 and class . Let and denote the first derivation and the type of . The inequalities
[TABLE]
hold, and the following conditions are equivalent
[TABLE]
Otherwise, i.e. when these conditions are not satisfied, there are inequalities
[TABLE]
with
[TABLE]
The first set of inequalities in this theorem can be read off immediately from the multiplication table for , see (2.1.1), and the equivalence of conditions (i)–(iii) is known from [3, Cor. 3.3]. Thus, what is new is the “otherwise” statement. The inequalities and are proved in Section 5, and the last assertions about equalities and congruences are proved in Section 6.
To parse the theorem it may be helpful to visualize an instance of it. For and the parameter can take values between [math] and , and can take values between [math] and ; thus the classes fit naturally in a grid; see Table 1.1. It follows from the theorem that at most of the fields in the grid are populated, and in the experiments that informed the statement of Theorem 1.1 we encountered all of them; we discuss this point further in Section 7.
[TABLE]
If and, therefore, is Cohen–Macaulay of codimension , then the defining ideal with is perfect of grade . We obtain Theorem 1.1 as a special case of results—Theorems 5.4 and 6.1—about grade perfect ideals in regular local rings.
Our main tool of investigation is linkage. It is known that every grade perfect ideal in is linked to an ideal such that the local ring is complete intersection; such ideals are called licci. Not every grade perfect ideal is licci, see for example [9, Prop. 3.5], but as a special case of Theorem 3.7 we obtain:
1.2 Theorem.
Every grade perfect ideal in a regular local ring is linked to a grade perfect ideal such that is complete intersection or Golod.
The paper is organized as follows. We recall the details of the classification from [4, 14] in Section 2. The motor of the paper is Section 3; it has four statements that track relations between the multiplicative structures on the Tor algebras of linked ideals. As a first application of these statements, Theorem 1.2 is proved in Section 3; the proofs of the four statements themselves are deferred to the Appendix. The applications to the classification scheme begin in Section 4 and continue in Sections 5 and 6, which contain the proof of Theorem 1.1. In Section 7 we provide a summary of the status of the realizability question.
2. Multiplicative structures in homology
Throughout this paper, denotes a commutative noetherian local ring with maximal ideal and residue field . For an ideal with , let be a minimal free resolution over and set
[TABLE]
if is regular, then these numbers are the first derivation and the type of . Notice that one has , which forces as has Euler characteristic [math].
2.1.
By a result of Buchsbaum and Eisenbud [7] the resolution has a structure of a commutative differential graded algebra. This structure is not unique, but the induced graded-commutative algebra structure on is unique. By [4] there exist bases
[TABLE]
such that the multiplication on is one of following:
[TABLE]
Here it is understood that all products that are not mentioned—and not given by those mentioned and the rules of graded commutativity—are zero. We say that , or , is of class if the multiplication on is given by in (2.1.1); similarly for and .
2.2.
To deal with the multiplicative structures on it is helpful to consider a few additional invariants; set
[TABLE]
where is defined by for and . Depending on the class of , the values of these invariants are
[TABLE]
2.3.
Recall that an ideal is called complete intersection of grade if it is generated by a regular sequence of length . For such an ideal the minimal free resolution of over is the Koszul complex on the regular sequence. In particular, a complete intersection ideal is perfect. A perfect ideal of grade is called almost complete intersection if it is minimally generated by elements.
For a grade perfect ideal one has , and the next conditions are equivalent; for the equivalence of (ii) and (iii) see the remark after [4, Def. 2.2].
- (i)
.
- (ii)
is of class .
- (iii)
is complete intersection.
- (iv)
and .
2.4.
Recall that an ideal is called Gorenstein of grade if it is perfect of grade with , in which case one has .
If is Gorenstein of grade , then is of class or of class with odd ; see the remark after [4, Def. 2.2].
We refer to [3, 1.4] for the following facts and precise references to their origins.
2.5.
Assume that is regular, and let be a grade perfect ideal.
- (a)
The ring is complete intersection if and only if is of class .
- (b)
The ring is Gorenstein and not complete intersection if and only if is of class with .
- (c)
The ring is Golod if and only if is of class .
3. Multiplication in Tor algebras of linked ideals
Let be a grade perfect ideal. Recall that an ideal is said to be directly linked to if there exists a complete intersection ideal of grade with . The ideal is then also a perfect ideal of grade with , and one has ; see Golod [10]. In particular, being directly linked is a reflexive relation. An ideal is said to be linked to if there exists a sequence of ideals such that is directly linked to for each . Evidently, being linked is an equivalence relation; the equivalence class of under this relation is called the linkage class of .
3.1 Proposition.
Let be a grade perfect ideal.
- (a)
If is of class , then it is directly linked to a grade perfect ideal with
[TABLE]
Moreover, is of class .
- (b)
If is of class , then it is directly linked to a grade perfect ideal with
[TABLE]
Moreover, is of class .
- (c)
If is of class with , then it is directly linked to a grade perfect ideal with
[TABLE]
In particular, is of class .
- (d)
If is of class , then it is directly linked to a grade perfect ideal with
[TABLE]
In particular, is of class or .
- (e)
If is of class , then it is directly linked to a grade perfect ideal with
[TABLE]
Proof.
See A.6. ∎
The next three propositions deal with rings of class the way Proposition 3.1 deals with rings of class , , and .
3.2 Proposition.
Let be a grade perfect ideal. If is of class with , then it is directly linked to a grade perfect ideal with
[TABLE]
Moreover, the following assertions hold:
- (a)
If , then .
- (b)
If , then is of class .
- (c)
If , then .
- (d)
If , then is of class .
- (e)
If , then is of class or .
- (f)
If and , then is of class .
- (g)
If is of class , then .
Proof.
See A.7. ∎
3.3 Proposition.
Let be a grade perfect ideal. If is of class with , then it is directly linked to a grade perfect ideal with
[TABLE]
Moreover, the following assertions hold:
- (a)
If , then .
- (b)
If , then is of class
- (c)
If , then .
- (d)
If , then is of class .
- (e)
If , then is of class or .
- (f)
If , then is of class .
- (g)
If and , then is of class and is of class .
Proof.
See A.8. ∎
3.4 Proposition.
Let be a grade perfect ideal. If is of class with , then it is directly linked to a grade perfect ideal with
[TABLE]
If , then and is complete intersection. Moreover, if or , then the following assertions hold:
- (a)
If , then .
- (b)
If , then is of class
- (c)
If , then .
- (d)
If , then is of class .
- (e)
If , then is of class or .
- (f)
If and , then is of class .
Proof.
See A.9. ∎
3.5.
For a grade perfect ideal the quantity is a measure of the size of the minimal free resolution of over . Indeed, one has
[TABLE]
we refer to this number as the total Betti number of . As one has and the least possible total Betti number is and attained if and only if is complete intersection; see 2.3.
The next corollary records the observation, already used in [4], that any grade perfect ideal with is linked to an ideal with lower total Betti number.
3.6 Corollary.
Let be a grade perfect ideal not of class . There exists a grade perfect ideal that is directly linked to and has
[TABLE]
Proof.
Immediate from Propositions 3.1–3.4. ∎
Theorem 1.2 from the introduction is a special case of the next result; see 2.5.
3.7 Theorem.
Every grade 3 perfect ideal in is linked to a grade perfect ideal of class or .
Proof.
Let be a grade perfect ideal, and assume that is not of class or . If , then is of class or per (2.2.1), so . It now follows from Proposition 3.1(b) and Proposition 3.2 that is linked to a grade perfect ideal with and . Thus it follows from Corollary 3.6 that every grade perfect ideal that is not of class or can be linked to a grade perfect ideal with smaller total Betti number, and ideals of class have the smallest possible total Betti number; see 3.5. ∎
4. Grade 3 perfect ideals generated by at most 5 elements
Under the assumption that is Gorenstein, the next result can be deduced from Avramov’s proof of [1, Thm. 2]; see also [3, 3.4.2]. The Gorenstein assumption is used to invoke a result of Buchsbaum and Eisenbud [7], but it follows from later work of Golod [10] it is superfluous; see also Brown [5, Intro. to Sec. 2]. Here we give a proof based on the results in Section 3.
4.1 Theorem.
Let be a grade 3 perfect ideal with .
- (a)
If is odd, then and is of class .
- (b)
If is even, then is of class .
- (c)
If , then is of class .
In particular, one has and .
Proof.
By 2.3 the ideal is not of class . If is of class or , then there exists by Proposition 3.1(a,b) a grade perfect ideal of class with ; that is, a Gorenstein ideal of class . Per 2.4 no such ideal exists, so is of class or , cf. (2.1.1).
(a): Assume that is odd. It is immediate from 2.3 and 2.4 that cannot be , so holds. To prove that the ideal is of class , assume towards a contradiction that it is of class . By (2.1.1) one has . If holds, then there exists by Proposition 3.2 a grade perfect ideal with and , which contradicts 2.4 as is even. If holds, then Proposition 3.4 yields a similar contradiction.
(b): Assume that is even. To prove that the ideal is of class , assume towards a contradiction that it is of class . It follows from Proposition 3.1(d) that there exists a grade perfect ideal with and , which contradicts 2.4 as is even. Thus is of class with , see (2.1.1), and we argue that equality holds. If , then is by Proposition 3.2 linked to a grade perfect ideal with . By 2.4 the ideal is of class which contradicts 3.2(g). If holds, then there exists by Proposition 3.3 a grade perfect ideal with and , which contradicts 2.4 as is even. Thus, is of class with .
Assume now that holds. By Proposition 3.4(a) the ideal is linked to a grade perfect ideal with and . By 2.4 and (2.1.1) one has , so is of class .
(c): The argument above shows that is of class with . As Proposition 3.4 yields . ∎
The results in Section 3 easily yield the codimension case of the fact that every almost complete intersection ideal is linked to a Gorenstein ideal; see [7, Prop. 5.2].
4.2 Remark.
Let be a grade perfect ideal with . It follows from Theorem 4.1, Proposition 3.1(e), Proposition 3.4, and 2.4 that is directly linked to a Gorenstein ideal with
[TABLE]
From the proof of the theorem in J. Watanabe’s [13] one can deduce that every almost complete intersection ideal is linked to a complete intersection ideal. Here is an explicit statement.
4.3 Proposition.
Let be a grade perfect ideal with . There exists a grade perfect ideal of class that is linked to in at most links if is odd and at most links if is even.
Proof.
First assume that is even. If , then it follows from Remark 4.2 that is directly linked to a Gorenstein ideal with , i.e. is of class ; see 2.3. Now let be an integer and assume that the statement holds for ideals with . If , then it follows from Remark 4.2 that is directly linked to a Gorenstein ideal with . By Proposition 3.1(b) there is an ideal that is directly linked to and has and . By assumption is linked to an ideal of class in at most links, so is linked to the same ideal in at most links.
Now assume that is odd. If , then it follows from Remark 4.2 that is directly linked to a Gorenstein ideal with , i.e. is of class ; see 2.3. If for some , then it follows from Remark 4.2 that is directly linked to a Gorenstein ideal with . By Proposition 3.1(b) there is an ideal that is directly linked to and has and . By what has already been proved, is linked to an ideal of class in at most links, so is linked to the same ideal in at most links. ∎
The next result is proved by Brown [5, Thm. 4.5]. For completeness we include a proof based on the results in Section 3.
4.4 Proposition.
Let be a grade 3 perfect ideal with and . If , then the following assertions hold:
- (a)
is of class if is odd.
- (b)
is of class if is even.
In particular, one has .
Proof.
By the assumptions and , the ideal is not of class , see 2.3, so the assumption implies that is of class , , or ; see (2.2.1).
If is of class , then there exists by Proposition 3.1(d) a grade perfect ideal of class or with , but by Theorem 4.1 no such ideal exists.
If is of class , then there exists by Proposition 3.1(a) a grade perfect ideal of class with , and , so is odd by Theorem 4.1.
Assume now that is of class . By (2.1.1) one has , so if holds, then there exists by Proposition 3.4 a grade perfect ideal with and , which contradicts 2.3. Thus one has , and it now follows from Proposition 3.3 that is linked to a grade perfect ideal with , and . By Theorem 4.1 one has , so by 3.3(g) the ideal is of class and is of class . Now it follows from Theorem 4.1 that is even. The in particular statement is now immediate, see (2.2.1). ∎
Proposition 3.1 applies to restrict the possible classes of five generated ideals.
4.5 Theorem.
Let be a grade 3 perfect ideal with . The following assertions hold:
- (a)
is of class only if .
- (b)
is of class only if .
- (c)
is of class only if .
Notice that by part (b) a five generated ideal of class is Gorenstein and hence of class ; see 2.4.
Proof.
(a): If is of class , then it is by Proposition 3.1(a) linked to a grade perfect ideal with
[TABLE]
If , then one has and, therefore, per 2.3; a contradiction. If , then one has so Proposition 4.4 yields ; a contradiction.
(b): If is of class , then it is by Proposition 3.1(b) linked to a grade perfect ideal with
[TABLE]
If holds, then one has and hence by Proposition 4.4; a contradiction.
(c): If is of class , then it is by Proposition 3.1(e) linked to a grade perfect ideal with and ; by 2.3 this forces . ∎
The next result was first proved by Sánchez [12].
4.6 Proposition.
Let be a grade perfect ideal. If and , then . In particular, is not of class .
Proof.
Assume towards a contradiction that and hold. Per (2.2.1) the ideal is of class or .
If is of class , then there exists by Proposition 3.4(a) a grade perfect ideal with and . By Theorem 4.1 one has then and, therefore, . Thus 3.4(c) yields , which contradicts 4.1.
If is of class , then it follows from Proposition 3.1(d) there exists a grade perfect ideal in of class or with
[TABLE]
By Theorem 4.5 one has and, therefore, , which contradicts 4.5. ∎
5. Class H: Bounds on the parameters of multiplication
Throughout this section, the local ring is assumed to be regular.
5.1 Lemma.
Let be a grade perfect ideal in . If is not contained in , then one has .
Proof.
Choose an element and set ; notice that is a -regular element. By the Auslander–Buchsbaum Formula one has , so the format of the minimal free resolution of as a -module is
[TABLE]
for some . Now it follows from [2, Thm. 2.2.3] that the minimal free resolution of as a -module has format . In particular, one has and . ∎
5.2.
Let be a grade perfect ideal. If is not contained in , then implies that is of class by Lemma 5.1 and 2.3. If is contained in , then holds if and only if is Gorenstein, i.e. of class or ; see 2.4. In particular, one has the following lower bounds on and depending on the class of .
[TABLE]
Lemma 5.1 together with results from [3] yield bounds on the parameters of multiplication for ideals of class H.
5.3 Proposition.
Let be a grade perfect ideal of class ; one has
[TABLE]
Moreover, the following conditions are equivalent:
[TABLE]
Notice that when these conditions are satisfied, one has .
Proof.
The inequalities (5.3.1) are immediate from (2.1.1). If is not contained in , then by Lemma 5.1 the inequalities in (5.3.2) agree with those in (5.3.1). If is contained in , then the inequalities (5.3.2) hold by [3, Thm. 3.1]. Moreover, for such an ideal the conditions – are equivalent by [3, Cor. 3.3].
Now let be an ideal that is not contained in ; in view of Lemma 5.1 it suffices to prove that conditions and are equivalent. By (5.2.1) one has , and we proceed by induction on . For one has by Lemma 5.1, and the equivalence of and is trivial, as is of class by Theorem 4.1. Now let , by Lemma 5.1 one has .
: Assume that holds. By Proposition 3.4(b) there exists a grade perfect ideal of class with
[TABLE]
By (5.3.1) one has , so holds. By the induction hypothesis, or by the equivalence of – for ideals contained in , one now has . As holds by 3.4(a), one has .
: Assume that holds. If one has , then there exists by Proposition 3.2(d) a grade perfect ideal of class with , , and . By (5.3.2) one has , so equality holds. It follows from Lemma 5.1 that is contained in , so conditions – are equivalent. However, as the equality does not hold; a contradiction. Thus, one has , and it follows from Proposition 3.4(d) that there exists a grade perfect ideal of class with
[TABLE]
By (5.3.1) one has , so holds. By the induction hypothesis, or by the equivalence of – for ideals contained in , one now has . As holds by 3.4(c), one has . ∎
The goal of this section is to establish part of Theorem 1.1 by proving the following statement:
5.4 Theorem.
Let be a grade perfect ideal of class . If or hold, then one has and .
The proof takes up the balance of the section; it is propelled by Propositions 3.2, 3.3, and 3.4. For example, an ideal with is by these results linked to an ideal with , and and are essentially determined by and , respectively. Thus restrictions on (or imply restrictions on (or ). Here is an outline: Let be a grade perfect ideal of class . By Proposition 5.3 one can assume that and hold, and it suffices to prove
(1) and (2)
By (5.2.1) one has and . If or holds, then both (1) and (2) follow from Theorem 4.1 and Proposition 4.4. A few other low values of and require special attention, and after that the proof proceeds by induction: (2) for implies (1) for , which in turn implies (2) for etc.
5.5 Lemma.
For every grade 3 perfect ideal with , one has .
Proof.
If is not of class , then per (2.2.1) one has , so assume that is of class . Towards a contradiction, assume that holds. By (5.3.1) one has
[TABLE]
If holds, then as by assumption it follows from Proposition 3.2 that there exists a grade perfect ideal in with
[TABLE]
By one has , so Proposition 4.4 yields ; a contradiction.
If holds, then as by assumption, Proposition 3.3(b,c) yields the existence a grade perfect ideal in of class with . As one has , see (5.2.1), and it follows from Theorem 4.1 and Proposition 4.4 that no such ideal exists.
If holds, then as by assumption, Proposition 3.4(b,c) yields the existence of a grade perfect ideal in of class with . As one has , see (5.2.1), and it follows from Theorem 4.1 and 4.4 that no such ideal exists. ∎
5.6 Lemma.
Let be an integer and assume that for every grade perfect ideal with one has . For every grade 3 perfect ideal with one then has .
Proof.
Let be an integer and a grade 3 perfect ideal with . Assume towards a contradiction that holds. As , it follows that is of class , see (2.2.1). Per (5.3.1) one has
[TABLE]
We treat the cases , , and separately.
Case 1. If holds, then as by assumption it follows from Proposition 3.2(d) that there exists a grade perfect ideal in of class with
[TABLE]
From and one gets
[TABLE]
so Proposition 5.3 yields . By one now has , which is a contradiction.
Case 2. If holds, then as by assumption it follows from Proposition 3.3(b,c) that there exists a grade perfect ideal in of class with , and that is a contradiction.
Case 3. If holds, then as by assumption it follows from Proposition 3.4(b,c) that there exists a grade perfect ideal in of class with , and that is a contradiction. ∎
5.7 Lemma.
Let be an integer and assume that for every grade 3 perfect ideal with one has . For every grade perfect ideal with one then has .
Proof.
Let be an integer and a grade 3 perfect ideal with . Assume towards a contradiction that
[TABLE]
holds. As , it follows that is of class , see (2.2.1), and (5.3.1) yields
[TABLE]
We treat the cases and separately.
Case 1. Assume that holds. Per one has , so by Proposition 3.3(b) there exists a grade perfect ideal in of class with
[TABLE]
In view of (5.3.2) one now has
[TABLE]
As one has by assumption, which forces . As one has , this contradicts Proposition 5.3.
Case 2. Assume that holds. Per (1) one has , so by and Proposition 3.4(b) there exists a grade perfect ideal in of class with
[TABLE]
In view of (5.3.2) one now has
[TABLE]
As , one has by assumption, so that forces . As one has , this contradicts. ∎
Proof of 5.4.
By (5.3.2) there are inequalities and . By the assumptions and Proposition 5.3 neither equality holds, so it is sufficient to show that and hold. Per (5.2.1) one has and ; to prove the assertion we show by induction on that the following hold:
- (a)
Every grade perfect ideal of class with has .
- (b)
Every grade perfect ideal of class with has .
For part (a) follows from Theorem 4.1 and (b) follows from Theorem 4.1 and Proposition 4.4.
For part (a) is Lemma 5.5, and (b) holds by Theorem 4.1 and Proposition 4.6.
Let and assume that (a) and (b) hold for all lower values of . By part (b) for the hypothesis in Lemma 5.6 is satisfied for , and it follows that (a) holds for . By part (a) for and the hypothesis in Lemma 5.7 is satisfied for , and it follows that (b) holds for . ∎
5.8 Corollary.
Let be a grade perfect ideal of class . If holds, then one has and .
Proof.
The assertion is immediate from Proposition 5.3 and Theorem 5.4. ∎
6. Class H: Extremal values of the parameters of multiplication
Throughout this section, the local ring is assumed to be regular. We establish the remaining part of Theorem 1.1 by proving the following:
6.1 Theorem.
For every grade perfect ideal of class the following hold
- (a)
If , then .
- (b)
If , then .
The proof follows the template from Section 5. It is, eventually, an induction argument with the low values of and handled separately in Theorem 4.1, Proposition 4.4, and Lemmas 6.2–6.4.
6.2 Lemma.
For every grade perfect ideal of class with and one has .
Proof.
By (5.2.1) and (5.3.1) one has and . The assumption together with (5.3.2) and Lemma 5.5 yield or . We treat odd and even values of separately.
Assume that is even. The goal is to show that is odd, i.e. not 0, 2, or 4.
- •
If , then there would by Proposition 3.2—part (f) if and part (d) if —exist a grade perfect ideal in of class with
[TABLE]
As is odd, it follows from Proposition 4.4 that no such ideal exists.
- •
If , then there would by Proposition 3.4—part (f) if and part (d) if —exist a grade perfect ideal in of class with
[TABLE]
As is odd, it follows from Proposition 4.4 that no such ideal exists.
- •
If , then there would by Proposition 3.4(b) exist a grade perfect ideal in of class with
[TABLE]
As is odd, it follows from Proposition 4.4 that no such ideal exists.
Assume now that is odd. The goal is to show that is even, i.e. not 1 or 3.
- •
If , then there would by Proposition 3.3—part (f) if and part (d) if —exist a grade perfect ideal in of class with
[TABLE]
As is odd, it follows from Proposition 4.4 that no such ideal exists.
- •
If , then there would by Proposition 3.3(b) exist a grade perfect ideal in of class with
[TABLE]
As is odd, it follows from Proposition 4.4 that no such ideal exists.∎
6.3 Lemma.
For every grade perfect ideal of class with and one has .
Proof.
By (5.2.1) and (5.3.1) one has and . The assumption together with (5.3.2) and Proposition 4.6 yields or . We treat odd and even values of separately.
Assume that is even. The goal is to show that is even, i.e. not 1 or 3.
- •
If , then there would by Proposition 3.4—part (f) if and part (b) if —exist a grade perfect ideal in of class with and . As is odd, this contradicts Theorem 4.1.
- •
If , then there would by Proposition 3.4(d) exist a grade perfect ideal in of class with and . As is odd, this contradicts Theorem 4.1.
Assume now that is odd. The goal is to show that is odd, i.e. not 0 or 2.
- •
If , then there would by Proposition 3.3(a) exist a grade perfect ideal in with
[TABLE]
It follows from (2.2.1), 2.3, and Theorem 4.5 that is of class . As is even, it follows from Lemma 6.2 that no such ideal exists.
- •
If , then there would by Proposition 3.3(a,c) exist a grade perfect ideal in with
[TABLE]
It follows from (2.2.1) that is of class . As is even, it follows from Lemma 6.2 that no such ideal exists.∎
6.4 Lemma.
For every grade perfect ideal of class with and one has .
Proof.
By (5.2.1) one has . Notice that if holds, then Theorem 4.1 yields . If , then Proposition 3.4(a) yields a grade perfect ideal in with
[TABLE]
It follows from Theorem 4.5 and (2.2.1) that is of class or .
If is of class , then 4.5(a) yields , and hence . Per (2.2.1) one thus has .
If is of class , then Lemma 6.2 yields . ∎
6.5 Lemma.
Let be an integer. Assume that for every grade perfect ideal of class with and , one has . For every grade perfect ideal of class with and , one then has .
Proof.
Let be an integer and a grade perfect ideal in of class with
[TABLE]
It needs to be shown and have opposite parity. By (5.3.1) one has ; we treat the cases , , and separately.
Case 1. Assume that holds. As by , there exists by Proposition 3.2(c) and a grade perfect ideal in with
[TABLE]
For one has by , so it follows from Proposition 3.2(d) that is of class ; for the same conclusion follows from (2.2.1) and Proposition 4.6. The assumptions now yield the congruence
[TABLE]
Case 2. Assume that holds. As by , there exists by Proposition 3.3(c) and a grade perfect ideal in with
[TABLE]
As , the ideal is by (2.2.1) of class , so the assumptions yield
[TABLE]
Case 3. Assume that holds. As by , there exists by Proposition 3.4(c) and a grade perfect ideal in with
[TABLE]
As , the ideal is by (2.2.1) of class , so the assumptions yield
[TABLE]
6.6 Lemma.
Let be an integer. Assume that for every grade perfect ideal of class with and one has .For every grade perfect ideal of class with and , one then has .
Proof.
Let be an integer and a grade perfect ideal in of class with
[TABLE]
It needs to be shown and have the same parity. By (5.3.1) one has . We treat inequality and equality separately.
Case 1. Assume that holds. As by , there exists by Proposition 3.3(b) a grade perfect ideal in of class H with
[TABLE]
As , the assumptions now yield
[TABLE]
Case 2. Assume that holds. As by , there exists by Proposition 3.4(b) a grade perfect ideal in of class with
[TABLE]
As , the assumptions now yield
[TABLE]
Proof of 6.1.
Per (5.2.1) one has and ; to prove the assertion we show by induction on that the following hold:
- (1)
For every grade perfect ideal of class with and one has .
- (2)
For every grade perfect ideal of class with and one has .
For the assertion (1) holds by Theorem 4.1 and (2) holds by Proposition 4.4.
For the assertion (1) is Lemma 6.2 and (2) is Lemma 6.3.
Let . As (2) holds for , the premise in Lemma 6.5 is satisfied for , and it follows that (1) holds. The assertion (2) is Lemma 6.4.
Let and assume that (1) and (2) hold for all lower values of . By part (2) for the premise in Lemma 6.5 is satisfied for , and it follows that (1) holds for . By the assertion (1) for and the premise in Lemma 6.6 is satisfied for , and it follows that (2) holds for . ∎
7. The realizability question
We sum up the contributions of the previous sections towards an answer to the realizability question [3, Question 3.8]. The focus is still on restricting the range of potentially realizable classes.
7.1 Class B.
Let be a grade perfect ideal of class . Theorem 4.1, 2.3, and 2.4 yield and . Moreover, if holds, then Theorem 4.5 yields , and if holds, then is odd by Proposition 4.4.
[FIGURE:]
7.2 Class T.
Let be a grade perfect ideal of class ; by 2.3 one has . If holds, then it follows from Theorem 4.1 that is odd and at least . If holds, then Proposition 4.4 and Proposition 4.6 yield .
7.3 Summary.
Table 7.1 illustrates which Cohen–Macaulay local rings of class are possible per Theorem 1.1. The table also shows the restrictions imposed by 7.1 and 7.2 on the existence of rings of class or .
7.4 Conjectures.
The statement of Theorem 1.1 was informed by experiments conducted using the Macaulay 2 implementation of Christensen and Veliche’s classification algorithm [8, 11]. Based on these experiments we conjecture that for and there are no further restrictions on realizability of codimension Cohen–Macaulay local rings of class , , or than those captured by 1.1, 7.1, and 7.2 and illustrated by Table 7.1.
Absent from Table 7.1 are rings of classes and . A Cohen–Macaulay local ring of codimension , first derivation , and type is Gorenstein if and only if holds, and they exist for odd ; see 2.4 and 2.5. For they are complete intersections, precisely the rings of class , and for they are of class . Per [3, Thm. 3.1] one has for rings of class that are not Gorenstein. Our experiments suggests that there are further restrictions.
Let be a grade perfect ideal of class and not Gorenstein. By 2.3, Theorem 4.1, and Theorem 4.5 one has . We conjecture that the following hold:
- (a)
If , then one has or .
- (b)
If , then one has .
The next proposition proves part (b) of this conjecture for six generated ideals.
7.5 Proposition.
Let be a grade 3 perfect ideal with and . If is of class , then .
Proof.
If is of class , then there exists by Proposition 3.1(b) a grade perfect ideal in with
[TABLE]
As , Corollary 5.8 yields , which forces . ∎
Appendix. Multiplication in Tor algebras of linked
ideals—proofs
A.1 Setup.
Let be a local ring with maximal ideal . Let be a grade perfect ideal; set
[TABLE]
Let be a minimal free resolution over and set
[TABLE]
Let , , and denote bases for , , and .
A.2 Multiplicative structures.
Adopt Setup A.1. The homology classes of the bases for , , and yield bases
[TABLE]
As recalled in 2.1, bases can be chosen such that the multiplication on is one of:
[TABLE]
A.3 Linkage.
Adopt Setup A.1. Let be a complete intersection ideal generated by a regular sequence and set . Let be the Koszul complex and be a lift of the canonical surjection to a morphism of DG algebras.
[TABLE]
Let denote the generators of . From the mapping cone of this morphism one gets, see e.g. [4, Prop. 1.6], a free resolution of over :
[TABLE]
The complex carries a multiplicative structure given by [4, Thm. 1.13] in terms of the basis
[TABLE]
Here we recall from [4, Thm. 1.13] the products that involve the last three vectors in the basis for .
[TABLE]
From the free resolution one gets a minimal free resolution of that we denote . From [4, (1.80)] one has :
[TABLE]
The identity yields
Set
[TABLE]
and denote the homology classes of the basis vectors as follows:
[TABLE]
Notice that (A.3.6) yields
[TABLE]
A.4 Linkage via minimal generators.
Adopt the setup established in A.1–A.3. If are part of a minimal system of generators for , then the homomorphism is given by for .
One has . More precisely, let , , and denote the three elements in , now [4, Prop. 1.6] yields
[TABLE]
so in the algebra one has
[TABLE]
Moreover, the rank of is the number of linearly independent products mod for . More precisely, if holds for some , then [4, Prop. 1.6] yields
[TABLE]
so in one has
[TABLE]
In particular, it follows from (A.4.1) that the only possible non-zero products among the basis vectors for that involve , , or are:
[TABLE]
A.5 Regular sequences.
Adopt Setup A.1. Let be a minimal set of generators for ; the homomorphism is given by . By a standard argument, see for example the proofs of [9, Prop. 2.2] or [6, Lem. 8.2], one can add elements from to modify the sequence of generators such that form a regular sequence. This corresponds to a change of basis on , where for is replaced by with coefficients . Notice that the homology classes in of the basis vectors do not change: one has . This means that given any basis for one can without loss of generality assume that the generators of corresponding to form a regular sequence. We tacitly do so in A.6–A.9.
For ease of reference, still in A.6–A.9, we recall from (2.2.1) the parameters that describe the multiplicative structures:
[TABLE]
The rest of this appendix is taken up by the arguments for Propositions 3.1–3.4.
A.6 Proof of Proposition 3.1.
Adopt Setup A.1.
(a): One may assume that the nonzero products of elements from (A.2.1) are
[TABLE]
Proceeding as in A.3, it follows from A.4 that is directly linked to an ideal whose Tor algebra has bases
[TABLE]
In particular, one has and . Further it follows from (A.4.3) and (A.4.4) that the nonzero products in that involve and are precisely
[TABLE]
This means that is at least , so is of class or ; see (A.5.1).
Assume towards a contradiction that is of class . Per (A.2.2) there are bases for and for with nonzero products
[TABLE]
Write and ; now (1) yields
[TABLE]
As the vectors and are linearly independent, so are the vectors and . That is, the matrix
[TABLE]
has rank , whence it follows that the product
[TABLE]
is nonzero, and that contradicts (1).
(b): One may assume that the nonzero products of elements from (A.2.1) are
[TABLE]
Proceeding as in A.3, it follows from A.4 that is directly linked to an ideal whose Tor algebra has bases
[TABLE]
In particular, one has and . Further it follows from (A.4.3) and (A.4.4) that the nonzero products in that involve , , and are precisely
[TABLE]
As the ideal is per (A.5.1) of class or , and the argument from the proof of part (a) applies to show that is not of class .
(c): One may assume that the nonzero products of elements from (A.2.1) are
[TABLE]
Proceeding as in A.3, it follows from A.4 that is directly linked to an ideal whose Tor algebra has basis (2). In particular, one has and . Further it follows from (A.4.3) and (A.4.4) that the nonzero products in that involve , , and are precisely
[TABLE]
Thus one has , in particular is of class ; see (A.5.1).
(d): One may assume that the nonzero products of elements from (A.2.1) are
[TABLE]
Proceeding as in A.3, it follows from A.4 that is directly linked to an ideal whose Tor algebra has bases
[TABLE]
In particular, one has and . Further it follows from (A.4.3) and (A.4.4) that the nonzero products in that involve and are precisely
[TABLE]
As by 2.4, one has , , and , so per (A.5.1) the ideal is of class , , or , and if is of class or , then holds. If is of class , then per (A.2.2) there are bases for , for , and for with nonzero products
[TABLE]
Write . It follows from the nonzero product that is nonzero. As per (A.5.1) it follows that there are two linearly independent products of the form which contradicts (3).
(e): One may assume that the nonzero products of elements from (A.2.1) are
[TABLE]
Proceeding as in A.3, it follows from A.4 that is directly linked to an ideal whose Tor algebra has bases
[TABLE]
with and .
A.7 Proof of Proposition 3.2.
Adopt Setup A.1 and set and . One may assume that the nonzero products of elements from (A.2.1) are
[TABLE]
Proceeding as in A.3, it follows from A.4 that is directly linked to an ideal whose Tor algebra has bases
[TABLE]
In particular, one has and . Further it follows from (A.4.3) and (A.4.4) that the nonzero products in that involve are precisely
[TABLE]
Set and ; evidently one has
[TABLE]
Notice that is not complete intersection, as one has ; cf. 2.3. That is, is of class , , , or .
(a): If holds, then (2) yields and per (A.5.1) it follows that the ideal is of class , , or .
If is of class , then there are per (A.2.2) bases for , for , and for with nonzero products
[TABLE]
By (A.5.1) and (2) one has , so assume towards a contraction that holds. Write and . By (1) and (3) one has
[TABLE]
and, therefore, or . From the equalities
[TABLE]
it follows that is nonzero for some , which contradicts the assumption ; see (1). Thus holds.
If is of class , then (A.5.1) and (2) yield , so holds.
If is of class , then there are per (A.2.2) bases for , for , and for with nonzero products
[TABLE]
Write . The product is by (1) nonzero, so (4) yields . From
[TABLE]
it follows that there are linearly independent products of the form , which forces ; see (1). Thus holds by (2).
(b): If holds, then (2) yields , and per (A.5.1) it follows that the ideal is of class . Moreover, holds by (a).
(c): Assume that holds. As one has , see (2), it follows per (A.5.1) that is of class or . By (2) it suffices to prove that holds.
If is of class , then per (A.5.1) one has so trivially holds.
If is of class , then as in the proof of part (a) there exist bases with the nonzero products given in (4). Write
[TABLE]
[TABLE]
The vectors and are linearly independent, while one has
[TABLE]
it follows that is nonzero. Thus, one has
[TABLE]
It follows that there are linearly independent products of the form , which forces ; see (1).
(d): If holds, then (2) yields , and per (A.5.1) it follows that the ideal is of class or . If is of class , then there are per (A.2.2) bases for and for with nonzero products
[TABLE]
It follow that for any element in , the map given by multiplication by has rank at most . However, multiplication by has rank ; see (1). Thus is not of class and hence of class ; finally (c) yields .
(e): Assume that holds. By (2) one has , so the ideal is per (A.5.1) of class , , or . If is of class then, as in the proof of part (d), there are bases with nonzero products as in (5). Write and . By (1) and (5) one has
[TABLE]
It follows that at least one of , , or is nonzero. From the expressions
[TABLE]
it now follows that there are two linearly independent products of the form , which contradicts the assumption ; see (1). Thus is not of class .
(f): It follows from (e) that the ideal is of class or . If is of class , then as in the proof of part (a) there exist bases with the nonzero products given in (3). Write and . Now (1) and (3) yield
[TABLE]
and, therefore, or . As one has
[TABLE]
it follows that is nonzero for some , which contradicts the assumption ; see (1). Thus is not of class .
(g): Assume that is of class . The subspace
[TABLE]
has rank . By (A.4.4) one has for all , so has rank at least .
A.8 Proof of Proposition 3.3.
Adopt Setup A.1 and set and . One may assume that the nonzero products of elements from (A.2.1) are
[TABLE]
Proceeding as in A.3, it follows from A.4 that is directly linked to an ideal whose Tor algebra has bases
[TABLE]
In particular, one has and . Further it follows from (A.4.3) and (A.4.4) that the nonzero products in that involve are precisely
[TABLE]
Set and ; evidently one has
[TABLE]
Notice that is not complete intersection, as one has by assumption and hence ; see 2.3 and 2.4. That is, is of class , , , or .
(a): If holds, then (2) yields , and per (A.5.1) it follows that the ideal is of class , , or .
If is of class , then one has , see (A.5.1) and (2), so assume towards a contraction that holds. By an argument parallel to the one given in the proof of Proposition 3.2(a), the nonzero product from (1) forces for some , which contradicts the assumption ; see (1). Thus holds.
If is of class , then (A.5.1) and (2) yield , so holds.
If is of class , then an argument parallel to the one given in the proof of Proposition 3.2(a) shows that the nonzero product from (1) forces linearly independent products of the form . This implies , see (1), whence holds by (2).
(b): If holds, then (2) yields , and per (A.5.1) it follows that the ideal is of class . Moreover, holds by (a).
(c): Assume that holds. As one has , see (2), it follows per (A.5.1) that is of class or . By (2) it is sufficient to prove that holds.
If is of class , then per (A.5.1) one has , so trivially holds.
If is of class , then the argument given in the proof of Proposition 3.2(c) applies to show that the nontrivial products and from (1) force linearly independent products of the form . This implies ; see (1).
(d): The proof of Proposition 3.2(d) applies.
(e): Assume that holds. By (2) one has , so the ideal is per (A.5.1) of class , , or . If is of class , then the argument given in the proof of Proposition 3.2(e) applies to show that the nonzero product from (1) forces two linearly independent products of the form , which contradicts the assumption ; see (1).
(f): It follows from (e) that the ideal is of class or . If is of class , then the argument in the proof of Proposition 3.2(f) applies to show that the nonzero product from (1) forces for some , and that contradicts the assumption ; see (1).
(g): Assume that and hold. The basis for is , and by (A.4.3) one has for all , so the three products , , and are non-zero. As is not of class , it follows that is of class ; see (A.2.2). Finally, (1) yields and .
A.9 Proof of Proposition 3.4.
Adopt Setup A.1 and set and . One may assume that the nonzero products of elements from (A.2.1) are
[TABLE]
Proceeding as in A.3, it follows from A.4 that is directly linked to an ideal whose Tor algebra has bases
[TABLE]
In particular, one has and . Further it follows from (A.4.3) and (A.4.4) that the nonzero products in that involve are precisely
[TABLE]
Set and ; evidently one has
[TABLE]
If then and it follows from 2.3 that is complete intersection. Thus there are bases for and for with non-zero products
[TABLE]
For linearly independent vectors
[TABLE]
the matrix
[TABLE]
has rank ; it follows that the product
[TABLE]
is nonzero. Thus, the products and are nonzero, so (1) yields .
Assuming now that or and hence or , it follows 2.3 that is not complete intersection. That is, is of class , , , or .
(a): If holds, then (2) yields and per (A.5.1) it follows that the ideal is of class , , or .
If is of class , then one has , see (A.5.1) and (2), so assume towards a contraction that holds. By an argument parallel to the one given in the proof of Proposition 3.2(a), the nonzero product from (1) forces for some , which contradicts the assumption ; see (1). Thus holds.
If is of class , then (A.5.1) and (2) yield , so holds.
If is of class , then an argument parallel to the one given in the proof of Proposition 3.2(a) shows that the nonzero product from (1) forces linearly independent products of the form . This implies , see (1), whence holds by (2).
(b): If holds, then (2) yields , and per (A.5.1) it follows that the ideal is of class . Moreover, holds by (a).
(c): Assume that holds. As one has , see (2), it follows per (A.5.1) that is of class or . By (2) it is sufficient to prove that holds.
If is of class , then per (A.5.1) one has , so trivially holds.
If is of class , then the argument given in the proof of Proposition 3.2(c) applies to show that the nontrivial products and from (1) force linearly independent products of the form . This implies ; see (1).
(d): The proof of Proposition 3.2(d) applies.
(e): Assume that holds. By (2) one has , so the ideal is per (A.5.1) of class , , or . If is of class , then the argument given in the proof of Proposition 3.2(e) applies to show that the nonzero product from (1) forces two linearly independent products of the form , which contradicts the assumption ; see (1).
(f): It follows from (e) that the ideal is of class or . If is of class , then the argument in the proof of Proposition 3.2(f) applies to show that the nonzero product from (1) forces for some , and that contradicts the assumption ; see (1).
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