Large lower bounds for the betti numbers of graded modules with low regularity
Adam Boocher, Derrick Wigglesworth
TL;DR
This paper establishes new lower bounds on the Betti numbers of finitely-generated graded modules over polynomial rings with low regularity, revealing significant constraints on their algebraic complexity.
Contribution
It provides the first explicit lower bounds for Betti numbers of modules with bounded regularity and high codimension, extending understanding of their minimal free resolutions.
Findings
Sum of Betti numbers is at least (2^c + 2^{c-1}) times 00(M).
For codimension c 8, Betti numbers 1 to 00(c/2) are bounded below by 2 times 00(M) times binomial coefficients.
Results apply to modules with regularity at most 2a-2, where a is the minimal degree of a first syzygy.
Abstract
Suppose that is a finitely-generated graded module of codimension over a polynomial ring and that the regularity of is at most where is the minimal degree of a first syzygy of . Then we show that the sum of the betti numbers of is at least . In addition, if then for each , we show .
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Large Lower Bounds for the Betti Numbers of Graded
Modules with Low Regularity
Adam Boocher
A. Boocher, University of San Diego, San Diego, California, USA
[email protected] and [email protected]
and
Derrick Wigglesworth
D. Wigglesworth, University of Arkansas, Fayetteville, Arkansas, U.S.A.
[email protected] and [email protected]
Abstract.
Suppose that is a finitely-generated graded module (generated in degree [math]) of codimension over a polynomial ring and that the regularity of is at most where is the minimal degree of a first syzygy of . Then we show that the sum of the betti numbers of is at least . Additionally, under the same hypothesis on the regularity, we establish the surprising fact that if then the first half of the betti numbers are each at least twice the bound predicted by the Buchsbaum-Eisenbud-Horrocks rank conjecture: for , .
1. Introduction
Let be a polynomial ring over a field and let be a finitely generated graded -module of finite length. The total betti number is defined to be the sum of the betti numbers of . This number has been of recent interest, most notably in the context of the Total Rank Conjecture which predicts that . If , this conjecture was recently proved by Walker [8], who also showed that equality holds if and only if is isomorphic to modulo a regular sequence – such modules are called complete intersections.
Evidently if is not a complete intersection, then and since must be even, it follows that . In fact, there is reason to believe that if is not a complete intersection then must be considerably larger than . It was asked by Charalambous, Evans, and Miller in [3] whether it is true that . They proved that this is the case for arbitrary graded modules when and for all when is multi-graded. We remark that if is not of finite length, then the natural extension is to claim that
[TABLE]
Such an extension has recently been obtained for monomial ideals in [2] where it was also proved that equality is possible for all . The aim of the present paper is to prove that (1.1) holds for arbitrary provided that the regularity of is small relative to the degrees of its first syzygies.
Theorem 1.1**.**
Let be a graded -module of codimension generated in degree [math] and let be the minimal degree of a first syzygy of . If , then
[TABLE]
Our result is an extension of work by Erman [5], where he proved, under the same hypothesis on the regularity, that Erman’s work proves a special case of the Buchsbaum-Eisenbud-Horrocks rank conjecture which states that . Naturally, Erman’s bound will imply that when the regularity hypothesis holds. Noting that , the stronger bound in Theorem 1.1 asserts that on average, each betti number is at least times . We achieve this bound by showing that except in a small number of cases (which arise with ) it is true that the first half of the betti numbers are at least .
Theorem 1.2**.**
Let be a graded -module of codimension generated in degree [math] and let be the minimal degree of a first syzygy of . If then for each , .
This result implies a rather strong connection between the regularity of and its first few betti numbers. In the Artinian (finite-length) case, since the regularity can be interpreted as the socle degree, we can understand this result as making more precise the idea that having a small number of generators will naturally lead to a high socle degree. Our theorem provides bounds on this relationship which are new (even in the Artinian case).
As an example, consider the following statements for quadrics: Suppose is an ideal generated by quadrics of codimension . If has precisely minimal generators then is a complete intersection with regularity . On the other hand, the ideal has minimal generators and then has regularity . Theorem 1.2 implies that for the regularity of to drop below , must have at least minimal generators.
It seems to us rather bizarre that this theorem (like Erman’s results) should depend almost completely on the numerics coming from Boij-Söderberg Theory. This mysterious behavior is also apparent in McCullough’s work in [7] concerning the relationship between the regularity of an ideal and the degrees of half of its syzygies. In this vein, our results can be interpreted as saying that the degree of the first syzygy and the number of syzygies in the first half of the resolution can in some cases force the regularity to be large. We remark that the regularity bound is actually relaxed enough to include many interesting geometric examples. In [5], Erman presents several examples of modules that satisfy including smooth curves embedded by linear systems of high degree, toric surfaces, and Artinian rings whose socle degree is relatively low.
We comment now on our methods and how they differ from those of Erman. We begin as he did with standard Boij-Söderberg techniques to write an arbitrary betti diagram as a rational combination of normalized pure betti diagrams, whose entries are each a function of positive integers . In sections 2 and 3 we show that the proofs of our main theorems reduce to finding lower bounds for . Like Erman we reduce these calculations to the study of a function of variables. It is here that our analysis differs substantially from that of Erman.
Since Erman was concerned with a uniform bound for all betti numbers, his proof (in our notation) shows that . As we mentioned above, our main strategy to prove Theorem 1.1 is to focus on the first half of the betti numbers and prove that they are at least twice the bound that Erman proved. Roughly speaking we then want to show that for small . Since this statement is not true for all (nor is it true if the codimension is less than ) our analysis necessarily proceeds in a delicate way. In addition, if , since Theorem 1.1 holds whereas 1.2 does not, independent techniques are developed to address these cases. What ultimately makes the proofs difficult is that even if one fixes all but one variable, it is not necessarily the case that is an increasing function, and thus finding its minimum requires some care. Moreover, there are a whole host of cases where our general method fails – these arise primarily when the difference between the regularity of and the generating degree of a first syzygy of is very small. The reduction via Boij-Söderberg theory necessitates that we consider all of these cases, as otherwise our results would be significantly weaker. These special cases complicate the structure of our proof as evidenced by the flowchart (Fig. 2) which demonstrates how all the pieces fit together.
2. Boij-Söderberg Basics
In this section we will review the relevant pieces of Boij-Söderberg theory. Rather than state the theory in its fullest generality, we present only the version we need for our results. We begin with an example.
Example 2.1**.**
Let and take to be an ideal generated by random quadrics. Set . Similarly, let be a matrix of random quadrics and let . Finally, let . The betti diagrams of and are given below:
[TABLE]
We point out that the first two diagrams are pure in the sense that each column has at most one nonzero entry. The last betti diagram is not pure since the column representing the second syzygy module has two nonzero entries. Further, note that each of the first two diagrams is a sub-diagram of the third diagram, in the sense that the locations of the nonzero entries of the first two fit inside the third diagram. This will be made explicit in what follows.
Finally, we notice the rather astonishing fact that the third betti diagram (thought of as a matrix) can be written as a positive rational linear combination of the first two diagrams:
[TABLE]
The above example is an instance of the following, which is a summary of the main results in Boij-Söderberg Theory.
“The betti diagram of an (arbitrary) finite-length module
can be written as a positive rational linear combination of pure diagrams.”
We now set and work with finitely generated graded -modules . Henceforth all of our modules will be assumed to be generated in degree [math]; allowing for shifting, this is tantamount to saying that is generated in a single degree. If is a finite length module and each syzygy module of is generated in a single degree then we will say that has a pure resolution (or that is pure). Note that we require pure modules have finite length. For a pure module we let be the sequence whose -th entry is the degree of the generators of the -th syzygy module of . This increasing sequence of integers is called the degree sequence of . By we will mean the number , which corresponds to the regularity of the module .
Remark 2.2**.**
A finite length module is pure with degree sequence if and only if for each , the graded betti numbers of satisfy
[TABLE]
Remarkably, the betti numbers of pure modules are determined up to scalar multiple. Indeed, if a finite length module is pure with degree sequence then there is a scalar so that for all , the following holds:
[TABLE]
This was first proven by Herzog and Kühl [6] and the equalities above are called the Herzog-Kühl equations. Note that since we have that . In order to prove Theorems 1.1 and 1.2 we will study the rational functions and establish the following two theorems.
Theorem 2.3**.**
Suppose that and is a degree sequence of length with satisfying . Then
Theorem 2.4**.**
Let be a degree sequence of length with and .
- •
If then for each ,
[TABLE]
- •
If , the same conclusion holds unless
* and or*
* and *
Remark 2.5**.**
When there are only 36 degree sequences satisfying the regularity hypothesis but to which Theorem 2.4 does not apply. The pure diagrams are those that are subdiagrams of one of the following diagrams:
[TABLE]
The content of Theorems 2.3 and 2.4 is purely numerical. Their connection to our main theorems on betti numbers is achieved via the beautiful results of Boij-Söderberg Theory, developed in [4, 1]. This theory shows that the betti diagram of an arbitrary finite length module can be written as a finite rational linear combination of pure diagrams.
Given a module , its betti numbers are often arranged into a betti-diagram – thought of as a matrix (typically with the convention that is in the th column and the th row). With this convention the regularity of is equal to the index of the bottom row in the diagram. If is a degree sequence of length then we define to be the betti diagram with entry in column and row . By the Herzog-Kühl equations (2.1), if is a pure module with degree sequence then the betti diagram of will be a scalar multiple of .
Example 2.6**.**
We associate to the degree sequence the following diagrams:
[TABLE]
We use stars to emphasize that we care about the positions of the nonzero entries in the diagram, then use to denote the diagram of numbers .
Given two diagrams and we say that is a sub-diagram of if for each nonzero entry of , the corresponding entry in is also nonzero. If is the betti diagram of a finitely generated module then there are a finite number of degree sequences such that is a subdiagram of . We now summarize the results of Eisenbud-Schreyer and (respectively) Boij-Söderberg [4, 1] which show that a finite length module (respectively, one of codimension ) can be decomposed as a sum of pure diagrams.
Theorem 2.7** (Main Theorem of Boij-Söderberg Theory [4, 1]).**
Let be a finitely generated -module with betti diagram . Suppose that . If is the set of all pure sub-diagrams of having between and columns (indexed by their degree sequences with lengths between and ) then there exist non-negative rational numbers such that
[TABLE]
In particular, this implies that and more generally,
3. Reduction to Theorem 2.4
In this section we explain how to deduce our main theorems from their numerical versions stated in Section 2. We will then assume Theorem 2.4 and use it to prove Theorem 2.3. For convenience, all four theorems are restated in the diagram below.
Main Theorems on Betti Numbers
Theorem 1.1. Let be a graded -module of codimension generated in degree [math] and let be the minimal degree of a first syzygy of . If then
Theorem 1.2. Let be a graded -module of codimension generated in degree [math] and let be the minimal degree of a first syzygy of . If then for each , .
Main Numerical Results
Theorem 2.3. Suppose that , and is a degree sequence of length , and satisfying . Then
Theorem 2.4. If and and then for each ,
If and either or , then the same conclusion holds.
The theorems on the left follow more or less immediately from the corresponding theorems on the right via Boij-Söderberg theory. With the exception of a small number of special cases when , Theorem 2.3 will follow from Theorem 2.4, the proof of which will be postponed until Section 4.
3.1. Proofs of Theorems 1.1 and
Proof of Theorem 1.1.
Suppose is generated in degree zero, and is the minimal degree of a first syzygy of . By Theorem 2.7 there exist nonnegative rational numbers such that
[TABLE]
where runs over all degree sequences of length whose betti diagrams, , are sub-diagrams of . Let be such a degree sequence. Then and as we have assumed , it follows that
[TABLE]
Hence we can apply Theorem 2.3. Since every degree sequence appearing in the sum has length at least , Theorem 2.3 implies that . Hence we have
[TABLE]
Proof of Theorem 1.2.
The scaffolding is exactly the same as in the previous proof. If then equation (3.1) and Theorem 2.4 imply for
[TABLE]
3.2. Proof of Theorem 2.3
Proof of Theorem 2.3 when Theorem
2.4 holds.
Suppose that is a degree sequence satisfying the hypotheses of Theorem 2.4. Then let us add up all of the in pairs. If is odd, there are an even number of ’s. When summing, we can group them in pairs . Now since and by Erman’s Theorem. In all other pairs, we combine Theorem 2.4 with Erman’s result, and conclude that . Moreover, since the assumption on indices in Theorem 2.4 includes , the last pair is at least . Thus
[TABLE]
When is even, we proceed by pairing terms exactly as before. In this case however, there is a central term in the sum (the term ) which has no companion. We thus have:
[TABLE]
Proof of Theorem 2.3 for
.
By Remark 2.5 there are only 36 degree sequences that satisfy and for which Theorem 2.4 does not apply. Using Macaulay2 we checked that the sum of in each of these cases is at least . The reader is directed to the file computations.m2 included in our arXiv posting for explicit code that can be used to verify this statement. ∎
Proof of Theorem 2.3 for
.
For each value of , we will verify that via a direct computation. Suppose first that so that the degree sequence . We change notation to emphasize the nonlinear parts of by instead writing it as , where can easily be computed from the ’s. We may assume and our regularity assumption says . We want to prove that . Using the Herzog-Kühl equations, this is equivalent to the polynomial inequality
[TABLE]
If so that the resolution is linear, then the assumption that implies the inequality holds. On the other hand if the resolution is not linear, we observe that the left hand side is clearly an increasing function of , so it suffices to consider the case that , whereby the inequality becomes
[TABLE]
Evidently, each of these terms is positive at least two are nonzero (since and are not both [math]), so the inequality holds as desired.
Repeating an identical analysis with (so that ) again results in a polynomial inequality for which the left hand side is an increasing function of . After considering the linear case separately, we set , and are left to verify the polynomial inequality
[TABLE]
This will hold provided not all of .
The proof strategy for is exactly the same and begins by setting , then using the Herzog-Kühl equations to get a polynomial inequality. The expression thus obtained is now too complicated to be analyzed by hand, though it’s still very manageable for a machine. By writing it as a polynomial in , one can verify that all of the coefficients (besides the constant term) are positive and therefore that left hand side is increasing as a function of . Again substituting , one obtains an expression and factors it (with a computer) to arrive at an inequality in which all terms on the left hand side are positive except for the constant term. A simple computer verification shows that the inequality it holds for all . ∎
Remark 3.1**.**
The file computations.m2 included in our arXiv posting contains code to verify the numerical statements in this paper.
4. Proof of Theorem 2.4
In this section we will prove Theorem 2.4, which is the last ingredient needed to complete the proofs of our main results. We endeavor to show that for suitable and , we have
[TABLE]
Thus it is natural to study the function . Of course this function depends on parameters, so a simplification is required before a reasonable analysis can be performed. We will define a function depending on five parameters such that
[TABLE]
Main Notation: Let and set . Given , we define a modification of as follows:
[TABLE]
Considering now a degree sequence, we will focus our attention on its nonlinear parts.
[TABLE]
Notice then that we have
[TABLE]
The reader is urged to ignore these equations and press on to the example that follows, which should clarify the idea (and resolve the ambiguity when ).
Example 4.1**.**
Suppose that and then the betti diagrams for and would be formatted as shown
[TABLE]
Visually, we have kept in the same place, but have shifted all of the earlier numbers to the top of the diagram and all of the later ones to the bottom. Notice that in this example . In the right-hand diagram there are visible jumps of size and on either side of the in position .
Lemma 4.2**.**
If is a degree sequence then for all
[TABLE]
Proof.
We prove a slightly more general statement. Let and suppose that is a degree sequence with . Then
[TABLE]
As all the terms in the product are positive, a sufficient condition for is that
[TABLE]
for all . If then this is equivalent to requiring
[TABLE]
Conversely, if then the inequality is . To conclude, we simply observe that all of these inequalities hold for , whence the result follows. ∎
We now compute
[TABLE]
Definition 4.3**.**
We define the function as the coefficient of in the above computation. The domain of is , , , , ,
[TABLE]
In the sequel we will refer to each of the three fractions in the above equation as a grouping. When there are no terms in the first grouping. Similarly, when there are no terms in the third grouping.
Our present goal is to show that is at least for a suitable range of inputs (e.g. ).
Lemma 4.4**.**
* is increasing as a function of :*
[TABLE]
Proof.
If , then is equal to times an additional factor which has the form for some , which is evidently at least 1. If , then in addition to this extra factor, the numerators of the terms in the first grouping in will be larger than the corresponding terms on the left hand side of the inequality. ∎
One might hope that is an increasing function of . This is not the case as can be seen in Figure 1. However, note that in the figure is increasing for . It is no coincidence that as the following lemma shows.
Lemma 4.5**.**
If and then
[TABLE]
That is, if is at most then is an increasing function of .
Proof.
Let . Using our assumption on the regularity, we have
[TABLE]
This in turn implies
[TABLE]
Further if then
[TABLE]
Finally we compute
[TABLE]
This will be at least provided
[TABLE]
which is equivalent to:
[TABLE]
This is the inequality we have shown above. ∎
Remark 4.6**.**
Notice that Figure 1 shows that we cannot improve the bound . Further, note that in this proof we used that and that this came from our assumption that . If we relax that bound, even by one, say to then it will not be true that is an increasing function of . For instance, consider the following two degree sequences (with ):
[TABLE]
At this point we present a flowchart that indicates ultimately how we will prove Theorem 2.4. We have just seen (Lemmas 4.4 and 4.5) two crucial observations about the function . Using these, a few elementary computations would allow us to establish Theorem 2.4 for the vast majority of degree sequences of pure diagrams. However, as mentioned in the introduction, our reduction via Boij-Söderberg theory requires that we consider all degree sequences of pure sub-diagrams of the betti diagram of and many of these degree sequences are not covered by the lemmas above.
We begin in the upper right of the chart addressing the case of linear resolutions. These correspond to the case when and are handled by the following lemma.
Lemma 4.7**.**
If , then .
Proof.
If , then
[TABLE]
On the other hand, if there are no terms in the first grouping. Since and , there is at least one term in the middle grouping and we have
[TABLE]
Our approach is now as follows: by Lemma 4.7 we may assume that . For fixed our regularity assumption provides a minimum possible value of : we have and thus . In light of Lemma 4.4, it’s natural to set . We can then apply Lemma 4.5 and decrease to its minimum possible value of . However we will only do this when , since we only want to consider degree sequences with ; our argument will need modifications when . Thus, for and , we now consider the function defined by making these substitutions.
[TABLE]
We remind the reader that our goal is to find a lower bound for and point out that at this point we have (for and ):
[TABLE]
Lemma 4.8**.**
* is an increasing function of : .*
Proof.
We consider the quotient
[TABLE]
We want this to be at least . When we cross-multiply and subtract we are left with the inequality:
[TABLE]
which is evident. ∎
In consideration of this, since for all we show, with a few minor exceptions, that for relevant inputs.
Lemma 4.9**.**
If either or , then .
Proof.
We simply compute
[TABLE]
This will be at least if and only if
[TABLE]
Now
[TABLE]
If then this is which will be nonnegative provided . Otherwise, if either or is at least then one of or will be at least . Thus if then
[TABLE]
Restricting our attention to the situation where , the lemmas we have established are sufficient to conclude that for the vast majority of relevant inputs. The remaining cases (still assuming that ) are treated via direct computation.
Computation 4.10**.**
[TABLE]
As is an increasing function of , these computations will allow us to obtain the desired lower bound on when . Indeed, either Lemma 4.9 applies or else and which must be one of the numbers above.
We close with one final computation as well as a discussion of what happens for . The reader may note that the values of in these computations are creeping upwards; this is the first indication for the hypothesis that be greater than in our main theorems.
Computation 4.11**.**
[TABLE]
We now close by handling the case . Note that implies that . We may assume that and the assumption that implies that we may assume . What remains is to determine when
[TABLE]
There is a finite set of inputs for which this lower bound fails, and these are the source of the 36 betti diagrams of pure modules which satisfy our regularity bound but to which Theorem 2.4 does not apply.
Lemma 4.12**.**
For all and , we have
[TABLE]
That is, for all , the function is increasing as a function of .
Proof.
As usual, we want to establish the following inequality.
[TABLE]
Cross-multiplying, simplifying, and factoring, we find that this equivalent to
[TABLE]
which is evident as and . ∎
Lemma 4.13**.**
If , then .
Proof.
We compute
[TABLE]
This is greater than if and only if , which is the case for at least . ∎
As before, some sporadic cases will be handled by a few direct computations.
Computation 4.14**.**
[TABLE]
We have need of one final computation that will reduce from infinite to finite the number of degree sequences of pure diagrams that do not satisfy the hypotheses of our theorem. Indeed, if the regularity bound is strengthened by one and we assume that , then the minimum possible value of is . We compute:
Computation 4.15**.**
For and , we have .
We are now ready to put the jigsaw puzzle together and prove Theorem 2.4. For the reader’s convenience, we have restated it below in an equivalent form.
Proposition 4.16**.**
Let be a degree sequence with and . Then for each , . If and either or , then the same conclusion holds.
Proof of Proposition 4.16.
The proof amounts to piecing together the lemmas and computations above and is depicted in the flowchart (Figure 2). A key point is that for a fixed degree sequence , while (and the associated nonlinear parts and ) depends on the value of , the sum of is a function only of the original degree sequence and not of . For , refer to the flow chart.
If the resolution is linear so that , then Lemma 4.7 applies to give the desired conclusion. If , then we apply Lemma 4.4 and decrease to its minimum possible value while maintaining our regularity assumption. Then, if , we apply Lemma 4.5, decreasing to get
[TABLE]
Since , either or regardless of the value of . Thus, in all cases we may apply Lemma 4.8 decreasing the value of and then apply Lemma 4.9 to conclude
[TABLE]
If , we still apply Lemma 4.4. Then we note that this implies . Now Lemmas 4.12 and 4.13 allows us to conclude
[TABLE]
Now if , the above argument fails only for those values of where (because Lemma 4.9 fails); when , the argument needs no modification. If and , then we apply Lemmas 4.4, 4.5, and 4.8 just as above only this time we use Computation 4.10 to conlcude
[TABLE]
If , then rather than decreasing to in applying Lemma 4.5, we set and use Computation 4.11.
[TABLE]
If , the chain of inequalities (4.2) still holds for and the logic from above still applies for . Thus, the only remaining case is and our assumptions imply . When (resp. ), apply Lemma 4.5 to decrease to 7 (resp. 4), then apply Computation 4.11 to get
[TABLE]
If , the proof differs only in a few places and these are depicted in the flow chart by two blue arrows. The arrow on the left hand side concerns the setting where and , which implies that and . This time we apply Lemma 4.5 and decrease to the value of , then apply Lemma 4.12 setting and use Computation 4.14
[TABLE]
The second blue arrow concerns the case that , and for finitely many degree sequences, our method fails here. If , then we apply Lemma 4.4 decreasing to the minimum possible value of . Next apply Lemma 4.5 and set . Noting that , we use computation 4.15 to obtain
[TABLE]
Acknowledgments
We thank Daniel Erman for inspiring this project as well as for the many conversations about Boij-Söderberg theory over the years. We thank Craig Huneke for the suggestion to look at how the sum of the betti numbers behaves with respect to these Boij-Söderberg decompositions. A portion of this research was conducted at the Fields Institute and the second author thanks them for their hospitality during that period. Finally, we are grateful for helpful conversations with David Eisenbud, Srikanth Iyengar, Anurag Singh, and Mark Walker.
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