The correctness of the SBF Algorithm is stated by the next theorem.
Proof. We construct an SBF submodule of generated in at most the degrees .
Let
be a strongly stable submodule of
with
and corner values sequence
. Set
, for
. From [
1] Lemma 4.5, we may assume that
, for all
; furthermore, by Lemma 1, we may suppose that each ideal
is a finitely generated Borel ideal such that
, whenever
. We construct
rearranging the blocks and the sub-blocks of the ideals
, for
.
We distinguish two cases: ; .
First, we consider the case .
Step 1. Construction of .
Let us consider the
-degree blocks in
M with the greatest number of monomials. If
is such a block, we choose
. In order to construct
, we proceed as follows. Consider the following set of monomials of degree
:
If , we set .
Otherwise, if , let be the greatest monomial of degree with not belonging to . If , is a set with the largest number of elements, and such that , we set .
If , then and . Let , and consider the set . If , then it will come into play in the construction of the -degree generators of , as we will see in the sequel. Otherwise, will not give any contribution for the computations of such generators. Let , i.e., . The set is a segment of monomials of degree . Indeed, one can observe that, if , then and .
In order to construct the
-degree generators of
, let us consider the set:
If , let . Otherwise, if , let be the greatest monomial of degree with not belonging to . Setting (), we consider the elements with the greatest cardinality and such that . If is such an element, for some , we set .
If the -degree sub-block of the strongly stable ideal is not empty, then it will come into play in the construction of . More specifically, if , for , then the segment , () will be considered in the construction of , if it is not empty.
Proceeding in this way, we obtain a strongly stable ideal
of
S which is generated in at most
degrees and such that each
-degree block
(
) is determined by the set:
It is relevant to point out that in some degree , a certain -degree sub-block of (; ) can arise, as in the -degree cases. Such segments will be involved in the computation of , as we will explain in a while.
Step 2. Construction of .
In order to construct , we manage the blocks and the sub-blocks not involved in the construction of .
First, we examine all the blocks , with , where t is the integer defined in Step 1. Among all these sets, we consider the ones that are maximal -degree blocks in M. If is such a set, for some , we choose . If for all , then we set .
Let
. Consider the sets:
where
and
are the sets defined in Step 1. If the set defined in (
3) is empty, let
. Otherwise, if
is the greatest monomial of degree
with
not belonging to
, we test all
with the greatest cardinality, and such that
. If
is such an element, let
. Reasoning as in Step 1, if
, i.e.,
, for some
, and,
(it has to be
), then the segment
comes out. Such a set will be considered in the construction of the
-degree generators of
. Similarly, if
and
, then the set
will come into play in the construction of
, if it is a non-empty set.
Finally, if , for all , then , and we can construct using the above arguments on .
In order to get
, setting
, we consider the set:
where
,
and
are the sets defined in the
-degree case of Step 1.
If the set in (
4) is empty, let
. Otherwise, if
is the greatest monomial of degree
with
not belonging to
, we test all
with the greatest cardinality and such that
. If
is such an element, we set
. Let
, i.e.,
, for some
. If
, then such a set will contribute to the construction of the
-degree generators of
(see Step 1, construction of
). Otherwise, it will not give any contribution for such generators. A similar reasoning, follows as in the previous
-degree case, if
.
Going on this way, we obtain a strongly stable ideal
of
S, which is generated in at most
degrees and such that each block
(
) is determined either by the set:
where the
-degree blocks
have not been involved in the construction of
, or by a certain
-degree sub-block arising in the construction of
. Moreover, the nonempty sub-blocks of
(
) that will arise during the creation of
will be involved in the calculation of
.
Now, let us examine the special segments that can appear during the construction of . Let us consider the -degree case described in Step 1. The set , with , gives a contribution to the construction of the -degree generators of the ideal () for which . In other words, we can construct a strongly stable ideal such that , with and . Note that means that .
Assume , . In such a case, may give a contribution to the -degree generators of (i.e., ). Note that such a case is achieved if has the greatest cardinality among all the blocks, the sub-blocks and the segments Z of M that are not yet involved in the construction of the -degree generators of , and such that is equal to the greatest monomial with not belonging to . If , or does not satisfy the conditions above, then we look for a block, a sub-block or a segment of M not yet involved in the construction of the ideals and satisfying the conditions above. If it does not exist, we set ; and so on; similarly if . Furthermore, the same reasoning can be iterated for the segments arising in degrees , .
Finally, proceeding in the same way as in Steps 1 and 2, due to the structure of M, all the monomial generators of M are swapped in a suitable way so that the monomial submodule is an SBF submodule such that and = .
Now, we consider the second case. Let .
We construct an SBF submodule , such that with , for all , and , with . The monomial submodule will be obtained by using the criterion described in Steps 1 and 2. Note that does not give any contribution to the computation of the extremal Betti numbers of , and . ☐
We close this section by considering some examples where the algorithm in Theorem 1 is used. First, we consider a complicated example suitably chosen in order to show that all the cases considered in Theorem 1 can really occur in a single concrete situation.
Example 8. Let
. Set
,
,
, and
,
,
, and
. Consider the monomial submodule
of
in
Table 17:
M is a strongly stable submodule with
,
and:
Indeed, the ideals and do not give any contribution to the computation of the extremal Betti numbers of M, i.e., .
Using Theorem 1, we construct a monomial module , with and .
Therefore, in order to construct , we manage the blocks and sub-blocks of the ideals in . More specifically, when we speak about the blocks (or sub-blocks) of M, we refer to the blocks (or sub-blocks) of the corner ideals .
Construction of
: Let us consider the three-degree blocks of
M with the greatest cardinality. From
Table 17, there exists
such that
. Hence, let:
Now, let us consider the set of monomials:
One can observe that
; whereas:
Denote by the greatest monomial of S of degree three with not belonging to . It is .
Note that, setting:
the set
is a segment of degree four. It will come into play in the characterization of the four-degree generators of the ideals
.
In order to construct
, let us consider the set:
Hence, since the greatest monomial of
S of degree five with
not belonging to
is
, we set:
Observe that:
will be used for the construction of the five-degree generators of
and
.
Table 18 represents the finitely generated Borel ideal
:
Construction of
: Let us consider the non-zero three-degree blocks of
M not involved in the construction of
, i.e.,
, for
. Since
, let:
In order to determine the four-degree generators of
, we have to take into account the sets:
Let
be the greatest monomial of
S of degree four with
not belonging to
. It is:
Note that the set:
is a segment of degree four, which comes into play for determine the four-degree generators of
.
In order to construct
, we consider the following set:
Since,
,
and moreover,
is the greatest monomial of
S of degree five with
not belonging to
; let:
Construction of
: In order to determine the three-degree generators (four-degree generators, five-degree generators, respectively) of
, we have to consider the non-zero three-degree blocks (four-degree blocks, five-degree blocks, respectively) of
M not involved in the construction of
and
, and moreover, in the case of the
-degree generators we should also consider the sub-blocks arising during the construction of
(see (
9), (
10)).
We have obtained a monomial submodule
of
(
Table 21) , where the ideals
(
) are:
is an SBF submodule of
generated in degrees
with
and
. Indeed, the corner matrix of
is: