Gröbner bases of syzygies and Stanley depth

The Stanley Conjecture on Intersections of Four Monomial Prime Ideals

“…It is easy to see that the corollary holds for n ≤ 2. If n ≥ 3 then sdepth S I ≥ 2 = depth I by [3,Theorem 3.4], which is enough as shows our Lemma 1.3. For the sake of the completeness we give below another proof applying the above proposition.…”

Monomial Ideals, Computations and Applications

“…However since m i KOEZ i is a free KOEZ i -submodule of M and since x j m D 0 for all j , this is only possible if KOEZ i D K. Thus sdepth.M / D 0.As a second application we obtain the following result of[62, Theorem 2.2], see also[31, Corollary 2.12].Theorem 28. This yields the desired conclusion.…”

“…In Sect. The first is that Stanley depth zero implies depth zero, the second is that the kth syzygy module of a Z n -graded module has Stanley depth at least k. The second result (with a more difficult argument) was first shown in [62]. This result has two nice consequences which we cite from [32].…”

“…For our next result we will use the following proposition which can be extracted from the work of Keller et al [119] and Gi et al [76] and as it is summarized in [62].…”

“…7 to compute the Stanley depth leads to interesting combinatorial considerations in even seemingly simple cases as for example in the case of the graded maximal ideal m of a polynomial ring in n variables. Here we use these extensions to show, following the arguments of [62], that the lower bound for the Stanley depth of a squarefree monomial ideal in a polynomial ring in n variables is approximately of size p n. In Sect. We present in Sect.…”