This material has been published in
J. Commut. Algebra 2 (2010),
327-357,
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Stanley decompositions and Hilbert depth in the Koszul complex
(22 pages)
Abstract.
Stanley decompositions of multigraded modules M over polynomials rings
have been discussed intensively in recent years. There is a natural
notion of depth that goes with a Stanley decomposition, called the
Stanley depth. Stanley
conjectured that the Stanley depth of a module M is always at least
the (classical) depth of M.
In this paper we introduce a weaker type of decomposition, which we
call Hilbert decomposition, since it only depends on the Hilbert
function of M, and an analogous notion of depth, called
Hilbert depth.
Since Stanley decompositions are Hilbert decompositions, the latter
set upper bounds to the existence of Stanley decompositions. The advantage
of Hilbert decompositions is that they are easier to find.
We test our new notion on the syzygy modules of the
residue class field of K[X1,...,Xn]
(as usual identified with K). Writing M(n,k) for
the k-th syzygy module, we show that the Hilbert depth of
M(n,1) is [(n+1)/2]. Furthermore, we show that,
for n > k >= [n/2],
the Hilbert depth of M(n,k) is equal to n-1.
We conjecture that the same holds for
the Stanley depth. For the range n/2 > k > 1, it seems impossible
to come up with a compact formula for the Hilbert depth. Instead, we
provide very precise asymptotic results as n becomes large.
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