Seminar Algebra & Geometry: Winfried Bruns
Winfried Bruns, Universität Osnabrück
Hilbert depth and Stanley decomposition
We report on joint work with Chr. Krattenthaler and J. Uliczka.
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 $M$. 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
We test our new notion on the syzygy modules of the residue
class field of $K[X_1,\dots,X_n]$ (as usual identified with
$K$). Writing $M(n,k)$ for the $k$-th syzygy module, we show
that the Hilbert depth of $M(n,1)$, namely the irrelevant
maximal ideal, is $\lfloor(n+1)/2\rfloor$. Furthermore, we show
that, for $n > k \ge \lfloor n/2\rfloor$, the (multigraded)
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.
Related ideals are the powers of the irrelevant maximal ideal.
For them the (standard graded) Hilbert depth can be computed