By E. Grosswald
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5 Fix a term order on k[x1 , . . , xn ]. Use the artinian property of term orders to show that every ideal has a unique reduced Gr¨ obner basis. Do the same for submodules of free S-modules under any TOP or POT order. 6 Find a Borel-ﬁxed ideal that is not the initial monomial ideal of any homogeneous prime ideal in k[x1 , . . , xn ]. Are such examples rare or abundant? 7 Prove that if I is Borel ﬁxed and < is any term order, then gin< (I) = I. 8 Let I = x1 x2 , x1 x3 and ﬁx the lexicographic term order on S = k[x1 , x2 , x3 ].
Moreover, if min(m ) = uj , then deguj (mi ) < deguj (mj ). Therefore i < j with our labeling assumption. This means that the following vector is a nonzero ﬁrst syzygy on I: xu · ej − m · ei ∈ M. 3. THE ELIAHOU–KERVAIRE RESOLUTION 31 Fix any term order on S r that picks the underlined term as the leading term for every j = 1, . . , r and u = 1, . . , uj ; the POT order induced by e1 > e2 > · · · > er will do, for instance. 1), as u and j run over all pairs satisfying u < uj , equals the reduced Gr¨ obner basis of M , and in particular, generates M .
All six ideals J above are Borel-ﬁxed. Let us conclude this section with one more generality on Gr¨obner bases: they work for submodules of free S-modules. Suppose that F = S β is a free module of rank β, with basis e1 , . . , eβ . There is a general deﬁnition of term order for F, which is a total order on elements of the form mei , for monomials m ∈ S, satisfying appropriate analogues of the multiplicative and artinian properties of term orders for S. Initial modules are deﬁned just as they were for ideals (which constitute the case β = 1).
Bessel Polynomials by E. Grosswald