Let K. denote the graded Koszul complex associated to
the
regular sequence (x0, …, xn)
in the graded polynomial ring A =
k[x0, …,
xn],
[mid ]xi[mid ] = 1 for all i, over
an
arbitrary field k. Let K′. denote the Koszul complex
associated to another regular sequence of homogeneous elements
(p0, …, pn) in
A.
In [5] we have studied ranks of graded chain complex
morphisms
f.[ratio ]K′.→K′. with the property
f0 = id. Let Ωk (respectively,
Ω′k) denote the kernel of the Koszul differential
d[ratio ]Kk→
Kk−1 (respectively, d′[ratio ]
K′k→
K′k−1),
and let fk[ratio ]
Ω′k→Ωk
denote the restriction of fk. The
main result was that Rank
(fk)>n−k.