Exact sequences.
Graded modules and their Hilbert series.
Exact sequences of graded modules give an alternating sum formula for Hilbert series.
Let lambda be a weighting on the variables, a natural number for each x_i.
With this, we can generalize the notions of
Then the easy theorem:
homog_lambda(I) + ideal(y) = init_lambda(I) + ideal(y).
Stupid blogger.com is stealing my angle brackets again.
Note that on the LHS the familiar lambda is (1,1,1,1,...,1),
whereas on the RHS the familiar lambda is (N^n, N^{n-1}, ..., N) where N is very large. (That picks out the lex-first term of any fixed polynomial, once N is big enough.)
