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Summary: ter, whose eventual aim is to prove Theorem 11.4 (after Example 11.40).
Along the way, we'll see how injective modules and injective resolutions
arise naturally, allowing their well-behaved homological behavior to rub off
onto irreducible resolutions. Also, we will attempt to dispel the common
belief that injective modules must necessarily be unwieldy behemoths, by
describing them combinatorially in the context of affine semigroup rings.
Let us illustrate the difference between free resolutions and injective
resolutions for the ideal I = x4
, x2
y2
, y4
from Example 11.3. The free
resolution of k[x, y]/I (i) covers the set of standard monomials modulo I
with all of N2
; (ii) uncovers the monomials in I using shifted copies of the
positive quadrant N2
; and finally (iii) excludes the monomials in I that
were uncovered too many times:
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