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Suppose $S$ is a polynomial ring, $I$ an ideal and $<$ some term order. Why is the Hilbert series of $S/I$ the same as the Hilbert series of $S/\mathrm{in}_<(I)$?

I truly suspect the answer is lurking in Chapter 15 of Eisenbud's Commutative Algebra (or even staring me in the face), but I can't put it together.

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You are right: this is Theorem 15.3 (Macaulay) in Eisenbud's book.

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