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We have the following condition:

For each $i=2,...,m$ multiplication by $f_{i}$ is injective on $S/(f_{1},...,f_{i-1})$, where $S=k[T_{0},...,T_{n}]$, $m \leq n$, and the $f_{i} \in S_{d_{i}}$ are non-zero homogeneous elements of degree $d_{i}>0$.

Given this, we wish to compute the Hilbert polynomial of $M=S/(f_{1},...,f_{m})$.

The general theory of Hilbert polynomials makes sense when I read it, but I am somewhat confused in how to approach this problem. Any help would be appreciated!

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Use induction by using the following exact sequence: $$0\to S/(f_{1},...,f_{i-1})\stackrel{f_i\cdot}\to S/(f_{1},...,f_{i-1})\to S/(f_{1},...,f_{i})\to 0.$$

But wait, the multiplication by $f_i$ is not a graded homomorphism! Then we have to correct the grading by a shift and the exact sequence of graded $S$-modules looks like $$0\to S/(f_{1},...,f_{i-1})(-d_i)\stackrel{f_i\cdot}\to S/(f_{1},...,f_{i-1})\to S/(f_{1},...,f_{i})\to 0.$$ Now you get the following relation for Hilbert series: $H_{S/(f_{1},...,f_{i})}(t)=(1-t^{d_i})H_{S/(f_{1},...,f_{i-1})}(t)$.

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