What is Castelnuovo-Mumford regularity of this algebra?

Let $M=\mathbb{C}[f_1,f_2,\ldots,f_r]$ is finitely generated algebra, $f_i \in S:=\mathbb{C}[x_1,x_2,\ldots,x_n],$ $\deg(x_i)=1, 1<\deg(f_i)<99.$ Suppose that minimal free resolution of $S$-module $M$ has form $$ 0 \longrightarrow S(-102) \longrightarrow S(-50) \oplus S(-20) \longrightarrow M \longrightarrow 0, $$ Is it true that the Castelnuovo-Mumford regularity of $M$ equals 100?

flag	asked Mar 9 2011 at 19:29 Melania 249●5

Well, the CM-regularity can be read off a resolution $\cdots \bigoplus S(-j)^{b_{1j}} \to \bigoplus S(-j)^{b_{0j}}\to M\to 0$ as $\max \{j-i : b_{ij}\neq 0\}$, so I guess you get 101. – J.C. Ottem Mar 9 2011 at 20:14

Looks to me like M is a sub-algebra of S. How is the S-module structure defined on M ? – Ralph Mar 9 2011 at 22:24

Yes, I wondered about this too. Perhaps you should clarify the question, Melania.. – J.C. Ottem Mar 9 2011 at 22:38

Yes, of cource $M$ is subalgebra of $S$. @J.C. Ottem. I thought that $M=S(-\deg(f_1)) \oplus S(-\deg(f_2)) \oplus \cdots \oplus S(-\deg(f_r)).$ Then the ansver is 100. – Melania Mar 10 2011 at 20:36