Associated graded algebra of an ideal generated by a regular sequence

The degree-$n$ monomials in the $f_i$ generate $J^n$, so the displayed graded map is surjective. To prove injectivity it suffices to show that every homogeneous relation ${\sum }_{\mathrm{\mid }I\mathrm{\mid }=n}{a}_I{f}^I=0$ has all $a_I\in J$: a relation lying in $J^{n+1}$ can be made zero by subtracting a degree-$n$ expression whose coefficients lie in $J$. We prove this coefficient assertion by induction on the sequence length $c$. For $c=0$, $J=0$ and the associated graded ring has only degree zero, where the map is the identity.

Suppose $c>0$ and the assertion holds for $J^{\mathrm{\prime }}=(f_1,\dots ,f_{c-1})$. Fix $n$ and write the relation as ${\sum }_{e=0}^l{H}_e{f}_c^e=0$, where $H_e={\sum }_{\mathrm{\mid }{I}^{\mathrm{\prime }}\mathrm{\mid }=n-e}{a}_{I^{\mathrm{\prime }},e}{f}^{I^{\mathrm{\prime }}}$ and $l\le n$. We induct on $l$. If $l=0$, the assertion for $J^{\mathrm{\prime }}$ puts all coefficients in $J^{\mathrm{\prime }}\subset J$. If $l>0$, reduction modulo $(J^{\mathrm{\prime }}{)}^{n-l+1}$ gives $f_c^l{H}_l\in (J^{\mathrm{\prime }}{)}^{n-l+1}$. By the coefficient assertion for $J^{\mathrm{\prime }}$ (also applicable to an expression lying in the next power), $f_c^l{a}_{I^{\mathrm{\prime }},l}\in J^{\mathrm{\prime }}$ for every $I^{\mathrm{\prime }}$. Since $f_c$ is a nonzerodivisor on $R\mathrm{/}{J}^{\mathrm{\prime }}$, every $a_{I^{\mathrm{\prime }},l}$ lies in $J^{\mathrm{\prime }}$.

Consequently $H_l\in (J^{\mathrm{\prime }}{)}^{n-l+1}$; express it as ${\sum }_{\mathrm{\mid }{I}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\mid }=n-l+1}{b}_{I^{\mathrm{\prime }\mathrm{\prime }}}{f}^{I^{\mathrm{\prime }\mathrm{\prime }}}$. Absorb $f_c{H}_l$ into $H_{l-1}$ and remove the term with exponent $l$. This gives a relation of the same degree with largest exponent $l-1$. Its modified coefficients are $a_{I^{\mathrm{\prime }\mathrm{\prime }},l-1}+f_c{b}_{I^{\mathrm{\prime }\mathrm{\prime }}}$ and its other coefficients are unchanged. The induction on $l$ puts all modified coefficients in $J$, hence all original ones in $J$, since $f_c{b}_{I^{\mathrm{\prime }\mathrm{\prime }}}\in J$. Together with the coefficients of $H_l$, this proves the assertion and completes the induction on $c$.

The coefficient assertion proves injectivity in every degree. In degree one, the isomorphism identifies $(R\mathrm{/}J{)}^c$ with $J\mathrm{/}{J}^2$ on the displayed classes. The symmetric algebra of this free module is the polynomial algebra, yielding the canonical symmetric-algebra identification. Only regularity of the ordered sequence and ideal-power generation were used.