more details.

The modules $H_i(\bf f)$ and $H^i(\bf f)$ depend only on the ideal $J$ generated by $f_1,f_2,\ldots,f_n$. Note however, that changing the order changes $\alpha_p: H_p({\bf f}){\longrightarrow}H^{n-p}({\bf f})$ by a sign.

In general cohomology is also [*dual*]{} to homology, in the following sense.

Let $P$ be a ring, $(F_{\bullet},\partial_{\bullet})$ a complex of free $P$-modules and $M$ a $P$-module. Then there exists a spectral sequence with $$E_2^{p,q}=Ext^p_P(H_q(F_{\bullet}),M)\Longrightarrow H^{p+q}(Hom_P(F_{\bullet},M)).$$

This is of course classical, see for example [@eilen section XVI.2]. It can be shown as follows: consider an injective resolution $(G^{\bullet},\delta^{\bullet})$ of $M$ and the double complex $C_{\bullet,\bullet}$ with terms $C_{p,q}=Hom_P(F_{q}, G^{p})$ and differentials induced from $\partial$ (going up. increasing $q$) and $\delta$ (going right, increasing $p$). The spectral sequence obtained by first taking $\delta$ and then $\partial$ degenerates at $E_2$ and has $H^p(Hom_P(F_{\bullet},M))$ at spot $(p,0)$ as homology. The other spectral sequence obtained by first taking $\partial$ and followed by $\delta$ has as $E_2$-term $$\begin{array}{c|cccc}
q&Hom_P(H_q, M)&Ext^1_P(H_q, M)&\ldots&Ext^p_P(H_q, M)\\
\vdots&\vdots&\vdots& &\vdots\\
1&Hom_P(H_1, M)&Ext_P^1(H_1, M)&\ldots&Ext^p_P(H_1, M)\\
0&Hom_P(H_0, M)&Ext_P^1(H_0, M)&\ldots&Ext^p_P(H_0, M)\\
\hline
&0&1&\ldots&p
\end{array}$$ where $H_i:=H_i(F_{\bullet})$.

By taking $M=P$, this spectral sequence can be used to connect the homology and the cohomology of a complex. As a corollary, we note the two 