special situations which lead to exact sequences.

\[corollaryspek\] (i) If the groups $Ext^k_P(H_i(F_{\bullet}),P)$ vanish for $k>1$, the spectral sequence collapses to exact sequences $$0{\longrightarrow}Ext^1_P(H_k(F_{\bullet}),P){\longrightarrow}H^{k+1}(F^{\bullet}){\longrightarrow}Hom_P(H_{k+1}(F_{\bullet}),P){\longrightarrow}0.$$ (“universal coefficient theorem”)\
(ii) If the homology groups $H_k:=H_k(F_{\bullet})$ vanish for $k>1$, we get a long exact sequence $$\cdots {\rightarrow}Ext^k_P(H_0,P){\rightarrow}H^k {\rightarrow}Ext^{k-1}_P(H_1,P)\stackrel{d_2}{{\rightarrow}} Ext^{k+1}_P(H_0,P){\rightarrow}\cdots$$ where $H^i:=H^i(F^{\bullet})$.

We specialise the above to the case of an almost complete intersection.

A sequence ${\bf f}=f_1,f_2,\ldots,f_n$ of elements in a ring $P$ is said to defines an [*almost complete intersection*]{} if $codim(V(J))=n-1$.

It follows from \[vanishing\] that in this case one has two non-vanishing Koszul homology groups, $H_0({\bf f})=P/J$ and one further module $H_1({\bf f})$.

\[aci\] Let ${\bf f}=f_1,f_2,\ldots,f_n$ define an almost complete intersection in $P$ and put $H_i:=H_i({\bf f})$. Then:\
(i) $$Ext^{n-1}_P(H_0,P)\cong H_1$$ (ii) There is an exact sequence $$0 {\longrightarrow}Ext^{n}_P(H_0,P) {\longrightarrow}H_0 {\longrightarrow}Ext^{n-1}_P(H_1,P) \stackrel{d_2}{{\longrightarrow}} Ext_P^{n+1}(H_0,P) {\longrightarrow}0$$ (iii) For $k \ge n$ there are isomorphisms $$Ext^{k}_P(H_1,P) \stackrel{d_2}{{\longrightarrow}} Ext_P^{k+2}(H_0,P)$$

The self-duality \[selfdual\] of the Koszul complex gives $H^{n-1}\left({\bf f}\right)=H_1\left({\bf f}\right)$ and $H^n\left({\bf f}\right)=H_0\left({\bf f}\right)$ as only non-vanishing Koszul cohomology groups. As both modules are supported on $V(J)$, it follows from Ischebeck’s lemma that $Ext^p_P(H_i({\bf f}),P)=0$, for $p<n-1$. The theorem follows then immediately from the above long exact sequence \[corollaryspek\] (ii).

One-dimensional almost complete 