Let $R$ be a regular local ring and let $I \subset R$ be an almost complete intersection ideal, that is, $\mu(I)=\text{ht}(I)+1$ where $\mu(I)$ is the number of minimal generators of $I$ and $ht(I)=\text{dim}(R)-\text{dim}(R/I)$. Is it true that $\text{depth}(R/I)=\text{dim}(R/I)-1$? If not, is there any other relation between almost complete intersections and depth?



Depth of almost complete intersection rings
In general, depth can be zero. Let me give an example.


Let $X\subset\mathbb{P}^n$ be a smooth variety of dimension $d$ which is subcanonical (this means $K\_X=O\_X(\*)$ for some integer $\*$) and NOT arithmetically normal (this means the map $H^0(O\_{\mathbb{P}^n}(l))\to H^0(O\_X(l))$ is not onto for some $l$).


Let $m=n-d$ and choose general $m$ hypersurfaces $f\_1,\ldots,f\_m$ vanishing on $X$ of large enough degree. Then, $V(f\_1,\ldots,f\_m)=X\cup Y$ where $Y$ is another $d$ dimensional variety and the $f\_i$s define a complete intersection. One has an exact sequence,
$$0\to K\_X(\*)\to O\_{X\cup Y}\to O\_Y\to 0,$$
for a suitable integer $\*$.


Let $R$ be the homogeneous co-ordinate ring of $\mathbb{P}^n$ and let $I$ the defining ideal of $X$. Taking global sections in the above sequence gives,
$$0\to H^0\_\*(K\_X)=A\to H^0\_\*(O\_{X\cup Y})=R/(f\_i)=S\to S/A\to 0.$$


One has a natural inclusion $R/I\to A$, by our assumption on $K\_X$ and this inclusion is not an isomorphism. The cokernel $T$ is non-zero of finite length. Composing this inclusion with the inclusion of $A\to S$, we get an inclusion, $R/I\to S$ and let the cokernel be $B$. Notice that $B$ is almost complete intersection and by snake lemma we have an exact sequence $0\to T\to B\to S/A\to 0$ and thus depth of $B$ is zero.


To get a regular local ring, we may localize at the irrelevant maximal ideal of $R$ without changing the final claim.


There are several ways of constructing $X$ as above. Let me describe one. Take a non-degenerate smooth complete intersection $X$ of dimension $d$ in $\mathbb{P}^{n+1}$ where $n>2d$. Then, take a generic projection to $\mathbb{P}^n$.


