First, let us recall some definitions.
Let $P=\mathbb P^d_k$ be a projective space over a field $k$. Let $X$
be a closed subvariety of $P$ of dimension $r$. We say that $X$ is a
complete intersection if it is defined (as a variety) by $d-r$
homogeneous polynomials $F_1,\dots,F_{d-r}$. This notion depends on
the embedding $X\rightarrow P$.Let $Y$ be a locally Noetherian scheme. Let $f:X\rightarrow Y$ be a
morphism of finite type. We say that $f$ is a local complete
intersection at $x$ if there exist a neighborhood $U$ of $x$, a scheme
$Z$, a regular immersion $i:U\rightarrow Z$, and a smooth morphism
$g:Z\rightarrow Y$ such that $f|_U=g\circ i.$
I would like to know when a complete intersection is also a local complete intersection, if we understand this to mean that the inclusion $X\rightarrow \mathbb P^n_k$ is a local complete intersection morphism. I suspect this is always true.
We know $\mathbb P^n_k$ is always locally Noetherian, so that part of the definition is satisfied. I think we can just take $Z=\mathbb P^n_k$ and $U=X$, since I believe the canonical injection will always be regular. I do not know how to prove this, though. This would mean $g$ would have to be the identity. It seems like this is non-smooth for non-smooth varieties.
These thoughts lead me to the following questions.
$1.$ If $X$ is a complete intersection, how can we prove that the injection $X\rightarrow \mathbb P^n_k$ is regular? I know we want to prove the kernel of the map on the stalks is generated by a regular sequence, but I don't know how to carry this out.
$2.$ Are complete intersections always local complete intersections? If not, what hypotheses must we place on $X$ so that this is true? With the choice of $X$ and $Z$ above, this boils down to asking when $g$ is smooth.
(I realize this is a fairly elementary question, but I want to be absolutely sure about the details.)
Best Answer
Your guess is correct.
First, the notion of smooth for a morphism is intuitively a property about its fibers. The identity map of a scheme is always smooth.
Regular immersion means that at every point of $X$, there is an affine neighborhood where it is cut out by a regular sequence. If $X \subset P^n$ is cut out by $d-r$ polynomials as you say, then you can just take the restrictions of those polynomials to that open set and they will be a regular sequence: since the open set $U$ contains a point of $X$, the dimension of $U \cap X$ is the same as dimension of $X$, and a sequence of length $d-r$ is regular if and only if the zero scheme it cuts out is of codimension $d-r$ (this is a property of Cohen-Macaulay rings, of which the coordinate ring of a smooth affine open set in projective space is a special case).
So in terms of generalizing, everything above would be okay if you replaced projective space with any Cohen-Macaulay scheme (let's say connected to avoid technicalities).