Sorry for the recent question spam, but I've been striving furiously over the last few days to solve a problem in Hartshorne (Algebraic Geometry) discussed in this question. The problem (II.8.4a) says:
Let $Y$ be a closed subscheme of codimension $r$ in $\mathbb{P}^n_k$ ($k$ algebraically closed). Then the homogeneous ideal of $Y$ is generated by $r$ elements in $S=k[x_0,...,x_n]$ if and only if $Y$ is equal to the scheme-theoretic intersection of $r$-many hypersurfaces (locally principal closed subschemes), i.e., $\mathcal{I}_Y=\mathcal{I}_{H_1}+\cdots+\mathcal{I}_{H_r}$ where these are ideal sheaves.
Hartshorne gives the hint to use the unmixedness theorem for $S$.
The direction $\implies$ is easy. I'm trying to prove the other (where we assume $Y$ is a scheme theoretic intersection of hypersurfaces).
In exploring this problem, I think I've twigged to how it's "supposed" to be solved, but it relies on something not obvious to me, but perhaps obvious to the sources I'm appealing to. The something is this:
Let $I_{H_1},...,I_{H_r}$ denote the homogeneous ideals of the hypersurfaces, $I_Y$ the saturated homogeneous ideal of their scheme theoretic intersection. Then the primary components of $\sum{I_{H_j}}$ corresponding to minimal primes, and the primary components of $I_Y$ corresponding to minimal primes coincide. In other words, these ideals only differ in their embedded components.
(Matt E seems to use something like the above in answering my question here).
Once this is established, the result follows easily from the unmixedness theorem, but again I'm not sure why this should be obvious. These ideals clearly have the same minimal primes, since they cut out the same irreducible subsets of $\mathbb{P}^n$, but as for why the minimal primary components should agree, I'm at a loss. Thanks in advance for any tips.
Let me try to give slightly different reasoning from what you already have (dealing with minimal primary components, etc.):
Call $I$ the ideal of the scheme-theoretic complete intersection, and $J$ the homogeneous ideal generated by the $r$ homogeneous forms defining the hypersurfaces. Since $J \subseteq I$, there is an exact sequence
$$0 \to I/J \to S/J \to S/I \to 0$$
We want to show that $I/J = 0$. By assumption, $S/J$ and $S/I$ define the same scheme, so $I$ and $J$ have the same saturation, hence $(I/J) : \mathfrak{m}^\infty = 0$, i.e., $I/J$ is a module of finite length (where $\mathfrak{m} = (x_0,...,x_n)$ is the irrelevant ideal). But $S/J$ is Cohen-Macaulay (and has $\dim > 0$), hence cannot contain a nonzero submodule of finite length (since e.g. $H^0_\mathfrak{m}(S/J)$, the largest finite length submodule of $S/J$, is $0$, as $\operatorname{depth} S/J > 0$).