## Posts Tagged ‘makefile’

### Split injections of free modules over local rings

January 17, 2011

The main latest change is the addition of the following lemma: Suppose given two free modules $F, F'$ over a local ring, of finite rank, and a morphism $\phi$ between them. Then $\phi$ is a split injection iff the base-change $F \otimes k \to F' \otimes k$ to the residue field is an injection. This is not too difficult to prove, but I realized today that Hartshorne uses it at a key point in proving that a nonsingular subvariety of a nonsingular variety is a local complete intersection. It is kind of glossed over there,  probably for good reasons, but this lemma is now explained in our book.

The makefile is also fixed so that running “make” actually resolves cross-references. Apparently, you run pdflatex twice after invoking bibtex, and not the other way around. That makes sense.