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.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: