Euler characteristic of modules

Suppose R is a ring that has the invariant basis number property, and assume M is a module that has a finite free resolution (that is, a free resolution of finite length by finitely generated free modules). Define the Euler characteristic of M to be the Euler characteristic of any given finite free resolution:

0\to F_n\to\cdots \to F_0\to M\to 0

has characteristic \sum_{i=0}^n (-1)^i {\rm rank}\, F_i.

  1. Show the above is well defined, that is, it doesn’t depend on the given free resolution.
  2. Show that the Euler characteristic of any module over a noetherian or over a commutative ring is always non-negative.
  3. Exhibit a module over some ring that has negative Euler characteristic.
Euler characteristic of modules