TITLE: How do you define the Euler Characteristic of a scheme?
QUESTION [22 upvotes]: If X is a variety over the complex numbers, one reasonable thing to do is to consider the associated analytic space $X_{an}$  and to take the topological Euler characteristic of that.
Is there a purely algebraic way to obtain this number?
If X is non-singular then one might define it as the integral of the top Chern class of its tangent bundle.
The reason I ask is that I'm currently reading Joyce's survey on Donaldson-Thomas invariants and I wanted to know if by any chance he were using some more sophisticated notion.
On related note: if X is a non-proper scheme over C, why is its Euler characteristic well-defined?

REPLY [8 votes]: One comment on the "excision" property for Euler characteristics of complex algebraic varieties: $\chi(X) = \chi(Z) + \chi(X\backslash Z)$.  There is a short proof of this in Fulton's Introduction to Toric Varieties, p. 142, which avoids the sticky triangulation issue and doesn't seem to be so well-known.  In fact, it's equivalent to the statement that $\chi(X) = \chi_c(X)$ for any algebraic variety (the latter being the Euler characteristic for compactly supported cohomology). 
This doesn't exactly help with the first question (about an algebraic definition), but it does show why the Euler characteristic is well-defined for any variety, per Arend's comment.