  Generalized Donaldson-Thomas invariants defined by Joyce and Song
arXiv:0810.5645 are rational numbers which `count' both $\tau$-stable and
$\tau$-semistable coherent sheaves with Chern character $\alpha$ on a
Calabi-Yau 3-fold $X$, where $\tau$ denotes Gieseker stability for some ample
line bundle on $X$. These invariants are defined for all classes $\alpha$, and
are equal to the classical Donaldson-Thomas invariant defined by Thomas
arXiv:math/9806111 when it is defined. They are unchanged under deformations of
$X$, and transform by a wall-crossing formula under change of stability
condition $\tau$. Joyce and Song use gauge theory and transcendental complex
analytic methods, so that their theory of generalized Donaldson-Thomas
invariants is valid only in the complex case. This paper will propose a new
algebraic method extending the theory to algebraically closed fields K of
characteristic zero, and partly to triangulated categories and for non
necessarily compact Calabi-Yau 3-folds under some hypothesis.
