Editing Canonical bundle (section)

==Singular case==
On a singular variety <math>X</math>, there are several ways to define the canonical divisor.  If the variety is normal, it is smooth in codimension one.  In particular, we can define canonical divisor on the smooth locus.  This gives us a unique [[Weil divisor]] class on <math>X</math>.  It is this class, denoted by <math>K_X</math> that is referred to as the canonical divisor on <math>X.</math>

Alternately, again on a normal variety <math>X</math>, one can consider <math>h^{-d}(\omega^._X)</math>, the <math>-d</math>'th cohomology of the normalized [[dualizing complex]] of <math>X</math>.  This sheaf corresponds to a [[Weil divisor]] class, which is equal to the divisor class <math>K_X</math> defined above.  In the absence of the normality hypothesis, the same result holds if <math>X</math> is S2 and [[Gorenstein ring|Gorenstein]] in dimension one.