Editing Superalgebra (section)

==Sign conventions==
When the '''Z'''<sub>2</sub> grading arises as a "rollup" of a '''Z'''- or '''N'''-[[graded algebra]] into even and odd components, then two distinct (but essentially equivalent) sign conventions can be found in the literature.<ref name="deligne">See [http://www.math.ias.edu/QFT/fall/bern-appen1.ps Deligne's discussion] of these two cases.</ref> These can be called the "cohomological sign convention" and the "super sign convention". They differ in how the antipode (exchange of two elements) behaves. In the first case, one has an exchange map
:<math>xy\mapsto (-1)^{mn+pq} yx</math>
where <math>m=\deg x</math> is the degree ('''Z'''- or '''N'''-grading) of <math>x</math> and <math>p</math> the parity. Likewise, <math>n=\deg y</math> is the degree of <math>y</math> and with parity <math>q.</math> This convention is commonly seen in conventional mathematical settings, such as differential geometry and differential topology.  The other convention is to take 
:<math>xy\mapsto (-1)^{pq} yx</math>
with the parities given as <math>p=m\bmod 2</math> and <math>q=n\bmod 2</math> the parity. This is more often seen in physics texts, and requires a parity functor to be judiciously employed to track isomorphisms. Detailed arguments are provided by [[Pierre Deligne]]<ref name="deligne"/>