A sigma, or skew, derivation is a natural generalisation of the notion of derivation depending on an algebra automorphism $$\sigma$$ which when equal to $$id = \sigma$$ reduces to the usual notion of a derivation. For a precise definition see here

Does there exist a notion of skew differential graded algebra in the literature? If so where do these objects arise?

EDIT: To confirm I am asking if there exists a graded analogue of skew derivation algebra. So an $$\mathbb{N}_0$$-graded algebra $$A = \bigoplus_{k \in \mathbb{N}_)0} A_k$$, together with a degree $$1$$ map $$d$$ satisfying $$d^2 = 0$$, and a skew analogue of the graded Leibniz rule: $$d(a \wedge b) = da \wedge \sigma(b) + (-1)^k \sigma(a)db, ~ a \in A_k$$

• @Najib: I have edited the question to include a definition of the object I am wondering about. Commented Apr 21, 2020 at 15:18
• Is this the kind of construction you're interested in? perso.univ-rennes1.fr/bernard.le-stum/Publications_files/… Commented Apr 21, 2020 at 15:54
• Is your wedge graded (anti)commutative? If so then this $d$ seems poorly defined. Commented Apr 21, 2020 at 16:20
• @Alex: Yes, it is assumed to be anti-commutative. I have adjusted the ansatz. Commented Apr 22, 2020 at 11:54
• @NicolaCiccoli: The same but with functioning references: arxiv.org/abs/1503.05022v1 Commented Apr 22, 2020 at 12:02

This edited version of the "skew Leibniz rule" has appeared in geometry: if $$\varphi: N\to M$$ is a map of (super) manifolds, a section $$X$$ of the pullback bundle $$\varphi^\star TM$$ is a linear map $$C^\infty(M)\to C^\infty(N)$$ satisfying precisely your skew-leibniz rule: $$X(fg)=X(f)\varphi^\star g +(-1)^{\deg X\deg f} (\varphi^\star f) X(g)$$
So if you put $$\mathbb N_0$$-gradings on your structure sheaves (what is $$\mathbb N_0$$? Positive integers?) and pick an $$X$$ of degree $$+1$$ which squares to zero you seem to arrive at your setup. (Identifying $$\varphi^\star$$ with $$\sigma$$ and $$C^\infty(M)$$ with $$A$$.)
On the algebraic side there is the (comparatively more obscure) notion of $$({\bf l},{\bf r})$$-coderivation of Berglund (Definition 3.2) which should be dual to your proposal in the case $${\bf l}={\bf r}=\sigma$$.
• Just like any normal person, I write $\mathbb{N}_0 := \{0,1,2,3,\dots\}$. Commented Apr 24, 2020 at 11:51