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I asked this question in MathStackExchange 10 days ago but get no response (not a vote nor a comment), so I'm copying it here below. The link to the original question is:

https://math.stackexchange.com/questions/4507710/parity-reversed-tangent-bundle-of-a-supermanifold-and-the-corresponding-q-struct


Given an ordinary manifold $X$, the pair $(X,\Omega^\bullet)=:\Pi TX$ is called the parity reversed tangent bundle of $X$, which is a supermanifold. The page 4 of AKSZ's The Geometry of the Master Equation and Topological Quantum Field Theory gives, as an example of $Q$-manifold, the parity reversed tangent bundle $\Pi TN$ ($N$ an ordinary manifold) equipped with the odd vector field $Q$ defined in local coordinates $$ Q = \eta^a\frac{\partial}{\partial x^a}, $$ where $x^a$'s are the coordinates on $N$ and $\eta^a$'s are the corresponding coordinates along the fibre. After this it writes that "Note that $\Pi TN$ can be defined and equipped with a $Q$-structure also in the case where $N$ is a supermanifold" without any further explanation.

The question is that I'm not sure what is the definition of $\Pi TN$ when $N$ is a supermanifold. I did come up with one definition myself, but it seems too trivial to be the proper one. The following is the definition that I've came up with:

'Let $\mathcal{M}=(M,C^\infty_\mathcal{M})$ be a supermanifold and $C_M^\infty$ (not $C^\infty_\mathcal{M}$) denote the ring of ordinary smooth functions on $M$. The parity reversed tangent bundle of $\mathcal{M}$ is the supermanifold $$ \Pi T\mathcal{M}= (M,C^\infty_\mathcal{M}\otimes_{C^\infty_M} \Omega^\bullet), $$ where $C^\infty_\mathcal{M}\otimes_{C^\infty_M} \Omega^\bullet$ is the tensor product of $C^\infty_\mathcal{M}$ and $\Omega^\bullet$ seen as sheaves on $M$. '

With this definition, the mentioned $Q$-structure is obvious, with the defining formula exactly the same as above. But this doesn't look correct to me, since the supermanifold has a super tangent bundle on its own right, while this definition uses only the tangent bundle of the base manifold (with the induced smooth structure): the derivations along odd coordinates are nowhere included, and I don't know how to fix this.

I'd appreciate it if anyone could tell me the true definition, or convince me that the definition I came up by myself is the correct one (if it really is). If the latter is not the case, it would be even better if hints about the $Q$-structure could be given, since the structure might be no longer obvious under the new definition.

Thanks in advance.

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