# What is the ring and does $\phi^*$ preserves module operations

I have posted this problem recently and I think that as it was too long, not many people took interest in it. So I am reposting it also I am giving you an earlier link :I had a problem in manifold which states:

Show that $$A^1:Man^{op} \to Mod$$ is a functor, where $$M$$ is a manifold, $$A^1(M)$$ is the $$1$$-form on $$M$$. and for $$\phi:M \to N$$, $$A^1(\phi)=\phi^*$$. This means to show that

a) $$A^1(M)$$ is a module.(What is the ring?)

b) $$\phi^*$$ preserves module operations(does the ring change?)(I feel yes! but not getting why!!)

c)$$(1_M)^*=1_{A^1(M)}$$

In part c) I feel that I have to show $$w \circ (\Bbb 1_M)_*=(\Bbb 1_{A^1(M)}w$$ as $$w \circ (\Bbb 1_M)_*=(\Bbb 1_M)^*w$$ but how!! Please help me in a), b) and c). Especially I feel that the ring in a) should be $$\Bbb R$$

• 1) $\Bbb R$ 2) ?? 3) tautological if you know the definition of pullback – user98602 Jan 24 at 15:00
• Sorry, I am messed in the conception. Can you please explain in details? This will be of great help, as manifolds goes above my head. – Gimgim Jan 26 at 5:01
• I suggest that you look at a smooth manifolds book, such as John Lee's "Introduction to Smooth Manifolds". – user98602 Jan 26 at 17:08
• Yes, I am reading that and Lee and Tu's book but this subject is hard at least to me. So it would of great help if you can expand me. Thanks anyway – Gimgim Jan 27 at 2:18