symmetric monoidal (∞,1)-category of spectra
Under the interpretation of modules as generalized vector bundles a free module corresponds to a trivial bundle.
There is the evident forgetful functor that sends each module to its underlying object .
Let be a ring. We discuss free modules over .
Let be a commutative ring, and let denote the free -module on a set .
The free -module functor is strong monoidal with respect to the Cartesian monoidal structure on sets, and the tensor product of -modules.
In other words, the free module construction turns set-theoretic products into tensor products. Thus, it preserves algebraic objects (such as monoid objects, Hopf monoid objects, etc.) and their homomorphisms. In particular, if is a monoid in the category of sets (and hence a bimonoid with the canonical comonoid structure) then is a bimonoid object in , which is precisely a -bialgebra. A group in the category of sets is a Hopf monoid, and hence is a Hopf algebra — this is precisely the group algebra of .
Let be a commutative ring.
Assuming the axiom of choice, the following are equivalent
(See also Rotman, pages 650-651.) Condition 1. immediately implies condition 2., since ideals of are the same as submodules of seen as an -module. Now assume condition 2. holds, and suppose is any nonzero element. Let denote multiplication by (as an -module map). We have a sequence of surjective -module maps
(where is the codiagonal map); by the Yoneda lemma, the composite map is of the form , where is the value of the composite at . Since is surjective, we have for some , so that is invertible. Hence is invertible, and this implies is monic. Therefore is a domain. From that, we infer that if and belong to a basis of an ideal , then
whence and are not linearly independent, so and as an -module is generated by a single element, i.e., is a principal ideal domain.
That condition 3. implies condition 1. is proved here.
See at projective resolution – Resolutions of length 1 for more.
flat object, flat resolution