Noncommutative Smooth Spaces

We will work in the category Al gk of associative unital algebras over a fixed base field k. If A € Ob(Algk), we denote by 1A € A the unit in A and by m A : A ⊗ A—→A the product. For an algebra A, we denote by opp the opposite algebra, i.e., the same vector space as A endowed with the multiplication m AoPP (a⊗b) :— m A (b⊗a). If A and B are two algebras, then A⊗ k B is again an algebra. Also, A ⋆ B denotes the free product of A and B over k, the coproduct in the category Al gk . By A-mod we denote the abelian category of left A-modules. Analogously, mod-A are right modules (the same as A opp-modules) and A-mod-A are bimodules over k, or, equivalently, A ⊗ k A opp -modules. We shall write ⊗ instead of ⊗ k - For a vector space V, we denote by Sym *(V) and ⊗*(V) resp. the free commutative associative (polynomial) and free associative (tensor) k-algebra respectively, generated by V.