Bounded polynomial-factorization categories and the cotangent module diagram

Assume the Axiom of Choice (AC) (The Axiom of Choice). Fix a map $A\to B$ of commutative unital rings (Commutative ring) and an infinite cardinal $\kappa$ at least the cardinalities of $A$ and $B$ and of every variable set occurring in the specified countable polynomial resolutions of $B$ over $A$ that are used below.

For an ordinal $\lambda \le \kappa$ let $A[\lambda ]$ denote the polynomial $A$-algebra on the variable set $\lambda$ (The polynomial ring $R[x_i:i\in I]$ as finitely supported coefficient families on monomials). A polynomial presentation of $B$ over $A$ (bounded by $\kappa$) is an $A$-algebra map $\pi \text{\hspace{0.17em}⁣}:A[\lambda ]\to B$ for some ordinal $\lambda \le \kappa$; since $A[\lambda ]$ is free on $\lambda$, such a $\pi$ is determined by the family $(\pi (x_{\alpha }){)}_{\alpha <\lambda }$ in $B$, so the collection of all bounded polynomial presentations is a set. Here “presentation” means a polynomial factorization of $A\to B$; the augmentation need not be surjective. This permits the coefficient-change functor for arbitrary ring squares, even when $B{\otimes }_A{A}^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$ is not surjective.

Define ${\mathcal{P}}_{B\mathrm{/}A}^{\kappa }$ to be the category whose objects are the bounded polynomial presentations $\pi \text{\hspace{0.17em}⁣}:A[\lambda ]\to B$ and whose morphisms $\pi \to {\pi }^{\mathrm{\prime }}$ are the $A$-algebra maps $\phi \text{\hspace{0.17em}⁣}:A[\lambda ]\to A[{\lambda }^{\mathrm{\prime }}]$ with ${\pi }^{\mathrm{\prime }}\circ \phi =\pi$, i.e. maps commuting with the augmentations. Every hom collection is a set (a morphism is determined by the images of the variables of its source, which are polynomials over $A$ in ${\lambda }^{\mathrm{\prime }}$ variables), so ${\mathcal{P}}_{B\mathrm{/}A}^{\kappa }$ is a small category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). The bounded polynomial-factorization category is its opposite ${\mathcal{C}}_{B\mathrm{/}A}^{\kappa }:=({\mathcal{P}}_{B\mathrm{/}A}^{\kappa }{)}^{\mathrm{o}\mathrm{p}},$ and we write $P\to B$ for the object corresponding to a presentation. The ordinal representatives and the specified presentations are transported into this model degreewise; conjugating all face and degeneracy maps by the resulting algebra isomorphisms preserves the simplicial identities.

The cotangent diagram is the contravariant functor $F_{B\mathrm{/}A}\text{\hspace{0.17em}⁣}:{\mathcal{C}}_{B\mathrm{/}A}^{\kappa }\to B\text{-}\mathrm{M}\mathrm{o}\mathrm{d},F_{B\mathrm{/}A}(P\stackrel{\pi }{\to }B)={\mathrm{\Omega }}_{P\mathrm{/}A}{\otimes }_{P}B,$ where ${\mathrm{\Omega }}_{P\mathrm{/}A}$ is the module of Kähler differentials (Universal Kähler differential module) and the tensor product is taken along $\pi$ (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums); on a morphism $\phi \text{\hspace{0.17em}⁣}:P\to P^{\mathrm{\prime }}$ of presentations, functoriality of Kähler differentials gives ${\mathrm{\Omega }}_{P\mathrm{/}A}{\otimes }_P{P}^{\mathrm{\prime }}\to {\mathrm{\Omega }}_{P^{\mathrm{\prime }}\mathrm{/}A}$ and hence a $B$-linear map $F_{B\mathrm{/}A}(P)\to F_{B\mathrm{/}A}(P^{\mathrm{\prime }})$. The corresponding arrow in ${\mathcal{C}}_{B\mathrm{/}A}^{\kappa }$ goes from $P^{\mathrm{\prime }}$ to $P$, so $F_{B\mathrm{/}A}$ is contravariant (Covariant functor, identity functor, composite functor, and contravariant functor). Since $P$ is a polynomial $A$-algebra, ${\mathrm{\Omega }}_{P\mathrm{/}A}$ is a free $P$-module and $F_{B\mathrm{/}A}(P)$ is a free $B$-module.

A simplicial polynomial resolution $P_{\bullet }\text{\hspace{0.17em}⁣}:{\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}}\to {\mathcal{P}}_{B\mathrm{/}A}^{\kappa }$ (with each $P_n$ a polynomial presentation) yields by composition with the inclusion a simplicial object of ${\mathcal{P}}_{B\mathrm{/}A}^{\kappa }$, i.e. a cosimplicial object of ${\mathcal{C}}_{B\mathrm{/}A}^{\kappa }$.

Finally, a commutative square of ring maps $\begin{array}{ccc}A& ⟶& B\\ \downarrow & & \downarrow \\ A^{\mathrm{\prime }}& ⟶& B^{\mathrm{\prime }}\end{array}$ induces, after replacing $\kappa$ by a common bound for the two squares, a functor ${\mathcal{P}}_{B\mathrm{/}A}^{\kappa }\to {\mathcal{P}}_{B^{\mathrm{\prime }}\mathrm{/}{A}^{\mathrm{\prime }}}^{\kappa }$ sending $\pi \text{\hspace{0.17em}⁣}:A[\lambda ]\to B$ to $P{\otimes }_A{A}^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$, where the target is the composite $P{\otimes }_A{A}^{\mathrm{\prime }}\to B{\otimes }_A{A}^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$, and therefore a functor ${\mathcal{C}}_{B\mathrm{/}A}^{\kappa }\to {\mathcal{C}}_{B^{\mathrm{\prime }}\mathrm{/}{A}^{\mathrm{\prime }}}^{\kappa }$; the variable set, hence the bound, is unchanged.

Use of AC. AC is used only to choose, once and for all, the transported models of the specified countable presentations inside the bounded category and to bound the union of the countably many specified variable sets by a single infinite cardinal $\kappa$; the subsequent definition of the diagram, the contravariance and the coefficient-change functor are choice-free. Since a polynomial ring on at most $\kappa$ variables over a ring of size at most $\kappa$ has size at most $\kappa$, the standard resolution fits at every stage and no universe axiom beyond the ambient set theory of Category, object, morphism, domain, codomain, identity, composition, and hom-collection is introduced.