Todd Trimble
Dippy disproof of infinitary extensivity of affine schemes

Proposition

The category of affine schemes is lextensive, but is not infinitary extensive.

The fact that the category of affine schemes is lextensive is well-known. The dual statement for the category of commutative rings comes down to some easily verified claims:

  • The category CRingCRing of commutative rings is of course finitely cocomplete.

  • Product projections π 1:R×SR\pi_1: R \times S \to R, π 2:R×SS\pi_2: R \times S \to S are epic, and their pushout is terminal.

  • The pushout in CRingCRing of two ring maps RSR \to S, RTR \to T is given by the tensor product S RTS \otimes_R T, with the evident commutative ring structure.

  • The pushing-out functor along a map RSR \to S, namely S R:CAlg RCAlg RS \otimes_R -: CAlg_R \to CAlg_R, preserves finite products (since finite products are reflected and preserved by the forgetful functor to Mod RMod_R, where they are biproducts and therefore coproducts that are preserved by tensoring).

However, the category of affine schemes is not infinitary extensive. Again, we consider what infinitary extensivity would mean for the dual category:

Take RR to be Noetherian. Suppose that a commutative ring SS over RR is flat as an RR-module, and that S R:CAlg RCAlg RS \otimes_R -: CAlg_R \to CAlg_R preserves products. Since the forgetful functor CAlg RMod RCAlg_R \to Mod_R preserves and reflects equalizers, it must be that S R:CAlg RCAlg RS \otimes_R -: CAlg_R \to CAlg_R also preserves equalizers and therefore all limits. We will apply this observation to certain wide pullbacks in CAlg RCAlg_R to deduce that SS is finitely generated as an RR-module.

For any RR-module MM, let M^=R×M\hat{M} = R \times M be the commutative RR-algebra where the identity lives in the summand RR and the product of any two elements in the summand MM is defined to be zero. Then for any collection of RR-modules M iM_i, the wide pullback in CAlg RCAlg_R of the wide cospan consisting of projection maps M^=R×MR\hat{M} = R \times M \to R is just iM i^\widehat{\prod_i M_i}. Given that S RS \otimes_R - preserves this wide pullback, it means that the canonical arrow

S R iM i^S× iS RM iS \otimes_R \widehat{\prod_i M_i} \to S \times \prod_i S \otimes_R M_i

is an isomorphism, and this forces the functor S R:Mod RMod RS \otimes_R -: Mod_R \to Mod_R, now regarding SS simply as an RR-module, to preserve the product iM i\prod_i M_i. So assuming SS is flat and S R:CAlg RCAlg RS \otimes_R -: CAlg_R \to CAlg_R preserves products, we see that S R:Mod RMod RS \otimes_R -: Mod_R \to Mod_R preserves products and therefore limits (again by flatness).

Now S R:Mod RMod RS \otimes_R -: Mod_R \to Mod_R is an accessible (in fact, a finitary) functor between locally presentable categories, so that under the assumption that it preserves limits, it must be a right adjoint. Its left adjoint is, as any left adjoint Mod RMod RMod_R \to Mod_R must be, given by a functor of the form N RN \otimes_R -. Hence we have

(N R)(S R)(N \otimes_R -) \dashv (S \otimes_R -)

and it follows that S Rhom R(N,)S \otimes_R - \cong \hom_R(N, -). Now the left side of this isomorphism preserves colimits, but hom R(N,)\hom_R(N, -) can preserve colimits only if NN is finitely generated and projective, i.e., a direct summand of some finite free RR-module R nR^n. Hence

SS RRhom R(N,R)S \cong S \otimes_R R \cong \hom_R(N, R)

is also a direct summand of R nR^n, and in particular is finitely generated.

The upshot is that if a commutative algebra SS over a Noetherian RR is flat but not finitely generated as an RR-module, then products of commutative RR-algebras are not stable under pushing out along the ring map RSR \to S. For the opposite category of affine schemes, it means that coproducts are not stable under pullback.

Created on August 7, 2013 05:45:30 by Todd Trimble