# Affine communication lemma and finite limits in the category of rings

Let $X$ be a scheme and $\mathrm{Spec}(B) = V \subseteq X$ be an open affine subset. When using the affine communication lemma (c.f. Theorem 6.3.2, Vakil's notes, Foundations of Algebraic Geometry), one often encounters the situation $$B = B_{f_1} \times_{B_{f_1f_2}} B_{f_2}$$ (or its generalization with $n$ localizations, which I did not bother writing) because we get an open affine cover $V = \bigcup_{i=1}^n \mathcal D(f_i)$ by distinguished open sets, and considering the map $$B = \mathcal O_X(V) \to \prod_{i=1}^n \mathcal O_X(\mathcal D(f_i))$$ (which is the product of the restriction maps), we see that the it is injective (by the sheaf axiom) and the image is $$\left\{ \left. (s_i)_{1 \le i \le n} \in \prod_{i=1}^n \mathcal O_X(\mathcal D(f_i)) \, \right| \, \forall i,j, \quad s_i |_{\mathcal D(f_i f_j)} = s_j |_{\mathcal D(f_i f_j)} \right\}.$$ This is the definition of the limit of the diagram $\mathcal O_X(\mathcal D(f_i)) \rightarrow \mathcal O_X(\mathcal D(f_i f_j)) \leftarrow \mathcal O_X(\mathcal D(f_j))$ in the category of rings.

The usual trick to lift properties of the $B_{f_i}$'s to $B$ (finitely generated something-algebra/module, reducedness, noetherianness, etc) is to use this computation where we take high powers of the $f_i$'s and cancel them using the fact that $(f_1,\cdots,f_n) = B$.

For instance, if the $B_{f_i}$'s are finitely generated $A$-algebras for some ring $A$ via a commutative diagram $A \to B \to B_f$ (with an arrow $A \to B_f$, I don't know how to draw commutative diagrams on MathJax... I can only do it in TikzCD! If anyone could tell me how to do that it would be nice too), then given $b \in B$, we can write the following in $B_{f_i}$ : let $a_1/f_i^{m_1},\cdots,a_n/f_i^{m_n}$ be generators of $B_{f_i}$ as an $A$-algebra, and $$\frac b1 = f\left( \frac{a_1}{f_i^{m_1}},\cdots, \frac{a_n}{f_i^{m_n}} \right)$$ so multiplying by a high power of $f_i$, we get $f_i^N b$ as a polynomial in the $f_i$'s and the $a_j$'s, then we pick $c_i$'s such that $c_i f_i^N = 1$ and sum up the equations. A similar technique allows one to lift the other properties mentioned above.

Question : is there any way to avoid these computations using some properties which lift correctly through the fiber product, or is this computation supposed to be there in each case? For instance, do any of the properties listed above satisfy something in the fashion "a finite limit of -insert property here- satisfies -property-"? (Perhaps this is too strong, but I am just looking for a possible general statement allowing me to avoid computing this every time).

For instance, in the case $n=1$, the map $B \to B_f$ has really nice properties ; it is easy to see that $B_f$ is a finitely generated $B$-algebra (using Rabinowitch's trick to write $B_f \simeq B[x]/(xf-1)$), so that if $B$ is a finitely-generated $A$-algebra/module, then so is $B_f$ by composing the maps. I don't know such general properties in the case $n \ge 2$.

P.S. : I must say I don't know exactly what I am looking for, but I have the feeling that repeating this computation is somehow the wrong way to do things, and that's what I'm trying to get rid of.

• I do not understand what you are asking. – Jason Starr Nov 28 '14 at 20:59
• @JasonStarr : I added details. Can you have a look again? – Patrick Da Silva Nov 29 '14 at 13:41
• My best guess at your intention is that you want to prove that certain properties of affine schemes are Zariski-local in nature, and you are asking how to eliminate certain repetitive aspects of the standard arguments. Is that correct? – S. Carnahan Nov 30 '14 at 2:26
• @S. Carnahan : Precisely. The reason why I can't eliminate these aspects is because I don't understand this fiber product, i.e. I have no results about it (that I know of, so I'm asking if there are any). As a result I am always running down the same annoying computations on the elements, and they all look really similar. – Patrick Da Silva Nov 30 '14 at 18:19
• The big issue is that the condition that the restrictions are equal do not define an ideal in the product of the rings (it only defines a subgroup with respect to $+$), so it makes things complicated. – Patrick Da Silva Nov 30 '14 at 18:44

I think that Section 11 on transfer principles in these notes of mine is what you're looking for. A general machinery abstracts the business of tracking all the $f_i$'s and the required high powers. These then don't have to be dealt with when applying the machinery.
The method rests on the observation that the structure sheaf $\mathcal{O}_{\operatorname{Spec} A}$ can be viewed as the localization $\underline{A}[\mathcal{F}^{-1}]$ of the constant sheaf $\mathcal{A}$ at the generic filter.
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