Definition of presheaf, given in Basic Algebraic geometry 2 Shafarevich

Question

I was reading Basic Algebraic geometry 2 by Shafarevich, In the definition of presheaf, $\mathscr F(\emptyset)=1$

To illustrate this definition they have given an following example:
suppose first that $A$ has no zerodivisors, and write $K$ for its field of fractions. In this case $A$ is a subfield of $K$. For an open set $U\subset\operatorname{Spec}A$ we denote $\mathscr O(U)$ the set of elements $u \in K$ such that for any point $x \in U$ we have an expression $ u=\frac{a}{b}$ with a,b $\in A$ and $b(x)\neq 0$ that is, b is not an element of the prime ideal $x$. Now $\mathscr O(U)$ is obviously a ring. Since all the rings $\mathscr O(U)$ are contained in K, we can compare them as subsets of one set. If $U\subset V$ then clearly $\mathscr O(V)$ $\subset$ $\mathscr O(U)$. We write $\rho_{U}^{V}$ for the inclusion $\mathscr O(V)$ $\hookrightarrow$ $\mathscr O(U)$. A trivial verification shows that we get a presheaf of rings.

According to definition of presheaf $\mathscr O(\emptyset)$ should be 1 but I am getting $\mathscr O(\emptyset)=K$

I could not figure out why: can anyone kindly explain what is wrong with this?

Thanks in advance

Answer 1

You’re right, this is a problem as written — Shafarevich has made some dubious choices of conventions, and here they are incompatible.

The simplest fix in the definition of presheaf. Shafarevich includes the condition “$\mathcal{F}(\emptyset) = 1$” — but most modern treatments don’t include this in the definition of presheaf. Instead it’s included in the definition of sheaf — depending on the version of the sheaf condition used, this condition may need to be added, or may already follow as a special case of the general sheaf condition.

Another fix (which makes this fit Shafarevich’s definition of presheaf, but should be done anyway even with the standard definition of presheaf) is to change this definition of $\mathcal{O}$: either crudely by saying “$\newcommand{\O}{\mathcal{O}}\O(U)$ is as Shafarevich defines for non-empty $U$, and is a singleton for $U = \emptyset$”; or slightly more cleanly by using the general definition Shafarevich gives a few paragraphs for “the case of an arbitrary ring $A$”.