Benefit of being a sheaf.

By saying that $\mathcal{O}$ is a sheaf on $\operatorname{Spec}(A)$ , we may easily use the arguments we have used to proved the locality and the gluing lemma.

For example, the proof we gave in Theorem , especially the part where we chose the idempotent $p_1$ , was a bit complicated.

Let us give another proof using the sheaf arguments. There exists a unique element $p\in A=\mathcal{O}(\operatorname{Spec}(A))$ which coincides with $1$ on $U_1=V(J)$ and with 0 on $U_2=V(I)$ . From the uniqueness we see that

$$p^2=p$$

holds since $p^2$ satisfies the same properties as $p$ . The rest of the proof is the same.

As the second easier example, we consider the following undergraduate problem.

Problem: Find the inverse of the matrix

$$\begin{pmatrix} 3& 5 \\ 1 & 2 \end{pmatrix}.$$

A student may compute (using ``operations on rows'') as follows.

$$\begin{pmatrix}3& 5 &\vert & 1 &0\\ 1 & 2 &\vert & 0 & 1 \end{pma... ... \to \begin{pmatrix}1& 5/3 &\vert & 1/3 &0\\ 1 & 2 &\vert & 0 & 1 \end{pmatrix}$$

$$\to \begin{pmatrix}1& 5/3 &\vert & 1/3 &0\\ 0 & 1/3 &\vert & -1/3 & 1... ...\to \begin{pmatrix}1& 5/3 &\vert & 1/3 &0\\ 0 & 1 &\vert & -1 & 3 \end{pmatrix}$$

$$\to \begin{pmatrix}1& 0 &\vert & 2 &-5\\ 0 & 1 &\vert & -1 & 3 \end{pmatrix}$$

The calculation is valid on $\operatorname{Spec}(\mathbb{Z}[1/3])$ .

Another student may calculate (using ``operations on columns'') as follows.

$$\begin{pmatrix}3& 5 &\vert & 1 &0\\ 1 & 2 &\vert & 0 & 1 \end{pma... ... \to \begin{pmatrix}3& 5/2 &\vert & 1 &0\\ 1 & 1 &\vert & 0 & 1/2 \end{pmatrix}$$

$$\to \begin{pmatrix}1/2& 5/2 &\vert & 1 &0\\ 0 & 1 &\vert & -1/2 & 1/2... ...\to \begin{pmatrix}1& 5/2 &\vert & 2 &0\\ 0 & 1 &\vert & -1 & 1/2 \end{pmatrix}$$

$$\to \begin{pmatrix}1& 1 &\vert & 2 &-5\\ 0 & 1 &\vert & -1 & 3 \end{pmatrix}$$

The calculation is valid on $\operatorname{Spec}(\mathbb{Z}[1/2])$ . Of course, both calculations are valid on the intersection $\operatorname{Spec}(\mathbb{Z}[1/2])\cap \operatorname{Spec}(\mathbb{Z}[1/3])=\operatorname{Spec}(\mathbb{Z}[1/6])$ .

The gluing lemma asserts that the answer obtained individually is automatically an answer on the whole of $\operatorname{Spec}(\mathbb{Z})$ . Of course, in this special case, there are lots of easier ways to tell that. But one may imagine this kind of thing is helpful when we deal with more complicated objects.