  The center $Z(\mathcal{A})$ of an abelian category $\mathcal{A}$ is the
endomorphism ring of the identity functor on that category. A localizing
subcategory of a Grothendieck category $\mathcal{C}$ is said to be stable if it
is stable under essential extensions. The set $\mathbf{L}^{st}(\mathcal{C})$ of
stable localizing subcategories of $\mathcal{C}$ is partially ordered under
reverse inclusion. We show $\mathcal{L} \mapsto Z(\mathcal{C}/\mathcal{L})$
defines a sheaf of commutative rings on $\mathbf{L}^{st}(\mathcal{C})$ with
respect to finite coverings. When $\mathcal{C}$ is assumed to be locally
noetherian, we also show that the sheaf condition holds for arbitrary
coverings.
