Let (X, OX) be a ringed space and let A —> OX be a surjection of sheaves of algebras. Let S ⊂ A be the subsheaf of local sections which map to invertible functions of OX. Then S(U) is a multiplicative subset of A(U) for every open U of X and we can factor the map as

A —> S-1A —> OX

If X is a locally ringed space, then the stalks of S-1A are local rings too.

This construction is occasionally useful. For example, consider a fibre product of schemes W = X x_S Y with projections maps p : W —> X, q : W —> Y, and structure morphism h : W —-> S. Then the map

A = p-1OXh-1OS q-1OY —-> OW

is not an isomorphism in general, but a localization: with the notation above we have S-1A = OW.

One thought on “Localize

  1. A related remark: this construction (without the initial surjectivity assumption) is also used by Illusie to show that the cotangent complex of a ring A coincides with the cotangent complex for the ringed topos X = Spec(A). (One uses the transitivity triangle + the fact that the map A —-> O_X is a localisation in the sense of your post, so has a trivial cotangent complex by a formal argument.)

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