How do I update a single value in a json document using jq?

Appologies if I've overlooked something very obvious; I've just found jq and am trying to use it to update one JSON value without affecting the surrounding data.

I'd like to pipe a curl result into jq, update a value, and pipe the updated JSON to a curl -X PUT. Something like

curl http://example.com/shipping.json | jq '.' field: value | curl -X PUT http://example.com/shipping.json

So far I've hacked it together using sed, but after looking at a few examples of the |= operator in jq I'm sure that I don't need these.

Here's a JSON sample--how would I use jq to set "local": false, while preserving the rest of the JSON?

{"shipping": {"local": true,"us": true,"us_rate": {"amount": "0.00","currency": "USD","symbol": "$"    }  }}