We show how to define discrete adelic groups which are related to
infinitely generated discrete Heisenberg groups. This can be done not only
for schemes of dimension $2$, but quite surprisingly for schemes of
dimension $1$ and even of dimension $0$. Connections with $K$-theory, direct
image problem and representation theory will be explained. It's useful,
but not obligatory to look into my talk “A new kind of functoriality in the Langlands theory” or its slides from the attached file.