Motivic cohomology is a hypothetical cohomology theory for algebraic schemes, including algebraic varieties, over a given field, that can be seen as the counterpart   in algebraic geometry to  the singular cohomology theory in topology. Its construction was completed for smooth varieties, but for singular ones the situation was not clear.

 

In this talk, I will sketch some recent attempts of mine to provide an algebraic-cycle-based functorial model for the motivic cohomology of singular algebraic schemes, via formal schemes and some ideas from derived algebraic geometry.

 

As this is very complicated, as an illustration I will give an example on the concrete case of the fat points, where the situation is simpler, but not still trivial.