The Fuchi is the small metal ring under the Tsuba (guard). It allows both to decorate the Katana, but also to maintain the nomenclature of it. Usually the Fuchi is matched with the Kashira (pommel) and the Tsuba in style, as if the whole was telling a story.
Everything you need to know: Manufacturing, costs, problems...
For real Katanas the Fuchi must be made of stainless steel, iron, copper or brass. They can then be decorated with gold and silver plating. By the way, this point is an indication for recognizing a poor quality Katana, if the parts are made of zinc or some other low-grade material then you should perhaps be wary.
The Fuchi is often sold with Kashira and sometimes even with Tsuba. A very good Fuchi with Kashira can be sold between 50 and $200.
The craftsman must be impeccable in the making of the Fuchi and in the assembly with the Katana so that this ring leaves no space with the wood, because of that the nomenclature will be less solid as a whole.