There are several variants (still in fluctuation) what should be called semicommutative formal scheme. In any case, these include the subcategory of semicommutative schemes and a subcategory of commutative formal schemes and some combinations of those. The usual formal schemes are special cases of ind-schemes, and similar the semicommutative formal schemes are certain ind-objects in the category of semicommutative schemes; however some additional constraints apply, the most important being that it can be represented by a filtered diagram? in which all connecting morphisms are closed immersions of semicommutative schemes.
The richest sort of examples come from formal noncommutative deformations of commutative schemes: one truncates the formal parameter to nilpotent and the subsequent truncations form a filtered diagram representing the semicommutative formal scheme.
Only some among the Kapranov's noncommutative formal schemes from
belong to the class of semicommutative formal schemes, and viceversa, there are some semicommutative (not necessarily formal) schemes which do not fit into Kapranov’s framework. But there is a significant number of common examples.
Last revised on March 25, 2010 at 15:22:52. See the history of this page for a list of all contributions to it.