semicommutative formal scheme

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

- M. Kapranov,
*Noncommutative geometry based on commutator expansions*, J. reine und angew. Math.**505**(1998), 73-118, math.AG/9802041.

belong to the class of semicommutative formal schemes, and viceversa, there are some semicommutative (not necessarily formal) schemes which do not fit into Kapranovâ€™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.