defining ideal of a topologizing subcategory

The notion of the defining sheaf of ideals of a closed subscheme inspires the notion of defining ideal of a topologizing subcategory SS of an abelian category AA as the endofunctor = SEnd(A)\mathcal{I}=\mathcal{I}_S\in End(A) which is the subfunctor of identity Id AId_A assigning to any MAM\in A the intersection of kernels Ker(f)Ker(f) of all morphisms f:MNf: M\to N where NOb(S)N\in Ob(S). One can show that if TST\subset S is an inclusion of topologizing subcategories, then S T\mathcal{I}_{S}\subset \mathcal{I}_{T}.

If RR is an associative unital ring and JRJ\subset R a left ideal in RR. Let T=[J]T = [J] be the smallest coreflective subcategory of AA containing R/JR/J. Then T(R)R\mathcal{I}_T(R)\subset R is a two-sided ideal, namely the maximal 2-sided ideal in JJ, which is explicitly, (J:R)={rRJrJ}(J:R) = \{r\in R\,|\, J r \subset J\}. Moreover, J TTJ_{T\circ T} where TTT\circ T is the square under the Gabriel multiplication agrees with (J:R) 2(J:R)^2.

See also conormal bundle.

  • A. L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, MIA 330, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9

  • V. A. Lunts, A. L. Rosenberg, Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings, MPI 1996-53 pdf

Created on May 12, 2011 16:34:06 by Zoran Škoda (