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Nonstable K-Theory of Regular Rings and Banaschewski Functions

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Nonstable K-Theory of Regular Rings and Banaschewski Functions Nonst. K-th., Banaschewski The Nonstable K-theory of regular rings realization problem and Banaschewski functions Known cases Banaschewski functions Friedrich Wehrung Vaught's and Dobbertin's results Universit´ede Caen LMNO, UMR 6139 D´epartement de Math´ematiques 14032 Caen cedex E-mail: [email protected] URL: http://www.math.unicaen.fr/~wehrung August 9-13, 2010 For X 2 FP(R), set [X ] := isomorphism class of X . Then define [X ] + [Y ] := [X ⊕ Y ]. V(R) := f[X ] j X 2 FP(R)g, endowed with addition, is a commutative monoid (encodes the nonstable K-theory of R). It is conical: α + β = 0 ) α = β = 0. The element [R] is an order-unit: 8α 9β 9n α + β = n[R]. The definition is left-right symmetric. ∼ + V(R) = Z = f0; 1; 2;::: g if R is a division ring. FP(R) and V(R) Nonst. K-th., Banaschewski For a unital (associative) ring R, set The FP(R) := fX right R-module j X fin. gen. projectiveg realization problem n = fX j (9n)(9Y )(X ⊕ Y = RR )g : Known cases Banaschewski functions Vaught's and Dobbertin's results Then define [X ] + [Y ] := [X ⊕ Y ]. V(R) := f[X ] j X 2 FP(R)g, endowed with addition, is a commutative monoid (encodes the nonstable K-theory of R). It is conical: α + β = 0 ) α = β = 0. The element [R] is an order-unit: 8α 9β 9n α + β = n[R]. The definition is left-right symmetric. ∼ + V(R) = Z = f0; 1; 2;::: g if R is a division ring. FP(R) and V(R) Nonst. K-th., Banaschewski For a unital (associative) ring R, set The FP(R) := fX right R-module j X fin. gen. projectiveg realization problem n = fX j (9n)(9Y )(X ⊕ Y = RR )g : Known cases Banaschewski functions Vaught's and For X 2 FP(R), set [X ] := isomorphism class of X . Dobbertin's results V(R) := f[X ] j X 2 FP(R)g, endowed with addition, is a commutative monoid (encodes the nonstable K-theory of R). It is conical: α + β = 0 ) α = β = 0. The element [R] is an order-unit: 8α 9β 9n α + β = n[R]. The definition is left-right symmetric. ∼ + V(R) = Z = f0; 1; 2;::: g if R is a division ring. FP(R) and V(R) Nonst. K-th., Banaschewski For a unital (associative) ring R, set The FP(R) := fX right R-module j X fin. gen. projectiveg realization problem n = fX j (9n)(9Y )(X ⊕ Y = RR )g : Known cases Banaschewski functions Vaught's and For X 2 FP(R), set [X ] := isomorphism class of X . Dobbertin's results Then define [X ] + [Y ] := [X ⊕ Y ]. It is conical: α + β = 0 ) α = β = 0. The element [R] is an order-unit: 8α 9β 9n α + β = n[R]. The definition is left-right symmetric. ∼ + V(R) = Z = f0; 1; 2;::: g if R is a division ring. FP(R) and V(R) Nonst. K-th., Banaschewski For a unital (associative) ring R, set The FP(R) := fX right R-module j X fin. gen. projectiveg realization problem n = fX j (9n)(9Y )(X ⊕ Y = RR )g : Known cases Banaschewski functions Vaught's and For X 2 FP(R), set [X ] := isomorphism class of X . Dobbertin's results Then define [X ] + [Y ] := [X ⊕ Y ]. V(R) := f[X ] j X 2 FP(R)g, endowed with addition, is a commutative monoid (encodes the nonstable K-theory of R). The element [R] is an order-unit: 8α 9β 9n α + β = n[R]. The definition is left-right symmetric. ∼ + V(R) = Z = f0; 1; 2;::: g if R is a division ring. FP(R) and V(R) Nonst. K-th., Banaschewski For a unital (associative) ring R, set The FP(R) := fX right R-module j X fin. gen. projectiveg realization problem n = fX j (9n)(9Y )(X ⊕ Y = RR )g : Known cases Banaschewski functions Vaught's and For X 2 FP(R), set [X ] := isomorphism class of X . Dobbertin's results Then define [X ] + [Y ] := [X ⊕ Y ]. V(R) := f[X ] j X 2 FP(R)g, endowed with addition, is a commutative monoid (encodes the nonstable K-theory of R). It is conical: α + β = 0 ) α = β = 0. The definition is left-right symmetric. ∼ + V(R) = Z = f0; 1; 2;::: g if R is a division ring. FP(R) and V(R) Nonst. K-th., Banaschewski For a unital (associative) ring R, set The FP(R) := fX right R-module j X fin. gen. projectiveg realization problem n = fX j (9n)(9Y )(X ⊕ Y = RR )g : Known cases Banaschewski functions Vaught's and For X 2 FP(R), set [X ] := isomorphism class of X . Dobbertin's results Then define [X ] + [Y ] := [X ⊕ Y ]. V(R) := f[X ] j X 2 FP(R)g, endowed with addition, is a commutative monoid (encodes the nonstable K-theory of R). It is conical: α + β = 0 ) α = β = 0. The element [R] is an order-unit: 8α 9β 9n α + β = n[R]. ∼ + V(R) = Z = f0; 1; 2;::: g if R is a division ring. FP(R) and V(R) Nonst. K-th., Banaschewski For a unital (associative) ring R, set The FP(R) := fX right R-module j X fin. gen. projectiveg realization problem n = fX j (9n)(9Y )(X ⊕ Y = RR )g : Known cases Banaschewski functions Vaught's and For X 2 FP(R), set [X ] := isomorphism class of X . Dobbertin's results Then define [X ] + [Y ] := [X ⊕ Y ]. V(R) := f[X ] j X 2 FP(R)g, endowed with addition, is a commutative monoid (encodes the nonstable K-theory of R). It is conical: α + β = 0 ) α = β = 0. The element [R] is an order-unit: 8α 9β 9n α + β = n[R]. The definition is left-right symmetric. FP(R) and V(R) Nonst. K-th., Banaschewski For a unital (associative) ring R, set The FP(R) := fX right R-module j X fin. gen. projectiveg realization problem n = fX j (9n)(9Y )(X ⊕ Y = RR )g : Known cases Banaschewski functions Vaught's and For X 2 FP(R), set [X ] := isomorphism class of X . Dobbertin's results Then define [X ] + [Y ] := [X ⊕ Y ]. V(R) := f[X ] j X 2 FP(R)g, endowed with addition, is a commutative monoid (encodes the nonstable K-theory of R). It is conical: α + β = 0 ) α = β = 0. The element [R] is an order-unit: 8α 9β 9n α + β = n[R]. The definition is left-right symmetric. ∼ + V(R) = Z = f0; 1; 2;::: g if R is a division ring. order-unit of M: any e 2 M such that (8x 2 M)(9n 2 N)(x ≤ ne). Every conical commutative monoid with order-unit is isomorphic to V(R), for some hereditary, unital ring R (Bergman 1974 in the finitely generated case, Bergman and Dicks 1978 in the general case). What can V(R) be? Nonst. K-th., Banaschewski The realization problem On a commutative monoid M, x ≤ y :, (9z)(x + z = y); Known cases algebraic preordering of M. Banaschewski functions Vaught's and Dobbertin's results Every conical commutative monoid with order-unit is isomorphic to V(R), for some hereditary, unital ring R (Bergman 1974 in the finitely generated case, Bergman and Dicks 1978 in the general case). What can V(R) be? Nonst. K-th., Banaschewski The realization problem On a commutative monoid M, x ≤ y :, (9z)(x + z = y); Known cases algebraic preordering of M. Banaschewski functions order-unit of M: any e 2 M such that Vaught's and (8x 2 M)(9n 2 N)(x ≤ ne). Dobbertin's results What can V(R) be? Nonst. K-th., Banaschewski The realization problem On a commutative monoid M, x ≤ y :, (9z)(x + z = y); Known cases algebraic preordering of M. Banaschewski functions order-unit of M: any e 2 M such that Vaught's and (8x 2 M)(9n 2 N)(x ≤ ne). Dobbertin's results Every conical commutative monoid with order-unit is isomorphic to V(R), for some hereditary, unital ring R (Bergman 1974 in the finitely generated case, Bergman and Dicks 1978 in the general case). Which monoids are representable, that is, appear as V(R) for a (von Neumann) regular ring R? A survey paper about this problem: P. Ara, The realization problem for von Neumann regular rings. Ring theory 2007, 21{37, World Sci. Publ., Hackensack, NJ, 2009 (also arXiv:0802.1872). \Conicality"(8x; y)(x + y = 0 ) x = y = 0) and \existence of an order-unit"(8x)(9n)(x ≤ ne) not sufficient. Another condition, whose necessity was proved by Goodearl and Handelman (1975), is The refinement condition: a0 + a1 = b0 + b1 ) there are ci;j (i; j 2 f0; 1g) such that ai = ci;0 + ci;1 and bi = c0;i + c1;i 8i < 2. The realization problem in the regular case Nonst. K-th., Banaschewski Fundamental problem (Goodearl 1995) The realization problem Known cases Banaschewski functions Vaught's and Dobbertin's results A survey paper about this problem: P. Ara, The realization problem for von Neumann regular rings. Ring theory 2007, 21{37, World Sci. Publ., Hackensack, NJ, 2009 (also arXiv:0802.1872). \Conicality"(8x; y)(x + y = 0 ) x = y = 0) and \existence of an order-unit"(8x)(9n)(x ≤ ne) not sufficient. Another condition, whose necessity was proved by Goodearl and Handelman (1975), is The refinement condition: a0 + a1 = b0 + b1 ) there are ci;j (i; j 2 f0; 1g) such that ai = ci;0 + ci;1 and bi = c0;i + c1;i 8i < 2.
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