Group Completions and Grothendieck Groups

Abstract For a balanced wall crossing in geometric invariant theory (GIT), there exist derived equivalences between the corresponding GIT quotients if certain numerical conditions are satisfied. Given such a wall crossing, I construct a perverse sheaf of categories on a disk, singular at a point, with half-monodromies recovering these equivalences, and with behaviour at the singular point controlled by a GIT quotient stack associated to the wall. Taking complexified Grothendieck groups gives a perverse sheaf of vector spaces: I characterize when this is an intersection cohomology complex of a local system on the punctured disk.

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Algebraic cobordism and Grothendieck groups over singular schemes

Homology Homotopy and Applications ◽

10.4310/hha.2010.v12.n1.a8 ◽

2010 ◽

Vol 12 (1) ◽

pp. 93-110 ◽

Cited By ~ 3

Author(s):

Shouxin Dai

Keyword(s):

Grothendieck Groups ◽

Algebraic Cobordism

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Tensor Products of Dimension Groups and K0 of Unit-Regular Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1986-032-0 ◽

1986 ◽

Vol 38 (3) ◽

pp. 633-658 ◽

Cited By ~ 17

Author(s):

K. R. Goodearl ◽

D. E. Handelman

Keyword(s):

Regular Ring ◽

Hausdorff Space ◽

Tensor Products ◽

Matrix Algebras ◽

Dimension Group ◽

Grothendieck Groups ◽

Dimension Groups ◽

Direct Limits ◽

Finite Products ◽

Regular Rings

We study direct limits of finite products of matrix algebras (i.e., locally matricial algebras), their ordered Grothendieck groups (K0), and their tensor products. Given a dimension group G, a general problem is to determine whether G arises as K0 of a unit-regular ring or even as K0 of a locally matricial algebra. If G is countable, this is well known to be true. Here we provide positive answers in case (a) the cardinality of G is ℵ1, or (b) G is an arbitrary infinite tensor product of the groups considered in (a), or (c) G is the group of all continuous real-valued functions on an arbitrary compact Hausdorff space. In cases (a) and (b), we show that G in fact appears as K0 of a locally matricial algebra. Result (a) is the basis for an example due to de la Harpe and Skandalis of the failure of a determinantal property in a non-separable AF C*-algebra [18, Section 3].

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The Equivariant Grothendieck Groups of the Russell-Koras Threefolds

Canadian Journal of Mathematics ◽

10.4153/cjm-2001-001-x ◽

2001 ◽

Vol 53 (1) ◽

pp. 3-32 ◽

Cited By ~ 1

Author(s):

J. P. Bell

Keyword(s):

Fixed Point ◽

Tangent Space ◽

Unique Fixed Point ◽

Linear Action ◽

Grothendieck Groups ◽

Smooth Affine

AbstractThe Russell-Koras contractible threefolds are the smooth affine threefolds having a hyperbolic *-action with quotient isomorphic to the corresponding quotient of the linear action on the tangent space at the unique fixed point. Koras and Russell gave a concrete description of all such threefolds and determined many interesting properties they possess. We use this description and these properties to compute the equivariant Grothendieck groups of these threefolds. In addition, we give certain equivariant invariants of these rings.

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Grothendieck Groups of Bass Orders

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-011-9 ◽

1971 ◽

Vol 23 (1) ◽

pp. 103-115

Author(s):

Klaus W. Roggenkamp

Keyword(s):

Special Class ◽

Grothendieck Group ◽

Commutative Rings ◽

Gorenstein Rings ◽

Direct Sum ◽

Maximal Orders ◽

Grothendieck Groups ◽

Epimorphic Image ◽

Carry Over ◽

Separable Algebras

Commutative Bass rings, which form a special class of Gorenstein rings, have been thoroughly investigated by Bass [1]. The definitions do not carry over to non-commutative rings. However, in case one deals with orders in separable algebras over fields, Bass orders can be defined. Drozd, Kiricenko, and Roïter [3] and Roïter [6] have clarified the structure of Bass orders, and they have classified them. These Bass orders play a key role in the question of the finiteness of the non-isomorphic indecomposable lattices over orders (cf. [2; 8]). We shall use the results of Drozd, Kiricenko, and Roïter [3] to compute the Grothendieck groups of Bass orders locally. Locally, the Grothendieck group of a Bass order (with the exception of one class of Bass orders) is the epimorphic image of the direct sum of the Grothendieck groups of the maximal orders containing it.

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