Geodesic
Operational $K$-theory
Documenta mathematica, Tome 20 (2015), pp. 357-399.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We study the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and prove that it has many geometric properties analogous to those of operational Chow theory. This operational $K$-theory agrees with Grothendieck groups of vector bundles on smooth varieties, admits a natural map from the Grothen­dieck group of perfect complexes on general varieties, satisfies descent for Chow envelopes, and is $\{A}^1$-homotopy invariant. $\smallskip $Furthermore, we show that the operational $K$-theory of a complete linear variety is dual to the Grothendieck group of coherent sheaves. As an application, we show that the $K$-theory of perfect complexes on any complete toric threefold surjects onto this group. Finally we identify the equivariant operational $K$-theory of an arbitrary toric variety with the ring of integral piecewise exponential functions on the associated fan.
Classification : 14C35, 14C15, 19E08, 14M25, 14L30 
@article{DOCMA_2015__20__a31,
     author = {Anderson, Dave and Payne, Sam},
     title = {Operational $K$-theory},
     journal = {Documenta mathematica},
     pages = {357--399},
     publisher = {mathdoc},
     volume = {20},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2015__20__a31/}
}
TY  - JOUR
AU  - Anderson, Dave
AU  - Payne, Sam
TI  - Operational $K$-theory
JO  - Documenta mathematica
PY  - 2015
SP  - 357
EP  - 399
VL  - 20
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DOCMA_2015__20__a31/
LA  - en
ID  - DOCMA_2015__20__a31
ER  - 
%0 Journal Article
%A Anderson, Dave
%A Payne, Sam
%T Operational $K$-theory
%J Documenta mathematica
%D 2015
%P 357-399
%V 20
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DOCMA_2015__20__a31/
%G en
%F DOCMA_2015__20__a31
Anderson, Dave; Payne, Sam. Operational $K$-theory. Documenta mathematica, Tome 20 (2015), pp. 357-399. http://geodesic.mathdoc.fr/item/DOCMA_2015__20__a31/