Available in electronic format
Available in print format		 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

The $ K$-theory of toric varieties

Author(s): G. Cortiñas; C. Haesemeyer; Mark E. Walker; C. Weibel
Journal: Trans. Amer. Math. Soc. 361 (2009), 3325-3341.
MSC (2000): Primary 19D55, 14M25, 19D25
Posted: December 31, 2008
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Recent advances in computational techniques for $ K$-theory allow us to describe the $ K$-theory of toric varieties in terms of the $ K$-theory of fields and simple cohomological data.

References:

1.
G. Cortiñas,
The obstruction to excision in $ K$-theory and cyclic homology,
Invent. Math. 454 (2006), 143-173. MR 2207785 (2006k:19006)

2.
G. Cortiñas, C. Haesemeyer, M. Schlichting and C. Weibel,
Cyclic homology, $ cdh$-cohomology and negative $ K$-theory,
Annals of Math. 167 (2008), 549-573.

MR 2415380

3.
G. Cortiñas, C. Haesemeyer and C. Weibel,
$ K$-regularity, $ cdh$-fibrant Hochschild homology, and a conjecture of Vorst,
J. Amer. Math. Soc., 21 (2008), no. 2, 547-561. MR 2373359

4.
G. Cortiñas, C. Haesemeyer and C. Weibel,
Infinitesimal cohomology and the Chern character to negative cyclic homology,
Preprint, available at http://www.math.uiuc.edu/K-theory/0824/, 2007.

5.
V. Danilov,
The geometry of toric varieties, Russian Math. Surveys 33 (1978), 97-154. MR 495499 (80g:14001)

6.
V. Danilov,
de Rham complex on toroidal variety, pp. 26-38 in Lecture Notes in Math. 1479, Springer, 1991. MR 1181204 (93i:14050)

7.
P. Deligne,
Théorie de Hodge: III, Publ. Math. IHES 44 (1974), 5-77. MR 0498552 (58:16653b)

8.
Ph. DuBois,
Complexe de de Rham filtré d'une variété singulière, Bull. Soc. Math. France 109 (1981), 41-81. MR 613848 (82j:14006)

9.
W. Fulton, Introduction to Toric Varieties, Annals Math. Study 131, Princeton Univ. Press, 1993. MR 1234037 (94g:14028)

10.
F. Guillén and V. Navarro-Aznar,
Un critère d'extension des foncteurs définis sur les schémas lisses, Publ. Math. IHES 95 (2002), 1-91. MR 1953190 (2004i:14020)

11.
J. Gubeladze,
Anderson's conjecture and the maximal class of monoids over which projective modules are free, Math. USSR Sb. 135 (1988), 169-181 (Russian). MR 937805 (89d:13010)

12.
J. Gubeladze, Nontriviality of $ SK_1(R[M])$, J. Pure Applied Alg. 104 (1995), 169-190. MR 1360174 (96j:19002)

13.
J. Gubeladze,
Higher $ K$-theory of toric varieties $ K$-Theory 28 (2003), 285-327. MR 2017618 (2004i:19004)

14.
J. Gubeladze,
Toric varieties with huge Grothendieck group, Adv. Math. 186 (2004), 117-124. MR 2065508 (2005d:14014)

15.
J. Gubeladze,
The nilpotence conjecture in $ K$-theory of toric varieties, Inventiones Math. 160 (2005), 173-216. MR 2129712 (2006d:14057)

16.
J. Gubeladze,
The Steinberg group of a monoid ring, nilpotence, and algorithms, J. Algebra, 307 (2007), 461-496. MR 2278067 (2007i:19003)

17.
J. Gubeladze,
Global coefficient ring in the nilpotence conjecture, Proc. Amer. Math. Soc. 136 (2008), no. 2, 499-503. MR 2358489

18.
C. Haesemeyer,
Descent properties of homotopy $ K$-theory,
Duke Math. J. 125 (2004), 589-620. MR 2166754 (2006g:19002)

19.
M.-N. Ishida,
Torus embeddings and de Rham complexes, pp. 111-145 in Adv. Stud. Pure Math., 11, North-Holland, Amsterdam, 1987. MR 951199 (89j:14028)

20.
J.-L. Loday,
Cyclic homology.
Springer-Verlag, Berlin, 1992. MR 1217970 (94a:19004)

21.
A. Suslin and V. Voevodsky,
Bloch-Kato conjecture and motivic cohomology with finite coefficients, pp. 117-189 in
The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), volume 548 of NATO Sci. Ser. C Math. Phys. Sci.. Kluwer, Dordrecht, 2000. MR 1744945 (2001g:14031)

22.
R. Swan,
Gubeladze's proof of Anderson's conjecture, Contemp. Math. 124, Amer. Math. Soc., Providence, RI, 1992. MR 1144038 (92m:13012)

23.
W. Vasconcelos, A note on normality and module of differentials, Math. Z. 105 (1968), 291-293. MR 0227165 (37:2750)

24.
C. Weibel and S. Geller,
Étale descent for Hochschild and cyclic homology, Comment. Math. Helvetici 66 (1991), 368-388. MR 1120653 (92e:19006)

25.
C. Weibel,
Module structures on the $ K$-theory of graded rings, J. Algebra 105 (1987), 465-483. MR 873680 (88f:18018)

26.
C. Weibel,
Homotopy algebraic $ K$-theory,
AMS Contemp Math. 83 (1989), 461-488. MR 991991 (90d:18006)

27.
C. Weibel,
An introduction to homological algebra,
Cambridge Univ. Press, 1994. MR 1269324 (95f:18001)

28.
C. Weibel,
Cyclic homology of schemes, Proc. AMS 124 (1996), 1655-1662. MR 1277141 (96h:19003)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 19D55, 14M25, 19D25

Retrieve articles in all Journals with MSC (2000): 19D55, 14M25, 19D25

Additional Information:

G. Cortiñas
Affiliation: Departamento de Matemática, FCEyN-UBA, Ciudad Universitaria Pab 1, 1428 Buenos Aires, Argentina - and - Departamento Álgebra, Fac. de Ciencias, Prado de la Magdalena s/n, 47005 Valladolid, Spain
Email: gcorti@dm.uba.ar

C. Haesemeyer
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, University of California, Los Angeles, Box 95155, Los Angeles, California 90095-1555
Email: chh@math.uiuc.edu, chh@math.ucla.edu

Mark E. Walker
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Lincoln, Nebraska 68588-0130
Email: mwalker5@math.unl.edu

C. Weibel
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08901
Email: weibel@math.rutgers.edu

DOI: 10.1090/S0002-9947-08-04750-8
PII: S 0002-9947(08)04750-8
Keywords: Algebraic $K$-theory, toric varieties
Received by editor(s): September 14, 2007
Posted: December 31, 2008
Additional Notes: The first author's research was partially supported by FSE and by grants ANPCyT PICT 03-12330, UBACyT-X294, JCyL VA091A05, and MEC MTM00958.
The third author's research was supported by NSF grant DMS-0601666.
The fourth author's research was supported by NSA grant MSPF-04G-184 and the Oswald Veblen Fund
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google