An Artin-Rees theorem in $K$-theory and applications to zero cycles
Author:
Amalendu Krishna
Journal:
J. Algebraic Geom. 19 (2010), 555-598
DOI:
https://doi.org/10.1090/S1056-3911-09-00521-9
Published electronically:
November 17, 2009
MathSciNet review:
2629600
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Abstract |
References |
Additional Information
Abstract:
For the smooth normalization $f : {\overline X} \to X$ of a singular variety $X$ over a field $k$ of characteristic zero, we show that for any conducting subscheme $Y$ for the normalization, and for any $i \in \mathbb {Z}$, the natural map $K_i(X, {\overline X}, nY) \to K_i(X, {\overline X}, Y)$ is zero for all sufficiently large $n$.
As an application, we prove a formula for the Chow group of zero cycles on a quasi-projective variety $X$ over $k$ with Cohen-Macaulay isolated singularities, in terms of an inverse limit of the relative Chow groups of a desingularization $\widetilde X$ relative to the multiples of the exceptional divisor.
We use this formula to verify a conjecture of Srinivas about the Chow group of zero cycles on the affine cone over a smooth projective variety which is arithmetically Cohen-Macaulay.
References
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References
- M. André, Homologie des algèbres commutatives, Die Grundlehren der mathematischen Wissenschaften, Band 206, Springer-Verlag, Berlin-New York, 1974. MR 0352220 (50:4707)
- L. Barbieri-Viale, Zero-cycles on singular varieties: torsion and Bloch’s formula, J. Pure Appl. Alg., 78, (1992), 1-13. MR 1154894 (93b:14021)
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- J. Cathelineau, Lambda structures in algebraic $K$-theory, $K$-Theory, 4, (1991), 591-606.
- C. Consani, $K$-theory of blow-ups and vector bundles on the cone over a surface, $K$-Theory, 7, (1993), 269-284. MR 1244003 (94i:19005)
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- M. Levine, C. Weibel, Zero cycles and complete intersections on singular varieties, 359, (1985), 106-120. MR 794801 (86k:14003)
- J-L Loday, Cyclic Homology, Grund. der math. Wissen. Series, 301, Springer Verlag, 1998. MR 1600246 (98h:16014)
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- A. Ramanathan, Schubert varieties are arithmetically Cohen-Macaulay, Invent. Math., 80, (1985), no. 2, 283-294. MR 788411 (87d:14044)
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- C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)
- C. Weibel, Negative $K$-theory of normal surfaces, Duke Math. J., 108, (2001), 1-35. MR 1831819 (2002b:14012)
Additional Information
Amalendu Krishna
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005 India
MR Author ID:
703987
Email:
amal@math.tifr.res.in
Received by editor(s):
May 31, 2008
Received by editor(s) in revised form:
September 28, 2008
Published electronically:
November 17, 2009
