The Grothendieck-Riemann-Roch theorem for group scheme actions. (English) Zbl 0951.14029

Köck, Bernhard

Ann. Sci. Éc. Norm. Supér. (4) 31, No. 3, 415-458 (1998).

This is a revised version of a previous paper by the author [B. Köck, “The Grothendieck-Riemann-Roch theorem in the higher \(K\)-theory of group scheme actions”, Habilitationsschrift (Karlsruhe 1995)]. Let \(K_{q}(G,X)\) denote the \(q\)th equivariant \(K\)-group of the \(G\)-scheme \(X\), where \(G\) is a flat group scheme. Let \(K(G,X)\) be the direct sum of all these \(K_{q}(G,X)\)’s. The objective is to compute the Euler characteristic, i.e. the push-forward homomorphism \(f_{\ast }:K(G,Y)\rightarrow K(G,X),\) where \(X\) and \(Y\) are \(G\)-schemes and \(f:Y\rightarrow X\) is a \(G\)-projective local complete intersection morphism. The first theorem is the equivariant Adams-Riemann-Roch formula proved here. The first states that \(\psi ^{j}f_{\ast }(y)=f_{\ast }(\theta ^{j}(f)^{-1}\psi ^{j}(y))\) in \(\widehat{K}(G,X)[ j^{-1}]\), where \(y\in K(G,Y), j\geq 1,\) and \(\theta ^{j}(f)\) is the \(j\)th equivariant Bott element associated with \(f\). The second theorem, the equivariant Grothendieck-Riemann-Roch formula, examines the behavior of a certain Chern character ch, and it is shown that ch\((f_{\ast }(y))=\)Gr\((f_{\ast })_{\mathbb Q}\)(Td\((f)\cdot \)ch\((y))\), where Td\((f)\) is the Todd class of \(f.\) This theorem, the equivariant version of a theorem proved for \(K_{0}\)-groups by Grothendieck and by Soulé for higher \(K\)-groups, depends on a pair of conjectures being true, one concerning the \(\lambda \)-structure on higher \(K\)-theory and the other concerning Grothendieck groups.
Finally, applications are given. As a corollary to the equivariant Adams-Riemann-Roch formula, it is shown that for \(A\) a commutative ring and \(H\) a subgroup of the finite group \(G\) the induced representation \(A\left[ G/H\right] \) is invariant under the Adams operations \(\psi ^{j}\) when viewed as an element of \(\widehat{K}_{0}(G,A)\left[ j^{-1}\right] .\)

Reviewer: Alan Koch (Austin)

MSC:

14L30	Group actions on varieties or schemes (quotients)
14C40	Riemann-Roch theorems
14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry
19E20	Relations of \(K\)-theory with cohomology theories

Keywords:

group scheme actions; equivariant geometry; equivariant Grothendieck-Riemann-Roch formula; Todd class; equivariant Adam-Riemann-Roch formula

References:

[1]	M. F. ATIYAH , Characters and cohomology of finite groups (Publ. Math. IHES, Vol. 9, 1961 , pp. 23-64). Numdam | MR 26 #6228 | Zbl 0107.02303 · Zbl 0107.02303 · doi:10.1007/BF02698718
[2]	M. F. ATIYAH and D. O. TALL , Group representations, \lambda -rings and the J-homomorphism (Topology, Vol. 8, 1969 , pp. 253-297). MR 39 #5702 | Zbl 0159.53301 · Zbl 0159.53301 · doi:10.1016/0040-9383(69)90015-9
[3]	H. BASS , Algebraic K-theory (Math. Lecture Note Series, Benjamin, New York, 1968 ). MR 40 #2736 | Zbl 0174.30302 · Zbl 0174.30302
[4]	P. BERTHELOT , A. GROTHENDIECK and L. ILLUSIE , Théorie des Intersections et Théorème de Riemann-Roch (Lecture Notes in Math., Vol. 225, Springer, New York, 1971 ). MR 50 #7133 | Zbl 0218.14001 · Zbl 0218.14001
[5]	R. BOLTJE , Mackey functors and related structures in representation theory and number theory , Report No. 327, Institut für Mathematik der Universität Augsburg, 1995 .
[6]	M. BRION and M. VERGNE , An equivariant Riemann-Roch theorem for complete, simplicial toric varieties (J. Reine Angew. Math., Vol. 482, 1997 , pp. 67-92). MR 98a:14067 | Zbl 0862.14006 · Zbl 0862.14006
[7]	D. BURNS and T. CHINBURG , Adams operations and integral Hermitian-Galois representations (Amer. J. Math., Vol. 118, 1996 , pp. 925-962). MR 98g:11126 | Zbl 0865.11076 · Zbl 0865.11076 · doi:10.1353/ajm.1996.0039
[8]	Ph. CASSOU-NOGUÈS and M. J. TAYLOR , Opérations d’Adams et Groupe des classes d’Algèbre de groupe (J. Algebra, Vol. 95, 1985 , pp. 125-152). MR 87a:11116 | Zbl 0603.12007 · Zbl 0603.12007 · doi:10.1016/0021-8693(85)90098-5
[9]	T. CHINBURG , B. EREZ , G. PAPPAS and M. J. TAYLOR , Riemann-Roch type theorems for arithmetic schemes with a finite group action (J. Reine Angew. Math., Vol. 489, 1997 , pp. 151-187). MR 98e:14006 | Zbl 0903.19001 · Zbl 0903.19001
[10]	C. W. CURTIS and I. REINER , Methods of representation theory with applications to finite groups and orders I (Pure Appl. Math., Wiley, New York, 1981 ). Zbl 0469.20001 · Zbl 0469.20001
[11]	M. DEMAZURE , Désingularisation des variétés de Schubert généralisées (Ann. Sci. École Norm. Sup., (4), Vol. 7, 1974 , pp. 53-88). Numdam | MR 50 #7174 | Zbl 0312.14009 · Zbl 0312.14009
[12]	M. DEMAZURE and A. GROTHENDIECK , Schémas en Groupes I, II, III (Lecture Notes in Math., Vol. 151-152-153, Springer, New York, 1970 ). MR 43 #223a
[13]	D. EDIDIN and W. GRAHAM , Equivariant intersection theory , preprint, alg-geom/9603008. · Zbl 0980.14004
[14]	W. FULTON and J. HARRIS , Representation theory (Grad. Texts in Math., Vol. 129, Springer, New York, 1991 ). MR 93a:20069 | Zbl 0744.22001 · Zbl 0744.22001
[15]	W. FULTON and S. LANG , Riemann-Roch algebra (Grundlehren Math. Wiss., Vol. 277, Springer, New York, 1985 ). MR 88h:14011 | Zbl 0579.14011 · Zbl 0579.14011
[16]	W. FULTON and R. MACPHERSON , Characteristic classes of direct image bundles for covering maps (Ann. of Math., Vol. 125, 1987 , pp. 1-92). MR 88f:14015 | Zbl 0628.55010 · Zbl 0628.55010 · doi:10.2307/1971288
[17]	D. R. GRAYSON , Exterior power operations on higher K-theory (K-theory, Vol. 3, 1989 , pp. 247-260). MR 91h:19005 | Zbl 0701.18007 · Zbl 0701.18007 · doi:10.1007/BF00533371
[18]	A. GROTHENDIECK , Sur quelques propriétés fondamentales en théorie des intersections , Séminaire C. Chevalley, 2e année, Anneaux de Chow et applications, Secr. Math. Paris, 1958 . Numdam | MR 21 #5637
[19]	A. GROTHENDIECK , Classes de Chern et représentations linéaires des groupes discrets , in J. Giraud et al., Dix exposés sur la cohomologie des schémas (Adv. Stud. Pure Math., North-Holland Publishing Company, Amsterdam, 1968 , pp. 215-305). MR 42 #280 | Zbl 0198.26101 · Zbl 0198.26101
[20]	A. GROTHENDIECK and J. A. DIEUDONNÉ , Eléments de Géométrie Algébrique I (Grundlehren Math. Wiss., Vol. 166, Springer, New York, 1971 ) ; ...II, III, (Publ. Math. IHES, Vol. 8, 1961 , Vol. 11, 1961 , Vol. 17, 1963 ). Numdam · Zbl 0203.23301
[21]	R. HARTSHORNE , Algebraic geometry (Graduate Texts in Math., Vol. 52, Springer, New York, 1977 ). MR 57 #3116 | Zbl 0367.14001 · Zbl 0367.14001
[22]	H. L. HILLER , \lambda -rings and algebraic K-theory (J. Pure Appl. Algebra, Vol. 20, 1981 , pp. 241-266). MR 82e:18016 | Zbl 0471.18007 · Zbl 0471.18007 · doi:10.1016/0022-4049(81)90062-1
[23]	J. C. JANTZEN , Representations of algebraic groups (Pure Appl. Math., Vol. 131, Academic Press, Boston, 1987 ). MR 89c:20001 | Zbl 0654.20039 · Zbl 0654.20039
[24]	M. KERVAIRE , Opérations d’Adams en Théorie des représentations linéaires des groupes finis (Enseign. Math., Vol. 22, 1976 , pp. 1-28). MR 54 #5325 | Zbl 0335.20003 · Zbl 0335.20003
[25]	B. KÖCK , Das Lefschetz- und Riemann-Roch-Theorem in der höheren äquivarianten K-Theorie Dissertation, Regensburg, 1989 . · Zbl 1137.14304
[26]	B. KÖCK , Chow motif and higher Chow theory of G / P (Manuscripta Math., Vol. 70, 1991 , pp. 363-372). Article | MR 91m:14077 | Zbl 0735.14001 · Zbl 0735.14001 · doi:10.1007/BF02568384
[27]	B. KÖCK , Das Adams-Riemann-Roch-Theorem in der höheren äquivarianten K-Theorie (J. Reine Angew. Math., Vol. 421, 1991 , pp. 189-217). Article | Zbl 0731.19005 · Zbl 0731.19005 · doi:10.1515/crll.1991.421.189
[28]	B. KÖCK , The Lefschetz theorem in higher equivariant K-theory (Comm. Algebra, Vol. 19, 1991 , pp. 3411-3422). MR 93g:19008 | Zbl 0760.14004 · Zbl 0760.14004 · doi:10.1080/00927879108824324
[29]	B. KÖCK , Higher K’-groups of integral group rings (K-Theory, Vol. 4, 1991 , pp. 177-187). MR 93a:19002 | Zbl 0752.16013 · Zbl 0752.16013 · doi:10.1007/BF01254547
[30]	B. KÖCK , Shuffle products in higher K-theory (J. Pure Appl. Algebra, Vol. 92, 1994 , pp. 269-307). MR 95c:19001 | Zbl 0797.19003 · Zbl 0797.19003 · doi:10.1016/0022-4049(94)90098-1
[31]	B. KÖCK , The Grothendieck-Riemann-Roch theorem in the higher K-theory of group scheme actions Habilitationsschrift, Karlsruhe, 1995 .
[32]	B. KÖCK , On Adams operations on the higher K-theory of group rings , in : G. Banaszak et al. (eds.), Algebraic K-theory (Poznań, 1995 ) (Contemp. Math., Vol. 199, Amer. Math. Soc., Providence, 1996 , pp. 139-150). Zbl 0857.19001 · Zbl 0857.19001
[33]	B. KÖCK , Adams operations for projective modules over group rings (Math. Proc. Cambridge Philos. Soc., Vol. 122, 1997 , pp. 55-71). MR 98g:13013 | Zbl 0884.19002 · Zbl 0884.19002 · doi:10.1017/S0305004196001351
[34]	B. KÖCK , Operations on locally free classgroups , preprint, Karlsruhe, 1997 . · Zbl 0980.11052
[35]	B. KÖCK , Riemann-Roch for tensor power , preprint, Karlsruhe, 1998 .
[36]	B. KOSTANT and S. KUMAR , The nil Hecke ring and cohomology of G/P for a Kac-Moody group G (Adv. Math., Vol. 62, 1986 , pp. 187-237). MR 88b:17025b | Zbl 0641.17008 · Zbl 0641.17008 · doi:10.1016/0001-8708(86)90101-5
[37]	B. KOSTANT and S. KUMAR , T-equivariant K-theory of generalized flag varieties (J. Differential Geom., Vol. 32, 1990 , pp. 549-603). MR 92c:19006 | Zbl 0731.55005 · Zbl 0731.55005
[38]	Ch. KRATZER , \lambda -Structure en K-théorie algébrique (Comment. Math. Helv., Vol. 55, 1980 , pp. 233-254). Zbl 0444.18008 · Zbl 0444.18008 · doi:10.1007/BF02566684
[39]	A. LASCOUX , Anneau de Grothendieck de la variété de drapeaux , in The Grothendieck Festschrift, vol. III (Progr. Math., Vol. 88, Birkhäuser, Boston, 1990 , pp. 1-34). MR 92j:14064 | Zbl 0742.14041 · Zbl 0742.14041
[40]	H. W. LENSTRA , Grothendieck groups of abelian group rings (J. Pure Appl. Algebra, Vol. 20, 1981 , pp. 173-193). MR 82g:16024 | Zbl 0467.16016 · Zbl 0467.16016 · doi:10.1016/0022-4049(81)90091-8
[41]	Y. I. MANIN , Lectures on the K-functor in algebraic geometry (Russian Math. Surveys, Vol. 24, No. 5, 1969 , pp. 1-89). MR 42 #265 | Zbl 0204.21302 · Zbl 0204.21302 · doi:10.1070/RM1969v024n05ABEH001357
[42]	R. MORELLI , The K theory of a toric variety (Adv. Math., Vol. 100, 1993 , pp. 154-182). MR 94j:14047 | Zbl 0805.14025 · Zbl 0805.14025 · doi:10.1006/aima.1993.1032
[43]	D. MUMFORD , Geometric invariant theory (Ergeb. Math. Grenzgeb., Vol. 34, Springer, New York, 1965 ). MR 35 #5451 | Zbl 0147.39304 · Zbl 0147.39304
[44]	D. QUILLEN , Higher algebraic K-theory : I , in H. Bass (ed.), Algebraic K-Theory I (Seattle, 1972 ) (Lecture Notes in Math., Vol. 341, Springer, New York, 1973 , pp. 85-147). MR 49 #2895 | Zbl 0292.18004 · Zbl 0292.18004
[45]	V. P. SNAITH , Invariants of representations , in J. F. Jardine and V. P. Snaith (eds.), Algebraic K-theory : Connections with geometry and topology (Lake Louise, 1987 ) (NATO Adv. Sci. Inst. Ser. C : Math. Phys. Sci., Vol. 279, Kluwer Acad. Publ., Dordrecht, 1989 , pp. 445-508). MR 91h:20010 | Zbl 0721.20004 · Zbl 0721.20004
[46]	C. SOULÉ , Opérations en K-Théorie algébrique (Canad. J. Math., Vol. 37, No. 3, 1985 , pp. 488-550). MR 87b:18013 | Zbl 0575.14015 · Zbl 0575.14015 · doi:10.4153/CJM-1985-029-x
[47]	H. SUMIHIRO , Equivariant completion II (J. Math. Kyoto Univ., Vol. 15, 1975 , pp. 573-605). Article | MR 52 #8137 | Zbl 0331.14008 · Zbl 0331.14008
[48]	G. TAMME , The theorem of Riemann-Roch , in M. Rapoport, N. Schappacher and P. Schneider (eds.), Beilinson’s conjectures on special values of L-functions (Perspect. in Math., Vol. 4, Academic Press, Boston, 1988 , pp. 103-168). MR 89d:14013 | Zbl 0664.14004 · Zbl 0664.14004
[49]	C. B. THOMAS , Characteristic classes and the cohomology of finite groups (Cambridge Stud. Adv. Math., Vol. 9, Cambridge Univ. Press, Cambridge, 1986 ). MR 88f:20005 | Zbl 0618.20036 · Zbl 0618.20036
[50]	R. W. THOMASON , Absolute cohomological purity (Bull. Soc. Math. France, Vol. 112, 1984 , pp. 397-406). Numdam | MR 87e:14018 | Zbl 0584.14007 · Zbl 0584.14007
[51]	R. W. THOMASON , Lefschetz-Riemann-Roch theorem and coherent trace formula (Invent. Math., Vol. 85, 1986 , pp. 515-543). MR 87j:14028 | Zbl 0653.14005 · Zbl 0653.14005 · doi:10.1007/BF01390328
[52]	R. W. THOMASON , Algebraic K-theory of group scheme actions , in W. Browder (ed.), Algebraic topology and algebraic K-theory (Princeton, 1983 ) (Ann. Math. Stud., Vol. 113, Univ. Press, Princeton, 1987 , pp. 539-563). MR 89c:18016 | Zbl 0701.19002 · Zbl 0701.19002
[53]	R. W. THOMASON , Une formule de Lefschetz en K-théorie équivariante algébrique (Duke Math. J., Vol. 68, 1992 , pp. 447-462). Article | MR 93m:19007 | Zbl 0813.19002 · Zbl 0813.19002 · doi:10.1215/S0012-7094-92-06817-7
[54]	R. W. THOMASON , Les K-groupes d’un schéma éclaté et une formule d’intersection excédentaire (Invent., Math., Vol. 112, 1993 , pp. 195-215). MR 93k:19005 | Zbl 0816.19004 · Zbl 0816.19004 · doi:10.1007/BF01232430
[55]	C. A. WEIBEL , Homotopy algebraic K-theory , in M. R. Stein and R. K. Dennis (eds.), Algebraic K-theory and algebraic number theory (Honolulu, 1987 ) (Contemp. Math., Vol. 83, Amer. Math. Soc., Providence, 1989 , pp. 461-488). MR 90d:18006 | Zbl 0669.18007 · Zbl 0669.18007