Computing the homology of Koszul complexes

Köck, Bernhard

doi:10.1090/S0002-9947-01-02723-4

Computing the homology of Koszul complexes
HTML articles powered by AMS MathViewer

by Bernhard Köck PDF

Trans. Amer. Math. Soc. 353 (2001), 3115-3147 Request permission

Abstract:

Let $R$ be a commutative ring and $I$ an ideal in $R$ which is locally generated by a regular sequence of length $d$. Then, each f. g. projective $R/I$-module $V$ has an $R$-projective resolution $P.$ of length $d$. In this paper, we compute the homology of the $n$-th Koszul complex associated with the homomorphism $P_1 \rightarrow P_0$ for all $n \ge 1$, if $d=1$. This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if $d=2$, we compute the homology of the complex $N \operatorname {Sym}^2 \Gamma (P.)$ where $\Gamma$ and $N$ denote the functors occurring in the Dold-Kan correspondence.

References

Kaan Akin, David A. Buchsbaum, and Jerzy Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), no. 3, 207–278. MR 658729, DOI 10.1016/0001-8708(82)90039-1
M. F. Atiyah and D. O. Tall, Group representations, $\lambda$-rings and the $J$-homomorphism, Topology 8 (1969), 253–297. MR 244387, DOI 10.1016/0040-9383(69)90015-9
Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
Albrecht Dold and Dieter Puppe, Homologie nicht-additiver Funktoren. Anwendungen, Ann. Inst. Fourier (Grenoble) 11 (1961), 201–312 (German, with French summary). MR 150183
Samuel Eilenberg, Abstract description of some basic functors, J. Indian Math. Soc. (N.S.) 24 (1960), 231–234 (1961). MR 125148
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
William Fulton and Serge Lang, Riemann-Roch algebra, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 277, Springer-Verlag, New York, 1985. MR 801033, DOI 10.1007/978-1-4757-1858-4
Daniel R. Grayson, Adams operations on higher $K$-theory, $K$-Theory 6 (1992), no. 2, 97–111. MR 1187703, DOI 10.1007/BF01771009
Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
Brenda Johnson and Randy McCarthy, Linearization, Dold-Puppe stabilization, and Mac Lane’s $Q$-construction, Trans. Amer. Math. Soc. 350 (1998), no. 4, 1555–1593. MR 1451606, DOI 10.1090/S0002-9947-98-02067-4
J. P. Jouanolou, Riemann-Roch sans dénominateurs, Invent. Math. 11 (1970), 15–26. MR 332789, DOI 10.1007/BF01389802
Bernhard Köck, The Grothendieck-Riemann-Roch theorem for group scheme actions, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 3, 415–458 (English, with English and French summaries). MR 1621405, DOI 10.1016/S0012-9593(98)80140-7
—, Riemann-Roch for tensor powers, Math. Z. 233 (2000), 755–801.
Ju. I. Manin, Lectures on the $K$-functor in algebraic geometry, Uspehi Mat. Nauk 24 (1969), no. 5 (149), 3–86 (Russian). MR 0265355
Christophe Soulé, Opérations en $K$-théorie algébrique, Canad. J. Math. 37 (1985), no. 3, 488–550 (French). MR 787114, DOI 10.4153/CJM-1985-029-x
R. W. Thomason and Thomas Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, DOI 10.1007/978-0-8176-4576-2_{1}0
Charles E. Watts, Intrinsic characterizations of some additive functors, Proc. Amer. Math. Soc. 11 (1960), 5–8. MR 118757, DOI 10.1090/S0002-9939-1960-0118757-0
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136