Loop spaces of the Q-construction

• Autorzy: A. Neeman
• Języki publikacji: EN

Abstrakty

EN

Giffen in [1], and Gillet-Grayson in [3], independently found a simplicial model for the loop space on Quillen's Q-construction. Their proofs work for exact categories. Here we generalise the results to the K-theory of triangulated categories. The old proofs do not generalise. Our new proof, aside from giving the generalised result, can also be viewed as an amusing new proof of the old theorems of Giffen and Gillet-Grayson.

Kategorie tematyczne

• 18E30: Derived categories, triangulated categories
• 19D99: None of the above, but in this section
• 18F25: Algebraic K -theory and L -theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2000
• Tom: 164
• Numer: 1
• Strony: 71-95

Daty

wydano

2000

otrzymano

1999-11-23

poprawiono

2000-01-19

Twórcy

autor

A. Neeman

• Centre for Mathematics and its Applications, The Australian National University, Canberra, ACT 0200, Australia

Bibliografia

• [1] C. H. Giffen, Loop spaces for the Q-construction, J. Pure Appl. Algebra 52 (1988), 1-30.
• [2] C. H. Giffen, Unitary algebraic Witt- and K-theories, preprint.
• [3] H. Gillet and D. R. Grayson, The loop space of the Q-construction, Illinois J. Math. 31 (1987), 574-597.
• [4] D. R. Grayson, Higher algebraic K-theory II (after D. Quillen), in: Algebraic K-Theory, Lecture Notes in Math. 551, Springer, 1976, 217-240.
• [5] D. R. Grayson, Exterior power operations on higher K-theory, K-Theory 3 (1989), 247-260.
• [6] J J. F. Jardine, The multiple Q-construction, Canad. J. Math. 39 (1987), 1174-1209.
• [7] A. Neeman, K-theory for triangulated categories I(A), Asian J. Math. 1 (1997), 330-417.
• [8] A. Neeman, K-theory for triangulated categories I(B), ibid. 1 (1997), 435-529.
• [9] A. Neeman, K-theory for triangulated categories II, ibid. 2 (1998), 1-125.
• [10] A. Neeman, K-theory for triangulated categories III(A), ibid. 2 (1998), 495-594.
• [11] A. Neeman, K-theory for triangulated categories III(B), ibid. 3 (1999), 555-606.
• [12] D. Quillen, Higher algebraic K-theory I, in: Algebraic K-Theory I, Lecture Notes in Math. 341, Springer, 1973, 85-147.
• [13] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293-312.