Celebratio Mathematica

P. Baum, A. Carey, and B.-L. Wang: “\( K \)-cycles for twisted \( K \)-homology,” pp. 69–98 in Nanjing special issue on K-theory, number theory and geometry, published as J. K-Theory 12 : 1. Issue edited by X. Guo, H. Qin, and G. Tang. Cambridge University Press, August 2013. MR 3126635 Zbl 1300.19003 incollection

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P. Baum, Carey, A., and B. Wang: \( K \)-homology and Fredholm operators I: Dirac operators. Preprint, 2016. ArXiv 1604.03502 techreport

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P. Baum, Carey, A., and B. Wang: \( K \)-homology and Fredholm operators II: Elliptic operators. Preprint, 2016. ArXiv 1604.03535 techreport

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P. Baum, Carey, A., and B. Wang: On the spectra of finite type algebras. Preprint, 2017. ArXiv 1705.01404 techreport

We review Morita equivalence for finite type \( k \)-algebras \( A \) and also a weakening of Morita equivalence which we call stratified equivalence. The spectrum of \( A \) is the set of equivalence classes of irreducible \( A \)-modules. For any finite type \( k \)-algebra \( A \), the spectrum of \( A \) is in bijection with the set of primitive ideals of \( A \). The stratified equivalence relation preserves the spectrum of \( A \) and also preserves the periodic cyclic homology of \( A \). However, the stratified equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence. A key example illustrating the distinction between Morita equivalence and stratified equivalence is provided by affine Hecke algebras associated to affine Weyl groups. Stratified equivalences lie at the heart of the ABPS conjecture, and lead to an explicit description of geometric structure in the smooth dual of a connected split reductive \( p \)-adic group.

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Paul Baum

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