Free products, cyclic homology, and the Gauss-Manin connection. (English) Zbl 1356.16008
Summary: We use the techniques of J. Cuntz and D. Quillen [J. Am. Math. Soc. 8, No. 2, 251–289 (1995; Zbl 0838.19001)] to present a new approach to periodic cyclic homology. Our construction is based on \((\Omega^*A)[t],\text{d}+t\cdot t_\Delta)\), a noncommutative equivariant de Rham complex of an associative algebra \(A\). Here \(\text{d}\) is the Karoubi-de Rham differential and ı\(_\Delta\) is an operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit construction of the Gauss-Manin connection, introduced earlier by E. Getzler, on the relative periodic cyclic homology of a flat family of associative algebras over a central base ring.
We introduce and study free-product deformations of an associative algebra, a new type of deformation over a not necessarily commutative base ring. Natural examples of free-product deformations arise from preprojective algebras and group algebras for compact surface groups.
We introduce and study free-product deformations of an associative algebra, a new type of deformation over a not necessarily commutative base ring. Natural examples of free-product deformations arise from preprojective algebras and group algebras for compact surface groups.
MSC:
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
| 53C05 | Connections (general theory) |
Citations:
Zbl 0838.19001Software:
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