A Chern-Weil formula for the Chern character of a perfect curved module. (English) Zbl 1471.13038
Recall that the Chern character is a ring homomorphism from the Grothendieck group of complex vector bundles on a finite CW complex \(X\) to the even part of its de Rham cohomology.
\[
\mathrm{ch}\,:\,KU^0(X)\,\rightarrow\,\bigoplus_i H^{2i}(X,\,\mathbb{C})\,.
\]
Given \(V\in KU^0(X)\), Chern-Weil theory provides an explicit formula for \(\mathrm{ch}(V)\) in terms of curvature.
The Chern character and Chern-Weil formula have purely algebraic analogs: Let \(k\) be a field of characteristic \(0\), and \(A\) be a commutative unital \(k\)-algebra. There is a Chern character map \[ \mathrm{ch}_0\,:\,K_0(A)\,\rightarrow\,\bigoplus_i H^{2i}_{DR}(A,\,k)\,, \] where \(K_0(A)\) is the Grothendieck group of projective modules over \(A\), and \( H^{\bullet}_{DR}(A,\,k)\) denotes the algebraic de Rham cohomology of \(A\). Notice that \(K_0(A)\) is defined for not necessarily commutative algebras. The non-commutaive Chern character, denoted by \(\mathrm{ch}_{\mathrm{HN}}\), is a map from \(K_0(A)\) to the negative cyclic homology \(\mathrm{HN}_0(A)\). One can generalize \(\mathrm{ch}_{\mathrm{HN}}\) via replacing \(A\) by a DG category \(\mathcal{A}\). Moreover, there is a variant of negative cyclic homology called negative cyclic homology of the second kind, denoted by \(\mathrm{HN}^{II}\), and one can obtain the Chern character of the second kind \[ \mathrm{ch}_{\mathrm{HN}}^{II}\,:\,K_0(\mathcal{A})\,\rightarrow\,\mathrm{HN}^{II}_0(\mathcal{A})\,. \]
The paper defines \(\mathrm{ch}_{\mathrm{HN}}^{II}\) for a curved DG category \(\mathcal{A}\). In particular, if \(\mathcal{A}\) is a curved algebra satisfying a certain smoothness condition, the paper provides a Chern-Weil-type formula for \(\mathrm{ch}_{\mathrm{HN}}^{II}\). This formula recovers the classical Chern-Weil formula for \(\mathrm{ch}_0\) when \(\mathcal{A}\) is a smooth commutative \(k\)-algebra. The paper contains several concrete examples as well as brief introductions to curved DG categories and Hochschild and cyclic homology of the second kind. Additionally, one of the two appendixes gives an interpretation of Hochschild (co)homology of the second kind in terms of twisted Hom and tensor product.
The Chern character and Chern-Weil formula have purely algebraic analogs: Let \(k\) be a field of characteristic \(0\), and \(A\) be a commutative unital \(k\)-algebra. There is a Chern character map \[ \mathrm{ch}_0\,:\,K_0(A)\,\rightarrow\,\bigoplus_i H^{2i}_{DR}(A,\,k)\,, \] where \(K_0(A)\) is the Grothendieck group of projective modules over \(A\), and \( H^{\bullet}_{DR}(A,\,k)\) denotes the algebraic de Rham cohomology of \(A\). Notice that \(K_0(A)\) is defined for not necessarily commutative algebras. The non-commutaive Chern character, denoted by \(\mathrm{ch}_{\mathrm{HN}}\), is a map from \(K_0(A)\) to the negative cyclic homology \(\mathrm{HN}_0(A)\). One can generalize \(\mathrm{ch}_{\mathrm{HN}}\) via replacing \(A\) by a DG category \(\mathcal{A}\). Moreover, there is a variant of negative cyclic homology called negative cyclic homology of the second kind, denoted by \(\mathrm{HN}^{II}\), and one can obtain the Chern character of the second kind \[ \mathrm{ch}_{\mathrm{HN}}^{II}\,:\,K_0(\mathcal{A})\,\rightarrow\,\mathrm{HN}^{II}_0(\mathcal{A})\,. \]
The paper defines \(\mathrm{ch}_{\mathrm{HN}}^{II}\) for a curved DG category \(\mathcal{A}\). In particular, if \(\mathcal{A}\) is a curved algebra satisfying a certain smoothness condition, the paper provides a Chern-Weil-type formula for \(\mathrm{ch}_{\mathrm{HN}}^{II}\). This formula recovers the classical Chern-Weil formula for \(\mathrm{ch}_0\) when \(\mathcal{A}\) is a smooth commutative \(k\)-algebra. The paper contains several concrete examples as well as brief introductions to curved DG categories and Hochschild and cyclic homology of the second kind. Additionally, one of the two appendixes gives an interpretation of Hochschild (co)homology of the second kind in terms of twisted Hom and tensor product.
Reviewer: Yining Zhang (Boulder)
MSC:
| 13D09 | Derived categories and commutative rings |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
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