Additive \(K\)-theory and crystalline cohomology
In recent years many new (co)homology theories of (non)commutative (operator) algebras have been discovered: The homological \(K\)-functor Ext of \textit{L. G. Brown}, \textit{R. G. Douglas} and \textit{P. A. Fillmore} [see Bull. Am. Math. Soc. 79, 973--978 (1973; Zbl 0277.46052)] having after all some interesting application in the structure theory of \(C^*\)-algebras [see the reviewer, Funkts. Anal. Prilozh. 9, No. 1, 63--64 (1975; Zbl 0347.46068)], Kasparov's \(KK\)-functor [see \textit{G. G. Kasparov}, Izv. Akad. Nauk SSSR, Ser. Mat. 44, 571--636 (1980; Zbl 0448.46051)], Connes' cyclic homology \(HC\)-functor [see \textit{M. Karoubi}, C. R. Acad. Sci., Paris, Sér. I 297, 381--384, 447--450, 513--516, 557--560 (1983; Zbl 0528.18008, Zbl 0528.18009, Zbl 0528.18010, Zbl 0532.18009)], and the Gel'fand-Fuks cohomology theory of infinite-dimensional Lie algebras. The most important point in these developments is replacing the structural function sheaf by an arbitrary (non)commutative algebra. The paper under review is a new account of this fact, where the algebra of functions on a manifold is replaced by a (non)commutative algebra. One considers for each algebra \(A\) over a field \(K\) of characteristic zero the standard bimodule resolution \[ R_* :\quad... \to A\otimes A\otimes A\to^{\delta}A\otimes A\to^{\delta}A\to 0 \] with the differential \(\delta (a_ 1\otimes...\otimes a_ k)=\sum^{k-1}_{i=1}(-1)^ ia_ i\otimes...\otimes a_ ia_{i+1}\otimes...ca_ k\), the standard complex \(C_*(A)\) of the Lie algebra \(\mathfrak{gl}_{\infty}(A)\) of infinite matrices of finite type over \(A\) with coefficients in the unit module, and finally a complex homomorphism \[ \phi: C_*(A)\to (R_*,\delta),\quad \phi ((m_ 1\otimes a_ 1)\wedge...\wedge (m_ k\otimes a_ k))=\sum_{i}\text{sgn}(i)\text{tr}(m_{i_ 1}...m_{i_ k}) a_{i_ 1}\otimes...\otimes a_{i_ k}. \] The image of \(\phi\) is the subcomplex \(\bar R_*\subset R_*\) consisting of chains that are invariant under the transfer map \(t\), \(t(a_ 1\otimes...\otimes a_ k)=(- 1)^{k-1}a_ k\otimes a_ 1\otimes...\otimes a_{k-1}\). Via \(\phi\), the additive \(K\)-functor group \[ K^+_*(A)=\lim_{\leftarrow} H_*(\mathfrak{gl}_{\infty}(A),K) \] is isomorphic [\textit{J.-L. Loday} and \textit{D. Quillen}, C. R. Acad. Sci. Paris, Sér. I 296, 295--297 (1983; Zbl 0536.17006); \textit{B. L. Tsygan}, Usp. Mat. Nauk 38, No. 2 (230), 217--218 (1983; Zbl 0518.17002)] to Connes' cyclic cohomology group \(HC^*(A)=H_*(\bar R_*,\delta).\) The authors show that \(K^+_*(A)\) are the higher derived functors of the functor \(K^+_ 1\) in the category of algebras: \[ K^+_ 1(R)\simeq R/[R,R],\quad H_*(R/[R,R])\simeq \lim_{\leftarrow} H_*(\mathfrak{gl}_{\infty}(A),\mathfrak{gl}_{\infty}(K);K). \] On the category of arrows \(A\to^{f}B\) one defines (Th. 1) the relative additive \(K^+\)-functor \(K^+_*(B,A)\) as the derived functors of \(K^+_ 0(B,A)=B/([B,B]+\text{Im}\, f)\). One has (Th. 2) an exact sequence \[ ... \to K^+_{i+1}(C,B) \to K^+_ i(B,A) \to K^+_ i\quad (C,A) \to K^+_ i(C,B) \to... \] associated with the pair of arrows \(A\to^{f}B\to^{g}C.\) The authors propose also (in Section 2) a construction independent from the Connes-Karoubi one of the Chern characters \(Ch: K_ n(A)\to K^+_{n+2i+1}(A)\), \(i\in\mathbb Z_+\). Then, the Bott periodicity morphism \(K^+_ i(A)\to K^+_{i-2}(A)\) exists, and hence, the groups \(K^+_ i\) with i of fixed parity form an inverse system. The authors define the cohomology of an algebra \(A\) as \[ H^{\text{odd}}(A)=\lim_{\leftarrow}K^+_{2i}(A) \oplus R^ 1\lim_{\leftarrow}K^+_{2i+1}(A),\quad H^{\text{ev}}(A)=\lim_{\leftarrow}K^+_{2i+1}(A) \oplus R^ 1\lim_{\leftarrow}K^+_{2i\quad}(A). \] In the case where \(A\) is commutative (Section 3), \(H^{\text{odd}}(A)+H^{\text{ev}}(A)\) is isomorphic to the crystalline cohomology of the spectrum of A and there exists a morphism \[ K^+_ n(A)\to \oplus_{N}H^*(X_{\text{cris}}, \mathcal O_ X/J^ N). \] If \(R\) is a free commutative differential graded algebra for which there exists an epimorphic quasiisomorphism \(R\to A\) the authors consider the de Rham complex \(R: R\to^{d}\Omega^ 1_ R\quad \to \quad...\) and prove that \(K^+_*(A,K)\simeq H_*(\Omega^*_ R/(K+d\Omega^*_ R))\). This just coincides with Connes' result in the case where \(\text{Spec}\,A\) is a nonsingular variety.
cohomology theories of noncommutative operator algebras
algebra of infinite matrices of finite type
Gel'fand-Fuks cohomology theory of infinite-dimensional Lie
Gel'fand-Fuks cohomology theory of infinite-dimensional Lie algebras
scientific article; zbMATH DE number 3937380
scientific article