Semi-infinite homological algebra
The paper starts with a review of some elements of the construction of semi-infinite cohomology via the Feigin standard complex, which is based on semi-infinite forms (forms of ``\(\infty/2\) degree'') [\textit{B. L. Feigin}, Russ. Math. Surv. 39, No. 2, 155-156 (1984); translation from Usp. Mat. Nauk 39, No. 2, 195-196 (1984; Zbl 0544.17009)]. The cohomology vector spaces \(H^{\infty/2+n} ({\mathfrak g}, M)\) \((n\in\mathbb{Z})\) of the Feigin complex are defined on a \(\mathbb{Z}\)-graded Lie algebra \({\mathfrak g}= \bigoplus_{n\in\mathbb{Z}} {\mathfrak g}_ n\) over a field \(k\) of characteristic 0, with \(\dim({\mathfrak g}_ n)<\infty\) for all \(n\in\mathbb{Z}\), and take values in a \(\mathbb{Z}\)-graded \({\mathfrak g}\)-module \(M\) such that the action law \(U({\mathfrak g})\times M\mapsto M\) is continuous. Here, \(U({\mathfrak g})\) is the universal enveloping algebra of \({\mathfrak g}\), endowed with the topology defined by the filtration associated with the grading inherited from that of \({\mathfrak g}\), and \(M\) is equipped with the discrete topology. It is shown that, besides the long exact sequence of cohomology vector spaces which follows, via the Snake Lemma, from a short exact sequence of \({\mathfrak g}\)-modules, there exist also two semi-infinite analogues of the Hochschild-Serre spectral sequence. Then the author introduces and studies two-sided derived functors of the functors of semivariants \(F:M\mapsto H^{\infty/ 2+0} ({\mathfrak g},M)\). He proves that, for each \(n\in\mathbb{Z}\), the functor \(M\mapsto H^{\infty/2+n} ({\mathfrak g},M)\) of semi-infinite cohomology coincides with the two-sided derived functor \(M\mapsto D^ n F(M)\) of the functor \(F\).
scientific article; zbMATH DE number 549683
scientific article