114

A . A . Voronov

Remark . The space Mf and modules similar to M will play a crucial role in the construction of semi-infinite cohomology as a derived functor.

Before proving Theorem 2.1, we will consider the spectral sequences associated with two filtrations on the Feigin standard complex. These sequences are both semi-infinite analogues of the Hochschild-Serre spectral sequence.

Theorem 2.2 For a Lie algebra g = b ф n with a semi-infinite structure and a Q-module Me(9, there exists a spectral sequence

with the following properties'.

( i ) ÊÎ'^ = H^(n, Л°"/^-'^(д/п)*(х)М)

( ii ) if n is an ideal, then £f ^ = Я^^^^-^^Сд/п, Ю(п, M)) = Я_Дд/п, Я^п, M) ®

( Ш ) £?;« = gr^Я°°/2■'^■'^(g, М)

( iv ) the differentials dr induce a sequence of epimorphisms E^'^ -^ E.^+i -^ ' ' ' for

r large enough, so that

lim E;''« = E5;« . Proof We want to define a filtration on the Feigin standard complex

^00 / 2 + . ._ ^ao/2+.(g^ Щ ^ М(Х)Л°^/2+.д* _

Set

ppçco / 2 + p + q . _ [coeC^'^^P^^lni . . . M,4-iCo = 0foreachni,. . .,n^ + ^en} ,

where p ^1, q^O and the multipHcation nœ is the Qifford algebra (interior) multiplication on semi-infinite forms. It is evident that

^00 / 2 + . 3 . . . 3 /7P 3 /ГР+1 3 . . . 3 i7l = 0 ,

that is, the filtration is decreasing and regular, and moreover,

\ \ fP — ^00/2+• ^ P

Using identities (iii) and (iv) in Proposition 2.6, one can check that the differential din C^i^^* is compatible with the filtration:

Jj7P^oo / 2 + P + g С /rP(^co/2+P + q+l

Thus , we have a filtered complex. It gives rise to a spectral sequence £f'^, whose first few terms we are now going to compute.

Computation ofE^'^. By definition,

£Р'Я ._ pp(^^l2 + p + q/pp+l^ccl2+p + q

= Л«п*®Л°"/2-'^(д/п)*®М

= {the qui cochain space of the standard complex of n} .