International Congress of Mathematicians

110 Dimitri TamarkinProof. Let F = j m .j*E. We have B = F ® öx „ î>x»- Note that F =limdir F Q , where F Q runs through the set of all free coherent submodules of F.We haveP(B,M) = liminv hom^» (F Q ® 0x „ V x^,i n *(M® 0x L))~ liminv F* cgi 0x „ t„,(M ® 0x D x ) ®i, x F=* liminv hom 0x „ (F a ,i m (M ® 0x V x )) ®v x L=* prohom(B,z n »(M cgi 0x T> x )) ® Px F.2.5.3.Let F, Af be as above. We have a natural morphismi : i m .P(B, M) ~ P(B, M) ® Vx vf >x -+ prohom(F, M ®v x ®(V x f°x ).The above lemmas imply that i is an isomorphism.2.5.4.Let M be a right F> x -module. SetU M (n) = V ch (M, M,..., M ; M ® V x ) •= prohom(j„,j;M H ", t„,(M ® 2> x )),it is a right pro-F> x -module. We will endow the collection UM with the structureof an operad in *-pseudotensor category. This means that we will define thecomposition mapsOj G Px(UM(n),UM(m);U M (n + m - 1)),z = l,...,n + m — 1, satisfying the operadic axioms. We need a couple of technicaltacts.2.5.5.Let i n : X —t X n be the diagonal embedding and p l n : X n -+ I be theprojections. Lemma 2.5.3. implies thatLemma 2.2i m U M (k) = V ch (M,...,M;M® Vx Vf n ).Lemma 2.3 For any T>x -modules M,S we have an isomorphismin*(M)®p j n*S^i m .(M®S).