By Moore R.L.

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The formal Hodge theorem shows that ker ∼ = H∗ (C). Consider the linear operator ker odd ⊕ C even → ker even ⊕ C odd , kodd ceven → 0 iodd ∗ kodd ieven · . ∂ +η ceven We thus get an isomorphism Det ker odd ˆ Det(C even ) → Det ker ⊗ even ˆ Det(C odd ) ⊗ This yields the isomorphisms Det ker odd ˆ Det s (C) → Det ker ⊗ even , and I : Det s (C) → Det(ker odd ) −1 ˆ Det ker ⊗ even → Dets H∗ (C) . Up to a permutation, this is the Euler isomorphism. More precisely, we have a commutative diagram Det s (C) I ϒ u Det(C) w Det s H∗ (C) ϒ EulC u w Det H∗ (C) .

Suppose that H is a finitely generated Abelian group. Denote by ı : H → N(H ) the natural morphism. If P , Q ∈ N(H ) then P |Q ⇐⇒ P |Q(ı(h) − 1), ∀h ∈ H. 29. Suppose H ∼ = Z ⊕ G where G is a finite Abelian group. Denote by t the generator of Z. Then T = It = 1 St = N 1 N g t, N := |G| g∈G The group algebra Q[H ] is isomorphic to the ring of Laurent polynomials Q[H ] ∼ = = Q[G][t, t −1 ] ∼ Then Ki [t, t −1 ]. N[H ] = Z[H ] + SZ[T , T −1 , (1 − T )−1 ]. The correspondence H → N(H ) is functorial.

Then Observe that η2 = 0 and ∂η TSπ1 ×X = det ∂ˆ + η : C even (R × X) → C odd (R ⊗ X) . With respect to the bases e0 ⊗ S0 (X) ∪ e1 ⊗ S1 (X) ∪ e0 ⊗ S2 (X) ∪ · · · of C even , and e1 ⊗ S0 (X) ∪ e0 ⊗ S1 (X) ∪ e1 ⊗ S2 (X) ∪ e0 ⊗ S3 (X) ∪ · · · of C odd , the operator ∂ˆ + η has the description n0 columns n1 columns n2 columns · · · n0 rows (t − 1)−1 −∂ 0 ··· n1 rows 0 (t − 1) ∂ ··· n2 rows .. 0 .. 0 .. (t − 1)−1 .. ··· where nk := #Sk (X). We deduce TSπ1 ×X = (t − 1)− k (−1) kn k = (t − 1)−χ(X) = TS 1 χ(X) .

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A Character of Jordan Regions by Properties Having no Reference to Their Boundaries (1918)(en)(7s) by Moore R.L.

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