By Sapagovas M.P.

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**Extra info for A Boundary Value Problem with a Nonlocal Condition for a System of Ordinary Differential Equations**

**Example text**

M# ) = 0. Thus the HC q(j! 3) is an isomorphism, and the second map is then the required splitting. 2. Assume given a commutative k-algebra R with a maximal ideal m ⊂ R, and a deformation AR of the algebra A over R. Then if Spec(R) is smooth, the R-modules HP q(AR /R) carry a natural connection. 36 D. Kaledin Proof (Sketch of a proof ). Consider the R ⊗ R-algebras AR ⊗ R and R ⊗ AR , and their restrictions to the ﬁrst inﬁnitesimal neighborhood of the diagonal in Spec(R ⊗ R) = Spec(R) × Spec(R).

For any E ∈ q Fun(Γ, k), the cohomology H (Γ, E) and the homology H q(Γ, E) are modules q over H (Γ, k). We also note, although it is not needed for the deﬁnition of cyclic homology, that for any functor γ : Γ → Γ between two small categories, we have the pullback functor γ ∗ : Fun(Γ, k) → Fun(Γ , k), and for any E ∈ Fun(Γ, k), we have natural maps q q H (Γ, E) → H (Γ , γ ∗ E). 1) H q(Γ , γ ∗ E) → H q(Γ, E), Moreover, the pullback functor γ ∗ has a left adjoint γ! : Fun(Γ , k) → Fun(Γ, k) and a right-adjoint f∗ : Fun(Γ , k) → Fun(Γ, k), known as the left and right Kan extensions.

2. Another way to view this structure is the following. 3). By adjunction, we have a natural map Δ τ# : j! M# → M# . Δ in this formula depends only on M ∈ A-bimod, and all the strucThen j! M# ture maps that turn M into the cyclic bimodule M# are collected in the map τ# . We can now deﬁne cyclic homology with coeﬃcients. The deﬁnition is rather tautological. We note that for any cyclic A-bimodule M# , or in fact, for any M# ∈ Sec(A-bimod# ), we can treat M# as a cyclic vector space by forgetting the bimodule structure on its components Mn .

### A Boundary Value Problem with a Nonlocal Condition for a System of Ordinary Differential Equations by Sapagovas M.P.

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