Problem set 2, due Thursday, October 6.

(1) For a morphism f : K^{·} → L^{·} of complexes (of, say, R modules for a ring R) write down explicitly a map of complexes f′ : K′ → L′ which is a term-wise split embedding and which is isomorphic to f in the homotopy category; in fact you can (and should) find f′ such that K = K′ and Coker(K′ → L′) ≅ Cone(f).

[The natural choice for L′ is called the cylinder of the map f.]

(2) Deduce from the axioms Tr1–Tr3 that a direct sum of two distinguished triangles is a distinguished triangle.

(3) Let A be a ℤ_{≥0} graded associative unital algebra over a field k, with A^{0} = k and dim_{k}(A^{i}) < ∞. Let h_{i}^{j} be the dimension of the j-th graded component in T or_{i}^{A}(k, k), and h^{j} = ∑(−1)^{i}h_{i}^{j} (check that the last sum is finite). Define formal Taylor series P (t) = ∑_{i} dim A^{i} t^{i}, Q(t) = ∑_{j} h^{j}t^{j}. Prove that P (t)Q(t) = 1.

(4) Let G be a group, and A be an abelian group A equipped a G action; recall that by the definition H^{i}(G; A) = Ext_{ℤ[G]}^{i}(ℤ, A).

(a) Define a bijection between the following sets.

(i) H^{1}(G; A).

(ii) A-torsors over G. [A torsor over a G module A (or A-torsor over G) is a set T together with a free transitive action of A, and an action of G compatible with the action of A (i.e. the identity g(a(t)) = g(a)(g(t)) holds for all g ∈ G, a ∈ A, t ∈ T).]

(iii) One sided inverse homomorphisms to the sujrection H → G up to conjugacy by elements in A; here H is the semi-direct product of A by G.

(iv) (Applies when the action of G on A is trivial) Hom_{groups}(G, A).

(b) H^{2}(G, A) can be naturally identified with the set of isomorphism classes of group extensions

{1} → A → G̃ → G → {1}

such that the induced conjugation action of G on A coincides with the given one. (In particular, if the action is trivial, we get the set of central extensions). (Hint: given an extension G̃ of G by A choose a set theoretic splitting g ↦ g̃ of the surjection G̃ → G. Then check that the map

(g_{1}, g_{2}) ↦ { \atop (g_{1}g_{2})}^{−1}g̃_{1}g̃_{2}

is a cocycle, and that changing a splitting of the map G̃ → G amounts to adding a coboundary to this cocycle.)

(5) Let A be an associative (unital) algebra over a field k, and M be a bimodule.

(a) A derivation with values in M is a map δ : A → M, such that δ(ab) = a δ(b) + δ(b) a. A derivation is called inner if δ(a) = am − ma for some m ∈ M. Show that HH^{1}(A; M) ≅ Der(A, M)/Inn(A.M) where Der(A, M) is the space of derivations, and Inn(A, M) is the subspace of inner derivations.

Deduce that if A = C^{∞}(X) is the algebra of smooth (real valued) functions on a smooth manifold X, then HH^{1}(A) is naturally identified with the space of vector fields on X.

(b) An extension of A by M is a short exact sequence 0 → M → Ã → A → 0, where Ã is equipped with an algebra structure so that Ã → A is an algebra homomorphism, m_{1}m_{2} = 0 for m_{1}, m_{2} ∈ M, and the induced A bimodule structure on M coincides with the given one.

(i) Show that for any such extension the group of automorphisms of Ã inducing identity on A and M is Der(A, M); and inner derivations correspond to inner automorphisms of Ã.

(ii) Identify the set of isomorphism classes of such extensions with

HH^{2}(A, M). (iii) A first order deformation of A is an extension of A by a regular

bimodule; it can also be thought of as a flat k[ϵ]/ϵ^{2}] algebra Ã with an isomorphism Ã/ϵÃ ≅ A. Given a first order deformation Ã and an A module N a lifting of N to Ã is a module over Ã which is flat over k[ϵ]/ϵ^{2}, and such that Ñ/ϵÑ ≅ N. Let Ã be a first order deformation, and γ ∈ HH^{2}(A) be the class attached to it by 5(b)ii. Show that a lifting for an A-module N exists iff the class in Ext^{2}(N, N) induced by γ vanishes.

(6) Show that composition of Ext’s equips the group cohomology H^{·}(G; R) = Ext_{R[G]}(R, R) (where R is a commutative ring with a trivial G action) with a commutative ring structure.

(Hint: assume a projective resolution P^{·} for the trivial module, and two its endomorphisms e_{1}, e_{2} are given; show that P^{·}⊗_{R}P^{·} is again a projective resolution for the trivial module, and find explicit endomorphisms e′_{1}, e′_{2} of P^{·} ⊗_{R} P^{·} which commute and represent the same cohomology classes as e_{1}, e_{2}).

Similarly, show that multiplication on the Hochschild cohomology of any associative algebra is commutative.

(7) Homological dimension of a ring R is defined to be the maximal i such that Ext_{R}^{i}(M, N) is nonzero for some M, N.

(a) Check that the homological dimension of R equals one if R is either ℤ or k[x] where k is a field.

(b) Show that if R has homological dimension 1 then a submodule of a projective R module is projective and a quotient of an injective module is injective.

(c) ^{1} Show that if R has homological dimension 1 then for every bounded above complex P^{·} of projectives there is a map of complexes f : P^{·} → C^{·}, such that f is a quasiisomorphism, and C^{·} has zero differential; and also for every bounded below complex I^{·} of injectives there is a map of complexes f : C^{·} → I^{·}, such that f is a quasiisomorphism, and C^{·} has zero differential.

1This problem was stated wrongly in the 1st version of the assignment; it can be handed in with the next assignment.