Problem set 3, due Thursday, October 13.

(1) Last problem from the previous assignment, with the corrected statement.

(2) Use the equality Ext^{i}(M, N) = Hom_{D(A)}^{i}(M, N) to show that Ext^{i}(M, N) (i > 1) is identified with the set of isomorphism classes of objects C^{·} in D(A) with a fixed isomorphism H^{0}(C^{·}) = M, H^{−(i−1)}(C^{·}) = N.

(3) Let A → B be a surjection of rings. Every B module carries then a natural structure of an A module. Show that for B-modules M, N there is a natural map Ext_{B}^{i}(M, N) → Ext_{A}^{i}(M, N) which is injective for i = 1, but may be not injective for i = 2.

(4) Projective dimension p dim(M) of a module M is defined to be the maxi-

mum (possibly infinity) of all i such that Ext^{i}(M, N) ≠ 0 for some N.

Let 0 → M_{1} → M_{2} → M_{3} → 0 be a short exact sequence. Show that if p dim(M_{2}) < p dim(M_{1}), then p dim(M_{3}) = p dim(M_{1}) + 1.

(5) Assume that a module M has a finite right resolution consisting of projective modules. Show that M is projective.

(6) Recall that the functor P ↦ Hom_{R}(P, R) gives an equivalence between the opposite category to left finitely generated projective modules and right finitely generated projective modules over a ring R.

Use the extension of this functor to complexes to show the following:

if R is a Noetherian ring, and M is a finitely generated module of finite projective dimension, then M is projective iff Ext_{R}^{i}(M, R) = 0 for i > 0. More generally, show that p dim(M) = max_{i} Ext^{i}(M, R) ≠ 0.

Give an example showing that the assumption p dim(M) < ∞ is essential in the last statement.

(7) Let A be a Frobenius algebra over a field k, i.e. dim_{k}(A) < ∞ and A ≅ A^{∗} as left A-modules. The full subcategory of perfect complexes D_{perf}^{b}(A) ⊂ D^{b}(A − mod^{fg}) = D^{b}(A) (where fg stands for “finitely generated”) is defined to consist of objects represented by finite complexes of (finitely generated) projective (equivalently, injective) modules.

Show that D_{perf}^{b}(A) is a thick triangulated subcategory.

Prove that the quotient D^{b}(A)/D_{perf}^{b}(A) is equivalent to the “stable category” S(A) defined in class: Ob(S(A)) = Ob(A), and Hom_{S(A)}(M, N) = Hom(M, N)/Hom_{0}(M, N) where Hom_{0} consists of maps which factor through a free module.

[Hint: It is easy to define a functor S(A) → D^{b}(A)/D_{perf}^{b}(A). Then from the definition of the quotient category check that it is a full embedding; then check that it is triangulated (triangulated structure on S(A) was defind in class); since A generated D^{b}(A) under cones and shifts, it follows that the essential image of S(A) is the whole D^{b}(A)/D_{perf}^{b}(A).

If you prefer, you may use the following explicit description for the inverse functor: for a bounded complex C^{·} there is a bounded above complex of projectives P^{·}, which is quasiisomorphic to C^{·}. Check that the assignment C^{·} ↦ (im(P^{−i} → P^{−i+1}))[−i] for i ≫ 0 induces a well-defined functor D^{b}(A)/D_{perf}^{b}(A) → S(A), which is inverse to the above].

(8) Compute Hom_{S(A)}^{i}(M, N) for all i in the following situation A = k[G] is the group algebra where k = ℤ/pℤ, G = ℤ/nℤ, p|n; M = N is the trivial module.

[Hint: Hom_{S(A)}(k,?) (where k denotes the trivial module) are not hard to compute from the definition (to warm up check that Hom_{S(A)}(k, k) = k). Then observe that N[i] for i > 0 can be described as the image of differential P^{−i} → P^{−i+1} in a projective resolution of N, so one can use an explicit resolution of N to compute N[i] (and similarly for negative shifts and injective resolutions).]

(9) Let A be an algebra over a field k. Let HH_{A}^{·} be the Hochschild complex computing Hochschild cohomology of A; for A-modules M, N let HH_{M,N}^{·} be the Hochschild complex computing Ext^{i}(M, N); thus HH_{A}^{i} = Hom_{k}(A^{⊗i}, A), HH_{M,N}^{i} = Hom_{k}(A^{⊗i} ⊗ M, N).

Show that the formulas

ϕ_{1} ∘ ϕ_{2}(a_{1} ⊗ ⋯ ⊗ a_{i+j}) = ϕ_{1}(a_{1} ⊗ ⋯ a_{i})ϕ_{2}(a_{i+1} ⊗ ⋯ ⊗ a_{n});

ϕ(ψ)(a_{1} ⊗ ⋯ ⊗ a_{i+j} ⊗ m) = ϕ(a_{1} ⊗ ⋯ a_{i})ψ(a_{i+1} ⊗ ⋯ ⊗ a_{n} ⊗ m) define, respectively, a structure of a DG algebra on HH_{A}^{·}, and a structure of a DG module over HH_{A}^{·} on HH_{M,N}^{·}.

Write down a similar formula defining a DG category whose objects are A-modules, and Hom(M, N) := HH_{M,N}^{·}.

Prove that the induced multiplication on Ext_{A⊗A^{op}}^{i}(A, A), and composition on Ext’s between A modules coincide with the standard one.

[Hint. To prove the statement about multiplication on Hochschild cohomology, show first that if C_{1}^{·}, C_{2}^{·} are two projective resolutions of the regular bimodule, then C_{1}^{·} ⊗_{A} C_{2}^{·} is also a projective resolution of the regular bimodule; and that if classes h_{1}, h_{2} in Hochschild cohomology are represented by maps h̃_{1} : C_{1}^{·} → A, h̃_{2} : C_{2}^{·} → A then h_{1}h_{2} is represented by h_{1} ⊗ h_{2} : C_{1}^{·} ⊗ C_{2}^{·} → A ⊗_{A} A = A.

Apply this to C_{1}^{·} = C_{2}^{·} = B^{·} being the Bar resolution; then (check and) use the following explicit formula for a quasiisomorphism B^{·} → B^{·} ⊗_{A} B^{·}:

a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} ⊗ a_{n+1} ↦ ∑_{i=0}^{n}(a_{0} ⊗ ⋯ ⊗ a_{i} ⊗ 1) ⨂(1 ⊗ a_{i+1} ⋯ ⊗ a_{n+1})]

(10) (Koszul duality) A quadratic algebra over a field k is a graded algebra given by a finite dimensional space of degree one generators V, and a space of relations R ⊂ V ⊗ V. The dual algebra A^{!} is the algebra with the dual space V^{∗} as a space of generators, and the orthogonal R^{⊥} as the space of relations.

(a) Show that A^{!} is a subalgebra of the cohomology algebra of A, H^{·}(A) = Ext_{A}^{·}(k, k).

(b) Define a differential on the space ⊕_{i} (A_{i}^{!})^{∗} ⊗ A, and show that cohomology of this complex has dimension 1 iff A^{!} = H^{·}(A). [Hint: Observe that

A^{j} = V^{⊗j}/ ∑_{p=0}^{j−2} V^{⊗p} ⊗ R ⊗ V^{j−p−2};

(A_{i}^{!})^{∗} = ⋂_{s=0}^{i−2} V^{⊗s} ⊗ R ⊗ V^{⊗i−s−2}) ].