Problem set 5, due Thursday, October 27.

(1) In this problem we work in the category of sheaves of abelian groups on ℝ. Let F_{i} = ℤ_{1/i}, i = 1, … be the skyscraper at 1/i; and let G be the extension by zero of the constant sheaf on the ray (0, ∞): a section of G on an open set U ⊂ ℝ is a locally constant ℤ valued function on U which vanishes at 0.

(a) Compute the stalk of ∏_{i=1}^{∞} F_{i} at 0. Conclude that the functor “stalk

at a given point” on the category of sheaves does not commute with direct products.

(b) Compute the stalk of ∏_{i=1}^{∞} G.

(c) Show that we have a surjective morphism G → F_{i}, however the product of these surjective morphisms is not surjective. Thus infinite products are not exact in the category of sheaves.

(2) (see Gelfand, Manin, exercise 4 to §IV.4) An oriented graph defines a category whose objects are vertices and morphisms are oriented paths. A (finite dimensional) representation of the graph over a field k is a functor from this category to a category of (finite dimensional) k vector spaces. In other words, a representation is the data of a vector space V_{α} for every vertex α, and a map ϕ_{α,β} : V_{α} → V_{β} given for every oriented edge from α to β.

Let Γ be a graph; assume that it has no oriented cycles, and every two vertices are connected by at most one edge (then Γ is called a quiver).

(a) Show that irreducible representations of Γ are in bijection with vertices of Γ, so that to vertex α there corresponds a representation V^{(α)} with dim V_{α}^{(α)} = 1, dim V_{β}^{(α)} = 0 for β ≠ α.

(b) Show that Ext^{1}(V^{(α)}, V^{(β)}) has dimension 1 if there is an oriented edge from α to β, and is zero otherwise.

(c) Let α be a vertex of Γ, such that no oriented edges end at Γ (one says α is a source). Let Γ′ be another graph with the same set of vertices, and same edges except that orientation of edges starting at α is changed. Show that Γ′ is also a quiver. Construct an equivalence between bounded derived categories of representations of Γ and Γ′. [See [Gelfand Manin] for formulas defining the equivalences.]

(3) Let D be a k linear triangulated category where k is a field, such that Hom^{·}(X, Y) is finite dimensional over k for every X, Y ∈ D.

(a) An object X ∈ D is called exceptional if Hom^{·}(X, X) = k · id. Let X ∈ D be exceptional; and let C ⊂ D be the smallest full triangulated subcategory containing X. Show that C is equivalent to the category of graded finite dimensional vector spaces; and that it is left admissible in D, i.e. the embedding functor has a left adjoint.

(b) A Serre functor^{1} for D is an auto-equivalence S : D → D together with an isomorphism Hom(X, Y) ≅ Hom(Y, S(X))^{∗} fixed for all X, Y ∈ D and functorial in X, Y.

1The name comes from Serre duality in algebraic geometry.

Assume that Serre functor exists for D Show that for any subcategory C ⊂ D the Serre functor sends ^{⊥}C to C^{⊥}.

(c) Assume that a Serre functor exists for D, and assume that C ⊂ D is a full triangulated subcategory. Show that C is left admissible iff it is right admissible.

(4) (Stable category AGAIN). Let again A be a Frobenius algebra over a field k, and D = D^{b}(A)/D_{perf}^{b}(A). Let M, N ∈ A − mod. Show that for i > 0 we have Hom_{D}(M, N[i]) = Ext_{A}^{i}(M, N); and for i < 0 we have Hom_{D}(M, N[i]) = T or_{−i}^{A}(M^{∗}, N) where M^{∗} is considered as a right A module.

[Hint: Check first that if M → P is an emebeding into a projective (= injective), then Hom_{D}(M, N) = Hom_{A−mod}(M, N)/im(Hom_{A−mod}(P, N)).

Then show by induction that Hom_{D}(M, N[i]) = Hom_{D}(B^{−i+1}, N) where B^{−i} ⊂ P^{−i} is the image of the differential in the corresponding term of a projective resolution for M; and compare Hom_{D}(Z^{−i+1}, N) with Ext^{i}(M, N). For the second part of the statement, apply a similar argument to a projective resolution of the right module M^{∗} (notice that Hom_{A}(M, N) = N ⊗_{A} M^{∗} for finitely dimensional modules M, N.]