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Math 501 - Problem Set 5 (due in class on Tue, Mar 26) Problem 1. Show directly (without using the bijection with Ext1 ) that the extension classes of the abelian group Z/(p) are represented by the split exact sequence and the short exact sequences of the form p i 0 → Z/(p) → Z/(p2 ) → Z/(p) → 0. where 1 ≤ i ≤ p − 1. Problem 2. Let A be an abelian category with enough injectives, and let A, B be objects of A. Given n-extensions (exact sequences) ξ : 0 → B → X 1 → · · · → X n → A → 0, and ξ 0 : 0 → B → Y 1 → · · · → Y n → A → 0, of A by B, we say that ξ is pre-equivalent to ξ 0 if there exists a commutative diagram ξ: / 0 B idB ξ0 : / 0 B / / X1 Y1 / ... / ... / / Xn Yn / / / A 0 idA A / 0. We define Yoneda equivalence to be the smallest equivalence relation on the set of n-extensions of A by B which contains the pre-equivalences and write ξ ∼ ξ 0 if ξ and ξ 0 are Yoneda equivalent. (1) Show that there is a natural bijection ∼ n-extensions of A by B = Θ: −→ ExtnA (A, B) up to Yoneda equivalence generalizing the statement for 1-extensions proved in class. The map Θ, called the Yoneda bijection, is constructed as follows. Given an n-extension ξ as above, we consider ξ as a right resolution ' B −→ (X 1 → X 2 → · · · → X n → A → 0) =: X • ' and choose an injective resolution B → I • . The identity map B → B extends to a map of complexes f : X • → I • which is unique up to homotopy. We apply the functor Hom(A, −) and pass to nth cohomology to obtain a canonical map Hom(A, A) → ExtnA (A, B). Define Θ(ξ) to be the image of idA under this map. (2) We extend the Baer sum defined in class for n = 1: For n > 1 we define the Baer sum of n-extensions ξ and ξ 0 to be the n-extension ` 0 → B → X 1 B Y 1 → X 2 ⊕Y 2 → · · · → X n−1 ⊕Y n−1 → X n ×A Y n → A → 0. where the leftmost and rightmost terms denote a fibered coproduct and fibered product, respectively. Show that this operation provides an abelian group structure on the set of n-extension classes such that the map Θ becomes a group isomorphism. Problem 3. Let A be an abelian category with enough projectives. Consider the abelian category Aop and define the map Θ = ΘAop as in Problem 2(1). (1) Given objects A, B of A, show that there is a natural identification of Yoneda equivalence classes of n-extensions of B by A in Aop with equivalence classes of n-extensions of A by B in A. Show that this identification is a group isomorphism with respect to the Baer sum. Show that there is a natural identification ExtnAop (B, A) ∼ = ExtnA (A, B) where both groups are computed via a projective resolution of P• → A. Deduce that we obtain a natural group isomorphism Θ0 which is uniquely determined by the commutativity of the diagram ΘAop n-extension classes / Extn op (B, A) op A ∼ = in A of B by A ∼ = ∼ = n-extension classes in A of A by B Θ0 / ExtnA (A, B) (2) Assume that A has enough projectives and injectives. Let A,B be ' ' objects of A, let P• → A be a projective resolution, and let B → I • be an injective resolution of B. We introduce the mapping complex Hom• (P• , I • ) to be the cochain complex given by M Homn (P• , I • ) = Hom(Pi , I j ) i+j=n 2 with differential on a map f of degree n given by d(f ) = dI ◦ f − (−1)n f ◦ dP . Show that there are natural quasi-isomorphisms ' ' Hom(P• , B) −→ Hom• (P• , I • ) ←− Hom(A, I • ) which can be used to balance Ext∗ in analogy to the balancing isomorphisms for Tor∗ constructed in class. Show that under the resulting identification of ExtnA (A, B), computed via a projective resolution of A, and ExtnA (A, B) computed via an injective resolution of B, the two maps Θ and Θ0 agree. Problem 4. Let A be an abelian category with enough injectives. Let A, B and C be objects of A. Given a map f : A → B and an n-extension ξ : C → X1 → · · · → Xn → B → 0 of B by C, we define an n-extension ξ ◦ f : 0 → C → X 1 → · · · → X n−1 → X n ×B A → A → 0. Given a map g : B → C and an m-extension ν : B → Y 1 → · · · → Y m → A → 0, of A by B, we define an m-extension ` g ◦ ν : 0 → C → Y 1 B C → Y 2 · · · → Y m → A → 0. Finally, given an n-extension ξ of B by C and an m-extension ν of A by B as above, we define an n + m-extension ξ ◦ ν : C → X1 → · · · → Xn → Y 1 → · · · → Y m → A → 0 by composing X n → B and B → Y 1 . (1) Use the Yoneda interpretation of Ext∗ to show that, for every n, m ≥ 0, these constructions provide a well-defined bilinear map n+m ExtnA (A, B) × Extm (A, C) A (B, C) → ExtA called Yoneda composition. 3 (2) Given objects A, B of A, we define the graded abelian group M Ext∗A (A, B) := ExtnA (A, B). n≥0 Show that Yoneda composition defines a category Ext∗ (A) with objects given by the objects of A and morphisms between objects A,B of A given by M HomExt∗ (A) (A, B) = Ext∗A (A, B) := ExtnA (A, B). n≥0 The statement to verify is the associativity of the composition law. Note that all morphism sets in this category carry the structure of a graded abelian group which is respected by the composition law. We say that Ext∗ (A) is enriched over graded abelian groups. Further note that the category A is obtained from Ext∗ (A) by discarding all morphisms of positive degree. (3) Deduce that, given an object A of A, the graded abelian group Ext∗A (A, A) carries a canonical associative graded Z-algebra structure. (4) Let k be a field. Consider the k[x, y]-module k ∼ = k[x, y]/(x, y). Explic∗ itly compute the graded algebra Extk[x,y] (k, k). (5) Bonus: Prove a statement which, for any abelian category A with enough projectives, allows you to directly compute the algebra Ext∗A (A, A) in terms of a projective resolution of A without reference to the Yoneda isomorphism. 4