A short proof of HRS-tilting

We give a short proof to the following tilting theorem by Happel, Reiten and Smal{\o} via an explicit construction: given two abelian categories $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{B}$ is tilted from $\mathcal{A}$, then $\mathcal{A}$ and $\mathcal{B}$ are derived equivalent.


Introduction
Let A be an abelian category. Recall that a torsion pair on A means a pair (T , F ) of full subcategories satisfying (T1). Hom A (T, F ) = 0 for all T ∈ T and F ∈ F; both subcategories T and F are closed under direct summands; (T2). for each object X ∈ A, there is a short exact sequence 0 −→ T −→ X −→ F −→ 0 for some T ∈ T and F ∈ F.
In a torsion pair (T , F ), it follows that the subcategory T is closed under extensions and factor objects; F is closed under extensions and sub objects. The torsion pair (T , F ) is called a tilting torsion pair, provided that each object in A embeds into an object in T . Dually the torsion pair (T , F ) is called a cotilting torsion pair, provided that each object in A is a factor object of an object in F ([3, Chapter I, section 3]).
Denote a complex in A by X • = (X n , d n X ) n∈Z where d n X : X n −→ X n+1 is the differential satisfying d n+1 X • d n X = 0; its shift X • [1] is a complex given by (X • [1]) n = X n+1 and d n X [1] = −d n+1 X . Denote by D(A) the (unbounded) derived category of A, D + (A), D − (A) and D b (A) the full subcategory consisting of bounded-below, bounded-above and bounded complexes, respectively ( [6,4]). We will always identify the abelian category A as the full subcategory of D(A) consisting of stalk complexes concentrated at degree zero ([4, p.40, Proposition 4.3]). Let (T , F ) be a torsion pair on A. Following [3, Chapter I, section 2], set B to be the full subcategory of D(A) consisting of complexes X • satisfying H 0 (X • ) ∈ T , H −1 (X • ) ∈ F and H i (X • ) = 0 for i = 0, 1. Note that T ⊆ B and F [1] ⊆ B. By [3, Chapter I, Proposition 2.1] the category B is the heart of certain t-structure on D(A) and thus by [1] it is an abelian category (also see [2]); moreover the pair (F [1], T ) is a torsion pair on B.
One might expect that the resulting new abelian category B is derived equivalent to A. In the case of Theorem the category B is said to be tilted from A. Note that the original theorem only claims the equivalence between the bounded derived categories and requires the existence of enough projective or injective objects. The quoted version is improved by Noohi ([5, Theorem 7.6]). We will give a short proof of the theorem via an explicit construction of the equivalence functor.

The Proof of Theorem
Throughout (T , F ) is a torsion pair on A and B is the resulting abelian category. We start with an easy observation. Proof of Theorem: Denote by K(A) the homotopy category of complexes in A, K(T ) (resp. K ex (A)) its full subcategory consisting of complexes in T (resp. exact complexes). The inclusion K(T ) ֒→ K(A) induces the following exact functor Since the torsion pair (T , F ) is tilting and T is closed under factor objects, we have for each X ∈ A a short exact sequence 0 −→ X −→ T 0 −→ T 1 −→ 0 with T i ∈ T . Note further that T is closed under extensions, we infer that the conditions in [4, p.42, Lemma 4.6 2)] are fulfilled, and thus for each complex . This implies that the functor F is dense and by [6,p.283, it is fully-faithful, that is, the functor F is an equivalence of triangulated categories. By Lemma 2.2 we may apply the dual argument to obtain a natural equivalence To see other equivalences, let * ∈ {+, −, b} and let K * (−) denote the corresponding homotopy categories. Note that in the argument above, for a complex X • ∈ K * (A) we may take a quasi-isomorphism X • −→ T • with T • ∈ K * (T ) (for the case * = +, just consult the proof in [4, p.43, 1)]; for the case * = −, because T is closed under factor objects one may replace T • by its good truncations; for the case * = b, consult the proof in [4, p.43, 1)] and note that since T is closed under factor objects, the argument therein is done within finitely many steps, consequently the obtained complex T • is bounded). Thus we construct the equivalences F * and G * as above. This proves the corresponding equivalences between the derived categories D * (−).
Finally we will show that the obtained equivalence F G −1 is compatible with the inclusion B ֒→ D(A). This is subtle. Given an object B ∈ B, since the torsion pair (F [1], T ) is cotilting, we have a short exact sequence in B, η : Note that the complex T • is the mapping cone of d and thus form the triangle ξ we obtain T • is isomorphic to B ([4, p.23, Propostion 1.1 c)]), in particular T • ∈ B. Note the following natural triangle and thus a short exact sequence Comparing the short exact sequences η and γ we obtain a unique isomorphism θ B : B ≃ T • in B. We claim that θ is natural in B and then we obtain a natural isomorphism between the inclusion functor B ֒→ D(A) and the composite In fact, given a morphism f : B −→ B ′ in B, choose an exact sequence η ′ : 0 −→ Form the complex T ′• and then obtain the short exact sequence γ ′ and the isomorphism θ B ′ as above. Identify G(T • ) with B, G(T ′• ) with B ′ . Since the functor G is fully-faithful, we have a chain map φ • : T • −→ T ′• such that G(φ • ) = f . This implies the following commutative exact diagram in B 0. From this it is direct to see that θ B ′ • f = φ • • θ B in B and thus in D(A). This finishes the proof.