# Morphism of Complexes Induces Corresponding Morphisms on Cohomology Objects – Thought of the Day 146.0

Let A = Mod(R) be an abelian category. A complex in A is a sequence of objects and morphisms in A

… → Mi-1 →di-1 Mi →di → Mi+1 → …

such that di ◦ di-1 = 0 ∀ i. We denote such a complex by M.

A morphism of complexes f : M → N is a sequence of morphisms fi : Mi → Ni in A, making the following diagram commute, where diM, diN denote the respective differentials:

We let C(A) denote the category whose objects are complexes in A and whose morphisms are morphisms of complexes.

Given a complex M of objects of A, the ith cohomology object is the quotient

Hi(M) = ker(di)/im(di−1)

This operation of taking cohomology at the ith place defines a functor

Hi(−) : C(A) → A,

since a morphism of complexes induces corresponding morphisms on cohomology objects.

Put another way, an object of C(A) is a Z-graded object

M = ⊕i Mi

of A, equipped with a differential, in other words an endomorphism d: M → M satisfying d2 = 0. The occurrence of differential graded objects in physics is well-known. In mathematics they are also extremely common. In topology one associates to a space X a complex of free abelian groups whose cohomology objects are the cohomology groups of X. In algebra it is often convenient to replace a module over a ring by resolutions of various kinds.

A topological space X may have many triangulations and these lead to different chain complexes. Associating to X a unique equivalence class of complexes, resolutions of a fixed module of a given type will not usually be unique and one would like to consider all these resolutions on an equal footing.

A morphism of complexes f: M → N is a quasi-isomorphism if the induced morphisms on cohomology

Hi(f): Hi(M) → Hi(N) are isomorphisms ∀ i.

Two complexes M and N are said to be quasi-isomorphic if they are related by a chain of quasi-isomorphisms. In fact, it is sufficient to consider chains of length one, so that two complexes M and N are quasi-isomorphic iff there are quasi-isomorphisms

M ← P → N

For example, the chain complex of a topological space is well-defined up to quasi-isomorphism because any two triangulations have a common resolution. Similarly, all possible resolutions of a given module are quasi-isomorphic. Indeed, if

0 → S →f M0 →d0 M1 →d1 M2 → …

is a resolution of a module S, then by definition the morphism of complexes

is a quasi-isomorphism.

The objects of the derived category D(A) of our abelian category A will just be complexes of objects of A, but morphisms will be such that quasi-isomorphic complexes become isomorphic in D(A). In fact we can formally invert the quasi-isomorphisms in C(A) as follows:

There is a category D(A) and a functor Q: C(A) → D(A)

with the following two properties:

(a) Q inverts quasi-isomorphisms: if s: a → b is a quasi-isomorphism, then Q(s): Q(a) → Q(b) is an isomorphism.

(b) Q is universal with this property: if Q′ : C(A) → D′ is another functor which inverts quasi-isomorphisms, then there is a functor F : D(A) → D′ and an isomorphism of functors Q′ ≅ F ◦ Q.

First, consider the category C(A) as an oriented graph Γ, with the objects lying at the vertices and the morphisms being directed edges. Let Γ∗ be the graph obtained from Γ by adding in one extra edge s−1: b → a for each quasi-isomorphism s: a → b. Thus a finite path in Γ∗ is a sequence of the form f1 · f2 ·· · ·· fr−1 · fr where each fi is either a morphism of C(A), or is of the form s−1 for some quasi-isomorphism s of C(A). There is a unique minimal equivalence relation ∼ on the set of finite paths in Γ∗ generated by the following relations:

(a) s · s−1 ∼ idb and s−1 · s ∼ ida for each quasi-isomorphism s: a → b in C(A).

(b) g · f ∼ g ◦ f for composable morphisms f: a → b and g: b → c of C(A).

Define D(A) to be the category whose objects are the vertices of Γ∗ (these are the same as the objects of C(A)) and whose morphisms are given by equivalence classes of finite paths in Γ∗. Define a functor Q: C(A) → D(A) by using the identity morphism on objects, and by sending a morphism f of C(A) to the length one path in Γ∗ defined by f. The resulting functor Q satisfies the conditions of the above lemma.

The second property ensures that the category D(A) of the Lemma is unique up to equivalence of categories. We define the derived category of A to be any of these equivalent categories. The functor Q: C(A) → D(A) is called the localisation functor. Observe that there is a fully faithful functor

J: A → C(A)

which sends an object M to the trivial complex with M in the zeroth position, and a morphism F: M → N to the morphism of complexes

Composing with Q we obtain a functor A → D(A) which we denote by J. This functor J is fully faithful, and so defines an embedding A → D(A). By definition the functor Hi(−): C(A) → A inverts quasi-isomorphisms and so descends to a functor

Hi(−): D(A) → A

establishing that composite functor H0(−) ◦ J is isomorphic to the identity functor on A.

# Conjuncted: Balance of Payments in a Dirty Float System, or Why Central Banks Find It Ineligible to Conduct Independent Monetary Policies? Thought of the Day

If the rate of interest is partly a monetary phenomenon, money will have real effects working through variations in investment expenditure and the capital stock. Secondly, if there are unemployed resources, the impact of increases in the money supply will first be on output, and not on prices. It was, indeed, Keynes’s view expressed in his General Theory that throughout history the propensity to save has been greater than the propensity to invest, and that pervasive uncertainty and the desire for liquidity has in general kept the rate of interest too high. Given the prevailing economic conditions of the 1930s when Keynes was writing, it was no accident that he should have devoted part of the General Theory to a defence of mercantilism as containing important germs of truth:

What I want is to do justice to schools of thought which the classicals have treated as imbeciles for the last hundred years and, above all, to show that I am not really being so great an innovator, except as against the classical school, but have important predecessors, and am returning to an age-long tradition of common sense.

The mercantilists recognised, like Keynes, that the rate of interest is determined by monetary conditions, and that it could be too high to secure full employment, and in relation to the needs of growth. As Keynes put it in the General Theory:

mercantilist thought never supposed as later economists did [for example, Ricardo, and even Alfred Marshall] that there was a self-adjusting tendency by which the rate of interest would be established at the appropriate level [for full employment].

It was David Ricardo, in his The Principles of Political Economy and Taxation, who accepted and developed Say’s law of markets that supply creates its own demand, and who for the first time expounded the theory of comparative advantage, which laid the early foundations for orthodox trade and growth theory that has prevailed ever since. Ricardian trade theory, however, is real theory relating to the reallocation of real resources through trade which ignores the monetary aspects of trade; that is, the balance between exports and imports as trade takes place. In other words, it ignores the balance of payments effects of trade that arises as a result of trade specialization, and the feedback effects that the balance of payments can have on the real economy. Moreover, continuous full employment is assumed because supply creates its own demand through variations in the real rate of interest. These aspects question the prevalence of Ricardian theory in orthodox trade and growth theory to a large extent in today’s scenario. But in relation to trade, as Keynes put it:

free trade assumes that if you throw men out of work in one direction you re-employ them in another. As soon as that link in the chain is broken the whole of the free trade argument breaks down.

In other words, the real income gains from specialization may be offset by the real income losses from unemployment. Now, suppose that payments deficits arise in the process of international specialization and the freeing of trade, and the rate of interest has to be raised to attract foreign capital inflows to finance them. Or suppose deficits cannot be financed and income has to be deflated to reduce imports. The balance of payments consequences of trade may offset the real income gains from trade.

This raises the question of why the orthodoxy ignores the balance of payments? There are several reasons, both old and new, that all relate to the balance of payments as a self-adjusting process, or simply as a mirror image of autonomous capital flows, with no income adjustment implied. Until the First World War, the mechanism was the gold standard. The balance of payments was supposed to be self-equilibrating because countries in surplus, accumulating gold, would lose competitiveness through rising prices (Hume’s quantity theory of money), and countries in deficit losing gold would gain competitiveness through falling prices. The balance of payments was assumed effectively to look after itself through relative price adjustments without any change in income or output. After the external gold standard collapsed in 1931, the theory of flexible exchange rates was developed, and it was shown that if the real exchange rate is flexible, and the so-called Marshall–Lerner condition is satisfied (i.e. the sum of the price elasticities of demand for exports and imports is greater than unity), the balance of payments will equilibrate; again, without income adjustment.

In modern theory, balance of payments deficits are assumed to be inherently temporary as the outcome of inter-temporal decisions by private agents concerning consumption. Deficits are the outcome of rational decisions to consume now and pay later. Deficits are merely a form of consumption smoothing, and present no difficulty for countries. And then there is the Panglossian view that the current account of the balance of payments is of no consequence at all because it simply reflects the desire of foreigners to invest in a country. Current account deficits should be seen as a sign of economic success, not as a weakness.

It is not difficult to question how balance of payments looks after itself, or does not have consequences for long-run growth. As far as the old gold standard mechanism is concerned, instead of the price levels of deficit and surplus countries moving in opposite directions, there was a tendency in the nineteenth century for the price levels of countries to move together in the same direction. In practice, it was not movements in relative prices that equilibrated the balance of payments but expenditure and output changes associated with interest rate differentials. Interest rates rose in deficit countries which deflated demand and output, and fell in surplus countries stimulating demand.

On the question of flexible exchange rates as an equilibrating device, a distinction first needs to be made between the nominal exchange rate and the real exchange rate. It is easy for countries to adjust the nominal rate, but not so easy to adjust the real rate because competitors may “price to market” or retaliate, and domestic prices may rise with a nominal devaluation. Secondly, the Marshall–Lerner condition then has to be satisfied for the balance of payments to equilibrate. This may not be the case in the short run, or because of the nature of goods exported and imported by a particular country. The international evidence over the past almost half a century years since the breakdown of the Bretton Woods fixed exchange rate system suggests that exchange rate changes are not an efficient balance of payments adjustment weapon. Currencies appreciate and depreciate and still massive global imbalances of payments remain.

On the inter-temporal substitution effect, it is wrong to give the impression that inter-temporal shifts in consumption behaviour do not have real effects, particularly if interest rates have to rise to finance deficits caused by more consumption in the present if countries do not want their exchange rate to depreciate. On the view that deficits are a sign of success, an important distinction needs to be made between types of capital inflows. If the capital flows are autonomous, such as foreign direct investment, the argument is plausible, but if they are “accommodating” in the form of loans from the banking system or the sale of securities to foreign governments and international organizations, the probable need to raise interest rates will again have real effects by reducing investment and output domestically.