This book teems with the author's joy in his subject. It is accordingly an idiosyncratic work, with a variety of human elements poking through the mathematics at nearly every corner.
The Heart of Cohomology is an expansion, and then some, of the author's lecture notes for the short course "Introduction to Derived Category," given in 2004 at the University of Antwerp. The title of the book is a play on words, given that derived categories' cores are also called hearts; it is attributed to Fred Van Ostayen of U. Antwerp, and Kato places a poem in Dutch-Flemish, by the early 20th century writer, Paul Van Ostayen, in the book's fly leaf; more about this later.
In his preface to The Heart of Cohomology the author singles out a number of unquestioningly seminal sources for his material, with Cartan-Eilenberg (Homological Algebra) and Godement (Topologie Algébrique et Théorie des Faisceaux) leading the field. Says Kato: "If you are capable of learning from either of these two books, I am afraid that The Heart of Cohomology (referred to hereafter as [THOC]) is not for you."
I beg to differ. Both of these books have figured, and figure still, in my own studies in major ways, (especially Cartan-Eilenberg), but I expect that [THOC] will soon begin to play a major role in my work, too. The reasons for this are two-fold: a great deal has happened since the early years of the subject, i.e. the middle 1950s, and a quick introduction to the subject "for the working mathematician," to borrow Saunders MacLane's phrase, is highly desirable. For many of us (especially non-expert fellow-travellers like me) a development of cohomology along autonomous lines is most welcome, since this would eliminate the need for sifting and selecting what to study in the (older) foundational texts whose larger focus would typically be algebraic geometry or algebraic topology. Furthermore, while the notion of a derived category goes back to Verdier's thesis under Grothendieck in the 1970s, it has recently become a subject of much greater interest, cutting across the older boundaries locating it inside algebraic geometry. Accordingly this aspect of Kato's book is most welcome indeed.
Before I address specific features of The Heart of Cohomology, and in view of the lingering peculiar position occupied by category theory in the mathematical community, some general remarks are perhaps in order. The year 1956 marks the appearance of the aforementioned Homological Algebra by Henri Cartan and Samuel Eilenberg, a work representative of the French school of algebraic topology and, presently, algebraic geometry, with the entire enterprise fitted into the philosophy of Bourbaki. Cartan and Eilenberg were both members (cf. Mashaal's book) as was Jean-Pierre Serre, whose gorgeous paper "Faisceaux Algébriques Cohérents" (FAC) had just appeared the year before in the Annals of Mathematics. Sheaf cohomology was the order of the day, be it in the Čech-esque style of Serre's FAC or in the evolving categories-and-functors style presented (for R-modules) in Cartan-Eilenberg and presently fitted to his own algebraic-geometric needs by Alexandre Grothendieck in "Sur quelques points d'algèbre homologique," published in Tôhoku Math. J. in 1957. The origin of the categorical treatment of sheaves and their cohomology can be traced to this magnificent paper, usually called Grothendieck's Tôhoku paper, and it is this categorical perspective that dominates contemporary treatments of sheaf cohomology and its outgrowths.
In Grothendieck's formulation, a presheaf of sets (or abelian groups, or R-modules, and so on) is a contravarient functor from the category of open sets of a topological space (e.g. an algebraic variety) to the target category of sets (or abelian groups, etc); a presheaf becomes a sheaf if it can be suitably locally defined, meaning that something like analytic continuation is in place. In fact, for sheaves of sections of analytic functions on a complex manifold we are literally dealing with analytic continuation. In general, the point is that there is a process in place for gluing local data to get global data. The upshot is that a sheaf is a generalization of the eminently useful notion of a fibre bundle. In both cases, the idea is that the fibres in question carry algebraic structure that varies continuously across the topological space that is its base (or pre-image). It is most useful in applications to play both ends against the middle, sheaves as topological spaces espaces étalés, actually) or as "sheafified" contravariant functors. (Specifically, a presheaf associates to a sheaf, unique up to isomorphism, by a process of sheafification, with the usual universal property in place).
Grothendieck's presentation of sheaves in the indicated categorical terms is the starting point for developing sheaf cohomology using the machinery presented in Cartan-Eilenberg and expanded in Tôhoku. The main idea is that a sheaf, F, of, say, abelian groups, over a topological space, X, allows an associated left exact functor Γ(X, F), the global sections functor, whose right derived functors, RnΓ(X, F), or, more properly, RnΓ(X, –), give the derived cohomology groups:
Hn(X, F) =RnΓ(X, F ).
One proves à la Cartan-Eilenberg that if
0 → F ′ → F → F ″ → 0
is a short exact sequence of sheaves (it's enough to have s.e.s.'s in the fibres) then there is a natural connecting homomorphism RnΓ(X, F ″) → Rn+1Γ(X, F ′) so that we get the all-important long exact sequence in cohomology. In other words, axiomatically speaking sheaf cohomology is quickly taking shape.
One of the primary motivations for this work by Grothendieck, soon the leader of the most famous seminar in the history of algebraic geometry, was to develop what is often called a Weil cohomology, the hope being that this machinery should be equal to the task of establishing André Weil's conjecture about geometric zeta-functions, including a counterpart to the Riemann Hypothesis. Grothendieck's pupil, Pierre Deligne, proved the latter result in the 1970's, in one of the most famous tours de force in algebraic geometry.
But sheaf cohomology, or even the more specialized and exotic étale cohomology, is just the start of the story. If derived functors are central to the construction of sheaf cohomology outlined above, they come off as somewhat prosaic in comparison to another idea of Grothendieck's, slated to form the focus of the thesis of another student of his, Jean-Louis Verdier. The thesis, written in the 1970s and titled "Des catégories dérivés des catégories abéliennes," only appeared in print in 1996, but its influence was felt already some two decades earlier, with Verdier's famous note "Catégories derivés, état 0," in SGA 4½ (i.e. in the Séminaire de Geométrie Algébrique du Bois-Raymond). As R.P. Thomas points out in a recent article ("Derived categories for the working mathematician," arXiv. math 0001045), the guiding objective behind derived categories is to have the categories themselves do the work of the erstwhile cohomology groups (says Thomas: "cohomology: bad; categories: good"), and, indeed, one gets at the all-important long exact sequences in a greatly simplified and more intrinsic fashion.
Specifically, starting with an abelian category A, one reaches its derived category D(A) by going first to the associated additive category of chain complexes from A, identified modulo chain homotopy, and then localizing at the "ideal" of quasi-isomorphisms. Here two complexes are quasi-isomorphic if they possess the same (ordinary) cohomology groups in all degrees, and localization at these quasi-isomorphisms essentially means that these chain maps are formally inverted: a theory of fractions is formally defined relative to this so-called localizing class of quasi-isomorphisms and the final result is the desired derived category. Thus, the object class of D(A) is just the class of chain complexes from A but the morphisms for D(A) behave more exotically, commuting with chain homotopy (as befits a replacement for a cohomology theory) and tagging quasi-isomorphisms, so to speak. To get at the desired mimicry of cohomology, one introduces the so-called truncation functors (being largely what they seem to be in this setting of chain complexes) and the central notion of a distinguished triangle. These take the place of the earlier short exact sequences and the truncation functors permit the (easy) construction of so-called cohomological functors which, when suitably applied, yield long exact sequences.
At this stage, however, another feature attending cohomology theories in general comes into play, namely, that of an object's resolution. Already in Cartan-Eilenberg we encounter projective and injective resolutions of R-modules as tools with which to compute (co)homology groups; arguably the most well-known specific instance of this approach is that of the calculation of group cohomology (cf. e.g., Serre's Corps Locaux). In the setting of derived categories resolutions are, if anything, more important, given that they not only facilitate computations but provide the important connection to standard cohomology theories indispensable in applications. Indeed, it is via Verdier's famous functor, aptly denoted R, that an object and its injective resolution are tied to the cohomology formulated by Grothendieck in the 1950's. Quite dramatically, the category of sheaves on a space X is abelian (and enjoys the property of having "enough injectives") and here we recover the R•Γ(X,–) formalism perfectly.
Most recently, however, derived categories, and, more generally, triangulated categories, have attracted even more attention due to their utility in physics. Recent (and famous) work by Maxim Kontsevich, for example, has centered on the role of derived categories in relation to mirror symmetry, the greater context being string theory. Consider, for instance, the following passage from Michael Douglas' 2002 arXiv article , "Dirichlet branes, homological mirror symmetry, and stability": "Kontsevich…proposed that [mirror symmetry] should be understood as an equivalence between the derived category of coherent sheaves on [a suitable Calabi-Yau 3-fold] M and the Fukaya category on W, a category whose objects are isotopy classes of Lagrangian submanifolds carrying flat connections and whose morphisms are elements of Floer cohomology." Douglas' own use of derived categories in studying stability (in the indicated sense) in physics has led to some exceedingly interesting work by Tom Bridgeland on stability conditions on derived and triangulated categories as such.
The point is that what Grothendieck began some fifty years ago has grown into a vast and autonomous part of mathematics, breaking free of the original boundaries set for it by algebraic geometry and algebraic topology. Indeed, one of my favorite books in this (or any) subject is Sheaves on Manifolds, by Masaki Kashiwara and Pierre Schapira, who aim to prepare the reader for the micro-local analysis of Mikio Sato, and we find, on p.2 of the Introduction, the following passage: "Sato introduced the functor… of specialization (along a submanifold M of a manifold X) and its Fourier transform, the functor… of microlocalization. These functors send the derived categories of sheaves on the normal and conormal bundles to M in X, respectively, and they allow us to analyze… a sheaf on a neighborhood of M, taking into account all normal (or conormal) directions to M." This is geometric analysis with a vengeance, and, to be sure, sheaf cohomology and the theory of derived categories can today be found everywhere in mathematics, from topology and analysis to algebra, number theory, and physics.
Thus it stands to reason that these things categorical, even as they are still sometimes called "abstract nonsense," with sheaves figuring prominently throughout, should be given a compact and accessible treatment in the literature, all in the worthy cause of preparing beginning researchers for what lies ahead. In my opinion, Kato's The Heart of Cohomology goes a long way to doing exactly that. In fact, Kato states explicitly that "[o]ne of the goals of [THOC] is to provide young readers with elemental aspects of the algebraic treatment of cohomologies." The book fits this bill very well, but I think its value is even greater in that its development of rather austere topics is both compact and readable, implying that [THOC] qualifies as a nice supplementary text to Cartan-Eilenberg, Godement, Tôhoku, Kashiwara-Schapira, Robin Hartshorne's Residues and Duality or Homological Algebra by S. I. Gel'fand and Yuri Manin (or any weighted linear combination of these sources).
The chapters in [THOC] are titled "Category," "Derived Functors," "Spectral Sequences," "Derived Categories," and "Cohomological Aspects of Algebraic Geometry and Algebraic Analysis." Each of the first four chapters is a vacle mecum of centrally important topics, presented with proofs in place, with thoroughness, and with style. I am particularly impressed with Kato's inclusion of over thrity pages on spectral sequences, given the notoriety of this subject, at least in many quarters.
Additionally, [THOC] evinces a distinct personal touch: Kato includes a handful of photographs (Yoneda, Deligne, Weil, Lubkin, Sato, Oka), with, shall we say, unusual captions; also, as mentioned at the start of this review, the book's flyleaf contains a poem by, apparently, Fred van Ostagen's relative, Paul, in Dutch-Flemish. Here is my translation:
Marc greets the objects in the morning.
Good day little boy with the bicycle on the vase with the flower.
good day chair by the table
good day bread on the table
good day fisherman-fish with the pipe
good day fisherman-fish with the cap
cap and pipe
of the fisherman-fish
Good day fish
good day fish
good day tiny little fishy mine.
I guess these poetic objects of Marc's morning are meant to suggest their categorical counterparts: a nice touch, I think.
Kato's The Heart of Cohomology is a pleasure to read and it'll be a pleasure to come back to it again and again. Just a couple of days ago, at a small gathering at my house, the playful question was raised of what book to bring to a desert island. I'm afraid that I picked Cartan-Eilenberg, much to everyone's amusement; I'd like to add that I'd slip my copy of The Heart of Cohomology inside my shirt, too.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.