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Lectures on Algebraic Cycles

Cambridge University Press
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It was Grothendieck himself, possibly when he was lecturing at Harvard in the 1960s, who popularized the phrase “abstract nonsense” for such things as homological algebra and specifically cohomology. Although I grew up with this slang in graduate school (and even before), I recall first learning of the use of this appellation by the founder himself in one of his own writings. I was studying étale cohomology in the late 1990s, and felt I should go to some of the root sources, specifically Grothendieck himself. I guess this realization, that he himself joked about this marvelous machine in this way, does take the sting out of it a bit: if Grothendieck himself uses the phrase, then we should be sanguine about it, too. In fact, we might even gloat quietly over the irony of it all, given that most of the time these condescending words are spoken necessarily ironically: there is no doubt that abstract nonsense has won the day.

Indeed, any one of us abstract nonsensers can certainly feel dramatically justified by the fact that, its many apparent detractors notwithstanding, cohomology is arguably the greatest mathematical discovery of the last half of the 20th century, at least as far as a methodology or tool kit goes. (I can’t really think of a competitor: just consider its sheer ubiquity and the trajectory of its evolution.)

Well, I am obviously a huge fan of all things Grothendieck and in fact find Bourbakist homological algebra all but irresistible. Starting as I did with “Tôhoku,” I was quickly led along what turned out to be quite a natural and well-travelled path (using a French road-map, as it turned out) from Cartan-Eilenberg back to Grothendieck’s Tôhoku paper, to Serre’s FAC and sheaf cohomology, through discussions of cohomologies scattered through, e.g., SGA n, 0 ≤ n < 7, and then on to such cool things as Weil cohomologies (which, indeed, are discussed pro forma in Hartshorne — so they’re within e/3 of being mainstream), derived and triangulated categories, and, finally, even a soupçon of motivic cohomology. I think my path to motives as sketched here is actually rather typical, at least for some one of my generation: I offer this as my rationale for this personal aside.

Yes, motives. When I first read about them they still had the status of conjectured objects, products of wishful if brilliant thinking on Grothendieck’s part, that he had proposed should be conjured up so as to capture in one big algebraic net (no pun intended) any number of things that behave eerily similarly, as it were. This all fitted in which his famous Standard Conjectures, and quickly took on the status of big game.

But they’re not conjectural any more: Vladimir Voevodsky developed a motivic cohomology that succeeded in taking care of Milnor’s conjecture and was awarded the 2002 Fields Medal for his work. Then, in 2009 Voevodsky announced a proof of the Bloch-Kato Conjectures at the IHES Grothendieck conference.

What does Bloch-Kato say? Wikipedia to the rescue: Apparently the (motivic) Bolch-Kato conjecture equates the torsion of a K-group of a field to the torsion of a Galois cohomology group for the field and thus establishes a bridge between K-theory and Galois cohomology (which is a special case of étale cohomology); it fits squarely within the context of nothing less than non-abelian class field theory. À propos, the aforementioned Milnor conjecture, taken separately, is just the 2-torsion case of the foregoing.

And this takes us to the book under review. Here is part of Spencer Bloch’s (yes, it’s the same guy, of course) prenote to Ch 5, titled, “The conjecture of Milnor-Bloch-Kato”:

The conjecture … states that the natural map [from the Milnor ring of the given field, working l-adically for a fixed prime l] to Galois cohomology is an isomorphism. [Bloch’s] own contribution to this is a proof [of surjectivity in degree n when the field has cohomological degree n]. For some years Voevodsky has been working on a very difficult program, using his own motivic theory and results of M. Rust to prove the conjecture in complete generality. The proof is now complete… (Bloch writes as of 2010.)

This is manifestly a very beautiful and very important result. And it is central to the book under review, the second edition of Spencer Bloch’s Lectures on Algebraic Cycles.

Originally written in 1980 and coming out of Bloch’s April 1979 lectures at Duke University, Lectures on Algebraic Cycles is a compact collection of nine presentations on some of the most evocative themes in modern mathematics, collected around the common notion of an algebraic cycle. Bloch characterizes these lectures as “the fruit of ten years reflection on algebraic cycles, that is formal linear combinations åni[Vi] of subvarieties Vi of a fixed smooth and projective variety X with integer coefficients ni,” and then takes off into the stratosphere. His Introduction of six pages starts off with Riemann-Roch and Abel-Jacobi (and we hit Abelian varieties right off), gets quickly to standard (but non-trivial) algebraic geometry, and then begins to address the individual lectures. Viz.:

The first three lectures and geometric in content, studying various aspects of the Abel-Jacobi construction in codimensions > 1 … Lectures 4 through 6 develop the algebraic side of the theory: the cohomology groups of [certain] K-sheaves… , the Gersten-Quillen resolution, and analogues for singular and étale cohomology theories… [and f]inally Lectures 8 and 9 take up from a number theoretic point of view the work of Lecture 3 on relative intermediate Jacobians for [certain] curves.

Bloch then goes on to hum some music to any number theorist’s ear by stating that

[w]e consider the case [of an] elliptic curve [E] and compute explicitly the class in [first cohomology with coefficients in R] associated to the curve … {(x, f(x), g(x)) | x Î E} for f, g rational functions on E. When the zeroes and poles of F are points of finite order, we show how to associate to f and g [a certain global section of E with values in K2] and how to associate to [such a section] a relative algebraic 1-cycle on E×P1×P1.

And then the zinger:

When E has complex multiplication by the ring of integers in an imaginary quadratic field of class number one, we construct [a global section of E over Q, working still with coefficients in K2] such that the image under the Abel-Jacobi map into H1(E,R) » C as the associated relative algebraic cycle multiplied by a certain simple constant (involving the conductor of the curve and [a] Gauss sum) is the value of the Hasse-Weil zeta function at s = 2.

This is simply stunning, and these are only remarks Bloch penned in 1980. The present book provides a retrospective of sorts, 30 years hence, of marvelous developments in this area and here is what Bloch has to say today about the Hasse-Weil L-functions occurring in Chapters 8 and 9:

[Taking Hom and Ext in the triangulated category of Voevodsky motives over Q and working at finite primes] one gets a trivialization over R of the tensor product of [certain] determinant lines … The various determinants have Q-structures … so one may examine … the ratio of the real trivialization and the rational structure … [U]sing Galois and l-adic cohomology [Fontaine, in 1992, and Perrin-Riou, in 1994] show that the integral conjecture in Bloch and Kato (1990) [= Volume 1 of the Grothendieck Festschrift] is equivalent to this ratio being given by … the first non-vanishing term in the Taylor expansion of the Hasse-Weil L-function associated to [a certain motive] M.

Well, to be sure, gorgeous and suggestive though these things are (certainly as seen through number theorists’ lenses), they are undoubtedly very sporty; Bloch himself says, right after the foregoing discussion, “Well, okay, there is a lot here we do not understand…” True enough. But this is largely descriptive of the base state we find ourselves in as mathematicians when we begin to peek behind the proverbial curtain: we’re dealing with avant garde material here and there is no other way than to endure the suffering. There is no royal road to arithmetic geometry either.

All this having been said, the book is browsable (to coin a phrase) by a professional or even a relative neophyte with a good deal of training under his belt, trying to make his way from a standard understanding of some algebraic geometry (Hartshorne’s first three chapters, as usual, and some Griffiths-Harris), algebraic topology in the sense of homological algebra (Weibel, for example), and a predilection for categorical algebra, to an idea of what’s happening at this particular frontier, particularly what magical things motives can do. Those who actually propose to read these Lectures on Algebraic Cycles more carefully (“a consummation devoutly to be wished”) should probably have a context in which to approach the whole business and prepare to do a bit of outside work: Bloch refers to Quillen’s work on K-theory, for example, and “Tate’s p[roof of then Tate conjecture for abelian varieties” in his brief pedagogical note in the Introduction. (A number theorist with a history of class field theory and arithmetic geometry would be happy as a clam.) In any case, these things are good to know, if the book is to serve its purpose.

And this purpose is well represented in the sequence of chapters: zero-cycles on surfaces → curves on threefolds and intermediate Jacobians → curves on threefolds: the relative case → K-theoretic and cohomological methods → torsion in the Chow group → complements on H2(K2) → Diophantine questions → relative cycles and zeta functions. Beautiful material, and Bloch puts some icing on the cake with a very pithy Preface to the second edition in which he performs a sequence of exercises in hindsight: as already indicated above, in his treatment of his fifth chapter he talks about Voevodsky’s work on the Milnor-Bloch-Kato conjecture, motivic methods pepper everything, and he even adds a “Coda: Motives in physics,” including the following phrase: “Dick Kreimer has been teaching me about Feynman amplitudes and perturbative calculations in quantum field theory.”

There is a great deal to be had here: these are very deep waters, but the fauna are gorgeous.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Date Received: 
Tuesday, October 5, 2010
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Spencer Bloch
New Mathematical Monographs 16
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Michael Berg

Preface to the second edition; Introduction; 1. 0-cycles on surfaces; Lecture 1 Appendix. On an argument of Mumford in the theory of algebraic cycles; 2. Curves on threefolds and intermediate Jacobians; 3. Curves on threefolds and intermediate Jacobians - the relative case; 4. K-theoretic and cohomological methods; 5. Torsion in the Chow group; 6. Complements on H2(K2); 7. Diophantine questions; 8. Relative cycles and zeta functions; 9. Relative cycles and zeta functions (continued); References; Index.

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Tuesday, November 23, 2010