An obvious question: why another graduate algebra book? Aren’t there all but too many already? There are the classics some one my age naturally gravitates to: B.L. Van der Waerden’s (once) *Modern(e) Algebra*, Serge Lang’s occasionally idiosyncratic but hugely influential *Algebra *(witness e.g. the notorious exercise on homological algebra in the first edition), Mac Lane-Birkhoff’s *Algebra* (not Birkhoff-MacLane), the marvelous series by Nathan Jacobson, and so forth. Somewhat more recently J.J. Rotman’s encyclopedic *Advanced Modern Algebra* appeared, and quite recently (and reviewed in this venue) Anthony Knapp produced the even more encyclopedic pair, *Basic Algebra, Advanced Algebra* (and both Rotman’s and Knapp’s books are also excellent, by the way — to no one’s surprise). Is there anything genuinely novel to be done when it comes to educating fledgling graduate students in this subject, given such an already well-populated and high-quality field?

Well, the answer is yes. And the title of the book under review, *Algebra: Chapter 0*, is already a clue to what the author, Paolo Aluffi, is up to. In a perhaps Bourbakian sense, the prevailing motivation and objective is to present the subject at hand in a manner that pays proper due to relatively new foundations, making for a rather different orientation and flavor for what ensues. Clearly, when it comes to algebra we can historically identify three such foundational paradigm shifts, so to speak: first, the shift from the original conceptions of Kronecker, Weber, Dedekind, and Hilbert, to the *abstrakte Algebra* of Emmy Nöther, who, by the way, was apt to give a huge portion of the credit the to Dedekind; second, the maneuver of basing not just algebra but nigh on all of mathematics on set theory, a move sometimes associated with Bourbaki in its (or his) heyday; and third, the rather recent incursions made by category theory into, again, nigh on everything, at least *in potentio*. For the latter it is not wrong to give the lion’s share of historical credit (or blame) to Grothendieck, both for this introduction of a sweeping categorical perspective in this area and for the development of a huge number of attendant techniques. What he did for algebraic geometry has manifestly taken on a largely autonomous character and has come to inform the foundations of any number of mainstream mathematical disciplines at this point in time.

Aluffi’s book accordingly aims at developing algebra, at the usual advanced undergraduate to beginning graduate level, on an explicitly category-theoretical foundation. But Aluffi tempers his revolutionary zeal by also giving set theory its due: his first chapter is titled, “Preliminaries: set theory and categories,” so (for those who still don’t have any time for categories and functors) it could be worse.

His treatment of these preliminaries is thorough as well as eminently accessible: Aluffi writes well, clearly and engagingly. This characterizes all of* Algebra: Chapter 0, *actually, and makes it easy to recommend the book enthusiastically even aside from the fact that I am a big fan of category theory to begin with. The sequence of subsequent chapters of the book is as follows:

- Ch. II, “Groups, first encounter,” taking one from the basic definitions through group actions, and then appending a section titled “Group objects in categories”;
- Ch. III, “Rings and modules,” including a welcome subsection differentiating between the notions of finite generation and finite type, and capped off by a section on complexes and homology (ending with the snake lemma);
- Ch. IV, “Groups, second encounter,” including e.g. the class formula, Sylow Theory, Jordan-Hölder (and Schreier), the extension problem (i.e. second cohomology, really, although Aluffi restricts himself to semi-direct products), and finite abelian groups;
- Ch. V, “Irreducibility and factorization in integral domains,” starting with chain conditions, ending with a thorough discussion of polynomial rings (and Fermat’s theorem on sums of squares as icing on the cake);
- Ch. VI, “Linear algebra,” covering “just about everything” (including e.g. the Euler characteristic and the Grothendieck group, presentations and resolutions, and sundry canonical forms);
- Ch. VII, “Fields,” including a decent dosage of algebraic geometry (surrounding the
*Nulstellensatz*), geometric impossibilities, and “[a] little Galois theory” — or more than a little, given that the last subsection of this chapter is titled, “Abelian groups as Galois groups over **Q**”;
- Ch. VIII, “Linear algebra: reprise,” introducing a functorial perspective into the affair, followed by coverage of limits and colimits, tensor products, Tor, Hom, Ext, intermingled with treatments of duality, projective and injective modules, and adjunction;
- Ch. IX, “Homological algebra,” taking the reader from “(the) necessary categorical preliminaries” to, in order, additive and abelian categories, “
omplexes and homology, again” (replete with the long exact sequence in cohomology), triangles, derived categories, a very thorough discussion of homotopy, derived functors, a return to (e.g. group) cohomology from this new perspective, double complexes, various things acyclic, Tor and Ext again, and finally a brief discussion of derived categories, triangulated categories, and spectral sequences, grouped together under the heading “Further topics.”</li>
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<p>It is clear, then, that Aluffi’s grand undertaking (we’re talking about a little over 700 pages!) is indeed a most useful and welcome labor: he has composed a coherent treatment of mainstream graduate algebra tied together with material from homological algebra that not all that long ago was dealt with separately and subsequently. As already suggested, the approach chosen in <em>Algebra: Chapter 0</em> properly reflects relatively recent changes in the way research in algebra is done: there is a far greater presence of homological algebraic methods early on, and, indeed, categories (even derived, triangular ones) come into the game much as a matter of course. In this sense <em>Algebra: Chapter 0</em> certainly breaks new ground, and does so with <em>élan</em>.</p>
<p>Finally, Aluffi also possesses the gift of a light touch: the book has a lot of humor in it. For example, his introduction of group theory to the presumably uninitiated starts with “Joke 1.1. Definition: A group is a groupoid with a single object,” and we find on p. 334 the (revealing) phrase, “Against our best efforts, we cannot resist extending these simple observations to more general complexes…” and it’s on to the Euler characteristic and the Grothendieck group. There are also a lot of good exercises and very useful (and pedagogically astute) footnotes. <em>Algebra: Chapter 0</em> is a very good book that should be used in a huge number of departments across the county (and beyond).</p>
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<p>Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.</p>