IHES
The Cartier Isomorphism by Luc Illusie
It was with great sadness that I learned of the passing of Pierre Cartier. An immensely talented mathematician has left us, someone I knew well starting in the 1960’s. We first met at a seminar by Grothendieck. I was deeply impressed by the depth and encyclopedic breadth of his knowledge, but I had no idea at that time that his work would play a pivotal role in my research for decades to come.
In the early 1970s, crystalline cohomology—a new cohomology theory envisioned by Alexander Grothendieck in the late 1960s, and developed by Pierre Berthelot in his doctoral thesis—was attracting a lot of attention from many geometers. A conjecture of Katz, predicting certain inequalities on p-adic valuations of the Frobenius, was particularly studied. I was astonished when I realized that the solution, under certain assumptions by Mazur, and later in full generality by Ogus, ultimately relied on what is now called the Cartier isomorphism—a reformulation by Katz of a construction originally published by Cartier in a 1957 note in the Comptes Rendus de l’Académie des Sciences. This isomorphism, for the de Rham complex of a smooth variety in positive characteristic, which links its components to its cohomology sheaves, would play a key role in differential calculus in characteristic p or mixed characteristic for years to come.
However, yet another construction of Cartier led to a decisive turning point in my research. This was
Cartier’s theory of typical curves, a more flexible and powerful approach to the classical theory of Dieudonné modules. A bold application of Cartier’s theory by Spencer Bloch to Quillen’s K-theory groups led to the construction of an explicit complex with remarkable structure and properties, particularly concerning the p-adic valuations of the Frobenius mentioned earlier. This complex computed the crystalline cohomology of smooth projective varieties of dimension less than p, for any p > 2. Following a suggestion of Deligne, I proposed a K-theory-free, more general construction of this complex, with no restrictive hypotheses, using purely differential geometric means. Here again, the Cartier isomorphism proved to be an essential tool. This complex, which I named the de Rham-Witt complex, is still generating a lot of research today.
Let me share one more memory, where the Cartier isomorphism was to strike yet again. I recall Cartier’s surprise and joy (perhaps tinged with slight frustration?) when, in 1986, I told him that Deligne and I had just discovered that a reinterpretation of his isomorphism in terms of deformation theory led to a purely algebraic proof of one of the fundamental theorems of Hodge theory—namely, the degeneration of the Hodge to de Rham spectral sequence for a complex smooth projective variety. Various questions and conjectures related to this result have been very much studied in recent years.
I much admired Cartier’s mathematical rigor, and the deeply moral rigor of which it was a manifestation, combined with his keen sense of humor and his genuine simplicity. Cartier was also a wonderful storyteller. I could listen to him for hours. He had juicy anecdotes about many subjects, particularly about the Bourbaki group. It was a joy to see him mimic Dieudonné’s Homeric fits of anger. I still hear his deep, warm voice.
Acknowledgement: I heartily thank Nick Katz and Richard Kruel for the English translation of my text “L’isomorphisme de Cartier.”
Photo credit: Jean-François Dars / IHES
