Cohomological theory of crystals over function fields.

*(English)*Zbl 1186.14002
EMS Tracts in Mathematics 9. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-074-6/hbk). viii, 187 p. (2009).

For schemes in positive characteristic \(p>0\) this books develops a new characteristic \(p\)-valued cohomological theory, the theory of \(A\)-crystals, and applies it to give a purely algebraic proof of a conjecture of Goss on the rationality of \(L\)-functions arising in the arithmetic of function fields.

In the late 1940s, Weil posed the challenge to create a cohomology theory for algebraic varieties \(X\) over a finite field \(k\). Such a theory should provide a tool for proving his conjectures on the Zeta-functions of such \(X\), namely its rationality (first part), the existence of a functional equation satisfied by it (second part), and finally certain estimates on its poles and zeroes (third part).

The first significant progress towards the Weil conjecture came, however, from another approach by Dwork, who resolved the first part and some cases of the second part by \(p\)-adic analytic methods. Only later Grothendieck, Deligne et. al. gave cohomological proofs of the full conjectures of Weil along the lines he had proposed. Key ingredients in these proofs are the cohomological theory of \(\ell\)-adic étale sheaves together with the Lefschetz trace formula.

Drinfeld initiated an arithmetic theory for objects over global fields of positive characteristic \(p\), called \(A\)-modules (with reference to a fixed Dedekind domain \(A\) that is finitely generated over the field \({\mathbb F}_p\) with \(p\) elements). Later on he introduced more general objects: elliptic sheaves, shtukas, and then Anderson created \(t\)-motives, which generalize to \(A\)-motives. The category of \(A\)-motives contains (via a contravariant embedding) that of Drinfeld \(A\)-modules, but has the advantage over it of having all the standard operations from linear algebra.

These \(A\)-motives bear many analogies to abelian varieties. For any place \(v\) of \(A\) one associates the \(v\)-adic Tate module to an \(A\)-motive. It carries a continuous action of the absolute Galois group of the finite base field \(k\). This Galois representation is completely described by the action of the Frobenius automorphism. Its main invariant is therefore the dual characteristic polynomial of Frobenius, an element in \(1+tA_v[t]\). If, more generally, we are given a family of \(A\)-motives over a base scheme \(X\) of finite type over \({\mathbb F}_p\) then following Weil we may form a suitable product, over the closed points of \(X\), of the above dual characteristic polynomials (suitably stretched) of Frobenius in the fibres, to obtain an \(L\)-function, a priori an element in \(1+tA[[t]]\). Inspired by Weil’s conjectures, Goss conjectured that such an \(L\)-function should be rational as well.

This was proved in 1996 by Taguchi and Wan by a method inspired by Dwork’s. With the paradigm of the development around the Weil conjectures, the main motivation for the authors of the present book was to develop a set of algebro geometric and cohomological tools to give a purely algebraic proof of Goss’ rationality conjecture. Namely, they develop a cohomological theory of so called \(A\)-crystals which has functors \(f^*\), \(\otimes^{\mathbb L}\), \({\mathbb R}f_*\) for proper \(f\), and \({\mathbb R}f_!\) for compactifiable \(f\), and for which they can prove, as the central technical result, the trace formula.

The definition of an \(A\)-crystal is as follows. Let as before \(X\) be an algebraic variety over a finite field \(k\), but now let \(A\) denote an arbitrary localization of a finitely generated \(k\)-algebra. Write \(C=\text{Spec}(A)\). A coherent \(\tau\)-sheaf over \(A\) on \(X\) is a pair \(\underline{\mathcal F}=({\mathcal F},\tau_{{\mathcal F}})\) consisting of a coherent sheaf \({\mathcal F}\) on \(X\times C\) and an \({\mathcal O}_{X\times C}\)-linear homomorphism \[ (\sigma\times\text{id})^*{\mathcal F}\overset{\tau_{\mathcal F}}{\longrightarrow}{\mathcal F}. \] With obvious morphisms one gets an abelian \(A\)-linear category \(\mathbf{Coh}_{\tau}(X,A)\). A coherent \(\tau\)-sheaf \(\underline{\mathcal F}=({\mathcal F},\tau_{{\mathcal F}})\) is called nilpotent if the iterated homomorphism \(\tau^n_{{\mathcal F}}:(\sigma^n\times\text{id})^*{\mathcal F}\to{\mathcal F}\) vanishes for some \(n>>0\). A homomorphism of coherent \(\tau\)-sheaves is called a nil-isomorphism if both its kernel and its cokernel are nilpotent. And now: The category \(\mathbf{Crys}_{\tau}(X,A)\) of \(A\)-crystals on \(X\) is the localization of \(\mathbf{Coh}_{\tau}(X,A)\) at the multiplicative system of nil-isomorphisms.

The point of inverting nil-isomorphisms is to allow the definition of an extension by zero functor \(j_!\). As the autors point out, the effect is that crystals behave more like constructible sheaves than like coherent sheaves. They construct a derived category of crystals and derived functors as indicated above. They associate what they call crystalline \(L\)-functions to complexes of crystals of finite Tor dimension, and for them they prove the relevant trace formula:

Theorem. Let \(f:Y\to X\) be a morphism of schemes of finite type over \(k\). Suppose that \(A\) is an integral domain that is finitely generated over \(k\). Then for any complex \(\underline{\mathcal F}^{\bullet}\) of finite Tor dimension of crystals on \(Y\), one has \[ L^{\text{crys}}(Y,\underline{\mathcal F}^{\bullet},t)=L^{\text{crys}}(X,{\mathbb R}f_!\underline{\mathcal F}^{\bullet},t). \] Using this, a new proof of Goss’ rationality conjecture is obtained. However, this book visibly contains very much more than just a new prove of this conjecture.

If \(A\) is finite, the \(\tau_{{\mathcal F}}\)-invariant sections of a coherent \(\tau\)-sheaf form a constructible étale sheaf of \(A\)-modules on \(X\), and this assignment passes to a functor \[ \mathbf{Crys}_{\tau}(X,A)\longrightarrow \text{Ét}_c(X,A). \] It follows from an old theorem of Katz that this is an equivalence of categories.

For smooth schemes \(X\), a theory similar to the one presented in this book was developed independently by Emerton and Kisin, emphasizing rather the indicated relation with \(p\)-adic étale cohomology, culminating in a Riemann-Hilbert type correspondence.

A prerequisite to reading this book is a good working knowledge in algebraic geometry as well as familiarity with homological algebra and derived categories (although, for the convenience of the reader, many or all the required categorial concepts are assembled in the second section of the book). From that point on it is largely self contained.

The book can be warmly recommended to anyone interested or working in the modern arithmetic of function fields. It is written with great didactical care. All ten chapters and many of the sections and subsections begin with a helpful motivating text. Examples are included.

The reviewer did not spot a single misprint. The layout is very attractive, it is a pleasure to have this book in hands.

In the late 1940s, Weil posed the challenge to create a cohomology theory for algebraic varieties \(X\) over a finite field \(k\). Such a theory should provide a tool for proving his conjectures on the Zeta-functions of such \(X\), namely its rationality (first part), the existence of a functional equation satisfied by it (second part), and finally certain estimates on its poles and zeroes (third part).

The first significant progress towards the Weil conjecture came, however, from another approach by Dwork, who resolved the first part and some cases of the second part by \(p\)-adic analytic methods. Only later Grothendieck, Deligne et. al. gave cohomological proofs of the full conjectures of Weil along the lines he had proposed. Key ingredients in these proofs are the cohomological theory of \(\ell\)-adic étale sheaves together with the Lefschetz trace formula.

Drinfeld initiated an arithmetic theory for objects over global fields of positive characteristic \(p\), called \(A\)-modules (with reference to a fixed Dedekind domain \(A\) that is finitely generated over the field \({\mathbb F}_p\) with \(p\) elements). Later on he introduced more general objects: elliptic sheaves, shtukas, and then Anderson created \(t\)-motives, which generalize to \(A\)-motives. The category of \(A\)-motives contains (via a contravariant embedding) that of Drinfeld \(A\)-modules, but has the advantage over it of having all the standard operations from linear algebra.

These \(A\)-motives bear many analogies to abelian varieties. For any place \(v\) of \(A\) one associates the \(v\)-adic Tate module to an \(A\)-motive. It carries a continuous action of the absolute Galois group of the finite base field \(k\). This Galois representation is completely described by the action of the Frobenius automorphism. Its main invariant is therefore the dual characteristic polynomial of Frobenius, an element in \(1+tA_v[t]\). If, more generally, we are given a family of \(A\)-motives over a base scheme \(X\) of finite type over \({\mathbb F}_p\) then following Weil we may form a suitable product, over the closed points of \(X\), of the above dual characteristic polynomials (suitably stretched) of Frobenius in the fibres, to obtain an \(L\)-function, a priori an element in \(1+tA[[t]]\). Inspired by Weil’s conjectures, Goss conjectured that such an \(L\)-function should be rational as well.

This was proved in 1996 by Taguchi and Wan by a method inspired by Dwork’s. With the paradigm of the development around the Weil conjectures, the main motivation for the authors of the present book was to develop a set of algebro geometric and cohomological tools to give a purely algebraic proof of Goss’ rationality conjecture. Namely, they develop a cohomological theory of so called \(A\)-crystals which has functors \(f^*\), \(\otimes^{\mathbb L}\), \({\mathbb R}f_*\) for proper \(f\), and \({\mathbb R}f_!\) for compactifiable \(f\), and for which they can prove, as the central technical result, the trace formula.

The definition of an \(A\)-crystal is as follows. Let as before \(X\) be an algebraic variety over a finite field \(k\), but now let \(A\) denote an arbitrary localization of a finitely generated \(k\)-algebra. Write \(C=\text{Spec}(A)\). A coherent \(\tau\)-sheaf over \(A\) on \(X\) is a pair \(\underline{\mathcal F}=({\mathcal F},\tau_{{\mathcal F}})\) consisting of a coherent sheaf \({\mathcal F}\) on \(X\times C\) and an \({\mathcal O}_{X\times C}\)-linear homomorphism \[ (\sigma\times\text{id})^*{\mathcal F}\overset{\tau_{\mathcal F}}{\longrightarrow}{\mathcal F}. \] With obvious morphisms one gets an abelian \(A\)-linear category \(\mathbf{Coh}_{\tau}(X,A)\). A coherent \(\tau\)-sheaf \(\underline{\mathcal F}=({\mathcal F},\tau_{{\mathcal F}})\) is called nilpotent if the iterated homomorphism \(\tau^n_{{\mathcal F}}:(\sigma^n\times\text{id})^*{\mathcal F}\to{\mathcal F}\) vanishes for some \(n>>0\). A homomorphism of coherent \(\tau\)-sheaves is called a nil-isomorphism if both its kernel and its cokernel are nilpotent. And now: The category \(\mathbf{Crys}_{\tau}(X,A)\) of \(A\)-crystals on \(X\) is the localization of \(\mathbf{Coh}_{\tau}(X,A)\) at the multiplicative system of nil-isomorphisms.

The point of inverting nil-isomorphisms is to allow the definition of an extension by zero functor \(j_!\). As the autors point out, the effect is that crystals behave more like constructible sheaves than like coherent sheaves. They construct a derived category of crystals and derived functors as indicated above. They associate what they call crystalline \(L\)-functions to complexes of crystals of finite Tor dimension, and for them they prove the relevant trace formula:

Theorem. Let \(f:Y\to X\) be a morphism of schemes of finite type over \(k\). Suppose that \(A\) is an integral domain that is finitely generated over \(k\). Then for any complex \(\underline{\mathcal F}^{\bullet}\) of finite Tor dimension of crystals on \(Y\), one has \[ L^{\text{crys}}(Y,\underline{\mathcal F}^{\bullet},t)=L^{\text{crys}}(X,{\mathbb R}f_!\underline{\mathcal F}^{\bullet},t). \] Using this, a new proof of Goss’ rationality conjecture is obtained. However, this book visibly contains very much more than just a new prove of this conjecture.

If \(A\) is finite, the \(\tau_{{\mathcal F}}\)-invariant sections of a coherent \(\tau\)-sheaf form a constructible étale sheaf of \(A\)-modules on \(X\), and this assignment passes to a functor \[ \mathbf{Crys}_{\tau}(X,A)\longrightarrow \text{Ét}_c(X,A). \] It follows from an old theorem of Katz that this is an equivalence of categories.

For smooth schemes \(X\), a theory similar to the one presented in this book was developed independently by Emerton and Kisin, emphasizing rather the indicated relation with \(p\)-adic étale cohomology, culminating in a Riemann-Hilbert type correspondence.

A prerequisite to reading this book is a good working knowledge in algebraic geometry as well as familiarity with homological algebra and derived categories (although, for the convenience of the reader, many or all the required categorial concepts are assembled in the second section of the book). From that point on it is largely self contained.

The book can be warmly recommended to anyone interested or working in the modern arithmetic of function fields. It is written with great didactical care. All ten chapters and many of the sections and subsections begin with a helpful motivating text. Examples are included.

The reviewer did not spot a single misprint. The layout is very attractive, it is a pleasure to have this book in hands.

Reviewer: Elmar Große-Klönne (Berlin)