Grothendieck duality for affine M 0 -schemes.

Grothendieck duality for affine M 0 -schemes. A. Salch March 2011

Outline Classical Grothendieck duality. M 0 -schemes. Derived categories without an abelian category of modules. Computing Lf and Rf and f! for the map from Spec Z to Spec M 0. The homology of the dualizing complex.

Duality for finite-dimensional vector spaces. Let s begin with a stupid observation: given a field k, let fdvect(k) be the category of finite-dimensional k-vector spaces. Then there is an equivalence of categories fdvect(k) fdvect(k) op given by sending a vector space V to its k-linear dual. One wants to know if there is a similar kind of duality for modules over an arbitrary commutative ring, or even more generally, quasicoherent modules over a scheme (or over the structure ring sheaf of an algebraic stack, locally ringed topos,...).

If X f Spec k is a scheme of finite type, then one has functors on the derived categories D + (X ) D + (X ) D + (X ) Lf Rf f! D + (k) D + (k) D + (k) and adjunctions Lf Rf f!.

If X f Spec k is a scheme of finite type, then one has functors on the derived categories D + (X ) D + (X ) D + (X ) Lf Rf f! D + (k) D + (k) D + (k) and adjunctions Lf Rf f!. One defines the dualizing complex associated to f as the (quasi-isomorphism class of the) chain complex of O X -modules f! (k[0]). On a sufficiently restricted category of O X -modules M, the functor hom D + (X )(M[0], f! (k[0])) is a nice duality functor.

The meaning of the adjunction Rf f! is the following isomorphism: Rf R hom OX (C, f! C ) = R hom k (Rf C, C ).

The meaning of the adjunction Rf f! is the following isomorphism: Rf R hom OX (C, f! C ) = R hom k (Rf C, C ). Let s try to construct a Grothendieck duality of this kind for the morphism Spec Z Spec M 0.

By M 0 we mean the initial object {0, 1} in the category of commutative monoids with zero element.

By M 0 we mean the initial object {0, 1} in the category of commutative monoids with zero element. Modules over M 0 are simply pointed sets. There is a natural tensor product of modules over M 0 given by M N = (M N)/(M N), the Cartesian product with the one-point union collapsed to a basepoint. If N is some commutative monoid with zero, i.e., some commutative M 0 -algebra, then a N -module M is simply a pointed set equipped with a unital, associative action map N M M. There is a relative tensor product M N N defined by the same coequalizer sequence as the relative tensor product for modules over rings.

One can base-change a M 0 -module M to a Z-module: Z[M]/( = 0). Similarly, one can base change commutative M 0 -algebras to commutative Z-algebras in a way that respects the module categories, etc.; and given a Z-module, one can forget the addition to get a M 0 -module, and similarly for commutative algebras. These operations define functors: Mod(Z) Mod(Z) f Mod(M 0 ) f Mod(M0 ) and an adjunction f f. We will take seriously Grothendieck s philosophy that one should work with module categories instead of with rings, schemes, etc., and we will treat this adjunction as defining a map Spec Z Spec M 0.

The object M 0 is sometimes taken as a model for F 1, that is, the geometry of topological spaces equipped with structure sheaves which look Zariski-locally like commutative monoids with zero is believed to be a good model for the geometry of F 1 -schemes. We will study M 0, its derived category, and Grothendieck duality for the morphism Spec Z Spec M 0, with the understanding that whatever we learn in this case will probably shed some light on any other model one chooses for F 1 -geometry. So what is D + (M 0 )? The category of M 0 -modules isn t abelian, so one can t form chain complexes of M 0 -modules.

Let R be a commutative ring. Here is a natural (to a topologist s view, at least) description of D + (R): begin with the category smod(r) of simplicial R-modules.

Let R be a commutative ring. Here is a natural (to a topologist s view, at least) description of D + (R): begin with the category smod(r) of simplicial R-modules. By the Dold-Kan theorem, the category smod(r) is equivalent to the category of chain complexes of R-modules concentrated in nonnegative degrees. A morphism of chain complexes is a quasi-isomorphism if and only if the equivalent morphism of simplicial R-modules is a weak equivalence in the projective model structure on smod(r); so Ho(sMod(R)) is the semi-triangulated derived category of chain complexes of R-modules concentrated in nonnegative degrees.

Now to get D + (R) we need to allow desuspensions of object in Ho(sMod(R)). We will do this by formally adjoining desuspensions of objects in smod(r): a bounded-below spectrum object in smod(r) consists of a functor Z Z F smod(r) which preserves finite homotopy limits and such that F (m, n) is contractible if m n and such that F (n, n) is contractible if n << 0. In practical terms: we have a bunch of diagrams F (n, n) 0 0 F (n + 1, n + 1) which are homotopy pullback squares, i.e., F (n, n) 0 F (n + 1, n + 1) is an exact triangle in the semitriangulated category of connective chain complexes of R-modules; so F (n + 1, n + 1) F (n, n)[1].

Write S + (smod(r)) for the category of bounded-below spectrum objects in smod(r). Passing from smod(r) to S + (smod(r)) formally inverts suspension, and Ho(S + (smod(r))) D + (R).

Write S + (smod(r)) for the category of bounded-below spectrum objects in smod(r). Passing from smod(r) to S + (smod(r)) formally inverts suspension, and Ho(S + (smod(r))) D + (R). What if we do this with R = M 0? Then smod(m 0 ) = spsets pssets, the category of pointed simplicial sets, which is Quillen-equivalent (has the same homotopy category) as the category of CW-complexes and CW maps. And when we invert suspension: S + (smod(m 0 )) is the category of bounded-below spectra. So D + (M 0 ) is triangulated-equivalent to the homotopy category of bounded-below spectra.

We have maps Mod(Z) Mod(Z) f Mod(M 0 ) f Mod(M0 ) defined earlier in this talk; what functors do they induce on the derived categories?

We have maps Mod(Z) Mod(Z) f Mod(M 0 ) f Mod(M0 ) defined earlier in this talk; what functors do they induce on the derived categories? D + (Z) D + (Z) Lf D + (M 0 ) Rf D + (M 0 ) Lf sends a spectrum to its singular chain complex, while Rf sends a chain complex C of abelian groups to the generalized Eilenberg-Maclane spectrum HC ; this has the property that π i (HC ) = H i (C ).

There exists a functor D + (Z) f! D + (M 0 ) which is a derived right adjoint to Rf ; its existence can be shown formally by methods of Neeman, or it can be explicitly constructed before passing to the homotopy category (f! (X ) is the function spectrum F (HZ, X ), by EKMM). Let s use the derived adjunction R hom Z (C [i], f! X ) = [Σ i Rf C, X ] to compute the homology groups of the dualizing complex f! M 0 = f! S. This is the Grothendieck duality one gets by pulling back (via f! ) the dualizing object for Spanier-Whitehead duality of spectra to some kind of dualizing complex of abelian groups.

Let C = Z[0]. Then the spectral sequence Ext s Z (Z, H tf! X ) R hom s t Z (C, f! X ) collapses, at E 2, on to the s = 0 line.

Let C = Z[0]. Then the spectral sequence Ext s Z (Z, H tf! X ) R hom s t Z (C, f! X ) collapses, at E 2, on to the s = 0 line. So: H t (f! X ) = Ext s Z (Z, H t f! X ) = [Σ t HZ, X ]. So the homology of the dualizing complex coming from a spectrum X is given by homotopy classes of maps from suspensions of HZ into X.

H t (f! X ) = [Σ t HZ, X ]. A theorem of T.Y. Lin: if X is a finite CW-complex, then [Σ i HZ, X ] = Ext 1 Z (Q, H 1+i(X )).

H t (f! X ) = [Σ t HZ, X ]. A theorem of T.Y. Lin: if X is a finite CW-complex, then [Σ i HZ, X ] = Ext 1 Z (Q, H 1+i(X )). If X S, this formula reads: { H i (f! Ext 1 S) = Z (Q, Z) if i = 1 0 if i 1.

H t (f! X ) = [Σ t HZ, X ]. A theorem of T.Y. Lin: if X is a finite CW-complex, then [Σ i HZ, X ] = Ext 1 Z (Q, H 1+i(X )). If X S, this formula reads: { H i (f! Ext 1 S) = Z (Q, Z) if i = 1 0 if i 1. g If we let g be the morphism Spec F p Spec M 0, then H (g! S) = 0, so the dualizing complex becomes trivial when pulled back to any closed point of Spec Z.

An adelic description. Let NF be the category of number fields, and let Ab be the category of abelian groups. These two functors NF Ab are naturally equivalent: K/Q H 1 (f! (MO K )) K/Q G(A K )\G(A K )/G(K), where G is the additive group scheme. Here MO K is the Moore spectrum of O K, a ring spectrum satisfying H (MO K ) = O K concentrated in degree 0. If K = Q this is the sphere spectrum.

Phantom maps. Notice that π (Σ 1 HZ) is concentrated in degree 1, while π 1 (S) = 0; so the non-nulhomotopic maps Σ 1 HZ S which are responsible for the homology of f! S, these maps are necessarily zero on applying stable homotopy groups. Hence they are phantom maps, non-nulhomotopic maps between spectra which are not detected by stable homotopy groups. These are important objects, and Freyd s generating hypothesis, one of the most important open conjectures in algebraic topology, is the statement that there are no phantom maps between finite CW-complexes. The fact that the homology of the Grothendieck dualizing complex for the morphism Spec Z Spec M 0 is isomorphic to a group of phantom maps, and in turn an adelic double quotient, is rather remarkable.