The ´ Etale Spectrum of a Commutative Ring

Sol_ApOq

u //Sol_A0pOq

v

Sol_BpO¹q ^{ρ //}Sol_ApO¹q ^//Sol_A0pO¹q.

SinceA0 is isomorphic to the polynomial ring Zrx1, . . . , xns, we can identify v with the map αⁿ:OⁿÑO¹ⁿ. Our assumption that α is an effective epimorphism now guarantees thatv is an effetive epimorphism. Sinceα is local, the right square in this diagram is a pullback, so that u is also an effective epimorphism. Our assumption that O is strictly Henselian guarantees that ρ is an effective epimorphism. It now follows by inspection of the above diagram thatρ is an effective epimorphism as desired.

1.2.3 The ´Etale Spectrum of a Commutative Ring

Let R be a commutative ring and let SpecR“ pX,OXqbe the associated affine scheme.

Then we can regard OX as a commutative ring object of the topos ShvSetpXq. This commutative ring object is local, but is usually not strictly Henselian. To remedy this, one can replace the Zariski spectrum SpecR by a slightly more sophisticated object, which we will refer to as the´etale spectrum ofR.

Definition 1.2.3.1 (The ´Etale Topos of a Commutative Ring). Let R be a commutative ring. We let CAlg^♥_R denote the category of commutative R-algebras, and CAlg^´et_R the full subcategory of CAlg_R spanned by the ´etale R-algebras. The (opposite of) the 8-category CAlg^´et_Ris equipped with a Grothendieck topology, where a family of mapstAÑA_αugenerates a covering sieve if and only if there exists some finite collection of indices α1, α2, . . . , αn

such that the mapAÑś

1ďiďnAαi is faithfully flat (this is an immediate consequence of Proposition A.3.2.1). We will refer to this Grothendieck topology as the´etale topology.

We letShv_SetpCAlg^´et_Rq denote the full subcategory of FunpCAlg^´et_R,Setq spanned by those functors which are sheaves with respect to the ´etale topology. We will refer toShv_SetpCAlg^´et_Rq as the´etale topos of R.

Proposition 1.2.3.2. Let R be a commutative ring, and let O : CAlg^´et_R Ñ Set be the forgetful functor (which assigns to each ´etale R-algebraA its underlying set). Then O is a sheaf for the ´etale topology, and can therefore be identified with a commutative ring object of the toposShv_SetpCAlg^´et_Rq. Moreover, O is strictly Henselian.

Proof. The assertion that O is an ´etale sheaf is equivalent to the assertion that for every

´

etaleR-algebraR¹ and every faithfully flat ´etale map R¹ ÑR², the diagram R¹ ÑR² ÑR²bR¹R²

1.2. DELIGNE-MUMFORD STACKS 103 is an equalizer in the category of sets (see Proposition A.3.3.1). We now show thatO is strictly Henselian. Suppose we are given a commutative ringAand a faithfully flat ´etale map AÑś

1ďiďnA_i. We wish to show that the induced mapθ:š

1ďiďnSol_AipOq ÑSol_ApOq is an epimorphism in the topos Shv_SetpCAlg^´et_Rq. To prove this, suppose we are given an

´

etale R-algebra R¹ and a point η P Sol_ApOqpR¹q, which we can identify with an algebra homomorphism A Ñ R¹. For 1 ďiď n, let R_i¹ “ A_ibAR¹, and let η_i denote the image of η in Sol_ApOqpR¹_iq. Then ś

1ďiďnR_i¹ is faithfully flat and ´etale over R, and is therefore generates a covering sieve on the objectR¹ P pCAlg^´et_Rq^op. Moreover, each of the points η_i can be lifted to a point η_iPSol_AipOqpR¹_iq, so thatθ is an epimorphism as desired.

Definition 1.2.3.3. LetR be a commutative ring, and let O be as in Proposition 1.2.3.2.

We will denote the ringed topos pShv_SetpCAlg^´et_Rq,Oq by Sp´etR, and refer to it as the´etale spectrum of the commutative ringR.

IfpX,OXqis a ringed topos, we let ΓpX;OXqdenote the commutative ring Hom_Xp1,OXq, where 1 denotes a final object ofX. We will refer to the construction pX,OXq ÞÑΓpX;OXq as theglobal sections functor..

Proposition 1.2.3.4. Let pX,OXq be a ringed topos for which OX is strictly Henselian, and let R be a commutative ring. Then the global sections functor induces an equivalence of categories

Map_1TopsHen

CAlg♥ppX,OXq,Sp´etRq ÑHom_CAlg♥pR,ΓpX;OXqq

(where the set on the right hand side is interpreted as a discrete category, having only identity morphisms).

Corollary 1.2.3.5. The global sections functor 1Top^sHen

CAlg^♥Ñ pCAlg^♥q^op pX,OXq ÞÑΓpX;OXq admits a right adjoint, given on objects byRÞÑSp´etR.

Remark 1.2.3.6. If R is a commutative ring and Sp´etR “ pX,OXq, then the unit map RÑΓpX;OXq is an equivalence. It follows that the construction RÞÑSp´etR determines a fully faithful functor from (the opposite of) the category of commutative rings to the 2-category 1Top^sHen

CAlg^♥.

In particular, we can regard the construction R ÞÑ Sp´etR as a contravariant functor from the category of commutative rings to the 2-category 1Top^sHen_CAlg♥ Ď1Top_CAlg♥.

Remark 1.2.3.7. LetR be a commutative ring, and letpX,OXqdenote the ´etale spectrum Sp´etR. For every ´etaleR-algebraA, leth^A: CAlg^´et_R ÑSet denote the functor corepresented byA, which we regard as an object of the toposX. Then there is a canonical equivalence of ringed topoi

pX_{hA,OX|_h^Aq »Sp´etA.

This can be deduced either from the universal properties of Sp´etR and Sp´etA (Proposition 1.2.3.4), or directly from the construction of the ´etale spectra.

Proof of Proposition 1.2.3.4. Fix a ring homomorphismφ:R ÑΓpX;OXq, and letCdenote the fiber product

Map_1TopsHen

CAlg♥ppX,OXq,Sp´etRq ˆ_Hom

CAlg♥pR,ΓpX;O^Xqqtφu.

We will show that the categoryC is trivial (that is, it is equivalent to the category having a single object and a single morphism).

Write Sp´etR “ pY,OYq. Let 1_X and 1_Y denote final objects of the topoi X and Y, respectively. For every ´etale R-algebra A, leth^APY ĎFunpCAlg^´et_R,Setq denote the functor corepresented byA, so that h^A fits into a pullback diagramσ_A:

h^{A //}

Sol_ApOYq

1_Y ^//Sol_RpOYq.

We also define an objectXAPX so that we have a pullback diagramτA: X_A ^//

Sol_ApOXq

1_X ^//Sol_RpOXq,

where the bottom horizontal map is determined by the ring homomorphism φ : R Ñ Hom_Xp1_X,OXq.

Let f “ pf_˚, αq be a map of ringed topoi from pX,OXq topY,OYq which induces the ring homomorphismφupon passage to global sections. For every ´etaleR-algebraA, the map α determines a natural transformation of diagrams f^˚σ_AÑτ_A, which gives in particular a map νf,A :f^˚h^A Ñ XA. If pf˚, αq belongs to C, then the maps νf,A are isomorphisms (Proposition 1.2.2.12). In particular, if we are given two objects pf_˚, αq and pf_˚¹, α¹q in C,

then for each APCAlg^´et_R there is a unique isomorphism θ_A:f^˚h^A»f^1˚h^A for which the

1.2. DELIGNE-MUMFORD STACKS 105