LECTURE II: δ-rings Fix a rime. In this lecture, we discuss some asects of the theory of δ-rings. This theory rovides a good language to talk about rings with a lift of Frobenius modulo. Some of the material discussed below can be found in [1, 2, 3]. 1. Definition and examles To motivate the definition of a δ-ring, note that if A is a commutative ring equied with a ma φ : A A that is a lift for Frobenius on A/, then for each element f A, we have an equation of the form φ(f) = f + δ in A. If A is -torsion free, then δ = δ(f) is uniquely determined by the receding formula, so we can regard δ( ) as an endomorhism of the set A. Moreover, the fact that φ( ) is a ring homomorhism can be encoded in terms of the behaviour of δ( ) under addition and multilication. If A is not necessarily -torsionfree, it is better record δ( ) instead of φ( ) as δ( ) records why φ( ) is a lift of Frobenius. This motivates the following: Definition 1.1 (Joyal). A δ-ring is a air (A, δ) where A is a commutative ring δ : A A is a ma of sets with δ(0) = δ(1) = 0, and satisfying the following two identities and δ(xy) = x δ(y) + y δ(x) + δ(x)δ(y) δ(x + y) = δ(x) + δ(y) + x + y (x + y) 1 = δ(x) + δ(y) i=1 1 ( ) x i y i. i There is an evident category of δ-rings. If the δ-structure on a ring A is clear from context, we often suress it from the notation and simly call A a δ-ring. In the literature, a δ-structure is often called a -derivation. Before giving examles, let us give the obligatory lemma justifying the revious discussion. Lemma 1.2. Let A be a commutative ring. (1) If δ : A A rovides a δ-structure on A, then the ma φ : A A given by φ(f) = f +δ(f) defines an endomorhism of A that is a lift of the Frobenius on A/. (2) When A is -torsionfree, the construction in (1) gives a bijective corresondence between δ-structures on A and Frobenius lifts on A. From now on, given a δ-ring A, we usually write φ : A A for the associated Frobenius ma. Note that the multilicative identity for δ can be written asymmetrically as This form is often convenient in comutations. δ(xy) = φ(x)δ(y) + y δ(x). (1) Proof. It is elementary to rove (1), i.e., to check φ is a ring homomorhism. Let us give the roof for additivity. φ(f + g) = (f + g) + δ(f + g) = (f + g) + δ(f) + δ(g) + f + g (f + g) = φ(f) + φ(g). 1
For (2), as A is -torsionfree, the formula φ(f) = f + δ uniquely defines δ = δ(f) given φ, so it suffices to show that the function δ( ) defined this way satisfies the relevant identities, which is easy to do by hand. The following lemma is very useful in comutations. Lemma 1.3. Let A be a δ-ring. Then φ : A A is a δ-ma, i.e., φ(δ(x)) = δ(φ(x)) for any x A. Proof. We give the roof in the -torsionfree case; the general case can be reduced to this one by Lemma 2.5 below. When A is -torsionfree, we have δ(x) = φ(x) x, so writing out δ(φ(x)) and using that φ is a ring homomorhism gives the required identity. Examle 1.4. As a -torsionfree ring A with a lift φ of Frobenius is a δ-ring, we obtain many easy examles such as: (1) The ring Z with φ being the identity ma. Exlicitly, we have δ(n) = n n. In fact, it is not difficult to see that this is the initial object in the category of δ-rings: the identities on δ in a δ-ring A force the ma Z A to be comatible with δ. (2) The ring Z[x] with φ determined by φ(x) = x + g(x) for any g(x) Z[x]. (3) For any erfect field k of characteristic, the ring W (k) of Witt vectors of k with φ being the standard lift of Frobenius. In this case, note that there is a unique lift of Frobenius, so W (k) admits only one δ-structure. (4) If A is a Z[1/]-algebra, then any endomorhism φ of A rovides a δ structure on A as the condition that φ lift Frobenius on A/ is vacuously true. It is slightly non-trivial to give examles of δ-rings with -torsion, but they do exist. We shall later give a systematic source of examles via the Witt vector construction. For now, we simly mention one examle: (5) There is a unique δ-structure on the ring Z[x]/(x, x) such that δ(x) = 0. The next lemma shows that there are no -ower torsion δ-rings and its roof justifies the terminology -derivation for the δ oerator: it lowers -adic order of vanishing by 1. Lemma 1.5. There is no nonzero δ-ring A such that n = 0 in A for some n 1. Proof. Assume such a δ-ring A exists. Then A is a Z () -algebra. We shall check the following: ( ) For any u Z () and m 1, we have δ(m u) = m 1 v for some v Z (). This imlies the lemma by induction: if m = 0 in A, then δ m ( m ), which is a unit in A by ( ), is also δ m (0) = 0, whence A = 0. To rove ( ), we first observe that ( ) holds true when u = 1 simly because φ is identity on Z: δ( m ) = φ(m ) m = m m = m 1 (1 m m ), which has the required form. For the general form of ( ), using (1) and the fact that φ must be the identity on owers of, we get δ( m u) = m δ(u) + u δ( m ) = m δ(u) + m 1 u w, where w is a unit in Z () by the revious case. Simlifying, this gives δ( m u) = m 1 (u w + δ(u)). So we must show that v = u w + δ(u) is in Z (). construction, while δ(u) lies in the Jacobson radical. 2 But this is clear: u and w are units by
Remark 1.6. Secifying a δ-structure on a ring A is the same as secifying the structure of a -tyical λ-ring. We do not elaborate on this here, but it motivates the following terminology: an element x A has rank 1 if δ(x) = 0. In this case, we have φ(x) = x. 2. The category of δ-rings We want to exlain some basic constructions with δ-rings. alternate ersective on the δ-ring structure. For this, it is useful to have an Construction 2.1 (The truncated Witt vector functor). For any ring A, the ring W 2 (A) of - tyical length 2 Witt vectors is defined as follows: we have W 2 (A) = A A as sets, and addition and multilication are defined via (x, y) + (z, w) := (x + z, y + w + x + z (x + z) ) and (x, y) (z, w) = (xz, x w + z y + yw). Ignoring the second comonent gives a ring homomorhism ɛ : W 2 (A) A. It is immediate from the definitions that secifying a δ-structure on A is the same as secifying a ring ma w : A W 2 (A) such that ɛ w = id: the corresondence attaches the ma w(x) = (x, δ(x)) to a δ-structure δ : A A on A. Remark 2.2. If A is a -torsionfree ring, then W 2 (A) can also be defined as the fibre roduct of the canonical ma A can A/ with the ma A can A/ φ A/, where φ is the Frobenius. Exlicitly, this identification is given by sending (x, y) W 2 (A) to the air (x + y, x) A A, noting that x + y and φ(x) agree in A/. We may thus view Sec(W 2 (A)) as obtained by glueing two coies of Sec(A) using the Frobenius on Sec(A/) Sec(A). Note that it is evident from this interretation that secifying a ma A W 2 (A) slitting the rojection down to A is the same as secifying a Frobenius lift on A. Lemma 2.3 (Limits and colimits). The category of δ-rings has all limits and colimits, and these commute with the forgetful functor to commutative rings. Proof. Fix a diagram {A i } of δ-rings. It is easy to see that there is a unique δ-structure on lim i A i comatible with the δ-structures on each A i via the rojection. For colimits, we use the descrition via the truncated Witt vectors. Given a diagram {A i } as before, the mas A i W 2 (A i ) from Construction 2.1 encoding the δ-structure are comatible in i. Taking colimits, we get a ma colim A i colim i W 2 (A i ). Comosing with the natural ma colim i W 2 (A i ) W 2 (colim i A) coming from functoriality of W 2 ( ), we get a ma colim i A i W 2 (colim i A i ). It is easy to see that comosing this ma with the rojection W 2 (colim i A i ) colim i A i gives the identity ma, so colim i A i acquires a δ-structure. One checks that this construction rovides the desired colimit. Remark 2.4. Combining Lemma 2.3 with the adjoint functor theorem, we learn that the forgetful functor from δ-rings to commutative rings has both a left adjoint and a right adjoint. The left adjoint rovides a notion of a free δ-ring Z{S} on a set S: aly the left adjoint to the olynomial ring Z[{x s } s S ]. The right adjoint is given by the Witt vector functor [3] (but we won t be using this fact in any essential way). In articular, for any commutative ring A, the ring W (A) of Witt vectors of A is naturally a δ-ring; this rovides many examles of δ-rings containing -torsion elements as W (A) for any non-reduced ring A of characteristic contains -torsion. Lemma 2.5 (Free δ-rings). The free δ-ring Z{x} on a variable x is the olynomial ring Z[x 0, x 1, x 2,...] with x = x 0 and δ(x i ) = x i+1. In articular, for any set S, the δ-ring Z{S} is -torsionfree. Proof. Consider the ring A := Z[x 0, x 1, x 2,...]. This ring admits a Frobenius lift φ given by φ(x i ) = x i +x i+1. As A is -torsionfree, this Frobenius lift has a unique associated δ-structure determined by δ(x i ) = x i+1. This shows that the object in the statement of the lemma is well-defined. To 3
identify A with the free δ-ring on x = x 0, we check the universal roerty. Fix a δ-ring S with some f S. Then there is a unique ma A S of commutative rings determined by x i δ i (f). Using the identities describing the behaviour of δ under multilication and addition on both A and S, it follows that A S is a ma of δ-rings. This ma carries x to f by construction, and is clearly the unique ma with this roerty. Remark 2.6. Using the existence of ushouts and free δ-rings, one can construct more δ-rings using generators and relations. For examle, there exists a free δ-ring Z{x, y}/(x 2 + y 3 + xy) δ on two variables x and y satisfying the equation x 2 + y 3 + xy = 0. It can be constructed as a ushout Z{z} z x 2 +y 3 +xy Z{x, y} z 0 Z Z{x, y}/(x 2 + y 3 + xy) δ, where we recall that ushouts in δ-rings are comuted on underlying rings. As free δ-rings are large, it can be rather tricky to analyse these ushouts. For examle, is the ushout above the 0 ring modulo? Let us next record the stability of δ-structures under some natural ring-theoretic oerations. Lemma 2.7 (Localizations of δ-rings). Fix a δ-ring A and S A a multilicative subset with φ(s) S. There is a unique δ-structure on the localization S 1 A comatible with the one on A. Proof. Assume first that A is -torsionfree. Then S 1 A is also -torsionfree. Since φ(s) S, the ma φ : A A extends uniquely to a ma ψ : S 1 A S 1 A. As φ lifts Frobenius on A/, ψ must lift Frobenius on S 1 A/, so it follows that ψ is associated to a δ-structure on S 1 A. The uniqueness of this structure and comatibility with the one on A is clear. In general, one chooses a surjection F A where F is free δ-ring. The reimage T F of S A is a multilicative subset such that φ(t ) T. The receding aragrah then gives a unique δ-structure on T 1 F comatible with the one on F. Base changing along F A then gives the desired δ-structure on S 1 A T 1 A as ushouts of δ-rings are comuted on underlying rings. Exercise 2.8. Let A be a δ-ring. (1) Assume that x A admits a n -th root. Then δ(x) n A. Deduce that if A is -adically searated, then any element x that admits n -th roots for all n 1 must have rank 1. (Hint: reduce to the free case and translate to a question about φ(x)).) (2) (Comletions) Fix a finitely generated ideal I A that contains. Then the I-adic comletion of A admits a unique δ-structure. It follows that Z with the δ-structure rescribed by requiring φ = id is the initial object in the category of -adically comlete δ-rings. Lemma 2.9 (Étale extensions of δ-rings). Fix a ma A B of -adically comlete and - torsionfree rings. Assume A is equied with a δ-structure and that A B is étale modulo. Then B has a unique δ-structure comatible with the one on A. Using the étale localization roerty of the Witt vectors (which is not too difficult in the - adically nilotent or comlete cases) and the interretation of δ-structures in Construction 2.1, one can dro the -torsionfreeness hyothesis in the above lemma (Rezk). Proof. As both A and B are -torsionfree, it suffices to show that B admits a unique Frobenius lift comatible with the one on A. By -adic comleteness, it suffices to do this modulo n for all 4
n. For n = 1, this is simly the well-known statement that the ushout of Frobenius on A is the Frobenius on B via the relative Frobenius B (1) := B A,F A B for A B. For larger n, one argues using the toological invariance of the étale site. Lemma 2.10 (Quotients of δ-rings). Fix a δ-ring A. Let I A be an ideal such that δ(i) I. Then A/I admits a unique δ-structure comatible with the one on A. Proof. It suffices to show that for x A and ɛ I, we have δ(x + ɛ) δ(x) mod I. This follows from the additivity formula for δ. 3. Perfect δ-rings The following class of δ-rings will be imortant for relating δ-rings to erfectoid rings. Definition 3.1. A δ-ring A is called erfect if the Frobenius φ : A A is an isomorhism. Our goal is to classify such rings as follows. Proosition 3.2 (Perfect δ-rings = erfect rings). The following categories are equivalent: (1) The category of erfect δ-rings that are -adically comlete. (2) The category of -adically comlete and -torsionfree rings that are erfect modulo. (3) The category of erfect rings of characteristic. The functor from (1) to (2) is the forgetful functor; the functor from (2) to (3) is A A/; the functor from (3) to (1) is A W (A). In other words, every -adically comlete erfect δ-ring has the form W (R) (with its natural Frobenius) for a erfect F -algebra R. In fact, the roof below does not use the definition of the Witt vector functor W ( ), and rovides an alternative way to think about it on erfect rings. One of the two main ingredients in the roof of Proosition 3.2 is the following. Lemma 3.3. Let A be a δ-ring and let x A with x = 0. Then φ(x) = 0. In articular, if φ is injective, then A is -torsionfree. Proof. We trivially have φ(x) = 0 in A[1/], so we may assume that A is a Z () -algebra. Alying δ to x = 0 shows that δ(x) + φ(x)δ() = 0. As δ() is a unit in Z (), it suffices to show that δ(x) = 0. But we have δ(x) = 1 δ(x) = 1 (φ(x) x ) = 2( φ(x) (x)x 1 ) = 0, where the last equality follows as x = 0. Remark 3.4. One can show that secifying a δ-structure on a ring A is the same as secifying a derived Frobenius lift on A, i.e., if A := A L Z Z/ denotes the derived mod reduction on A, then secifying a δ-structure on A is equivalent to giving an endomorhism φ : A A together with a homotoy between the comosite A φ A can A and the comosite A can A Frob A. To see this, one must show that W 2 (A) can be described as the fibre roduct of A can A Frob can A. This holds true (Remark 2.2) when A is -torsionfree, and the general case follows by left Kan extensions. (Details omitted) Using this interretation, Lemma 3.3 has a slightly more concetual interretation: as the - torsion of A identifies with π 1 (A), the lemma follows from the fact that Frobenius acts trivially on the higher homotoy grous of a simlicial commutative ring. The other ingredient is the following well-known lemma. Lemma 3.5. Let A be a erfect F -algebra. Then the cotangent comlex L A/F vanishes. Consequently, the following categories are equivalent: 5
(1) Perfect F -algebras. (2) For fixed n 1, the category of flat Z/ n -algebras à with Ã/ being erfect. (3) -adically comlete and -torsionfree Z -algebras à with Ã/ erfect. There are obvious functors in the direction (3) (2) and (2) (1). To go from (1) to (3), one may exlicitly use the Witt vector functor A W (A), though we shall not need its recise structure. Proof. The Frobenius endomorhism of any F -algebra A induces the 0 ma on its cotangent comlex: this is clear for olynomial F -algebras, and thus follows in general as Frobenius is functorial (and because there is a functorial simlicial resolution of any F -algebra by olynomial algebras). On the other hand, if A is erfect, then Frobenius is an isomorhism, so it must act as an isomorhism on the cotangent comlex as well. Combining these two observations shows that L A/F 0. The equivalence of (1) and (2) follows from standard relations between the cotangent comlex and deformation theory. The equivalence of (2) and (3) follows as the category of -adically comlete and -torsionfree Z -algebras can be described as the inverse limit of the categories of flat Z/ n -algebras. Proof of Proosition 3.2. Lemma 3.3 ensures that the forgetful functor goes from (1) to (2). The equivalence of (2) and (3) comes from Lemma 3.5. To get from (3) to (1), fix a erfect ring A, and let à denote the corresonding object in (2), so à is a -adically comlete and -torsionfree ring. As A à is a functor, the Frobenius on A lifts to a unique automorhism of Ã, so à comes equied with a Frobenius lift. As à is -torsionfree, this defines the δ-structure on Ã, giving an object in (1). Using the uniqueness of lifts, it is easy to check that these constructions rovide mutually inverse equivalences. Using Proosition 3.2, we can give a concetual construction of the Teichmuller ma R W (R) and the Teichmuller exansion of an element f W (R) for R erfect. Construction 3.6 (The Teichmuller exansion). Let R be a erfect F -algebra, and let W (R) be its ring of Witt vectors. Using the characterization of the latter as the unique -adically comlete and -torsionfree ring lifting R, let us describe a normal form for elements of W (R). First, we show that the rojection W (R) R has a unique multilicative section R W (R) denoted x [x]. It is enough to define such a section for W (R)/ n R. For this, given x R, choose y W (R)/ n lifting x 1/n R. Using the elementary observation that if a = b mod k then a = b mod k+1 (in any commutative ring), it follows that y n W (R)/ n is well-defined (i.e., indeendent of choices) and lifts x R. We set [x] = y; the multilicativity of x [x] is immediate from the construction. For uniqueness, consider two multilicative lifts [ ], [ ] : R W (R)/ n. Then for any a W (R), we can write [a] = [a] + b for some b W (R)/ n. Raising to the n -th ower and using multilicativity then shows that [a n ] = [a n ] in W (R)/ n. As the -ower ma on R is bijective, it follows that [x] = [x] for all x R, as wanted. Now given any f W (R), if we write f R for its image, then f = [f] mod, so we can write f = [f] + f 1 for a unique f 1 W (R) (where uniqueness of f 1 is due to -torsionfreeness of W (R)). Alying the same reasoning to f 1 and continuing, we find that f admits a unique -adic exansion i=0 [a i] i, called the Teichmuller exansion of f. Exercise 3.7. Let R be a erfect F -algebra. Show that f W (R) has rank 1 exactly when f = [a] for some a R. (Hint: first show that the Frobenius lift φ : W (R) W (R) is simly given by i=0 [a i] i i=0 [a i ]i.) References [1] J. Borger, Course on Witt vectors, lambda-rings, and arithmetic jet saces, available at htts://maths-eole. anu.edu.au/~borger/classes/coenhagen-2016/index.html 6
[2] A. Buium, Arithmetic analogues of derivations [3] A. Joyal, δ-anneaux et vecteurs de Witt [4] C. Rezk, Étale extensions of λ-rings 7