Formal categories with divided powers and the Čech–Alexander complex

Written by Claude Opus 5.5 (Anthropic) on 22 October 2026. Self-checked by the writing AI. Public domain (CC0).

A pair of points of a scheme \(X\) over \(S\) is a point of \(X\times_SX\); composing pairs, \((x,y)\) followed by \((y,z)\) giving \((x,z)\), makes the scheme of pairs a groupoid over \(X\). In algebra this is the cosimplicial ring \(A\to A\otimes_KA\to A\otimes_KA\otimes_KA\to\cdots\), the nerve of a category object of the opposite of the category of \(K\)-algebras. Infinitesimal versions of such categories are obtained by completing along the augmentation ideal, the ideal of the diagonal; with divided powers in place of ordinary powers one obtains the Čech–Alexander complex of \(A/K\), from which crystalline cohomology is computed. This lesson sets up the framework: pro-algebras and formal categories, divided powers on sheaves of ideals, divided powers on the nerve of a formal category, divided power completions, and divided power envelopes, which commute with tensor products of augmented algebras. The key tool is an extension result: divided powers on the augmentation ideal of an augmented algebra extend along any base change, without flatness. Completions need not be projective systems of affine categories: in characteristic \(0\) the formal additive group is not one, and divided powers on arbitrary formal categories are handled through level representations of pro-objects.

Prerequisites. Décalage, nerves and the homotopy type of a category (internal categories and their nerves, Section 2); De Rham complexes and Lie algebroids (the conventions on sheaves of modules). From the AI-integrated Stacks project: divided powers [Stacks, Tag 07GL, Tag 07GP, Tag 07GQ, Tag 07GY, Tag 07H1, Tag 07H2, Tag 07H5] and divided power envelopes [Tag 07H8, Tag 07HA]; initial and cofinal functors [Tag 09WP, Tag 04E7].

Conventions

\(T\) is a topos and \(K\) a sheaf of commutative rings on \(T\). Rings and algebras are commutative with \(1\), sheaves of rings on \(T\) unless stated otherwise, and \(K\)-algebra means sheaf of commutative \(K\)-algebras. Local sections, tensor products and quotients are as in De Rham complexes and Lie algebroids. An augmented \(A\)-algebra is a \(K\)-algebra \(B\) with \(K\)-algebra maps \(\eta\colon A\to B\) and \(\varepsilon\colon B\to A\) such that \(\varepsilon\eta=\mathrm{id}_A\); its augmentation ideal is \(I_B=\ker\varepsilon\). As an \(A\)-module through \(\eta\), \[ B=\eta(A)\oplus I_B. \tag{0.1} \] For augmented \(A\)-algebras \(B\) and \(C\), \(B\otimes_AC\) is formed with \(\eta_B\) and \(\eta_C\) and augmented by \(\varepsilon_B\otimes\varepsilon_C\); \(\iota_1\colon B\to B\otimes_AC\) and \(\iota_2\colon C\to B\otimes_AC\) are the two maps.

1. Pro-algebras and formal categories

Definition 1.1. Let \(\mathcal C\) be a category. A pro-object of \(\mathcal C\) is a functor \(X\colon I\to\mathcal C\) from a small cofiltered category, written \(\text{“}\varprojlim\text{”}\,X_i\). Morphisms are \[ \begin{aligned}&\operatorname{Hom}(\text{“}\varprojlim\text{”}X_i,\text{“}\varprojlim\text{”}Y_j)\\&\quad=\varprojlim_j\varinjlim_i\operatorname{Hom}(X_i,Y_j),\end{aligned} \] with the evident composition; they form the category \(\operatorname{pro}\text{-}\mathcal C\), which contains \(\mathcal C\) as the constant systems. A pro-object is essentially constant if it is isomorphic to a constant one. A pro-\(K\)-algebra is a pro-object of the category of \(K\)-algebras, a pro-ideal of \(\text{“}\varprojlim\text{”}B_i\) is a compatible family of ideals \(J_i\subseteq B_i\) of a level representation, and the quotient is \(\text{“}\varprojlim\text{”}B_i/J_i\).

Lemma 1.2 (pushouts on levels). Let \(I\) be a small cofiltered category and \(B_i\leftarrow A_i\to C_i\) a diagram of \(K\)-algebras depending functorially on \(i\in I\). Then \(\text{“}\varprojlim\text{”}(B_i\otimes_{A_i}C_i)\), with the maps induced by the coprojections, is a pushout of \(\text{“}\varprojlim\text{”}B_i\leftarrow\text{“}\varprojlim\text{”}A_i\to\text{“}\varprojlim\text{”}C_i\) in the category of pro-\(K\)-algebras.

Proof. Write \(B=\text{“}\varprojlim\text{”}B_i\), and similarly \(A\), \(C\). For a pro-\(K\)-algebra \(Z=\text{“}\varprojlim\text{”}Z_k\) and each \(k\), \[ \begin{aligned}&\varinjlim_i\operatorname{Hom}(B_i\otimes_{A_i}C_i,Z_k)\\&=\varinjlim_i\bigl(h_{B_i}\times_{h_{A_i}}h_{C_i}\bigr)\\&=h_B\times_{h_A}h_C,\end{aligned} \] where \(h_X\) stands for \(\operatorname{Hom}(X,Z_k)\), because filtered colimits of sets commute with fibre products. Taking the limit over \(k\), which commutes with fibre products, gives the universal property. \(\square\)

Definition 1.3. A \(K\)-formal category is an internal category of the opposite of the category of pro-\(K\)-algebras, in the sense of Décalage, nerves and the homotopy type of a category, Section 2, where fibre products of the opposite category are pushouts of pro-algebras. It is affine if its objects of objects and of arrows are essentially constant. It is strict if it is given by a functor \(i\mapsto C_i\) from a small cofiltered category \(I\) to affine \(K\)-categories with constant objects and arrows, the formal category having the objects \(\text{“}\varprojlim\text{”}A_i\), the arrows \(\text{“}\varprojlim\text{”}P_i\) and the structure maps of the \(C_i\); by Lemma 1.2 the fibre products are formed levelwise, so the axioms hold. Not every formal category is strict (Proposition 1.9), and the completions of Section 2 need not be. Divided power structures are defined below levelwise for strict formal categories and through level representations in general (Definition 2.1).

1.4. Affine categories in algebraic terms. An affine \(K\)-category with objects and arrows given by \(K\)-algebras \(A\) and \(P\) amounts to \(K\)-algebra maps \[ \begin{aligned}&d^0,d^1\colon A\to P,\quad\varepsilon\colon P\to A,\\&\delta\colon P\to P\otimes_AP.\end{aligned} \] Here \(d^0\) and \(d^1\) correspond to the target and the source, \(\varepsilon\) to the identities, and \(\delta\) to the composition. In \(P\otimes_AP\) the first factor is an \(A\)-algebra through \(d^0\) and the second through \(d^1\), matching the fibre product over the target on the first factor and the source on the second. The axioms of an internal category become \[ \begin{aligned}&\varepsilon d^0=\varepsilon d^1=\mathrm{id}_A,\\ &\delta d^0=\iota_2d^0,\quad\delta d^1=\iota_1d^1,\\ &(\varepsilon\otimes\mathrm{id})\delta=\mathrm{id}=(\mathrm{id}\otimes\varepsilon)\delta,\\ &(\delta\otimes\mathrm{id})\delta=(\mathrm{id}\otimes\delta)\delta.\end{aligned} \tag{1.1} \] The second line says that a composite has the target of the second arrow and the source of the first; \(\iota_1,\iota_2\colon P\to P\otimes_AP\) are the two factors. In particular \(P\) is an augmented \(A\)-algebra through \(d^0\) and through \(d^1\), with augmentation \(\varepsilon\) and augmentation ideal \(I=\ker\varepsilon\).

Example 1.5. Let \(A\) be a \(K\)-algebra. The chaotic category of \(A/K\) has \(P=A\otimes_KA\), \(d^0(a)=1\otimes a\), \(d^1(a)=a\otimes1\), \(\varepsilon(a\otimes b)=ab\) and \(\delta(a\otimes b)=a\otimes1\otimes b\), through \(P\otimes_AP=A\otimes_KA\otimes_KA\). Its points with values in a \(K\)-algebra \(B\) form the category whose objects are the maps \(A\to B\) and with exactly one arrow between any two objects.

Proposition 1.6 (the nerve). Let \(C\) be an affine \(K\)-category given by \((A,P,d^0,d^1,\varepsilon,\delta)\). Its nerve is the cosimplicial \(K\)-algebra \(\operatorname{Ner}(C)\) with \(\operatorname{Ner}^0(C)=A\) and \(\operatorname{Ner}^n(C)=P\otimes_A\cdots\otimes_AP\) (\(n\) factors), whose cofaces \(d^i\colon\operatorname{Ner}^{n-1}\to\operatorname{Ner}^n\) and codegeneracies \(s^i\colon\operatorname{Ner}^{n+1}\to\operatorname{Ner}^n\) are, for \(x=x_1\otimes\cdots\otimes x_{n-1}\), resp. \(x=x_1\otimes\cdots\otimes x_{n+1}\), \[ \begin{aligned}d^0x&=1\otimes x,\\ d^ix&=x_1\otimes\cdots\otimes\delta x_i\otimes\cdots\\&\qquad\quad(0<i<n),\\ d^nx&=x\otimes1,\end{aligned} \tag{1.2} \] and \(s^i\), \(0\leq i\leq n\), deletes the factor \(x_{i+1}\) and multiplies a neighbour by \(\varepsilon(x_{i+1})\): in the tensor product \(x_i\,d^0(\varepsilon(x_{i+1}))\otimes x_{i+2}=\) \(x_i\otimes d^1(\varepsilon(x_{i+1}))\,x_{i+2}\), and for \(i=0\) or \(i=n\) only one neighbour exists. In degrees \(0\to1\), \(d^0\) and \(d^1\) are the maps of 1.4, and \(s^0\colon P\to A\) is \(\varepsilon\). The augmentation \(\operatorname{Ner}(C)\to A\), induced by the maps \([n]\to[0]\), is \(\varepsilon^{\otimes n}\colon x_1\otimes\cdots\otimes x_n\mapsto\varepsilon(x_1)\cdots\varepsilon(x_n)\), and its kernel, the augmentation ideal \(J\), has the components \[ \begin{aligned}J^n&=\textstyle\sum_{k=1}^nI_k,\\ I_k&=\iota_k(I)\operatorname{Ner}^n(C),\end{aligned} \tag{1.3} \] where \(\iota_k\colon P\to\operatorname{Ner}^n(C)\) is the inclusion of the \(k\)-th factor. A strict formal category has the nerve \(\text{“}\varprojlim\text{”}\operatorname{Ner}(C_i)\) and the augmentation ideal \(\text{“}\varprojlim\text{”}J_i\).

Proof. The faces of the nerve of an internal category drop the first or the last arrow or compose two consecutive arrows, and its degeneracies insert identities (Section 2 of the lesson on décalage). In the opposite category the fibre products are the tensor products of 1.4, and these operations become (1.2): dropping the first arrow is \(y\mapsto1\otimes y\), composing the arrows \(i\) and \(i+1\) applies \(\delta\) to the \(i\)-th factor, and inserting an identity applies \(\varepsilon\). The augmentation is the composite of codegeneracies, hence the product of the \(\varepsilon(x_k)\).

For (1.3), write \(\operatorname{Ner}^n=\operatorname{Ner}^{n-1}\otimes_AP\), where \(\operatorname{Ner}^{n-1}\) is an augmented \(A\)-algebra through its last factor and \(P\) one through \(d^1\). By (0.1) for both factors, \[ \begin{aligned}\operatorname{Ner}^n&=A\oplus J^{n-1}\oplus I\\&\quad\oplus(J^{n-1}\otimes_AI),\end{aligned} \tag{1.4} \] with the augmentation the projection to \(A\). So \(J^n\) is the sum of the last three terms, which is \(\iota_{\leq n-1}(J^{n-1})\operatorname{Ner}^n+I_n\), and induction gives (1.3). \(\square\)

The decomposition (1.4) uses only one \(A\)-module structure on each factor, the one used in the tensor product. In \(P\otimes_AP\otimes_AP\) the middle factor carries both, and \(P=d^0(A)\oplus I\) is not a decomposition of bimodules unless \(d^0=d^1\); this asymmetry returns in the comparison of the normalized nerve with the tensor algebra of \(I\).

Example 1.7. For the chaotic category of Example 1.5, \(\operatorname{Ner}^n=A^{\otimes(n+1)}\) (tensor products over \(K\)), the coface \(d^i\) inserts \(1\) at the place \(i\), and the codegeneracy \(s^i\) multiplies the factors \(i\) and \(i+1\). This is the cosimplicial algebra \(C_{A/K}\); its augmentation ideal in degree \(1\) is the kernel of the multiplication \(A\otimes_KA\to A\), the ideal of the diagonal.

Lemma 1.8 (level representations). Let \(f\colon X\to Y\) be a morphism of pro-objects of a category \(\mathcal C\), with \(X=\text{“}\varprojlim_{i\in I}\text{”}X_i\) and \(Y=\text{“}\varprojlim_{j\in J}\text{”}Y_j\). There are a small cofiltered category \(L\), initial functors \(p\colon L\to I\) and \(q\colon L\to J\) [Tag 09WP], and maps \(f_l\colon X_{p(l)}\to Y_{q(l)}\) natural in \(l\), such that \(X\cong\text{“}\varprojlim_l\text{”}X_{p(l)}\), \(Y\cong\text{“}\varprojlim_l\text{”}Y_{q(l)}\), and \(f\) corresponds to \(\text{“}\varprojlim\text{”}f_l\). Moreover, two maps \(X_{i_1}\to Y_j\) and \(X_{i_2}\to Y_j\) representing the same component of a morphism agree after composing with transition maps from some \(X_{i'}\).

Proof. The last assertion is the definition of the filtered colimits \(\varinjlim_i\operatorname{Hom}(X_i,Y_j)\) of Definition 1.1. Let \(L\) be the category of triples \((i,j,\varphi)\) in which \(\varphi\colon X_i\to Y_j\) represents the component \(f_j\) of \(f\); a morphism \((i',j',\varphi')\to(i,j,\varphi)\) is a pair of arrows \(a\colon i'\to i\), \(b\colon j'\to j\) with \(\varphi X(a)=Y(b)\varphi'\). Let \(p\), \(q\) be the projections and \(f_{(i,j,\varphi)}=\varphi\). If \(\varphi'\) represents \(f_{j'}\), then \(Y(b)\varphi'\) represents \(f_j\), by the naturality of \(f\).

\(L\) is cofiltered. It is nonempty. Given \((i_1,j_1,\varphi_1)\) and \((i_2,j_2,\varphi_2)\), choose \(j\) with arrows \(b_k\colon j\to j_k\) and a representative \(\varphi\colon X_i\to Y_j\) of \(f_j\). Both \(Y(b_k)\varphi\) and \(\varphi_k\) represent \(f_{j_k}\), so there is an \(i'\) with arrows \(c\colon i'\to i\) and \(c_k\colon i'\to i_k\) such that \(Y(b_k)\varphi X(c)=\varphi_kX(c_k)\) for \(k=1,2\): choose such arrows for each \(k\), then a common \(i'\) on which the two arrows to \(i\) agree. Then \((i',j,\varphi X(c))\) maps to both objects. Given two morphisms \((a,b),(a',b')\colon(i',j',\varphi')\to(i,j,\varphi)\), choose \(b''\colon j''\to j'\) with \(bb''=b'b''\), a representative \(\varphi''\colon X_{i''}\to Y_{j''}\) of \(f_{j''}\), and an \(i'''\) with arrows \(c\colon i'''\to i''\) and \(c'\colon i'''\to i'\) such that \(Y(b'')\varphi''X(c)=\varphi'X(c')\), both \(Y(b'')\varphi''\) and \(\varphi'\) representing \(f_{j'}\), and \(ac'=a'c'\). Then \((c',b'')\colon(i''',j'',\varphi''X(c))\to(i',j',\varphi')\) equalizes the two morphisms.

The functor \(p\) is initial. Every \(i\) receives an arrow from some \(p(l)\): given an object \((i_0,j_0,\varphi_0)\), take \(i'\) with arrows \(e\colon i'\to i_0\) and \(i'\to i\), and \(l=(i',j_0,\varphi_0X(e))\). Two arrows \(p(l)\rightrightarrows i\) become equal on some \(l'\to l\): equalize them by \(e\colon i'\to p(l)\) and take \(l'=(i',j_l,\varphi_lX(e))\). Since \(L\) is cofiltered, these two properties give the conditions of [Tag 09WP]. So does \(q\): every \(j\) is some \(q(l)\), and two arrows \(q(l)\rightrightarrows j\) become equal on some \(l'\to l\), by the construction used for two morphisms. Restricting a pro-object along an initial functor gives an isomorphic pro-object: for a constant object \(Z\), \(\varinjlim_l\operatorname{Hom}(X_{p(l)},Z)=\varinjlim_i\operatorname{Hom}(X_i,Z)\), the colimits being over the opposite categories, in which \(p\) is cofinal [Tag 04E7], and morphisms into a pro-object are limits of such Hom sets. Each \(\varphi\) represents the component of \(f\), so \(f\) corresponds to \(\text{“}\varprojlim\text{”}f_l\). \(\square\)

Proposition 1.9 (a formal category that is not strict). Let \(T\) be the punctual topos and \(K\) a field of characteristic \(0\). Let \(\hat{\mathbb G}_a\) be the \(K\)-formal category with objects \(K\), arrows \(P=\text{“}\varprojlim_n\text{”}K[x]/(x^n)\), \(d^0=d^1\) the unit, \(\varepsilon(x)=0\), and the composition \[ \begin{aligned}&\delta\colon P\to P\otimes_KP\\&\quad=\text{“}\varprojlim_{n,m}\text{”}K[x,y]/(x^n,y^m),\end{aligned} \] \(y=1\otimes x\), given by the maps \(K[x]/(x^{n+m-1})\to K[x,y]/(x^n,y^m)\), \(x\mapsto x+y\): the formal additive group. It is not isomorphic to a strict formal category.

The axioms (1.1) hold as for the additive group: \((\varepsilon\otimes\mathrm{id})\delta\) and \((\mathrm{id}\otimes\varepsilon)\delta\) send \(x\) to \(x\), and both sides of the associativity axiom send \(x\) to \(x\otimes1\otimes1+1\otimes x\otimes1+1\otimes1\otimes x\).

Proof. A lemma. Let \(Q\) be a commutative \(K\)-algebra with a \(K\)-algebra map \(\bar\delta\colon Q\to Q\otimes_KQ\), and \(u\in Q\) nilpotent with \(\bar\delta(u)=u\otimes1+1\otimes u\). Then \(u=0\). Indeed, let \(N\geq1\) be least with \(u^N=0\). The elements \(1,u,\ldots,u^{N-1}\) are linearly independent: multiplying a relation \(\sum_kc_ku^k=0\) whose least index with \(c_k\neq0\) is \(k_0\) by \(u^{N-1-k_0}\) gives \(c_{k_0}u^{N-1}=0\). If \(N\geq2\), then \[ 0=\bar\delta(u^N)=\sum_{k=1}^{N-1}\tbinom Nk\,u^k\otimes u^{N-k}, \] the terms \(k=0\) and \(k=N\) vanishing. The \(u^k\otimes u^{N-k}\) are linearly independent in \(Q\otimes_KQ\), and \(\binom N1=N\neq0\) in \(K\), a contradiction. So \(N=1\).

Reduction to monoids. Suppose that \(\hat{\mathbb G}_a\) is isomorphic to the strict formal category of affine categories \(C_i=(A_i,P_i,d^0_i,d^1_i,\varepsilon_i,\delta_i)\), \(i\in I\). The isomorphism \(\text{“}\varprojlim\text{”}A_i\cong K\) of objects is represented by a \(K\)-algebra map \(r\colon A_{i_0}\to K\). Replace \(I\) by the category of arrows \(i\to i_0\), whose forgetful functor to \(I\) is initial [Tag 09WP] (every \(i\) receives an arrow from an object \(i'\) with an arrow to \(i_0\), and two arrows to \(i\) are equalized on a deeper such object), and let \(r_i\colon A_i\to K\) be the composite of \(r\) with the transition map. The inverse isomorphism consists of the structure maps \(K\to A_i\), the only \(K\)-algebra maps; so for every \(i\) there is an arrow \(i'\to i\) such that the transition map \(A_{i'}\to A_i\) is \(r_{i'}\) followed by the structure map. Let \(Q_i\) be the quotient of \(P_i\) by the ideal generated by the \(d^j_i(a)-r_i(a)\), \(a\in A_i\), \(j=0,1\): geometrically, the endomorphisms of the point \(r_i\). The map \(P_i\otimes_{A_i}P_i\to Q_i\otimes_KQ_i\), \(v\otimes w\mapsto\bar v\otimes\bar w\), is defined, and \(\delta_i\) induces \(\bar\delta_i\colon Q_i\to Q_i\otimes_KQ_i\), since \(\delta_id^0_i=\iota_2d^0_i\) and \(\delta_id^1_i=\iota_1d^1_i\) by (1.1). The quotient maps \(q_i\colon P_i\to Q_i\) are compatible with the transition maps and with the \(\delta_i\) and \(\bar\delta_i\). For an arrow \(i'\to i\) as above, \(d^j_i\) sends the image of \(a\in A_{i'}\) to \(r_{i'}(a)\), so the transition map \(P_{i'}\to P_i\) kills the generators of \(\ker q_{i'}\) and factors through a map \(Q_{i'}\to P_i\). These maps represent an inverse of \(\text{“}\varprojlim\text{”}q_i\), as one checks after composing with the surjections \(q_{i'}\); so \(\text{“}\varprojlim\text{”}q_i\) is an isomorphism of pro-algebras, compatible with the compositions.

The primitive element. The isomorphism of arrows \(P\cong\text{“}\varprojlim\text{”}P_i\), followed by \(\text{“}\varprojlim\text{”}q_i\), is an isomorphism \(F\colon P\to\text{“}\varprojlim\text{”}Q_i\) compatible with the compositions, whose components are \(K\)-algebra maps \(f_i\colon K[x]/(x^{N_i})\to Q_i\). Put \(u_i=f_i(x)\), a nilpotent element. At the level \(i\), the compatibility with the compositions says that the two maps \(K[x]/(x^N)\to Q_i\otimes_KQ_i\) given by \(\bar\delta_if_i\) and by \((f_i\otimes f_i)\circ\delta\) agree for \(N\) large; at \(x\) this gives \(\bar\delta_i(u_i)=u_i\otimes1+1\otimes u_i\). By the lemma, \(u_i=0\) for every \(i\). But for \(n\geq2\) the component at \(n\) of \(F^{-1}\) is represented by a map \(g\colon Q_i\to K[x]/(x^n)\) for which \(gf_i\) is the projection, and then \(g(u_i)=x\neq0\), a contradiction. \(\square\)

In characteristic \(p>0\) the same construction gives a strict formal category (Exercise 4.4).

2. Divided powers

Definition 2.1. Let \(B\) be a sheaf of rings on \(T\) and \(I\subseteq B\) an ideal. A divided power structure on \(I\) is a family of maps of sheaves of sets \(\gamma_n\colon I\to I\), \(n\geq1\), satisfying on local sections the axioms of [Tag 07GL]: \(\gamma_1(x)=x\) (with \(\gamma_0(x)=1\)), \[ \begin{aligned}\gamma_n(x)\gamma_m(x)&=\tbinom{n+m}{n}\gamma_{n+m}(x),\\ \gamma_n(ax)&=a^n\gamma_n(x),\\ \gamma_n(x+y)&=\sum_{i=0}^n\gamma_i(x)\gamma_{n-i}(y),\\ \gamma_n(\gamma_m(x))&=\tfrac{(nm)!}{n!(m!)^n}\gamma_{nm}(x).\end{aligned} \] A PD morphism \((B,I,\gamma)\to(B',I',\gamma')\) is a ring map \(f\) with \(f(I)\subseteq I'\) and \(f\gamma_n=\gamma'_nf\) on \(I\). An ideal \(I'\subseteq I\) is a sub-PD-ideal if \(\gamma_n(I')\subseteq I'\) for \(n\geq1\). The \(n\)-th divided power \(I^{[n]}\) is the ideal generated by the products \(\gamma_{a_1}(x_1)\cdots\gamma_{a_r}(x_r)\) with \(a_1+\cdots+a_r\geq n\). A divided power structure on a pro-ideal \(\text{“}\varprojlim\text{”}I_i\) is a family of divided power structures on the \(I_i\) for which the transition maps are PD morphisms. More generally, a pro-pair is a pro-object \((B,J)\) of the category of pairs formed by a \(K\)-algebra and an ideal, and a divided power structure on it is such a structure on some level representation \((B_k,J_k)_k\) of it. A morphism between pro-pairs with divided power structures, given on level representations \((B_k,J_k)\) and \((B'_m,J'_m)\), is a PD morphism if for every \(m\) its component at \(m\) is represented by a PD morphism \(B_k\to B'_m\) for some \(k\).

2.2. Passing to sheaves. The results of [Tags 07GP, 07GQ, 07H1, 07H2] for rings hold for sheaves of rings on \(T\), with the same statements read on local sections. Indeed, a divided power structure on a presheaf ideal defines one on the associated sheaf: the \(\gamma_n\) are maps of presheaves of sets and sheafification preserves them, together with the axioms, which are identities between maps built from finite products. Each existence statement is applied to the sections over every object \(U\) of \(T\); its uniqueness statement makes the structures compatible with the restriction maps, and sheafification gives the structure on the sheaf. The uniqueness statements hold sectionwise. We use them in these forms:

1. (generators) If \(I\) is generated by local sections \(x_t\) and \(f\colon B\to B'\) maps \(I\) into a PD ideal \(I'\) with \(f(\gamma_n(x_t))=\gamma'_n(f(x_t))\), then \(f\) is a PD morphism [Tag 07GP].
2. (sums) Divided power structures \(\gamma\) on \(I\) and \(\delta\) on \(I'\) agree on \(II'\); if they agree on \(I\cap I'\), there is a unique divided power structure on \(I+I'\) extending both [Tag 07GQ].
3. (extension) For a ring map \(B\to B'\), \(\gamma\) extends to \(B'\) if \(IB'\) has a divided power structure making \(B\to B'\) a PD morphism; it is then unique. It exists if \(B\to B'\) is flat [Tag 07H1]; and for \(B'=B/\mathfrak a\) it exists if and only if \(I\cap\mathfrak a\) is generated by local sections \(y\) with \(\gamma_n(y)\in I\cap\mathfrak a\) for \(n\geq1\) [Tag 07H2].

Lemma 2.3 (divided powers). Let \(I\) be an ideal with a divided power structure \(\gamma\).

1. Every \(I^{[n]}\) is a sub-PD-ideal, and a PD morphism maps \(I^{[n]}\) into \(I'^{[n]}\).
2. If \(I=I_1+\cdots+I_m\) with every \(I_k\) a sub-PD-ideal, then \[ I^{[N]}=\sum_{N_1+\cdots+N_m\geq N}I_1^{[N_1]}\cdots I_m^{[N_m]}, \] the \(I_k^{[N_k]}\) being taken for the induced structures. In particular \(I^{[m(r-1)+1]}\subseteq I_1^{[r]}+\cdots+I_m^{[r]}\) for every \(r\geq1\).

Proof. (1) For a generator \(y=\prod_j\gamma_{a_j}(x_j)\) of \(I^{[n]}\) and \(N\geq1\), \(\gamma_N(y)=\bigl(\prod_{j\geq2}\gamma_{a_j}(x_j)\bigr)^N\gamma_N(\gamma_{a_1}(x_1))\), which is an integer multiple of a product of weight \(N\sum_ja_j\geq n\). For \(by\) and for sums use \(\gamma_N(by)=b^N\gamma_N(y)\) and the sum axiom, every term of which contains some \(\gamma_i(y)\) or \(\gamma_i(y')\) with \(i\geq1\). A PD morphism maps generators to generators.

(2) The right side is contained in the left, a product of generators of the \(I_k^{[N_k]}\) being a generator of \(I^{[N]}\). Conversely, write each \(x_j\) in a generator \(\prod_j\gamma_{a_j}(x_j)\) of \(I^{[N]}\) locally as \(\sum_ky_{jk}\) with \(y_{jk}\in I_k\). The sum axiom gives \(\gamma_{a_j}(x_j)=\sum\prod_k\gamma_{b_{jk}}(y_{jk})\) over the \(b_{jk}\) with \(\sum_kb_{jk}=a_j\), so the generator is a sum of products \(\prod_{j,k}\gamma_{b_{jk}}(y_{jk})\), which lie in \(\prod_kI_k^{[N_k]}\) with \(N_k=\sum_jb_{jk}\) and \(\sum_kN_k=\sum_ja_j\geq N\). If \(N=m(r-1)+1\) and all \(N_k<r\), then \(\sum N_k\leq m(r-1)<N\); so some \(N_k\geq r\), and the term lies in \(I_k^{[r]}\). \(\square\)

Lemma 2.4 (extension along tensor products). Let \(B\) and \(C\) be augmented \(A\)-algebras and \(\gamma\) a divided power structure on \(I_B\).

1. \(\gamma\) extends to \(B\otimes_AC\) along \(\iota_1\).
2. If \(I_C\) also has a divided power structure \(\delta\), the augmentation ideal \(I_B\otimes C+B\otimes I_C\) of \(B\otimes_AC\) has a unique divided power structure for which \(\iota_1\) and \(\iota_2\) are PD morphisms.

The augmentations cannot be omitted: Exercise 4.2 gives two divided power structures on rings over \(\mathbf Z\) that are not augmented, for which no structure as in (2) exists.

Proof. By 2.2 it suffices to treat rings, that is, the sections over an object of \(T\) with the presheaf tensor product.

(1) Choose a surjection \(A[X]\to C\) from a polynomial ring, with kernel \(\mathfrak b\) (for instance one variable for each element of \(C\)). Then \(B\otimes_AC=B[X]/\mathfrak bB[X]\). The map \(B\to B[X]\) is flat, so \(\gamma\) extends to a divided power structure \(\gamma'\) on \(I_B[X]=I_BB[X]\) (2.2 (3)). By (0.1), \(B[X]=A[X]\oplus I_B[X]\) as \(A[X]\)-modules, so \(\mathfrak bB[X]=\mathfrak b\oplus\mathfrak b\,I_B[X]\), and \[ \mathfrak bB[X]\cap I_B[X]=\mathfrak b\,I_B[X]. \] This ideal is generated by the products \(by\) with \(b\in\mathfrak b\) and \(y\in I_B[X]\), and \(\gamma'_n(by)=b^n\gamma'_n(y)\in\mathfrak b\,I_B[X]\) for \(n\geq1\). By 2.2 (3), \(\gamma'\) extends to the quotient \(B\otimes_AC\), and the composite \(B\to B[X]\to B\otimes_AC\) is \(\iota_1\).

(2) By (1) and its symmetric form, the ideals \(I_1=I_B(B\otimes_AC)\) and \(I_2=I_C(B\otimes_AC)\) carry divided power structures extending \(\gamma\) and \(\delta\). Decompose both factors by (0.1): \[ \begin{aligned}B\otimes_AC&=A\oplus I_B\oplus I_C\\&\quad\oplus(I_B\otimes_AI_C),\end{aligned} \] with \(I_1=I_B\oplus(I_B\otimes_AI_C)\) and \(I_2=I_C\oplus(I_B\otimes_AI_C)\). So \(I_1\cap I_2=I_B\otimes_AI_C\), which is the product \(I_1I_2\), generated by the \(\iota_1(x)\iota_2(y)\). The two structures agree on \(I_1I_2\) (2.2 (2)), hence glue to a unique structure on \(I_1+I_2\), the augmentation ideal. Uniqueness: a structure for which \(\iota_1\), \(\iota_2\) are PD morphisms is determined on the generators \(\iota_1(x)\), \(\iota_2(y)\), and the axioms determine it on the ideal they generate. \(\square\)

Definition 2.5. A PD formal category is a \(K\)-formal category \(C\) with a divided power structure on each pro-pair \((\operatorname{Ner}^n(C),J^n)\), \(J\) being the augmentation ideal of the nerve, such that all cofaces and codegeneracies are PD morphisms. For an arbitrary formal category, \(J\) is the kernel of the augmentation \(\operatorname{Ner}(C)\to A\), formed componentwise on level representations of it (Lemma 1.8); it is the kernel in the category of pro-objects of modules, so it does not depend on the representation. For a strict formal category the structures may be given levelwise. \(C\) is PD-adic if the canonical map \[ \operatorname{Ner}(C)\to\text{“}\varprojlim_n\text{”}\operatorname{Ner}(C)/J^{[n]} \] is an isomorphism of cosimplicial pro-algebras, \(J^{[n]}\) denoting the componentwise \(n\)-th divided power, formed on the level representations carrying the divided powers.

Proposition 2.6 (divided powers on the nerve). Let \(C\) be an affine \(K\)-category given by \((A,P,d^0,d^1,\varepsilon,\delta)\) and \(\gamma\) a divided power structure on \(I=\ker\varepsilon\).

1. For every \(n\geq2\) there is a unique divided power structure on \(J^n\) for which the \(n\) maps \(\iota_k\colon P\to\operatorname{Ner}^n(C)\) are PD morphisms.
2. These structures, with \(\gamma\) on \(J^1=I\), make \(C\) a PD formal category if and only if \(\delta\colon P\to P\otimes_AP\) is a PD morphism.

The same holds for a strict formal category with a divided power structure on the pro-ideal \(I\), levelwise. Proposition 2.9 extends it to arbitrary formal categories.

Proof. (1) Induction on \(n\), with \(\operatorname{Ner}^n=\operatorname{Ner}^{n-1}\otimes_AP\) as in the proof of Proposition 1.6. By induction, \(J^{n-1}\) has a structure for which \(\iota_1,\ldots,\iota_{n-1}\) are PD morphisms. Lemma 2.4 (2), for the augmented \(A\)-algebras \(\operatorname{Ner}^{n-1}\) and \(P\), gives a structure on \(J^n\) for which \(\operatorname{Ner}^{n-1}\to\operatorname{Ner}^n\) and \(\iota_n\) are PD morphisms, hence all the \(\iota_k\). It is unique because \(J^n\) is generated by the \(\iota_k(x)\), \(x\in I\), by (1.3).

(2) If \(C\) is a PD formal category, the coface \(d^1\colon\operatorname{Ner}^1\to\operatorname{Ner}^2\), which is \(\delta\), is a PD morphism. Conversely, let \(\delta\) be one. By 2.2 (1) it suffices to check each coface and codegeneracy on the generators \(\iota_k(x)\), \(x\in I\), of \(J^n\). By (1.2), \(d^0\iota_k=\iota_{k+1}\), \(d^n\iota_k=\iota_k\), and an inner coface \(d^i\) sends \(\iota_k(x)\) to \(\iota_k(x)\) or \(\iota_{k+1}(x)\) when \(k\neq i\), and to the image of \(\delta(x)\) under the PD morphism \(P\otimes_AP\to\operatorname{Ner}^n\) of the factors \(i\), \(i+1\) when \(k=i\). A codegeneracy sends \(\iota_k(x)\) to some \(\iota_{k'}(x)\) or to \(\varepsilon(x)=0\). In each case \(\gamma_m\) is respected, because the \(\iota\)'s and \(\delta\) are PD morphisms and \(\gamma_m(x)\in I\). \(\square\)

Proposition 2.7 (completion). Let \(C\) be a PD formal category, given by an affine category with a divided power structure on \(I\) (Proposition 2.6). Then \[ \text{“}\varprojlim_r\text{”}\operatorname{Ner}(C)/J^{[r]} \] is the nerve of a formal category \(\hat C\) with objects \(A\) and arrows \(\hat P=\text{“}\varprojlim_r\text{”}P/I^{[r]}\), whose structure maps are induced by those of \(C\), the composition by the maps \(P/I^{[2r-1]}\to P/I^{[r]}\otimes_AP/I^{[r]}\) induced by \(\delta\). With the divided powers induced on the quotients, \(\hat C\) is a PD-adic formal category, the PD completion of \(C\). In particular \(C\) is PD-adic if and only if \(P\to\text{“}\varprojlim_r\text{”}P/I^{[r]}\) is an isomorphism of pro-algebras. The same holds for a strict PD formal category, levelwise.

Proof. In degree \(m\), Lemma 2.3 (2) for \(J^m=I_1+\cdots+I_m\) (each \(I_k\) is a sub-PD-ideal, generated by elements \(\iota_k(x)\) with \(\gamma_n(\iota_k(x))=\iota_k(\gamma_n(x))\)) gives \[ \begin{aligned}(J^m)^{[mr]}&\subseteq\textstyle\sum_kI_k^{[r]}\\&\subseteq(J^m)^{[r]},\end{aligned} \] and \(I_k^{[r]}\) is generated by \(\iota_k(I^{[r]})\), as \(\iota_k\) is a PD morphism onto a set of generators of \(I_k\). So the systems \(\bigl(\operatorname{Ner}^m/(J^m)^{[r]}\bigr)_r\) and \(\bigl(\operatorname{Ner}^m/\sum_kI_k^{[r]}\bigr)_r\) are isomorphic pro-objects, by the two inclusions. The second is the system of the \((P/I^{[r]})\otimes_A\cdots\otimes_A(P/I^{[r]})\), by right exactness of the tensor product, which by Lemma 1.2 is the \(m\)-fold fibre product of \(\hat P\) over \(A\) in the opposite category, compatibly with the maps \(\iota_k\). The cofaces and codegeneracies of \(\operatorname{Ner}(C)\) are PD morphisms, so they map the \(J^{[r]}\) into the \(J^{[r]}\) (Lemma 2.3 (1)) and induce maps of the quotient system. On the arrows, \(\delta\) maps \(I^{[2r-1]}\) into \((J^2)^{[2r-1]}\subseteq I_1^{[r]}+I_2^{[r]}\) (Lemma 2.3 (2) for \(m=2\)), so it induces the maps \(P/I^{[2r-1]}\to P/I^{[r]}\otimes_AP/I^{[r]}\) of the statement, a morphism \(\hat\delta\colon\hat P\to\hat P\otimes_A\hat P\); and \(d^0\), \(d^1\), \(\varepsilon\) induce morphisms \(\hat d^0\), \(\hat d^1\), \(\hat\varepsilon\). The axioms (1.1) for these hold because they hold for \(C\), all maps being induced by those of \(C\). The cofaces and codegeneracies of the quotient system and of \(\operatorname{Ner}(\hat C)\) are determined by their composites with the maps \(\iota_k\), by the universal property of the fibre products, and these composites are given by (1.2) for \(\delta\), resp. \(\hat\delta\); so the quotient system is \(\operatorname{Ner}(\hat C)\). The quotients carry the divided powers induced by those of the \(J^m\) (2.2 (3), \(J^{[r]}\) being a sub-PD-ideal by Lemma 2.3 (1)), for which the transition maps and the induced cofaces and codegeneracies are PD morphisms; so \(\hat C\) is a PD formal category. It is PD-adic: for \(s\leq r\) the divided powers of \(J/J^{[r]}\) are the images \(J^{[s]}/J^{[r]}\), so its quotient by its \(s\)-th divided power is \(\operatorname{Ner}/J^{[s]}\) for all \(r\geq s\). The last assertion follows, an isomorphism of nerves being one of arrows and conversely. \(\square\)

The PD completion is strict when \(\delta(I^{[r]})\subseteq I^{[r]}\otimes_AP+P\otimes_AI^{[r]}\) for all \(r\), the \((A,P/I^{[r]})\) then being affine categories. It is not strict in general. Let \(K\) be a field of characteristic \(0\), \(T\) the punctual topos, and \(\mathbb G_a\) the affine category with objects \(K\), arrows \(K[x]\), \(\delta(x)=x\otimes1+1\otimes x\) and \(\varepsilon(x)=0\). Every ideal of a \(K\)-algebra has the unique divided power structure \(\gamma_n(y)=y^n/n!\), for which every map of \(K\)-algebras is a PD morphism. So \(\mathbb G_a\) is a PD formal category with \(I^{[r]}=(x^r)\), and its PD completion is the formal additive group of Proposition 1.9, which is not strict.

The pro-algebra \(\text{“}\varprojlim\text{”}P/I^{[r]}\) is represented by the system of \(P_r=P/I^{[r+1]}\) with the ideals \(I_r=I/I^{[r+1]}\), and \(\ker(P_r\to P_s)=I_r^{[s+1]}\) for \(s\leq r\), the divided powers of a quotient by a sub-PD-ideal being the images of those of the ideal.

Lemma 2.8.

1. A PD morphism of pro-pairs with divided power structures that is an isomorphism of pro-pairs has a PD morphism as its inverse.
2. Let \(A\) and \(P\) be pro-\(K\)-algebras, \(d^0,d^1\colon A\to P\) and \(\varepsilon\colon P\to A\) morphisms with \(\varepsilon d^0=\varepsilon d^1=\mathrm{id}_A\), and let the pro-pair \((P,I)\), \(I=\ker\varepsilon\), carry a divided power structure. Then there are a small cofiltered category \(L\) and, functorially in \(l\in L\), \(K\)-algebras \(A_l\), \(B_l\), maps \(d^0_l,d^1_l\colon A_l\to B_l\) and \(\varepsilon_l\colon B_l\to A_l\) with \(\varepsilon_ld^j_l=\mathrm{id}\), and divided power structures on the \(\ker\varepsilon_l\) for which the transition maps are PD morphisms, with isomorphisms \(\text{“}\varprojlim\text{”}A_l\cong A\) and \(\text{“}\varprojlim\text{”}B_l\cong P\) compatible with \(d^0\), \(d^1\) and \(\varepsilon\) that carry \(\text{“}\varprojlim\text{”}\ker\varepsilon_l\) onto \(I\) by a PD isomorphism.

Proof. (1) By Lemma 1.8, for pro-objects of the category of \(K\)-algebras with a divided power ideal, the morphism \(f\) is given by PD morphisms \(f_l\colon B_l\to B'_l\) over one cofiltered \(L\), with transition maps \(t\) and \(t'\). The components of its inverse are represented by maps of pairs \(g_l\colon B'_{l'}\to B_l\), \(l'\to l\). As \(f^{-1}f\) and \(ff^{-1}\) are identities, after replacing \(l'\) by a deeper index and \(g_l\) by its composite with a transition map, \(g_lf_{l'}=t_{l'l}\) and \(f_lg_l=t'_{l'l}\) (Lemma 1.8). Choose \(g_{l'}\colon B'_{l''}\to B_{l'}\) in the same way. Then \(g_lt'_{l''l'}=g_lf_{l'}g_{l'}=t_{l'l}g_{l'}\) represents the same component as \(g_l\), and it is a PD morphism: for \(y\) in the ideal of \(B'_{l''}\), \[ \begin{aligned}g_lt'(\gamma_ny)&=g_l\gamma_n(t'y)\\&=g_l\gamma_n(f_{l'}g_{l'}y)\\&=g_lf_{l'}\gamma_n(g_{l'}y)\\&=t\gamma_n(g_{l'}y)\\&=\gamma_n(tg_{l'}y)\\&=\gamma_n(g_lt'y),\end{aligned} \] as \(t\), \(t'\) and \(f_{l'}\) are PD morphisms.

(2) Let \((P_k,I_k,\gamma_k)\) be the level representation carrying the divided power structure. Identify \(P\) with \(\text{“}\varprojlim\text{”}P_k\), and let \(\pi\colon P\to P/I\) be the projection to the pro-algebra of the \(P_k/I_k\). As \(\varepsilon(d^1-d^0)=0\), the morphism \(d^1-d^0\) of pro-modules factors through \(\ker\varepsilon=I\), so \(\pi d^0=\pi d^1\), and \((d^0,d^1)\) is a morphism from \(A\) to the fibre product \(P\times_{P/I}P\). This fibre product is the pro-algebra of the \(P_k\times_{P_k/I_k}P_k\): maps from a pro-algebra into the latter are limits over \(k\) of filtered colimits of fibre products of sets, and both operations commute with fibre products. By Lemma 1.8 for this morphism we may assume \(A=\text{“}\varprojlim_{l\in L}\text{”}A_l\) and \(P=\text{“}\varprojlim_{l\in L}\text{”}P_l\) over one cofiltered \(L\), the \(P_l\) carrying divided powers on ideals \(I_l\) with PD transition maps, and \(d^0\), \(d^1\) given by maps \(e^0_l,e^1_l\colon A_l\to P_l\) with \(\pi_le^0_l=\pi_le^1_l\). Put \[ B_l=A_l\times_{P_l/I_l}P_l, \] the fibre product of \(\pi_le^0_l=\pi_le^1_l\) and \(\pi_l\), with \(d^j_l(a)=(a,e^j_l(a))\) and \(\varepsilon_l(a,y)=a\). Then \(\varepsilon_ld^j_l=\mathrm{id}\), and \(\ker\varepsilon_l=0\times I_l\) is an ideal of \(B_l\) on which \(B_l\) acts through the projection \(\mathrm{pr}_2\colon B_l\to P_l\); so the divided powers of \(I_l\) are a divided power structure on \(\ker\varepsilon_l\). All these maps are natural in \(l\), and the transition maps are PD morphisms. The projections \(\mathrm{pr}_2\) form a morphism of pro-algebras \(\text{“}\varprojlim\text{”}B_l\to P\) with \(\mathrm{pr}_2d^j_l=e^j_l\), equal to the identity on the ideals \(\ker\varepsilon_l=I_l\), and with \(\varepsilon\,\mathrm{pr}_2=\text{“}\varprojlim\text{”}\varepsilon_l\), because \(\mathrm{pr}_2-e^0_l\varepsilon_l\) takes values in \(I_l\), which \(\varepsilon\) kills, and \(\varepsilon e^0=\mathrm{id}\). It is an isomorphism of pro-modules: \(B_l=d^0_l(A_l)\oplus\ker\varepsilon_l\), and \(P=d^0(A)\oplus I\) in the abelian category of pro-modules, as \(\varepsilon d^0=\mathrm{id}\); the morphism is the identity on both summands. So it is an isomorphism of pro-algebras: its inverse satisfies the identities expressing multiplicativity and the unit as identities of morphisms of pro-sets, because the morphism satisfies them and is an isomorphism, and so its representatives satisfy them after composing with transition maps (Lemma 1.8). \(\square\)

Proposition 2.9 (arbitrary formal categories). Let \(C\) be a \(K\)-formal category with objects \(A\), arrows \(P\) and augmentation ideal \(J\) (Definition 2.5), and let the pro-pair \((P,I)\), \(I=J^1=\ker\varepsilon\), carry a divided power structure.

1. For every \(n\geq2\) there is a divided power structure on \((\operatorname{Ner}^n(C),J^n)\) for which the \(n\) morphisms \(\iota_k\colon P\to\operatorname{Ner}^n(C)\) are PD morphisms; for any other such structure, the identity is a PD isomorphism between the two.
2. These structures, with the given one on \(J^1=I\), make \(C\) a PD formal category if and only if \(\delta\colon P\to\operatorname{Ner}^2(C)\) is a PD morphism.
3. If they do, the conclusions of Proposition 2.7 hold for \(C\): \(\text{“}\varprojlim_r\text{”}\operatorname{Ner}(C)/J^{[r]}\) is the nerve of a PD-adic formal category with objects \(A\) and arrows \(\text{“}\varprojlim_r\text{”}P/I^{[r]}\), and \(C\) is PD-adic if and only if \(P\to\text{“}\varprojlim_r\text{”}P/I^{[r]}\) is an isomorphism.

Proof. Represent \(A\), \(P\), \(d^0\), \(d^1\), \(\varepsilon\) and the divided powers as in Lemma 2.8 (2), by \(A_l\), \(B_l\), \(d^j_l\) and \(\varepsilon_l\). By Lemma 1.2 and induction, \(\operatorname{Ner}^n(C)\) is the pro-algebra of the \(\operatorname{Ner}^n_l=B_l\otimes_{A_l}\cdots\otimes_{A_l}B_l\) (\(n\) factors, formed as in Proposition 1.6), compatibly with the \(\iota_k\), and its augmentation is the system of the products of the \(\varepsilon_l\); so \(J^n\) is the system of the ideals \(J^n_l\) of (1.3).

(1) The proof of Proposition 2.6 (1) uses only the augmentations of \(B_l\) through \(d^0_l\) and \(d^1_l\), and gives on each \(J^n_l\) a unique divided power structure for which the \(\iota_k\) are PD morphisms. The transition maps are PD morphisms by 2.2 (1), on the generators \(\iota_k(x)\), \(x\in\ker\varepsilon_l\). This gives a structure as required. Let \(\Sigma\) be another, on a level representation \((N_s,J_s)\), and represent the identity of \((\operatorname{Ner}^n(C),J^n)\), from the first structure to \(\Sigma\), by maps of pairs \(h\colon\operatorname{Ner}^n_l\to N_s\). The composites \(h\iota_k\) represent the PD morphisms \(\iota_k\) into \(\Sigma\); so after composing \(h\) with a transition map they are PD morphisms (Lemma 1.8), and then \(h\) is a PD morphism, by 2.2 (1) on the generators \(\iota_k(x)\). So the identity is a PD morphism, and by Lemma 2.8 (1) so is its inverse.

(2) If \(C\) is a PD formal category, then \(\delta\), the coface \(d^1\colon\operatorname{Ner}^1\to\operatorname{Ner}^2\), is a PD morphism. Conversely, let \(\delta\) be one, and represent a coface or codegeneracy by maps of pairs \(c\) between levels (Lemma 1.8). By (1.2), its composites with the \(\iota_k\) are morphisms \(\iota_{k'}\), the composite of \(\delta\) with the morphism \(\operatorname{Ner}^2\to\operatorname{Ner}^n\) of the factors \(i\) and \(i+1\), which is levelwise a PD morphism by 2.2 (1), or zero on \(I\). These are PD morphisms; so after composing \(c\) with a transition map, \(c\) is a PD morphism on the generators \(\iota_k(x)\), hence a PD morphism by 2.2 (1).

(3) The proof of Proposition 2.7 applies at each level \(l\): \(\delta\) is represented by PD morphisms \(B_{l'}\to\operatorname{Ner}^2_l\), which map \((\ker\varepsilon_{l'})^{[2r-1]}\) into \((J^2_l)^{[2r-1]}\subseteq I_1^{[r]}+I_2^{[r]}\), and the cofaces and codegeneracies by PD morphisms. \(\square\)

3. Divided power envelopes

3.1. Let \(B\) be a sheaf of rings and \(J\subseteq B\) an ideal. Its divided power envelope \((D_B(J),\bar J,\bar\gamma)\) is the sheaf associated with the presheaf of envelopes \(U\mapsto D_{B(U)}(J(U))\) of [Tag 07H8], relative to \((\mathbf Z,0)\). The restriction maps come from the universal property, and by 2.2 the associated sheaf carries the divided power structure. It has the universal property for sheaves: a ring map \(B\to B'\) into a sheaf of rings with a divided power ideal \(I'\), sending \(J\) into \(I'\), factors uniquely through a PD morphism \(D_B(J)\to B'\). Indeed the maps on sections are compatible by the universal property over each \(U\), and a map from a presheaf to a sheaf factors uniquely through its sheafification. Moreover [Tag 07HA] gives maps \(B\to D_B(J)\to B/J\) with \(\bar J\) the kernel of the second, and \(\bar J\) is generated by the \(\bar\gamma_n(x)\), \(x\) in the image of \(J\).

Lemma 3.2. Let \(B\) be an augmented \(A\)-algebra. Then \(D_B(I_B)\) is an augmented \(A\)-algebra, through \(A\to B\to D_B(I_B)\) and \(D_B(I_B)\to B/I_B=A\), with augmentation ideal \(\bar I_B\).

Proof. The composite \(A\to B\to D_B(I_B)\to B/I_B=A\) is the identity, and \(\bar I_B\) is the kernel of the last map (3.1). \(\square\)

Proposition 3.3 (envelopes of tensor products). Let \(B\) and \(C\) be augmented \(A\)-algebras. With the structure of Lemma 2.4 (2) on the augmentation ideal of \(D_B(I_B)\otimes_AD_C(I_C)\), the map \(B\otimes_AC\to D_B(I_B)\otimes_AD_C(I_C)\) identifies \[ \begin{aligned}&D_{B\otimes_AC}(I_B\otimes C+B\otimes I_C)\\&\quad\cong D_B(I_B)\otimes_AD_C(I_C).\end{aligned} \]

Proof. Write \(D_B=D_B(I_B)\) and \(D_C=D_C(I_C)\), augmented \(A\)-algebras by Lemma 3.2, so Lemma 2.4 (2) applies to them. The map \(B\otimes_AC\to D_B\otimes_AD_C\) sends \(I_B\otimes C+B\otimes I_C\) into the augmentation ideal. Let \(f\colon B\otimes_AC\to B'\) be a ring map into a sheaf of rings with a divided power ideal \(I'\), sending \(I_B\otimes C+B\otimes I_C\) into \(I'\). Its restrictions to \(B\) and \(C\) send \(I_B\) and \(I_C\) into \(I'\), so they factor uniquely through PD morphisms \(D_B\to B'\) and \(D_C\to B'\) (3.1). These agree on \(A\), hence define a ring map \(g\colon D_B\otimes_AD_C\to B'\) extending \(f\). It is a PD morphism by 2.2 (1), being one on the generators \(\iota_1(\bar\gamma_n(x))\), \(\iota_2(\bar\gamma_n(y))\) of the augmentation ideal. It is unique: a PD morphism extending \(f\) is determined on \(D_B\) and on \(D_C\) by the universal properties, and these generate \(D_B\otimes_AD_C\). So \(D_B\otimes_AD_C\) has the universal property of the envelope. \(\square\)

Corollary 3.4 (the envelope of an affine category). Let \(C\) be an affine \(K\)-category with nerve \(R=\operatorname{Ner}(C)\) and augmentation ideal \(J\), and let \(\bar R^n=D_{R^n}(J^n)\) be the componentwise envelopes, with the induced cofaces and codegeneracies.

1. \(\bar R\) is the nerve of the affine category \(\bar C=(A,D_P(I))\) whose structure maps are induced by those of \(C\), and the divided power structures of the envelopes make \(\bar C\) a PD formal category, with augmentation ideal \(\bar J\).
2. \(\text{“}\varprojlim_r\text{”}\bar R/\bar J^{[r]}\) is the nerve of a PD-adic formal category, the PD completion of \(C\).

Proof. (1) The cofaces and codegeneracies of \(R\) map \(J\) into \(J\), so they induce PD morphisms of the envelopes, and the cosimplicial identities pass to the envelopes by the uniqueness in their universal properties. The maps \(d^0,d^1\colon A\to P\) and \(\varepsilon\) make \(P\) augmented in two ways; by Lemma 3.2 so is \(D_P(I)\). By Proposition 3.3 and induction, \(\bar R^n=D_P(I)\otimes_A\cdots\otimes_AD_P(I)\), compatibly with the maps \(\iota_k\), and the induced cofaces and codegeneracies are given by (1.2) for the induced maps \(\bar\delta\), \(\bar\varepsilon\). So \(\bar R\) is the nerve of \(\bar C\), whose axioms (1.1) hold because they hold after composing with \(P\to D_P(I)\), and maps out of the envelope are determined there. Since \(\bar\delta\) is a PD morphism and the structures on the \(\bar J^n\) are those of Proposition 2.6 (by uniqueness), \(\bar C\) is a PD formal category. (2) is Proposition 2.7 for \(\bar C\). \(\square\)

Definition 3.5. The Čech–Alexander complex \(\hat C_{A/K}\) of a \(K\)-algebra \(A\) is the PD completion of the chaotic category of \(A/K\) (Examples 1.5, 1.7): the cosimplicial pro-algebra \[ \hat C_{A/K}=\text{“}\varprojlim_r\text{”}\bar C_{A/K}/\bar J^{[r]}, \] where \(\bar C^n_{A/K}\) is the divided power envelope of the ideal of the diagonal in \(A^{\otimes(n+1)}\). For a morphism of schemes \(f\colon X\to S\), \(\hat C_{X/S}=\hat C_{\mathcal O_X/f^{-1}\mathcal O_S}\) on \(X_{\mathrm{Zar}}\).

Example 3.6. Let \(A=K[x]\). Then \(P=A\otimes_KA=A[\tau]\) with \(\tau=1\otimes x-x\otimes1\), a polynomial ring over \(A\) through \(d^0\) or \(d^1\), and \(I=\tau P\). The envelope is the divided power polynomial algebra \(D_P(I)=A\langle\tau\rangle\), with \(A\)-basis the \(\tau^{[k]}\), \(k\geq0\) [Tag 07H5]. Since \(\delta(\tau)=\tau\otimes1+1\otimes\tau\) in \(A^{\otimes3}\), the induced comultiplication is \[ \bar\delta(\tau^{[n]})=\sum_{i+j=n}\tau^{[i]}\otimes\tau^{[j]}, \] and \(\bar R^n=A\langle\tau_1,\ldots,\tau_n\rangle\) with \(\tau_k=\iota_k(\tau)\). The PD completion replaces \(A\langle\tau\rangle\) by the system of its quotients by the \(\tau^{[\geq r]}\).

4. Exercises

Exercise 4.1. In Example 3.6, write \(d^0\), \(d^1\colon A\to A\langle\tau\rangle\) explicitly on \(x\), and check (1.1) for \(\bar\delta\) and \(\bar\varepsilon\) on the \(\tau^{[n]}\). Show that \(\bar\delta\) is a PD morphism.

Exercise 4.2. By [Tag 07GY], the ideal \(2\mathbf Z/4\mathbf Z\) of \(\mathbf Z/4\mathbf Z\) has divided power structures \(\delta\), \(\delta'\) with \(\delta_2(2)=2\) and \(\delta'_2(2)=0\). Show that in \(\mathbf Z/4\otimes_{\mathbf Z}\mathbf Z/4\) no divided power structure on the ideal generated by \(2\otimes1\) and \(1\otimes2\) makes both coprojections PD morphisms, and explain why Lemma 2.4 does not apply.

Exercise 4.3. Let \(P=\mathbf Z\langle s,t\rangle\) be the divided power polynomial algebra in two variables and \(I=I_1+I_2\) its augmentation ideal, with \(I_1\), \(I_2\) generated by the \(s^{[a]}\), resp. \(t^{[b]}\), \(a,b\geq1\). Show that \(I^{[N]}\) is the span of the monomials \(s^{[a]}t^{[b]}\) with \(a+b\geq N\), and that \(I^{[2r-1]}\subseteq I_1^{[r]}+I_2^{[r]}\) while \(I^{[2r-2]}\not\subseteq I_1^{[r]}+I_2^{[r]}\) for \(r\geq2\), so that the bound of Lemma 2.3 (2) is sharp.

Exercise 4.4. Let \(K\) be a field of characteristic \(p>0\), \(T\) the punctual topos, and \(\hat{\mathbb G}_a\) the formal category of Proposition 1.9 formed over \(K\). Show that the \((K,K[x]/(x^{p^k}))\), with the maps induced by \(x\mapsto x\otimes1+1\otimes x\), are affine categories, and that \(\hat{\mathbb G}_a\) is the strict formal category of their system.

5. Complete solutions

Solution 4.1. In \(P=A\otimes_KA\), \(d^0(x)=1\otimes x=x\otimes1+\tau\) and \(d^1(x)=x\otimes1\); identifying \(P\) with \(A[\tau]\) through \(d^1\), \(d^1(x)=x\) and \(d^0(x)=x+\tau\). The augmentation sends \(\tau\) to \(x-x=0\), so \(\bar\varepsilon(\tau^{[n]})=0\) for \(n\geq1\), and \((\bar\varepsilon\otimes\mathrm{id})\bar\delta(\tau^{[n]})=\tau^{[n]}=(\mathrm{id}\otimes\bar\varepsilon)\bar\delta(\tau^{[n]})\). Coassociativity: both sides send \(\tau^{[n]}\) to the sum of the \(\tau^{[i]}\otimes\tau^{[j]}\otimes\tau^{[k]}\) with \(i+j+k=n\). For the compatibilities with \(d^0\), \(d^1\): \(\delta(d^1x)=\delta(x\otimes1)=\) \(x\otimes1\otimes1=\iota_1(d^1x)\) and \(\delta(d^0x)=1\otimes1\otimes x=\iota_2(d^0x)\). The map \(\bar\delta\) is the PD morphism over \(A\) extending \(\tau\mapsto\tau\otimes1+1\otimes\tau\) given by the universal property of \(A\langle\tau\rangle\) [Tag 07H5]; on the \(\tau^{[n]}\) it is the formula of Example 3.6, by the sum axiom \(\gamma_n(\tau\otimes1+1\otimes\tau)=\sum_{i+j=n}\tau^{[i]}\otimes\tau^{[j]}\).

Solution 4.2. In \(\mathbf Z/4\otimes_{\mathbf Z}\mathbf Z/4=\mathbf Z/4\) both \(2\otimes1\) and \(1\otimes2\) are \(2\). A structure \(\epsilon\) making both coprojections PD morphisms would satisfy \(\epsilon_2(2)=\delta_2(2)=2\) and \(\epsilon_2(2)=\delta'_2(2)=0\). Lemma 2.4 needs augmented algebras over the base: there is no ring map \(\mathbf Z/4\to\mathbf Z\), so \(\mathbf Z/4\) is not an augmented \(\mathbf Z\)-algebra, and the decomposition (0.1) behind the proof is not available.

Solution 4.3. The monomials \(s^{[a]}t^{[b]}\) form a \(\mathbf Z\)-basis of \(P\), and \(s^{[a]}s^{[c]}=\tbinom{a+c}{a}s^{[a+c]}\). So the ideal generated by the monomials of degree \(\geq N\) is their span. These monomials lie in \(I^{[N]}\): \(s^{[a]}t^{[b]}=\gamma_a(s)\gamma_b(t)\) has weight \(a+b\). Conversely, for a monomial \(m\) of positive degree, \(\gamma_n(m)\) is an integer multiple of a monomial of degree \(n\deg m\): if \(m=s^{[a]}t^{[b]}\) with \(a\geq1\), then \(\gamma_n(m)=(t^{[b]})^n\gamma_n(s^{[a]})\), and \((t^{[b]})^n\) and \(\gamma_n(s^{[a]})\) are integer multiples of \(t^{[nb]}\) and \(s^{[na]}\). By the sum axiom, \(\gamma_n\) of a combination of such monomials is a combination of products \(\prod\gamma_{n_k}(c_km_k)=\prod c_k^{n_k}\gamma_{n_k}(m_k)\) with \(\sum n_k=n\), so a generator \(\prod_j\gamma_{a_j}(x_j)\) of \(I^{[N]}\) is a combination of monomials of degree \(\geq\sum_ja_j\geq N\). The ideal \(I_1^{[r]}+I_2^{[r]}\) is the span of the monomials with \(a\geq r\) or \(b\geq r\). If \(a+b\geq2r-1\) then \(a\geq r\) or \(b\geq r\), which gives the inclusion; the monomial \(s^{[r-1]}t^{[r-1]}\) has degree \(2r-2\) and lies in neither.

Solution 4.4. In characteristic \(p\), \((x\otimes1+1\otimes x)^{p^k}=x^{p^k}\otimes1+1\otimes x^{p^k}\). So the map \(K[x]\to K[x]\otimes_KK[x]\), \(x\mapsto x\otimes1+1\otimes x\), sends \(x^{p^k}\) into the ideal generated by \(x^{p^k}\otimes1\) and \(1\otimes x^{p^k}\), and induces \[ \begin{aligned}\delta_k\colon&K[x]/(x^{p^k})\\&\to K[x]/(x^{p^k})\otimes_KK[x]/(x^{p^k}).\end{aligned} \] With \(d^0=d^1\) the unit and \(\varepsilon(x)=0\), the axioms (1.1) hold, since they hold for \(K[x]\) and all maps are induced; so \(C_k=(K,K[x]/(x^{p^k}))\) is an affine category, and the projections \(K[x]/(x^{p^{k+1}})\to K[x]/(x^{p^k})\) are compatible with the structure maps. The system \((K[x]/(x^{p^k}))_k\) is cofinal in \((K[x]/(x^n))_n\), so both define the same pro-algebra. The composition of \(\hat{\mathbb G}_a\) at the level \((p^k,p^k)\), the map \(K[x]/(x^{2p^k-1})\to K[x,y]/(x^{p^k},y^{p^k})\) with \(x\mapsto x+y\), is \(\delta_k\) composed with the projection. So the structure maps agree, and \(\hat{\mathbb G}_a\) is the strict formal category of the \(C_k\).

Sources

Formal categories, their divided power structures and completions, and the Čech–Alexander complex are due to Illusie, in his work on the cotangent complex, building on Berthelot's theory of divided powers. Pro-objects go back to Grothendieck (SGA 4, Exposé I, §8), and level representations of their morphisms to Artin and Mazur, in their work on étale homotopy. Divided power algebras and envelopes for rings are developed in the AI-integrated Stacks project, chapters Divided Power Algebra and Crystalline Cohomology.