Affine-space fibrations that are not locally trivial

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

Let \(R\) be a ring. An \(\mathbb A^n\)-fibration over \(R\) is a finitely generated flat \(R\)-algebra \(B\) such that \(B\otimes_R\kappa(\mathfrak p)\) is a polynomial ring in \(n\) variables over the residue field \(\kappa(\mathfrak p)\) for every prime \(\mathfrak p\) of \(R\). Geometrically, \(\operatorname{Spec}B\to\operatorname{Spec}R\) is flat and each of its fibres is affine \(n\)-space over the residue field of its point. Locally trivial bundles with fibre \(\mathbb A^n\) have these properties. A question going back to Dolgachev and Weisfeiler asks for a converse, in the form recorded in [Gupta-survey, Question 3]: if \(R\) is a regular noetherian domain, is every \(\mathbb A^n\)-fibration over \(R\) the symmetric algebra of a projective \(R\)-module, and in particular a polynomial ring when \(R\) is a regular local ring? For \(n=1\) the answer is yes, and for \(n=2\) and a discrete valuation ring containing \(\mathbb Q\) it is yes by a theorem of Sathaye [Gupta-survey, Section 4].

This lesson shows that the answer is no for \(n=3\). The polynomial \(H\) of A fourfold whose cylinder is affine space makes the polynomial ring in five variables a smooth \(\mathbb A^3\)-fibration over the plane, through the map \((p,H)\), and makes the hypersurface \(H=0\) a smooth \(\mathbb A^3\)-fibration over the line \(\operatorname{Spec}k[p]\). Neither fibration is locally trivial, and the second remains nontrivial over the discrete valuation ring \(k[p]_{(p)}\). The examples are from [OpenAI-cancellation, Section 7].

We use from this course Sections 2–4 of A fourfold whose cylinder is affine space, Theorem 9.1 of Cancellation fails in dimension four and Theorem 3.1 of A stable coordinate in five variables. From the course on projective modules we use Lemmas 1.2, 1.3 and Corollary 2.2 of Locally polynomial algebras, which rests on the Quillen–Suslin theorem, and Lemma 1.1 of Horrocks' theorem and the Quillen–Suslin theorem. From commutative algebra we use the definitions of formally smooth and smooth ring maps in Sections 1 and 4 of Formally smooth, unramified and étale ring maps; Propositions 3.1, 3.2 and Theorem 3.3 of Tor and flat modules; Grothendieck's slicing lemma, Theorem 5.2 of Faithful flatness and the local criterion for flatness; and Theorem 5.1 (local detection) of Localization, local properties and support. The field \(k\) has characteristic zero.

Basic references are [OpenAI-cancellation] and [Gupta-survey].

1. Fibrations and locally trivial bundles

An \(R\)-algebra \(B\) is a Zariski-locally trivial \(\mathbb A^n\)-bundle if there are \(g_1,\ldots,g_m\in R\) generating the unit ideal and isomorphisms of \(R_{g_i}\)-algebras \(B_{g_i}\cong R_{g_i}[X_1,\ldots,X_n]\). The principal open sets form a basis of the topology of the quasi-compact space \(\operatorname{Spec}R\), so this says that \(\operatorname{Spec}B\) is a product with \(\mathbb A^n\) over each member of an open covering of \(\operatorname{Spec}R\). A finitely generated Zariski-locally trivial \(\mathbb A^n\)-bundle is an \(\mathbb A^n\)-fibration, and it is locally polynomial: \(B_{\mathfrak m}\cong R_{\mathfrak m}[X_1,\ldots,X_n]\) for every maximal ideal \(\mathfrak m\) of \(R\) (Exercise 6.1).

Over a polynomial ring over a field, locally polynomial algebras are polynomial rings: by Corollary 2.2 of Locally polynomial algebras, a finitely presented algebra over \(F[y_1,\ldots,y_r]\), \(F\) a field, which is locally polynomial in \(n\) variables is a polynomial ring in \(n\) variables. Finitely generated algebras over a noetherian ring are finitely presented. So over such a base, an \(\mathbb A^n\)-fibration that is not a polynomial ring fails to be a polynomial ring after localizing at some maximal ideal, and it is not locally trivial.

2. Smoothness and flatness of a hypersurface

Lemma 2.1. Let \(R\) be a ring, \(f\in R[X_1,\ldots,X_n]\) and \(S=R[X_1,\ldots,X_n]/(f)\). If the images of \(\partial f/\partial X_1,\ldots,\partial f/\partial X_n\) generate the unit ideal of \(S\), then \(S\) is a smooth \(R\)-algebra.

Proof. The algebra \(S\) is finitely presented. Choose \(c_1,\ldots,c_n,h\in R[X]\) with \(\sum_jc_j\,\partial f/\partial X_j=1+fh\). Let \(A\) be an \(R\)-algebra, \(J\subseteq A\) an ideal with \(J^2=0\), and \(\bar u\colon S\to A/J\) a homomorphism of \(R\)-algebras. Choose \(a_1,\ldots,a_n\in A\) lifting the images \(\bar u(X_j)\). Then \(f(a)\in J\), and \(e=\sum_jc_j(a)\,(\partial f/\partial X_j)(a)\), which is \(1+f(a)h(a)\), is a unit, since \(f(a)h(a)\in J\) has square zero. Put \(\varepsilon_j=-c_j(a)\,e^{-1}f(a)\in J\). In \(R[X,Y]\), the difference of \(f(X+Y)-f(X)\) and \(\sum_j(\partial f/\partial X_j)(X)\,Y_j\) lies in the ideal generated by the products \(Y_iY_j\); as \(J^2=0\), \[ \begin{aligned} f(a+\varepsilon)&=f(a)+\sum_j(\partial f/\partial X_j)(a)\,\varepsilon_j\\ &=f(a)-e^{-1}f(a)\,e=0 . \end{aligned} \] So \(X_j\mapsto a_j+\varepsilon_j\) defines a homomorphism of \(R\)-algebras \(S\to A\) lifting \(\bar u\). Hence \(S\) is formally smooth over \(R\), and smooth. \(\square\)

Lemma 2.2. Let \(R\) be a noetherian ring and \(f\in R[X_1,\ldots,X_n]\) such that for every prime \(\mathfrak p\) of \(R\) the image of \(f\) in \(\kappa(\mathfrak p)[X_1,\ldots,X_n]\) is nonzero. Then \(S=R[X_1,\ldots,X_n]/(f)\) is flat over \(R\).

Proof. Let \(\mathfrak Q\) be a prime of \(S\), \(\mathfrak q\) its preimage in \(R[X]\) and \(\mathfrak p=\mathfrak q\cap R\). The free \(R\)-module \(R[X]\) is flat, so by Propositions 3.1, 3.2 and Theorem 3.3 of Tor and flat modules the local homomorphism \(R_{\mathfrak p}\to R[X]_{\mathfrak q}\) of noetherian local rings is flat. Its closed fibre \(R[X]_{\mathfrak q}/\mathfrak pR[X]_{\mathfrak q}\) is a localization of the domain \((R/\mathfrak p)[X]\subseteq\kappa(\mathfrak p)[X]\) at a prime ideal. The image of \(f\) in it is nonzero, hence a nonzerodivisor, and \(f\) lies in the maximal ideal \(\mathfrak qR[X]_{\mathfrak q}\). By the slicing lemma, \(S_{\mathfrak Q}=R[X]_{\mathfrak q}/(f)\) is flat over \(R_{\mathfrak p}\).

Let \(N'\to N\) be an injective map of \(R\)-modules and \(K\) the kernel of \(N'\otimes_RS\to N\otimes_RS\), an \(S\)-module. For a prime \(\mathfrak Q\) of \(S\), with \(\mathfrak p\) as above, localization is exact and \((N'\otimes_RS)_{\mathfrak Q}=N'_{\mathfrak p}\otimes_{R_{\mathfrak p}}S_{\mathfrak Q}\), so \(K_{\mathfrak Q}\) is the kernel of \(N'_{\mathfrak p}\otimes_{R_{\mathfrak p}}S_{\mathfrak Q}\to N_{\mathfrak p}\otimes_{R_{\mathfrak p}}S_{\mathfrak Q}\), which is zero. By local detection \(K=0\). So \(S\) is flat over \(R\). \(\square\)

3. A residual coordinate

Recall from A fourfold whose cylinder is affine space that \(P=k[p,s,u,F,J]\), \[ \begin{gathered} x=s^2+u^3+p^2F,\\ y=s+x(x-u^3),\qquad z=sx+p^2J,\\ H=x^2F-(1+2sx)J-p^2J^2-pu, \end{gathered} \] and \(A=P/(H)\). We regard \(P\) as the polynomial ring in \(s,u,F,J\) over \(k[p]\). For a prime \(\mathfrak p\) of \(k[p]\) let \(H_{\mathfrak p}\) be the image of \(H\) in \(P\otimes_{k[p]}\kappa(\mathfrak p)=\kappa(\mathfrak p)[s,u,F,J]\).

Proposition 3.1. For every prime \(\mathfrak p\) of \(k[p]\) there are \(G_1,G_2,G_3\in\kappa(\mathfrak p)[s,u,F,J]\) such that \(H_{\mathfrak p},G_1,G_2,G_3\) are algebraically independent over \(\kappa(\mathfrak p)\) and \[ \kappa(\mathfrak p)[s,u,F,J]=\kappa(\mathfrak p)[H_{\mathfrak p},G_1,G_2,G_3]. \]

Proof. If \(p\notin\mathfrak p\), then \(\kappa(\mathfrak p)\) is a \(k[p,p^{-1}]\)-algebra. By the proof of Proposition 2.2 of the cylinder lesson, \(P[p^{-1}]=k[p,p^{-1}][x,y,z,H]\) with \(x,y,z,H\) algebraically independent over \(k[p,p^{-1}]\). Tensoring with \(\kappa(\mathfrak p)\) over \(k[p,p^{-1}]\) gives the claim, with \(G_1,G_2,G_3\) the images of \(x,y,z\).

If \(p\in\mathfrak p\), then \(\mathfrak p=(p)\) and \(\kappa(\mathfrak p)=k\). By (4.3) of the cylinder lesson \(H\equiv L\) modulo \(p\), and by (4.1) and (4.2) there, \(P/pP=k[s,u,M,L]\) with \(s,u,M,L\) algebraically independent (Exercise 6.3 there). Take \(G_1,G_2,G_3\) to be the images of \(s,u,M\). \(\square\)

So \(H\) is a residual coordinate of \(P\) over \(k[p]\): it becomes a coordinate over every residue field of \(k[p]\).

Lemma 3.2. The partial derivatives \(\partial H/\partial s\), \(\partial H/\partial u\), \(\partial H/\partial F\), \(\partial H/\partial J\) generate the unit ideal of \(P\).

Proof. Let \(\mathfrak j\) be the ideal they generate. The derivation \(\Delta\) of Lemma 3.1 of the cylinder lesson is a \(k[p]\)-derivation of \(P\) with \(\Delta(H)=p^3\). By the chain rule, \[ \begin{aligned} p^3&=\Delta(H)=\Delta(s)\frac{\partial H}{\partial s}+\Delta(u)\frac{\partial H}{\partial u}\\ &\quad +\Delta(F)\frac{\partial H}{\partial F}+\Delta(J)\frac{\partial H}{\partial J}\in\mathfrak j . \end{aligned} \] Differentiation in \(s,u,F,J\) commutes with reduction modulo \(p\). Modulo \(p\), \(H\equiv x_0^2F-(1+2sx_0)J\) with \(x_0=s^2+u^3\), so \(\partial H/\partial F\equiv x_0^2\), \(\partial H/\partial J\equiv-(1+2sx_0)\), and \[ \begin{aligned} &4s^2\frac{\partial H}{\partial F}-(1-2sx_0)\frac{\partial H}{\partial J}\equiv4s^2x_0^2\\ &\quad +(1-2sx_0)(1+2sx_0)=1 , \end{aligned} \] the determinant of the change of variables (4.1) of the cylinder lesson. So \(1+pc\in\mathfrak j\) for some \(c\in P\), and \(1=(1+pc)(1-pc+p^2c^2)-c^3p^3\in\mathfrak j\). \(\square\)

4. Two smooth A3-fibrations

Theorem 4.1. Make \(P\) an algebra over \(k[p,q]\) through \(q\mapsto H\).

  1. \(P\) is a smooth \(\mathbb A^3\)-fibration over \(k[p,q]\): it is a flat and smooth \(k[p,q]\)-algebra, and for every prime \(\mathfrak y\) of \(k[p,q]\) the ring \(P\otimes_{k[p,q]}\kappa(\mathfrak y)\) is a polynomial ring in three variables over \(\kappa(\mathfrak y)\).
  2. \(A\) is a smooth \(\mathbb A^3\)-fibration over \(k[p]\).

Proof. (1) As a \(k[p,q]\)-algebra, \(P=k[p,q][s,u,F,J]/(q-H)\). Let \(\mathfrak y\) be a prime of \(k[p,q]\), \(\kappa=\kappa(\mathfrak y)\), \(b\) the image of \(q\) in \(\kappa\), and \(\mathfrak p=\mathfrak y\cap k[p]\). The map \(k[p]/\mathfrak p\to\kappa\) is injective, so \(k[p]\to\kappa\) factors through the field \(\kappa(\mathfrak p)\). The isomorphism \(\kappa(\mathfrak p)[T_0,\ldots,T_3]\to\kappa(\mathfrak p)[s,u,F,J]\), \(T_0\mapsto H_{\mathfrak p}\), \(T_i\mapsto G_i\), of Proposition 3.1 stays an isomorphism after tensoring with \(\kappa\). So \(\kappa[s,u,F,J]=\kappa[\bar H,\bar G_1,\bar G_2,\bar G_3]\) with algebraically independent generators, where \(\bar H\) is the image of \(H\). Hence \[ \begin{aligned} P\otimes_{k[p,q]}\kappa&=\kappa[s,u,F,J]/(\bar H-b)\\ &\cong\kappa[\bar G_1,\bar G_2,\bar G_3], \end{aligned} \] a polynomial ring in three variables. In particular the image \(b-\bar H\) of \(q-H\) in \(\kappa[s,u,F,J]\) is nonzero, so \(P\) is flat over \(k[p,q]\) by Lemma 2.2. It is finitely generated, and it is smooth by Lemmas 2.1 and 3.2, because \(\partial(q-H)/\partial v=-\partial H/\partial v\) for \(v=s,u,F,J\).

(2) As a \(k[p]\)-algebra, \(A=k[p][s,u,F,J]/(H)\). For a prime \(\mathfrak p\) of \(k[p]\), Proposition 3.1 gives \[ \begin{aligned} &A\otimes_{k[p]}\kappa(\mathfrak p)\\ &\quad=\kappa(\mathfrak p)[s,u,F,J]/(H_{\mathfrak p})\\ &\quad\cong\kappa(\mathfrak p)[G_1,G_2,G_3], \end{aligned} \] and \(H_{\mathfrak p}\ne0\). So \(A\) is flat by Lemma 2.2, and smooth by Lemmas 2.1 and 3.2, the partial derivatives of \(H\) generating the unit ideal of \(A\) because they do in \(P\). \(\square\)

5. Neither fibration is locally trivial

Theorem 5.1.

  1. For every maximal ideal \(\mathfrak m\ne(p)\) of \(k[p]\), \(A_{\mathfrak m}\) is a polynomial ring in three variables over \(k[p]_{\mathfrak m}\). But \(A_{(p)}=A\otimes_{k[p]}k[p]_{(p)}\) is not a polynomial ring in three variables over the discrete valuation ring \(k[p]_{(p)}\).
  2. \(P\otimes_{k[p,q]}k[p,q]_{(p,q)}\) is not a polynomial ring in three variables over \(k[p,q]_{(p,q)}\).

Proof. (1) If \(p\notin\mathfrak m\), then \(p\) is a unit of \(k[p]_{\mathfrak m}\), and Proposition 2.2 of the cylinder lesson gives \(A_{\mathfrak m}=A[p^{-1}]\otimes_{k[p,p^{-1}]}k[p]_{\mathfrak m}\), which is \(k[p]_{\mathfrak m}[x,y,z]\). Suppose that \(A_{(p)}\) were a polynomial ring in three variables over \(k[p]_{(p)}\). Then the finitely presented \(k[p]\)-algebra \(A\) would be locally polynomial in three variables, and Corollary 2.2 of Locally polynomial algebras would give \(A\cong k[p][X_1,X_2,X_3]\), a polynomial ring in four variables over \(k\). This contradicts Theorem 9.1 of Cancellation fails in dimension four.

(2) Let \(R=k[p,q]_{(p,q)}\). Reduction modulo \(q\) turns \(R\) into \(k[p]_{(p)}\), and, since \(q\) maps to \(H\) and \(P/HP=A\), it turns \(P\otimes_{k[p,q]}R\) into \(A\otimes_{k[p]}k[p]_{(p)}=A_{(p)}\). An isomorphism of \(R\)-algebras \(P\otimes_{k[p,q]}R\cong R[X_1,X_2,X_3]\) would reduce to an isomorphism of \(k[p]_{(p)}\)-algebras \(A_{(p)}\cong k[p]_{(p)}[X_1,X_2,X_3]\), contrary to (1). \(\square\)

Corollary 5.2. Neither \((p,H)\colon\mathbb A^5\to\mathbb A^2\) nor \(p\colon\operatorname{Spec}A\to\mathbb A^1\) is a Zariski-locally trivial \(\mathbb A^3\)-bundle. None of the \(\mathbb A^3\)-fibrations \(P\) over \(k[p,q]\), \(A\) over \(k[p]\) and \(A_{(p)}\) over \(k[p]_{(p)}\) is isomorphic to the symmetric algebra of a projective module. So the question recalled at the beginning has a negative answer for \(n=3\), over the regular local ring \(k[p]_{(p)}\) and over the polynomial rings \(k[p]\) and \(k[p,q]\).

Proof. \(A_{(p)}\) is an \(\mathbb A^3\)-fibration over \(k[p]_{(p)}\): it is flat by Proposition 3.2 of Tor and flat modules, and its fibre rings over the two primes of \(k[p]_{(p)}\) are fibre rings of \(A\). A Zariski-locally trivial \(\mathbb A^3\)-bundle is locally polynomial (Exercise 6.1), which Theorem 5.1 excludes at the maximal ideals \((p,q)\) and \((p)\).

Let \(B\) be one of the three algebras over its base \(R_0\), and suppose \(B\cong S(Q)\) with \(Q\) a projective \(R_0\)-module. By Lemma 1.3 of Locally polynomial algebras, \(Q\) is finitely presented, so for a maximal ideal \(\mathfrak m\) of \(R_0\) the module \(Q_{\mathfrak m}\) is free, by Lemma 1.1 of Horrocks' theorem and the Quillen–Suslin theorem. Its rank is three: \(S(Q\otimes\kappa(\mathfrak m))\cong B\otimes\kappa(\mathfrak m)\) is a polynomial ring in three variables, so \(Q\otimes\kappa(\mathfrak m)\cong\kappa(\mathfrak m)^3\) by Lemma 1.2 of Locally polynomial algebras. Hence \(B_{\mathfrak m}\cong S(R_{0,\mathfrak m}^3)\) is a polynomial ring in three variables over \(R_{0,\mathfrak m}\) at every maximal ideal \(\mathfrak m\), contrary to Theorem 5.1 at \(\mathfrak m=(p,q)\), respectively \((p)\). \(\square\)

6. Exercises

Exercise 6.1. Let \(B\) be a finitely generated \(R\)-algebra which is a Zariski-locally trivial \(\mathbb A^n\)-bundle. Show that \(B\) is an \(\mathbb A^n\)-fibration and that \(B_{\mathfrak m}\cong R_{\mathfrak m}[X_1,\ldots,X_n]\) for every maximal ideal \(\mathfrak m\) of \(R\).

Exercise 6.2. Show that \(P[p^{-1}]\cong k[p,p^{-1},q][X_1,X_2,X_3]\) as \(k[p,p^{-1},q]\)-algebras, so that \((p,H)\) is a trivial bundle over the open set \(p\ne0\) of \(\mathbb A^2\).

Exercise 6.3. Show that \(A[w]\cong k[p][X_1,X_2,X_3,X_4]\) as \(k[p]\)-algebras. Deduce that over the discrete valuation ring \(V=k[p]_{(p)}\), the algebra \(A_V=A\otimes_{k[p]}V\) satisfies \(A_V[w]\cong V[X_1,\ldots,X_4]\), although \(A_V\) is not a polynomial ring over \(V\).

Exercise 6.4. For \(c\in k\), show that \(P/(H-c)\) is an \(\mathbb A^3\)-fibration over \(k[p]\) and that \((P/(H-c))[w]\) is a polynomial ring in five variables over \(k\).

7. Solutions

6.1. For a maximal ideal \(\mathfrak m\), choose \(i\) with \(g_i\notin\mathfrak m\). Then \(B_{\mathfrak m}=(B_{g_i})_{\mathfrak m}\cong R_{\mathfrak m}[X_1,\ldots,X_n]\), a free \(R_{\mathfrak m}\)-module, so \(B\) is flat by Theorem 3.3 of Tor and flat modules. For a prime \(\mathfrak p\), choose \(i\) with \(g_i\notin\mathfrak p\). Then \(\kappa(\mathfrak p)\) is an \(R_{g_i}\)-algebra, and \(B\otimes_R\kappa(\mathfrak p)=B_{g_i}\otimes_{R_{g_i}}\kappa(\mathfrak p)\), which is isomorphic to \(\kappa(\mathfrak p)[X_1,\ldots,X_n]\).

6.2. By the proof of Proposition 2.2 of the cylinder lesson, \(P[p^{-1}]=k[p,p^{-1}][x,y,z,H]\) with \(x,y,z,H\) algebraically independent over \(k[p,p^{-1}]\). Since \(q\) maps to \(H\), this is \(k[p,p^{-1},q][x,y,z]\).

6.3. The automorphism \(\Phi=\exp(w\Delta)\) of the cylinder lesson fixes \(p\) and \(w\) and induces an isomorphism of \(k[p]\)-algebras \(A[w]\cong T\), by (3.2) there, and \(T=k[p][s,u,M,e]\) with \(s,u,M,e\) algebraically independent over \(k[p]\), by Theorem 4.1 there. Tensoring with \(V\) gives \(A_V[w]\cong V[s,u,M,e]\). If \(A_V\) were a polynomial ring over \(V\), it would have three variables, since \(A_V\otimes_Vk=A/pA\) is a polynomial ring in three variables over \(k\); this is excluded by Theorem 5.1.

6.4. \(P/(H-c)=P\otimes_{k[p,q]}k[p,q]/(q-c)\), and \(k[p,q]/(q-c)=k[p]\). Finite generation and flatness are preserved by base change (Proposition 3.2 of Tor and flat modules). For a prime \(\mathfrak p\) of \(k[p]\) with preimage \(\mathfrak y\) in \(k[p,q]\), \(\kappa(\mathfrak y)=\kappa(\mathfrak p)\) and the fibre ring \((P/(H-c))\otimes_{k[p]}\kappa(\mathfrak p)=P\otimes_{k[p,q]}\kappa(\mathfrak y)\) is a polynomial ring in three variables by Theorem 4.1. By Theorem 3.1 of A stable coordinate in five variables, \(P[w]=k[H,f_2,\ldots,f_6]\) with \(H,f_2,\ldots,f_6\) algebraically independent, so \(P[w]/(H-c)\cong k[\bar f_2,\ldots,\bar f_6]\) by Exercise 4.3 there; and \(P[w]/(H-c)=(P/(H-c))[w]\).

References