Locally polynomial algebras

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

A finitely presented algebra \(A\) over a ring \(K\) is locally polynomial if \(A_{\mathfrak m}\) is a polynomial ring over \(K_{\mathfrak m}\) for every maximal ideal \(\mathfrak m\) of \(K\). Geometrically, \(\operatorname{Spec}A\to\operatorname{Spec}K\) is then a family of affine spaces which is trivial over a neighbourhood \(\operatorname{Spec}K_s\), \(s\notin\mathfrak m\), of every maximal ideal \(\mathfrak m\), by Lemma 2.1(3) of the automorphisms lesson. This lesson proves the theorem of Bass, Connell and Wright: such an algebra is the symmetric algebra of a finitely generated projective module, so the fibre space comes from a vector bundle. With the Quillen–Suslin theorem it follows that over a polynomial ring over a field, every locally polynomial algebra is a polynomial ring.

We use Patching extended modules (Lemma 1.2, Proposition 3.1), Horrocks' theorem and the Quillen–Suslin theorem (Lemma 1.1, Theorem 3.2) and Automorphisms of symmetric algebras (Lemma 2.1, Proposition 4.4 and the notation of Section 3); Theorem 2.2 (localizing \(\operatorname{Hom}\)) of Localization, local properties and support; and Schanuel's lemma, Lemma 1.1 of Projective dimension and the Auslander–Buchsbaum formula. Rings and algebras are commutative, and \(K\) is a fixed ring.

1. Modules over two localizations

Lemma 1.1. Let \(s_0,s_1\in K\) with \(Ks_0+Ks_1=K\).

  1. For every \(K\)-module \(E\), every element of \(E_{s_0s_1}\) is the sum of the images of an element of \(E_{s_0}\) and an element of \(E_{s_1}\).
  2. Let \(M_i\) be a \(K_{s_i}\)-module (\(i=0,1\)) and \(\varphi\colon(M_0)_{s_1}\to(M_1)_{s_0}\) an isomorphism. Then the set \(M\) of pairs \((m_0,m_1)\in M_0\times M_1\) with \(\varphi(m_0)=m_1\) in \((M_1)_{s_0}\) is a \(K\)-module, and the projections induce isomorphisms \(M_{s_i}\to M_i\) compatible with \(\varphi\).
  3. If \(E_{s_0}\) and \(E_{s_1}\) are finitely presented, then \(E\) is finitely presented.
  4. If \(S\subseteq K\) is multiplicative, every finitely presented \(K_S\)-module is isomorphic to \(N_S\) for some \(s\in S\) and some finitely presented \(K_s\)-module \(N\).

Proof. (1) Write the element as \(x/(s_0s_1)^n\) and \(1=cs_0^n+ds_1^n\). Then \(x/(s_0s_1)^n=cx/s_1^n+dx/s_0^n\), the images of \(cx/s_1^n\in E_{s_1}\) and \(dx/s_0^n\in E_{s_0}\).

(2) Consider \(M\to M_0\). If \((m_0,m_1)\in M\) and \(m_0=0\), then \(m_1\) is zero in \((M_1)_{s_0}\), so \(s_0^km_1=0\) for some \(k\), and \((m_0,m_1)\) is zero in \(M_{s_0}\). Given \(m_0\in M_0\), write \(\varphi(m_0)=m_1'/s_0^k\); then \((s_0^km_0,m_1')\in M\), so \(m_0\) is the image of \((s_0^km_0,m_1')/s_0^k\). Hence \(M_{s_0}\to M_0\) is bijective, and symmetrically, using \(\varphi^{-1}\), so is \(M_{s_1}\to M_1\). The two identifications differ on \(M_{s_0s_1}\) exactly by \(\varphi\).

(3) Choose finitely many elements of \(E\) whose images generate \(E_{s_0}\) and \(E_{s_1}\). The cokernel of the map \(K^n\to E\) they define vanishes after inverting \(s_0\) and after inverting \(s_1\); an element killed by powers of \(s_0\) and of \(s_1\) is zero, so the map is onto. Its kernel \(Z\) satisfies: \(Z_{s_i}\) is the kernel of a surjection from a finitely generated free module onto the finitely presented \(E_{s_i}\), hence finitely generated, by Schanuel's lemma comparing with a finite presentation. The same argument shows that \(Z\) is finitely generated.

(4) A finite presentation matrix over \(K_S\) has entries with a common denominator \(s\in S\); the same matrix over \(K_s\) presents a module \(N\) with \(N_S\) isomorphic to the given one. \(\square\)

Lemma 1.2. If \(M,N\) are \(K\)-modules and \(S(M)\cong S(N)\) as \(K\)-algebras, then \(M\cong N\).

Proof. Let \(\phi\colon S(M)\to S(N)\) be an isomorphism and \(t=\varepsilon_N\circ\phi|_M\colon M\to K\). With the translation \(\tau_{-t}\) of \(S(M)\), the isomorphism \(w=\phi\circ\tau_{-t}\) maps \(x\in M\) to \(\phi(x)-t(x)\in I_N\). So \(\varepsilon_N\circ w=\varepsilon_M\), hence also \(\varepsilon_M\circ w^{-1}=\varepsilon_N\), and \(w(I_M)=I_N\), \(w(I_M^2)=I_N^2\). Thus \(M\cong I_M/I_M^2\cong I_N/I_N^2\cong N\). \(\square\)

Lemma 1.3. Let \(A\) be a finitely presented algebra and \(\varepsilon\colon A\to K\) an algebra homomorphism with kernel \(I\). Then \(I/I^2\) is a finitely presented \(K\)-module. In particular, if \(A\cong S(N)\) for some \(K\)-module \(N\), then \(N\) is finitely presented.

Proof. Write \(A=K[X_1,\ldots,X_p]/(f_1,\ldots,f_q)\). After the substitution \(X_i\mapsto X_i+\varepsilon(x_i)\) we may assume \(\varepsilon(x_i)=0\); then \(f_j(0)=0\), and \(I\) is generated by \(x_1,\ldots,x_p\). So \(I/I^2\) is generated by their classes, and a linear form \(\lambda=\sum a_iX_i\) maps to zero exactly when \(\lambda\in(X)^2+(f_1,\ldots,f_q)\), i.e. when \(\lambda\) is the linear part of an element \(\sum g_jf_j\). That linear part is \(\sum g_j(0)\,\ell_j\), where \(\ell_j\) is the linear part of \(f_j\), because \(f_j\) has no constant term. So \(I/I^2\cong K^p/(\ell_1,\ldots,\ell_q)\). For the second assertion, let \(\psi\colon S(N)\to A\) be an isomorphism and \(\varepsilon=\varepsilon_N\circ\psi^{-1}\). Then \(I=\psi(I_N)\), so \(N\cong I_N/I_N^2\cong I/I^2\). \(\square\)

2. The theorem

Theorem 2.1 (Bass–Connell–Wright). Let \(A\) be a finitely presented \(K\)-algebra such that for every maximal ideal \(\mathfrak m\) of \(K\), \(A_{\mathfrak m}\cong S(N_{\mathfrak m})\) as \(K_{\mathfrak m}\)-algebras for some \(K_{\mathfrak m}\)-module \(N_{\mathfrak m}\). Then \(A\cong S(N)\) for a finitely presented \(K\)-module \(N\).

Proof. If \(K=0\), then \(A=0=S(0)\); let \(K\ne0\). We apply Quillen induction (Proposition 3.1 of the patching lesson) to the property \(\mathcal P(L)\): \(A_L\cong S(N)\) as \(L\)-algebras for some finitely presented \(L\)-module \(N\), for \(L\in\operatorname{Loc}(K)\).

Specialization. \(S(N)\) and finite presentation are compatible with base change.

Local validity. This is the hypothesis, together with Lemma 1.3.

Finiteness. Let \(A_S\cong S(N)\) with \(N\) finitely presented over \(K_S\). By Lemma 1.1(4), \(N\cong N'_S\) with \(N'\) finitely presented over some \(K_s\), \(s\in S\). The \(K_s\)-algebras \(A_s\) and \(S(N')\) are finitely presented and become isomorphic over \(K_S\), so by Lemma 2.1 of the automorphism lesson (over the ring \(K_s\)) they are isomorphic over \(K_{st}\) for some \(t\in S\).

Sheaf condition. Let \(L\in\operatorname{Loc}(K)\) and \(s_0,s_1\in L\) with \(Ls_0+Ls_1=L\); everything below takes place over \(L\), and to simplify notation we write \(K\) for \(L\). We are given finitely presented \(K_{s_i}\)-modules \(M_i\) and isomorphisms \(u_i\colon S(M_i)\to A_{s_i}\).

Step 1: a global module. Over \(K_{s_0s_1}\), \(S((M_0)_{s_1})\cong A_{s_0s_1}\cong S((M_1)_{s_0})\), so Lemma 1.2 gives an isomorphism \(\varphi\colon(M_0)_{s_1}\to(M_1)_{s_0}\). Lemma 1.1(2) and (3) give a finitely presented \(K\)-module \(M\) with \(M_{s_i}=M_i\), the identifications differing by \(\varphi\). Put \(B=S(M)\); the \(u_i\) become isomorphisms \(u_i\colon B_{s_i}\to A_{s_i}\).

Step 2: an affine comparison. The element \(\theta=(u_0)_{s_1}^{-1}(u_1)_{s_0}\) lies in \(GA_M(K_{s_0s_1})\). By Proposition 4.4 of the automorphism lesson, \(\theta=(v_0)_{s_1}\,h\,(v_1)_{s_0}\) with \(v_0\in G_{M,0}(K_{s_0})\), \(h\in Af_M(K_{s_0s_1})\) and \(v_1\in GA_M(K_{s_1})\). Put \(w_0=u_0v_0\) and \(w_1=u_1v_1^{-1}\), isomorphisms \(B_{s_i}\to A_{s_i}\). Then \[ (w_0)_{s_1}^{-1}(w_1)_{s_0}=(v_0)_{s_1}^{-1}\,\theta\,(v_1)_{s_0}^{-1}=h . \]

Step 3: the linear part of \(A\). Let \(B^{\le1}=K\oplus M\subseteq B\). Since \(h\) is affine, \(h(B^{\le1}_{s_0s_1})=B^{\le1}_{s_0s_1}\), so the submodules \(w_i(B^{\le1}_{s_i})\subseteq A_{s_i}\) have the same localization in \(A_{s_0s_1}\). Let \(A^{\le1}\) be the set of \(a\in A\) with \(a\in w_0(B^{\le1}_{s_0})\) in \(A_{s_0}\) and \(a\in w_1(B^{\le1}_{s_1})\) in \(A_{s_1}\). Since localization is exact, the localizations of \(M_i'=w_i(B^{\le1}_{s_i})\) at \(s_{1-i}\) are submodules of \(A_{s_0s_1}\), and they are equal. By Lemma 1.2 of the patching lesson, \(A\) is the fibre product of \(A_{s_0}\) and \(A_{s_1}\) over \(A_{s_0s_1}\), so \(A^{\le1}\) is the module obtained from \(M_0'\) and \(M_1'\) by Lemma 1.1(2) with \(\varphi\) the identity; hence \((A^{\le1})_{s_i}\to A_{s_i}\) is injective with image \(M_i'=w_i(B^{\le1}_{s_i})\). It contains \(K\cdot1\), and \(K\to A\) is injective, because it is so after inverting \(s_0\) and \(s_1\). Let \(N=A^{\le1}/K\). Then \(w_i\) induces \(N_{s_i}\cong B^{\le1}_{s_i}/K_{s_i}=M_{s_i}\), and \(N\) is finitely presented by Lemma 1.1(3).

Step 4: a splitting. The maps \(\rho_i=\varepsilon\circ w_i^{-1}\colon(A^{\le1})_{s_i}\to K_{s_i}\), with \(\varepsilon\) the augmentation of \(B\), are retractions of the inclusions of \(K_{s_i}\). Their difference on \(A^{\le1}_{s_0s_1}\) kills \(K\), so it is \(\delta\circ\pi\) with \(\pi\colon A^{\le1}\to N\) and \(\delta\in\operatorname{Hom}(N_{s_0s_1},K_{s_0s_1})\). As \(N\) is finitely presented, \(\operatorname{Hom}(N_{s_0s_1},K_{s_0s_1})=\operatorname{Hom}_K(N,K)_{s_0s_1}\), and by Lemma 1.1(1), \(\delta=\alpha+\beta\) with \(\alpha\) coming from \(\operatorname{Hom}(N_{s_0},K_{s_0})\) and \(\beta\) from \(\operatorname{Hom}(N_{s_1},K_{s_1})\). The retractions \(\rho_0-\alpha\pi\) and \(\rho_1+\beta\pi\) agree on \(A^{\le1}_{s_0s_1}\). By the exact sequence of Lemma 1.2 of the patching lesson for the module \(K\), there is for each \(a\in A^{\le1}\) a unique \(\rho(a)\in K\) whose images in \(K_{s_0}\) and \(K_{s_1}\) are the values of these two retractions at \(a\). This defines a \(K\)-linear \(\rho\colon A^{\le1}\to K\), and \(\rho(c)=c\) for \(c\in K\), since this holds after inverting \(s_0\) and \(s_1\). Then \(A^{\le1}=K\cdot1\oplus\ker\rho\) and \(\ker\rho\cong N\).

Step 5: the isomorphism. Let \(t\colon S(N)\to A\) be the algebra map extending \(N\cong\ker\rho\subseteq A\). For each \(i\), \(w_i^{-1}\circ t_{s_i}\colon S(N_{s_i})\to S(M_{s_i})\) sends \(N_{s_i}\) into \(K_{s_i}\oplus M_{s_i}\), and its component in \(M_{s_i}\) is the isomorphism \(N_{s_i}\cong M_{s_i}\) of Step 3. So it is the composite of the graded isomorphism defined by that linear map with a translation, an isomorphism. Hence \(t_{s_0}\) and \(t_{s_1}\) are isomorphisms, and so is \(t\), its kernel and cokernel being killed by powers of \(s_0\) and of \(s_1\). This proves \(\mathcal P(K)\), the sheaf condition.

Quillen induction now gives \(\mathcal P(K)\). \(\square\)

Corollary 2.2. Let \(A\) be a finitely presented \(K\)-algebra with \(A_{\mathfrak m}\cong K_{\mathfrak m}[X_1,\ldots,X_n]\) for every maximal ideal \(\mathfrak m\). Then \(A\cong S(P)\) for a finitely generated projective \(K\)-module \(P\) of rank \(n\). If every finitely generated projective \(K\)-module of rank \(n\) is free, then \(A\cong K[X_1,\ldots,X_n]\). This holds for \(K=F[y_1,\ldots,y_r]\) with \(F\) a field.

Proof. Theorem 2.1 gives \(A\cong S(P)\) with \(P\) finitely presented. For each \(\mathfrak m\), \(S(P_{\mathfrak m})\cong S(K_{\mathfrak m}^n)\), so \(P_{\mathfrak m}\cong K_{\mathfrak m}^n\) by Lemma 1.2. Every prime \(\mathfrak q\) lies in a maximal ideal \(\mathfrak m\), and \(P_{\mathfrak q}\) is a localization of \(P_{\mathfrak m}\), hence free of rank \(n\). By Lemma 1.1(2) of the Horrocks lesson, \(P\) is projective, of rank \(n\). If \(P\cong K^n\), then \(A\cong S(K^n)=K[X_1,\ldots,X_n]\). For \(K=F[y_1,\ldots,y_r]\), freeness is the Quillen–Suslin theorem. \(\square\)

3. Exercises

Exercise 3.1. For \(K=\mathbb Z\) and \(N=\mathbb Z/2\), describe \(S(N)\) as a quotient of \(\mathbb Z[X]\) and decide whether it is locally polynomial.

Exercise 3.2. For \(A=K[X,Y]/(X-Y^2)\) with the augmentation \(X,Y\mapsto0\), compute \(I/I^2\) by the method of Lemma 1.3.

Exercise 3.3. Let \(K=\mathbb R[x,y,z]/(x^2+y^2+z^2-1)\) and let \(P\) be the kernel of the map \(K^3\to K\), \((a,b,c)\mapsto ax+by+cz\). Show that \(P\oplus K\cong K^3\), that \(S(P)[T]\cong K[X_1,X_2,X_3]\), and that \(S(P)\) is locally polynomial in two variables. Show that if \(P\) is not free, then \(S(P)\) is not isomorphic to \(K[X_1,X_2]\).

Exercise 3.4. Show that Corollary 2.2 applies to \(K=F[y]\) with \(F\) a field without using the Quillen–Suslin theorem.

4. Solutions

3.1. \(S(\mathbb Z/2)=\mathbb Z[X]/(2X)\). It is not locally polynomial at \((2)\): a polynomial ring over \(\mathbb Z_{(2)}\) is a domain, but \(2\cdot X=0\) with \(2\ne0\) and \(X\ne0\) in the localization. (At primes \(p\ne2\), the localization is \(\mathbb Z_{(p)}\), polynomial in zero variables.)

3.2. Here \(q=1\), \(f_1=X-Y^2\) with linear part \(\ell_1=X\), so \(I/I^2\cong K^2/(K\cdot e_1)\cong K\), generated by the class of \(Y\). Indeed \(A\cong K[Y]\).

3.3. The element \((x,y,z)\) is mapped to \(1\), so \(K^3=K\cdot(x,y,z)\oplus P\). Then \(S(P)[T]=S(P)\otimes S(K)\), which is \(S(P\oplus K)\cong S(K^3)\). \(P\) is a direct summand of \(K^3\), hence finitely presented and projective, and \(P_{\mathfrak m}\) is free (Lemma 1.1(1) of the Horrocks lesson) of rank \(2\), since \(P_{\mathfrak m}\oplus K_{\mathfrak m}\cong K_{\mathfrak m}^3\). So \(S(P)\) is finitely presented and \(S(P)_{\mathfrak m}=S(P_{\mathfrak m})\cong K_{\mathfrak m}[X_1,X_2]\). If \(S(P)\cong K[X_1,X_2]=S(K^2)\), Lemma 1.2 would give \(P\cong K^2\).

3.4. \(F[y]\) is a principal ideal domain, so finitely generated projective modules are free by Lemma 1.2(1) of the Horrocks lesson.

References