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Projective generation of ideals in polynomial extensions. (English) Zbl 1448.13018

For a commutative Noetherian ring (containing \(\mathbb{Q}\)), set \(\mathcal{P}_n (R)\) the set of (isomorphism classes of) projective \(R\) modules of rank \(n\). The paper is mainly devoted prove two results.
The first one states that if \(R\) is an affine \(k\)-algebra of dimension \(n \geq 3\) (\(k\) is a characteristic zero field), \(L \in \mathcal{P}_1 (R)\) and \(I \subset R[t]\) a local complete intersection ideal of height \(n\) such that \(\mathrm{ht}(I(0)) \geq n\), and one also assume that there exists \(Q \in \mathcal{P}_n (R)\) with determinant \(L\) and a surjection \(Q[t] \rightarrow I/I^2 \cap (t)\), then there exists \(P \in \mathcal{P}_n (R[t])\) with determinant \(L[t]\) and a surjection \(P \rightarrow I\).
The second result is the following one. Let \(R\) be a reduced affine \(k\)-algebra of dimension \(n \geq 3\) and \(L \in \mathcal{P}_1 (R)\). Let \(I \subset R[t]\) be an ideal of height \(n\) such that \(\mathrm{ht}(I(0)) \geq n\). Assume that \((I, \omega_I)\) belongs to the Euler class group \(\mathrm{E}(R[t],L)\) when \(n\geq 4\) and \((I, \omega_I)\) is in the restricted Euler class group of \(R[t]\) when \(n=3\). Also assume that there exists \(Q \in \mathcal{P}_n (R)\) with determinant \(L/tL\) and an isomorphism \(f: L/tL \rightarrow \bigwedge^n (Q)\) such that \(e(Q,f) =(I(0), \omega_{I(0)})\) in \(\mathrm{E\mathrm{}}(R,L/tL)\). Then, there exists \(P \in \mathcal{P}_n (R[t])\) with determinant \(L\) and an isomorphism \(f_1: L\rightarrow \bigwedge^n (P)\) such that \(e(P,f_1)=(I,\omega)\) in \(\mathrm{E}(R[t],L)\).
The authors also deduce a corollary that improves results of the references [S. M. Bhatwadekar and M. K. Das, Int. Math. Res. Not. 2015, No. 4, 960–980 (2015; Zbl 1342.13012)] and [S. M. Bhatwadekar and R. Sridharan, \(K\)-Theory 15, No. 3, 293–300 (1998; Zbl 0951.13006)] in the paper.

MSC:

13C10 Projective and free modules and ideals in commutative rings
13B25 Polynomials over commutative rings
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References:

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