Modules of principal parts on the projective line (Q1880456)

The author studies the splitting type of the \(k\)-th order principal parts \(\mathcal P^k_{\mathbb P^1}\) on the projective line \(\mathbb P^1\) twisted by the \(n\)-th tensor product \(\mathcal O_{\mathbb P^1}(n)\) of the {tautological} bundle. When the characteristic is \(0\) the author gives an explicit construction of the well known splitting \(\bigoplus_{i=0}^k \mathcal O_{\mathbb P^1}(n-k)\) as a left module. The author also studies the case \(k=1\) when \(\mathcal P^1_{\mathbb P^1}(\mathcal O_{\mathbb P^1}(n))\) is considered both as a left and as a right module. He obtains the quite surprising result that the splitting depends both on whether the twisted principal parts are considered as a left or a right module, and on the characteristic of the base field. More precisely, as a right module \(\mathcal P^1_{\mathbb P^1}(\mathcal O_{\mathbb P^1})\) splits as \(\mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(n-2)\). As a left module it splits as \(\mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(n-2)\) when the characteristic divides \(n\) and as \(\mathcal O_{\mathbb P^1}(n-1)\oplus \mathcal O_{\mathbb P^1}(n-1)\) when the characteristic does not divide \(n\).