Lemma 36.16.2 (0BQS)—The Stacks project

Lemma 36.16.2. Let $A$ be a ring. Let $X = \mathbf{P}^ n_ A$. For every $a \in \mathbf{Z}$ there exists an exact complex

\[ 0 \to \mathcal{O}_ X(a) \to \ldots \to \mathcal{O}_ X(a + i)^{\oplus {n + 1 \choose i}} \to \ldots \to \mathcal{O}_ X(a + n + 1) \to 0 \]

of vector bundles on $X$.

Proof. Recall that $\mathbf{P}^ n_ A$ is $\text{Proj}(A[X_0, \ldots , X_ n])$, see Constructions, Definition 27.13.2. Consider the Koszul complex

\[ K_\bullet = K_\bullet (A[X_0, \ldots , X_ n], X_0, \ldots , X_ n) \]

over $S = A[X_0, \ldots , X_ n]$ on $X_0, \ldots , X_ n$. Since $X_0, \ldots , X_ n$ is clearly a regular sequence in the polynomial ring $S$, we see that (More on Algebra, Lemma 15.30.2) that the Koszul complex $K_\bullet $ is exact, except in degree $0$ where the cohomology is $S/(X_0, \ldots , X_ n)$. Note that $K_\bullet $ becomes a complex of graded modules if we put the generators of $K_ i$ in degree $+i$. In other words an exact complex

\[ 0 \to S(-n - 1) \to \ldots \to S(-n - 1 + i)^{\oplus {n \choose i}} \to \ldots \to S \to S/(X_0, \ldots , X_ n) \to 0 \]

Applying the exact functor $\tilde{\ }$ functor of Constructions, Lemma 27.8.4 and using that the last term is in the kernel of this functor, we obtain the exact complex

\[ 0 \to \mathcal{O}_ X(-n - 1) \to \ldots \to \mathcal{O}_ X(-n - 1 + i)^{\oplus {n + 1 \choose i}} \to \ldots \to \mathcal{O}_ X \to 0 \]

Twisting by the invertible sheaves $\mathcal{O}_ X(n + a)$ we get the exact complexes of the lemma. $\square$

The Stacks project

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