# The Stacks Project

## Tag 0BQS

Lemma 35.15.2. 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 26.13.2. 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 15.27.2) 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 26.8.4 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. $\square$

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\begin{lemma}
\label{lemma-construct-the-next-one}
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 $X$.
\end{lemma}

\begin{proof}
Recall that $\mathbf{P}^n_A$ is $\text{Proj}(A[X_0, \ldots, X_n])$, see
Constructions, Definition \ref{constructions-definition-projective-space}.
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 \ref{more-algebra-lemma-regular-koszul-regular})
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 \ref{constructions-lemma-proj-sheaves} 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.
\end{proof}

Comment #2544 by Pieter Belmans (site) on May 16, 2017 a 7:14 am UTC

It should say vector bundles, not vectorbundles.

Comment #2577 by Johan (site) on May 25, 2017 a 7:51 pm UTC

Thanks Pieter. See here.

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