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Tag 0BQS

Chapter 35: Derived Categories of Schemes > Section 35.15: An example generator

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 vectorbundles 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. $\square$

    The code snippet corresponding to this tag is a part of the file perfect.tex and is located in lines 3221–3231 (see updates for more information).

    \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 vectorbundles 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}

    Comments (1)

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

    It should say vector bundles, not vectorbundles.

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