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    Finiteness of cohomology for coherent sheaves on a projective scheme

    T-COHFIN

    Properness plus coherence makes cohomology finite: coherent cohomology on a projective scheme is finitely generated, hence finite-dimensional over a field.Suggest a slogan

    Let \(A\) be Noetherian, \(X\) projective over \(A\), and \(\mcf \in \Coh(X)\). Then each \(H^i(X, \mcf)\) is a finitely generated \(A\dash\)module. Over a field, every \(h^i(X,\mcf)\) is finite and vanishes for \(i > \dim X\).

    (Proof shape)

    Reduce to \(X = \PP^n_A\) by pushing forward along the closed immersion, which changes no cohomology. Any coherent \(\mcf\) on \(\PP^n\) sits in \(0 \to \mcr \to \mce \to \mcf \to 0\) with \(\mce\) a finite sum of twists \(\OO(-d_i)\), and \(\mcr\) is again coherent. The twists have finitely generated cohomology by the explicit computation, and descending induction on \(i\) from Grothendieck vanishing runs the long exact sequence down.

    (Hypotheses)

    Properness is what fails first and it fails already in degree \(0\): on \(X = \AA^1_k\), \(H^0(X,\OO) = k[x]\) is infinite dimensional. So finiteness is a statement about proper schemes, not about nice sheaves, and the correct general form is for \(f: X \to \Spec A\) proper with \(\mcf\) coherent.

    Coherence is the second hypothesis, and it fails on quasicoherent sheaves for the same reason: \(\bigoplus_{d \geq 0} \OO(d)\) on \(\PP^n\) has infinite dimensional \(H^0\).

    The practical payoff is that \(h^0, h^1, \ldots\) are numbers, so \(\chi\) exists and Riemann–Roch can be a statement about integers.

    Why is $H^i(X, \mcf)$ finite dimensional?
    Which hypothesis in Serre finiteness fails first?
    If $X \subseteq \PP^n_k$ is a closed subscheme and $\mcf$ is a coherent $\OO_X$-module, why is $\Gamma(X, \mcf)$ finite dimensional over $k$?

    Dependencies

    • T-IJW1KThe cohomology of $\OO_{\PP^n}(d)$uses
    • PR-COHLESThe long exact sequence, and the ideal sequence as a computational deviceuses

    Backlinks

    • T-CARTSERREFiniteness of coherent cohomology (Cartan--Serre)related-to
    • T-COHSVANSerre vanishing for large twistsuses
    • D-COHEULERThe Euler characteristic and the Hilbert polynomialuses

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