Coherent sheaf

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

A coherent sheaf on a ringed space $(X,\def\cO{ {\mathcal O}}\cO)$ is a sheaf of modules $\def\cF{ {\mathcal F}}\cF$ over a sheaf of rings $\cO$ with the following properties:

1) $\cF$ is a sheaf of finite type, that is, it is locally generated over $\cO$ by a finite number of sections; and

2) the kernel of any homomorphism of sheaves of modules $\cO^p\mid_U\to \cF\mid_U$ over an open set $\cF\mid_U$ is a sheaf of finite type.

If in an exact sequence $0\to \cF_1\to\cF_2\to\cF_3\to 0$ of sheaves of $\cO$-modules two of the three sheaves $\cF_i$ are coherent, then the third is coherent as well. If $\def\phi{\varphi}\phi:\cF\to\def\cS{ {\mathcal S}}\cS$ is a homomorphism of coherent sheaves of $\cO$-modules, then ${\rm Ker}\;\phi$, ${\rm Coker}\;\phi$, ${\rm Im}\;\phi$ are also coherent sheaves. If $\cF$ and $\cS$ are coherent, then so are $\cF\otimes_\cO \cS$ and ${\rm Hom}_\cO(\cF,\cS)$ [Se].

A structure sheaf $\cO$ is called a coherent sheaf of rings if $\cO$ is coherent as a sheaf of modules over itself, which reduces to condition 2). If $\cO$ is a coherent sheaf of rings, then a sheaf of $\cO$-modules $\cF$ is coherent if and only if every point of the space $X$ has a neighbourhood $U$ over which there is an exact sequence of sheaves of $\cO$-modules:

$$\cO^p\mid_U\to\cO\mid_U\to\cF\mid_U\to 0,$$ [Se]. Furthermore, under this condition ${\rm Ext}_\cO^p(\cF,\cS)$ is coherent for any coherent sheaves $\cF$, $\cS$ and for all $p$ (see [BaSt]).

The fundamental classes of ringed spaces with a coherent structure sheaf $\cO$ are: analytic spaces over algebraically closed fields [Ab], Noetherian schemes and, in particular, algebraic varieties [Se]. A classical special case is the sheaf $\cO$ of germs of holomorphic functions in a domain of $\C^n$; the statement that it is coherent is known as the Oka coherence theorem [GuRo], [Fu]. The structure sheaf of a real-analytic space is not coherent, in general.

See also Coherent analytic sheaf; Coherent algebraic sheaf.

References

[Ab]	S.S. Abhyankar, "Local analytic geometry", Acad. Press (1964) MR0175897 Zbl 0205.50401
[BaSt]	C. Banica, O. Stanasila, "Algebraic methods in the global theory of complex spaces", Wiley (1976) (Translated from Rumanian) MR0463470 Zbl 0334.32001
[Fu]	B.A. Fuks, "Special chapters in the theory of analytic functions of several complex variables", Amer. Math. Soc. (1965) (Translated from Russian) MR0188477 Zbl 0146.30802
[GuRo]	R.C. Gunning, H. Rossi, "Analytic functions of several complex variables", Prentice-Hall (1965) MR0180696 Zbl 0141.08601
[Se]	J.-P. Serre, "Faisceaux algébriques cohérents" Ann. of Math., 61 (1955) pp. 197–278 MR0068874 Zbl 0067.16201