## Nonnegative additive functions on coherent sheaves

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

Let $(X,\mathcal{O}_X)$ be a Noetherian integral scheme and let $g$ be a (numerical) additive nonnegative function from coherent $\mathcal{O}_X$-modules to $[0,\infty)$. This question may be well known to the expert but I couldn't find a reference: is $g$ a constant multiple of generic rank? If true, do you know of any reference for this?

Notes:

1. If $X=\mathrm{Spec}\:R$ is affine with $R$ an integral domain, then a proof can be found in Northcott-Reufel, Theorem 2, p. 303. There are also other proofs.
2. If $X$ is a projective variety over a field, I think I can prove it, but I don't know any reference for this case.

I have a feeling this question must have been answered in K-theory.

-
Basically you ask for homomorphisms $G_0(X) \to \mathbb{R}_+$. – Martin Brandenburg Dec 13 2011 at 8:48

4
To Martin: no, he is asking for homomorphisms G0(X)→ℝ that are non-negative on classes of coherent sheaves. – Angelo Dec 13 2011 at 9:40

## 1 Answer

I suppose that "additive" means that "additive over short exact sequences". If so, this is does not seem too hard, at least if $X$ is separated.

By noetherian induction, you may assume that for all proper integral subscheme $Y$ of $X$, the restriction of $g$ to $Y$ is given by a multiple of the generic rank at $Y$. But every coherent sheaf with support on $Y$ can be given by a successive extension of coherent sheaves of $\mathcal O_Y$-modules; hence the restriction of $g$ to sheaves supported on $Y$ is given by a multiple of the length of the stalk at the generic point of $Y$. On the other hand, for each $n > 0$ denote by $Y_n$ the subscheme defined by the $n^{\mathrm th}$ power of the sheaf of ideals of $Y$; the length of the stalk of $\mathcal O_{Y_n}$ at the generic point of $Y$ is unbounded, but by the positivity of $g$ the value of $g(\mathcal O_{Y_n})$ is bounded by $g(\mathcal O_X)$. Hence this multiple is 0, and $g$ is 0 on all torsion sheaves. In particular, if $F \to G$ is a generic isomorphism of coherent sheaves, $g(F) = g(G)$.

But on the other hand if $F$ is a coherent sheaf of generic rank $r$, there exists homomorphisms $F \to G$ and $\mathcal O_X^r \to G$ of coherent sheaves that are generic isomorphisms; hence $g(F) = rg(\mathcal O_X)$. The conclusion follows.

-
This is great, thank you! – Mahdi Majidi-Zolbanin Dec 13 2011 at 14:01

2
Perhaps in line 9, you meant "is bounded by $g(\mathcal O_X)$" instead of $\mathcal O_Y$. – Hailong Dao Dec 13 2011 at 16:32
To Hailong: yes, thank you. I edited the post. – Angelo Dec 13 2011 at 18:42
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 39, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.916340708732605, "perplexity_flag": "head"}