# Is the following a sufficient condition for flatness?

Hi.

Let $f\rightarrow S$ be an open morphism of reduced finite dimensional complex spaces (or a universally open morphism of locally noetherian excellents without embedded components or reduced schemes) with fibers of dimension $n$.

If $f^{*}G$ is torsion free for all torsion free coherent sheaves $G$ on $S$, then is it true that $f$ is flat ?

Thank you.

• Kaddar, just to clarify, does "torsion-free" mean each stalk has positive depth, or the (seemingly stronger, if I'm not mistaken) condition that there is a dense Zariski-open $j:U \rightarrow S$ on which $G$ is a vector bundle s.t. $G \rightarrow j_{\ast}j^{\ast}(G)$ has trivial kernel (or equiv., $G$ is subsheaf of $j_{\ast}(E)$ for such a $U$ and some vector bundle $E$ on $U$)? Also, do you insist on the hypothesis only for $G$ given over all of $S$, or are you willing to make the hypothesis over all opens of $S$ too (as otherwise hard to localize the problem)? – BCnrd Jun 20 '10 at 1:53

I think so. It seems to me that the two definitions proposed by Brian are equivalent. Let us use the following: a coherent sheaf $F$ on a reduced space $X$ is torsion-free if whenever $a$ is a non-zero divisor in the local ring ${\cal O}_{X,p}$, multiplication by $a$ is injective on the stalk $F_p$. Or, for every $p\in X$ the only associated primes of $F_p$ as an ${\cal O}_{X,p}$-module are the minimal primes of ${\cal O}_{X,p}$.
Now, let $f\colon X \to S$ be a morphism, $p \in X$, $q := f(p)$, $A := {\cal O}_{S,q}$, $B := {\cal O}_{X,p}$. Let $M$ be the maximal ideal of $A$; this is the stalk of the torsion free sheaf $I_{S,q}$ (the sheaf of ideals of $q$ in $S$); set $k := A/M$. The fact that $f$ is open implies that every minimal prime of $B$ maps to a minimal prime in $A$.
By Grothendieck's local criterion of flatness, it is enough to show that $\mathop{\rm Tor}_1^A(k, B) = 0$, which is equivalent to saying that the natural homomorphism $M \otimes_A B \to B$ is injective. However, this homomorphism is injective outside of the inverse image of the fiber on the maximal ideal of $A$, which does not contain any of the minimal primes of $B$, so is nowhere dense. Since $M \otimes_A B$ is a torsion-free $B$-module, it can not contain a non-zero submodule supported on a nowhere dense closed subset; so the kernel has to be trivial, and this proves the result.
{"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": 1, "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, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9878526926040649, "perplexity": 105.35602620648484}, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256227.38/warc/CC-MAIN-20190521042221-20190521064221-00072.warc.gz"}