# “reduction” of a module

Let $X$ be a scheme over a field $k$. There is a well-known antiequivalence between locally free sheaves of $\mathcal{O}_X$-modules and vector bundles over $X$. Given a module $\mathcal F$ and a trivialisation $\{U_i\}$ with transition functions given by matrices $f_{ij}$ with values in $\mathcal{O}_{U_i \cap U_j}$ one takes $\mathbb{A}_{U_i}^n$ -s and glues them according to transition functions given by the adjoint matrices $f_{ij}^*$. This antiequivalence can also be described as taking the relative spectrum of the sheafification of a presheaf $U \mapsto Sym(\mathcal{F}(U))$.

More generally, given a coherent sheaf $\mathcal F$ and working locally, over an open $U$ one has $\mathcal{O}_X^n(U) \xrightarrow{f} \mathcal{O}_X^m(U) \to \mathcal{F}(U) \to 0$ and one can construct a scheme $Y_U$ as follows: let $Y_U$ be the subscheme of $\mathbb{A}_U^m$ defined by the ideal generated by $(\sum f_{1i} x_i, \ldots, \sum f_{ni} x_i)$. Then one can glue $Y_U$-s together and get $Y$. The scheme $Y$ has a natural structure of a linear space over $X$, i.e. there are maps $+: Y \times_X Y \to Y$, $\cdot: \mathbb{A}_k \times Y \to Y$ that satisfy the $k$-module axioms. A morphism of linear spaces over $X$ is a map of schemes that preserves these operations. To $Y$ one can associate the sheaf which is the sheafification of the presheaf $U \mapsto Hom(Y|_U, \mathbb{A}_k \times X)$, this defines an antiequivalence between the category of coherent sheaves on $X$ and the category of linear spaces over $X$.

The reduction functor is a functor from the category of schemes to the category of reduced schemes. In the latter category the product is the usual product of schemes followed by the application of reduction functor. Therefore, tautologically, $(-)_\mathrm{red}$ preserves products.

Assume $X$ reduced. In general, the linear space $X_\mathcal{F}$ associated to an $\mathcal{O}_X$-module $\mathcal F$ is not reduced. Since linear spaces are defined using diagrams (with products), and $(-)_\mathrm{red}$ is a product-preserving functor, $(X_\mathcal{F})_\mathrm{red}$ is a linear space and hence corresponds to some $\mathcal{O}_X$-module, call it $\mathcal{F}_\mathrm{red}$.

Is it possible to give a direct'' description the correspondence $\mathcal{F} \to \mathcal{F}^{red}$ without referring to the procedure described above?

Is it possble to characterise those sheaves of modules that correspond to reduced linear spaces?

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In the first paragraph, you probably mean $\mathrm{Sym}(F(U)^*)$. –  Martin Brandenburg Feb 10 '12 at 15:27
The first paragraph starts with a wrong statement. In general a coherent sheaf might not be generated by global sections. And I cannot believe that there is some antiequivalence between coherent sheaves and linear spaces. –  Martin Brandenburg Feb 10 '12 at 15:30
Regarding Martin's point, maybe you need to assume $X$ has an ample line bundle. –  Mahdi Majidi-Zolbanin Feb 10 '12 at 16:16
@Martin: Sorry, I meant locally free resolution. –  Dima Sustretov Feb 10 '12 at 16:38
I have updated the question. –  Dima Sustretov Feb 10 '12 at 16:43