# For a family of short exact sequences of coherent sheaves, can we define the splitting subscheme?

Let $$k$$ be an algebraically closed field of characteristic zero. Let $$X$$ be a projective scheme over $$k$$. We can talk about short exact sequences of coherent sheaves on $$X$$. Suppose we have a family of SES's parametrized by $$T$$, a scheme of finite type over $$k$$, that is a short exact sequence of coherent sheaves on $$X\times_kT$$ $$0\to E\to F\to G\to0$$ such that

1. $$E,F,G$$ are coherent sheaves on $$X\times_kT$$, flat over $$T$$;
2. for each closed point $$t\in T$$, the fibre over $$X_t:=X\times_k\mathrm{Spec}(\kappa(t))$$ is exact (As commented, this condition can be derived from flatness and exactness of $$0\to E\to F\to G\to0$$. ) $$0\to E|_{X_t}\to F|_{X_t}\to G|_{X_t}\to0\qquad(*)$$

My question is about the ''splitting locus'' $$\{t\in T:t\textrm{ is a closed point, }(*) \textrm{ is splitting}\}.$$ Now it is only defined as a subset of closed points. I wonder if this can be enhanced as a subscheme.

• Your condition 2. follows from 1. Commented Apr 17, 2022 at 17:32
• I wonder if the natural map $\mathrm{Ext}^1_{X \times T}(G, E) \to H^0(\mathcal{E}xt^1_{X \times T}(G,E))$ can be used here. That should at least detect some kind of local splitting on $X \times T$, if not fiberwise splitting (maybe we need $R^1\pi_{2*}$?). Commented Apr 17, 2022 at 22:19