 Cohomology and base change

Cohomology and base change asks whether changing the base of a family commutes with taking its sheaf cohomology. For a scheme $X$ over a ring $A$, an $A$-algebra $B$, and a quasi-coherent sheaf $\mathcal{F}$, the natural comparison isHere ${\mathcal{F}}_B$ is the pullback. The left side first computes cohomology and then changes scalars; the right side changes the geometric base first.

Flat base change gives an isomorphism for every $q\ge 0$ when $A\to B$ is flat and $X\to \mathrm{Spec}A$ is quasi-compact and quasi-separated. On separated schemes, the mechanism is a Čech complex from a finite affine cover: flat tensor product preserves the kernels and images needed to compute its cohomology.^[ref]

Without flatness, tensor product need not commute with cohomology. The two-term complex $k[t]\stackrel{t}{\to }k[t]$ in degrees zero and one has zero degree-zero cohomology, but tensoring with $k[t]/(t)$ makes the differential zero and produces a nonzero degree-zero group. This illustrates the algebraic obstruction; proper-family base-change theorems impose additional hypotheses to control it.