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Let $f\colon Y \to X$ be a morphism of schemes, the inverse image in $K$-theory always fit into a long exact sequence $$ \cdots \to K_i(f)\to K_i(X) \xrightarrow {f^*} K_i(Y)\to \cdots $$ where the ...

Let $V\to X$ be a vector bundle (over say a scheme). Then the cohomology of its projectivisation is $$\text{H}^*(\mathbf{P}V)\ =\ \text{H}^*(X)[t]/(t^{n+1}+c_1(V)t^n+\cdots+c_n(V))$$ as an algebra, ...