I havee such a question on construction of the Koszul complex (further we are concerned about K-theoretical Euler class). I was thinking of introducing the Koszul complex, and the existing of elements in $K^0(X)$ groups. Here is the picture
Such an example is a part of https://pages.uoregon.edu/ddugger/kgeom.pdf (page 113). I don't know exactly how we can deduce existing of the element in $K^0(B,B-s^{-1}(0))$. If anyone has studied any similar approach or can explain how (maybe in geometrical way) one can get elements that later are used to build the K-theory Euler class, it would be great to understand what is going on here.
Moreover, any other papers or suggestions about approaches are more than welcome. Thanks in advance.
You should read Definition 5.1 in the notes you've linked to: they explain the definition of $K$-theory classes associated to a bounded complexes of finitely-generated projectives. It's a little different than some more traditional definitions, but Proposition 5.4 shows that the definitions are the same.
To see that the $K$-theory class defined by the Koszul complex is supported away from $s^{-1}(0)$, note that (i) the complex is exact when pulled back to $B - s^{-1}(0)$, and (ii) exact complexes are zero in $K$-theory (see, e.g., Exercise 5.6).