Post #1815131

It's kind of funny, I remember learning algebraic geometry and wondering why people would care about the derived pushforward R*f. I don't think I ever got an answer to that specific question -- I just learned so many reasons to care about derived functors broadly that now R*f seems inevitable... BUT ☝️ I finally have a 𝑟𝑒𝑎𝑙𝑙𝑦 good reason that I think would have satisfied younger me! If you're interested in sheaves then you're interested in their global sections, as well as the cohomology groups of your sheaf. The problem is that computing cohomology is hard! Say that f : X → Y is a map which we'll think of as taking a "complicated" space X to a "simple" space Y. Then for a sheaf ℱ you might hope to recover the cohomology H*(X,ℱ) in terms of the cohomology of the pushforward fℱ on the simpler space Y... The 𝐋𝐞𝐫𝐚𝐲 𝐒𝐩𝐞𝐜𝐭𝐫𝐚𝐥 𝐒𝐞𝐪𝐮𝐞𝐧𝐜𝐞 tells you this dream is a reality! ~If~ you keep track of the higher pushforwards too! Indeed, one can relate the the cohomology groups Hⁱ⁺ʲ(X,ℱ) to the cohomology with coefficients in the higher pushfowards Hⁱ(Y, Rʲfℱ)! So as soon as you find yourself interested in computing sheaf cohomology, you ℎ𝑎𝑣𝑒 to be interested in derived pushforward in order to gain access to this powerful computational tool!