???so... ???we want x -> y -> y to be a cokernel diagram ... but moreover also xz -> yz -> yz for any object z ... ??or something?? ...
??so consider those objects w st ... ?? ??the kernel of [xz,w] <- [yz,w] is the entire domain ... ??so that is, for any morphism yz -> w, xz -> yz -> w vanishes ... ?? ??or something ?? ...
??let's try an example ... modules of k[x] ... morphism
??might be confused but not working out the way i think i want it to yet ... ???...
all right, so let's try pairs of vector spaces, and (1,1) -> (1,0) ...
object w = (a,b) that "thinks" that that map is zero, in the sense that ... ?? it looks like zero under homming into w ... ??so (a <- a, b <- 0) looks like the zero map ... ??so obviously a is zero ... ??or something ... ???so is this making sense ?? ... ??but so is the conceptual interpretation making sense here ?? ... ?? martin suggests that "modding out by an ideal" and "localizing wrt a function" should fit in here ... which seems sensible, except that i seem to keep getting confused when i think about it ... something seems backwards ... let's see... ideal (1,0) >-> (1,1) ... ???so i guess that i did have it backwards??? ??also composite function (1,1) -> (1,0) -> (1,1) ...
??so it seems good now ?? ... ???...
??so what about map from unit sheaf to skyscraper sheaf ??? ....
or ... map from unit graded module of [polynomials in n+1 degree 1 variables] to minimal quotient ... ?? or something ... ??hmm, or did i sort of screw up here ?? ... the coequalizer is zero ... inverting the map from the minimal quotient to zero ... making the identity map of the minimal quotient the coequalizer ... setting the original map to zero ... ???...
1+1 -> L -> cok(1+1 -> L) -> 0 ...
??something about setting cokernel map to zero vs setting cokernel object to zero ... ???maybe equivalent ??? ....
so did i mishear martin, or something ?? ... ???...
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