I don't know if this question has been asked before,this year i take a course in algebraic geometry in the first 5 paragraph of chapter 2 of hartshorne.
Our professor doesn't use a lot the language of category theory, but he advised us to learn it, for example in the course he asked us to show that kernel commute with stalk in classical way a by proving that a map is well defined and isomorphism, but i see that this result is particular case of limit commute with kernel.
another example is proving stalk of inverse image using adjoint.
so my question is :
-what important part that should be learned.
-some mathematicians treat isomorphism like equality for example some define exact sequence such that kernel equal image but some define exact sequence as complex and canonical morphism between kernel and image is isomorphism,what your advice about that.
And thanks
There aren’t really any “parts” of category theory to choose from at the level you’re asking about. All of basic category theory, plus the theory of abelian categories and to a lesser extent toposes, is very useful in algebraic geometry. If you’re reading Hartshorne already, then an advanced introductory category theory text, such as Riehl or Mac Lane, is probably appropriate for you.
Regarding your second bullet, consider the sequence $A\stackrel{f}{\to}B\stackrel{g}{\to}C$ to be one such that the canonical morphism $A/\mathrm{ker}(f)\to \mathrm{ker}(g).$ If you’re working in a general abelian category, the kernel of $g$ can be said to be equal to the image of $f$ as subobjects of $B$, which means that any pair of presentations of the two as monomorphisms into $B$ are isomorphic via an isomorphism respecting the given monomorphisms. That is, a subobject is an equivalence class of monomorphisms, taken up to isomorphism. There’s thus no actual difference between equality, in this sense, and the canonical map being an isomorphism.