Andrea Gagna
Reputation
589
Top tag
Next privilege 1,000 Rep.
Create new tags
 Apr 19 revised When is $N\otimes_A B \to N$ an isomorphism? edited body Apr 12 revised Flat Modules are Filtered Colimits of Free Modules added 2 characters in body Apr 12 comment Flat Modules are Filtered Colimits of Free Modules @Rachmaninoff : I added a sketch of proof of "flat as presheaf implies flat as module". The proof uses the most explicit description of flat module that I'm aware of (and so it is not very elegant or "formal"). Apr 12 revised Flat Modules are Filtered Colimits of Free Modules added 657 characters in body Apr 12 revised Flat Modules are Filtered Colimits of Free Modules Part added Apr 11 revised Flat Modules are Filtered Colimits of Free Modules Bad mistake spotted in the comments. Apr 11 comment Flat Modules are Filtered Colimits of Free Modules Ah, you're right! Sorry, it's clearly a bad mistake. It also shows that the functors relative to the $\mathbb Z$-modules $\mathbb Z/n\mathbb Z$, for any $n>2$ integer, cannot be flat. Apr 11 comment Flat Modules are Filtered Colimits of Free Modules For a very handy and explicit description, it may be useful to take a look at remark 1 in this section about flat functors on nlab. Apr 11 revised Flat Modules are Filtered Colimits of Free Modules deleted 1 character in body Apr 11 comment Flat Modules are Filtered Colimits of Free Modules A functor $F \colon A \to \mathbf{Set}$ is flat iff the opposite of the comma category $(e\downarrow F)$ is filtered (by $e$ I denote a terminal object in $\mathbf{Cat}$). Equivalently, regarding $F$ as a presheaf over $A^{\text op}$, iff its category of elements (à la Grothendieck), which I denoted $A^{\text op}/F$, is filtered. So one actually must check that $(e\downarrow F)$ is co-filtered. Apr 8 comment Categories of defintion for sites on spaces and sites on schemes It's generally called relative point of view and attributed to Grothendieck. Take a look at the following pages: wiki, nlab, MSE Apr 6 revised Flat Modules are Filtered Colimits of Free Modules edited body Apr 6 answered Flat Modules are Filtered Colimits of Free Modules Mar 18 comment Requirement for having a left-adjoint functor Also theorem 3.1.5 of Handook of categorical algebra 1 - Basic category theory by F. Borceux. Mar 3 comment Synthetic differential geometry and algebraic geometry You may want to take a close look here. Dec 19 accepted When is $N\otimes_A B \to N$ an isomorphism? Dec 15 awarded Self-Learner Nov 30 comment Will learning 'classical geometry' a la Hilbert's incedence geometry help me appreciate the results of classical algebraic geometry? I'd like to iterate the question: will learning classical algebraic geometry help with modern algebraic geometry (following Grothendieck's school) or even derived algebraic geometry? I feel like without the enlightening notion of sheaf and without the powerful tools of commutative algebra, one could easily get lost in the deceptive intuition provided by classical algebraic geometry. Nov 16 comment Are split exact sequences exact in the opposite direction? @Exterior If $f, g, r, \ell$ form a biproduct diagram (what Freyd calls direct sum system, see §2.4), then $(r, \ell)$ is exact basically by definition. Freyd's Proposition 2.42 is actually a necessary condition, as well; thus in order for our four morphisms to be a biproduct diagram, we need both the sequences of the OP to be exact. Furthermore, Proposition 2.68 tells us that for a given $\ell$ there exists $r$ such that these give rise to a biproduct diagram; moreover, as $r$ must be the kernel of $\ell$, this is uniquely determined up to automorphism of $C$. Nov 14 awarded Revival