Reputation
589
Top tag
Next privilege 1,000 Rep.
Create new tags
Badges
2 16
Impact
~4k people reached

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