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Jun
19
revised Funny identities
deleted 20 characters in body
Jun
19
revised Summing a exponential series
displaystyle
Jun
19
comment Notation for a derived function that map from power set to power set
@Thomas: Upper stars indicate contravariance. Some people would write $f_*$ for this. I would write $f_!$, because $f_!$ is left adjoint to $f^* = f^{-1}$.
Jun
19
comment Basic understanding of Spec$(\mathbb Z)$
Eisenbud and Harris's Geometry of schemes spends quite a few pages on $\operatorname{Spec} \mathbb{Z}$. Perhaps you'd like to take a look at that. What you say is more or less correct.
Jun
19
answered short exact sequences and direct product
Jun
19
comment Notation for a derived function that map from power set to power set
@Brian: That seems more likely to confuse than not. In general, given a function $f : X \to Y$, one gets a map of function spaces $Z^f : Z^Y \to Z^X$ going the other way!
Jun
19
comment Coordinate ring of general linear group
Yes, that is precisely what the coordinate ring is. I think one can argue that this is the most simple presentation as follows: the dimension of $\textrm{GL}(n, k)$ is $n^2$, but it is not affine space, so it must have at least $n^2 + 1$ generators and at least one relation...
Jun
19
revised Question about Bertrand Curve
added 77 characters in body
Jun
17
comment Monoidal categories, but not in SET
There are certain general principles which can be followed to lift definitions from the non-enriched case to the enriched case. (It is in fact very easy to describe categories enriched over any monoidal category.) I am sure it is possible in principle to describe a monoidal enriched category.
Jun
17
comment Is there an epsilon-delta definition of the second derivative?
@tampis: You must use at least the value of $f'(x)$. Otherwise how do you distinguish between functions with the same second derivative but not the same first derivative?
Jun
16
answered A particular isomorphism between Hom and first Ext.
Jun
16
revised A particular isomorphism between Hom and first Ext.
edited tags
Jun
16
comment Algebraic varieties and Hausdorff spaces
The set is not the cartesian product either, if you think of a variety as a scheme. It has a Zariski topology, like any other variety/scheme.
Jun
16
comment Different integrals for $\mathbb{C} \to \mathbb{C}$ functions
It's quite obvious that a line integral and an area integral are different things. Why should any one of them be more "right"? (The main one used in complex analysis is the line integral.)
Jun
15
answered Co/counter variancy of the Yoneda functor
Jun
15
comment Co/counter variancy of the Yoneda functor
"Contravariance", not "counter variancy". Neither of the functors you name is the Yoneda embedding. The covariant Yoneda embedding is the functor $C \mapsto \textrm{Hom}(-, C)$. The contravariant Yoneda embedding is the functor $C \mapsto \textrm{Hom}(C, -)$.
Jun
15
comment Modules over a functor of points
@Georges: I tried to read it some months ago, but I gave up quite quickly. I got as far as learning their definition of scheme and getting some vague idea of how they were equivalent to the definition via locally ringed spaces.
Jun
15
comment Modules over a functor of points
The correspondence is a little bit more sophisticated than what you seem to be suggesting. A vector bundle is, first and foremost, a scheme. There is a functorial construction that builds an $\mathscr{O}_X$-module out of a vector bundle and vice-versa, but they are not literally the same kinds of objects. (An $\mathscr{O}_X$-module is not a scheme.)
Jun
15
comment Modules over a functor of points
Not all $\mathscr{O}_X$-modules are vector bundles. Not even if quasicoherent.
Jun
15
comment My first course in algebraic geometry: two simple questions
(1) No. It is not just homeomorphic but identified with the corresponding standard open set in $V$ by a fixed isomorphism $X \cong V$. (2) Yes. You can check that this notion is independent of the choice of $V$ and of the choice of isomorphism $X \cong V$.