proof – Can Mathematica check if I applied an algebraic transformation correctly?

Mathematica offers countless functions for algebraic manipulations. But is there a way to use it check if I hadn't made any mistakes in my own algebraic manipulations.

More specifically, my trigonometric identities are a bit rusty and I would like to see Mathematica warn me if I make stupid mistakes like replacing $ cos ( alpha + beta) $ by $ cos ( alpha) sin ( beta) + sin ( alpha) cos ( beta) $ (Where should it be? minusNot plus). Is that possible?

algebraic groups – limitation to the maximum torus

Let me say that I'm pretty sure that all the things I'm going to ask are obviously wrong for an expert because I don't have much faith in this stuff. So take this question as a counterexample to uncomplicated things 🙂

Accept $ G $ is a semisimple split affine algebraic group about $ k $,
For an algebraic group we refer to $ ad (G) $ the associated lie algebra.
We know that $ G $ is determined by the associated master date, which is structured as follows:

  1. To let $ T $ Be a maximum torus (divided by PS). Any representation of $ T $ Divisions in one-dimensional repetitions, determined by a "weight" $ alpha in display (T) ^ * $,
  2. Thus, $ res_T (ad G) $ gives us a (finite) list of weights in $ ad (T) ^ * $ that determines the root date.

Note that the weights displayed can also be calculated $ K_0 (res_T ((ad G))) $ at the level of the Grothendieck groups.

(1) Can you understand who is $ (ad G) $ in the $ K_0 (Rep G) $? In other words, when two groups are the same $ K_0 $are the element $ (Display G), (display G & # 39;) $ equal?

In the case of finite groups over complex numbers, this is easy to see $ H to G $ is injective iff for each representation $ V neq 0 $ from $ H $. $ ind_H ^ G V neq 0 $,

This can also be checked on Grothendieck groups. Record the dot product $ K_0 (Rep G) $ given by $ langle (V), (W) rangle = dim_C Hom (V, W) $, Then $ langle (ind) V, W rangle = langle V, (res) W rangle $what determines $ (ind) $ as an adjoint in the linear algebra of res because the dot product is determined positively.

(2) Does this criterion apply in the general context? Can you only check this criterion for Grothendieck groups?

If so, let me name a map with neutral tannacic categories $ C to D $, Pseudo-injective if the adjoint $ D to C $ Sends non-zero objects to non-zero objects (somehow the adjunct's "kernel" is zero), logically for cards between their Grothendieck rings.

Now I'm saying that $ T $ is a maximum torus is like saying that $ Rep G to Rep T $ is maximally pseudo-injective of $ Rep G $ to a category of the form $ Rep (k ^ *) ^ { otimes n} $, Note that $ K_0 (Rep (k ^ *)) simeq Z (x, x ^ {- 1}) $ in char zero and $ mathbb {Z} (x) / (x ^ {p-1} -1) $ in char p (maybe? Not sure about char p)

Is it true that $ K_0 (Rep G) to K_0 (Rep T) $ is a maximum pseudo-objective map $ K_0 (Rep G) $ into a ring of shape $ Z (x_1 ^ { pm 1}, ldots, x_n ^ { pm 1}) $ (and analog for char p)?

If 1 is wrong, there is a possibility that all of these cards are from maximum tori. The question in this case is:

Do you think there is a possibility that in the general case $ K_0 (Rep G) $ and $ (ad G) in K_0 (Rep G) $ Determine group?

In fact, there is currently no indication of a maximum torus.

Of course, you also have to remember the K_0 of the fiber optic radio $ Rep G to Vec_k $what gives a card $ K_0 (Rep G) to mathbb {Z} $ remember the dimension of things.

If 1 is true, there should be some maximum pseudo-objective maps of $ K_0 (Rep G) $ to a ring of this shape that does not originate from maximally tori, otherwise one could reconstruct it $ G $ from $ K_0 (Rep G) $, and this does not even apply to finite groups (knowing K_0 in this case is like knowing the character table).

In this case the question is:

Can you find additional dates to remember? $ Rep G $ – that can be formulated abstractly for a general group – if a map can be understood $ K_0 (Rep G) to mathbb {Z} (x_1 ^ { pm 1}, ldots, x_n ^ { pm 1}) $ comes from a maximum torus?

Agal Algebraic Geometry – Can a non-singular curve be embedded in a singular surface?

To let $ (X, x) $ be an isolated, normal Gorenstein surface singularity. To let $ C subset X $ to be a curve in $ X $ and go through the singular point $ x $ the surface. Is it possible for the curve $ C $ is not singular, d. H. are there any examples, though $ C $ is not singular?

Number theory – p-adic algebraic number

To let $ K / mathbb Q $ be a finite extension, $ mathfrak P $ to be a maximum ideal of $ mathcal O_K $ above a prime number $ p $ from $ mathbb Z $, Consider a sequence $ (r_n) _ {n in mathbb N} $ from $ K $ that converges too $ 0 $ for the $ v_ mathfrak P $ Rating. Then, $ (r_n) _ {n in mathbb N} $ converges to $ 0 $ in the $ mathbb C_p $ a completion of $ overline { mathbb Q} $, Accept that $ sum_ {n ge0} r_n in mathbb C_p $ is algebraically over $ mathbb Q $, Designate with $ K_ mathfrak P $ the completion of $ K $ for th $ v_ mathfrak P $ Rating. Consider now $ beta = sum_ {n ge0} r_n in K_ mathfrak P $, The question is $ beta $ algebraically over $ mathbb Q $?

Thanks in advance for any answer or hint.

Algebraic curves – embedding a compact Riemann surface in a projective space

Trying to fill in details of a proof I learned some time ago: $ X $ a compact Riemann surface. We want to show that the degree of a divisor is big enough $ D $, the map
$ X rightarrow mathbb {P} (H ^ 0 (X, mathcal {O} _D) ^ *) $is the card $ x mapsto (f mapsto f (x)) $ is an embedding. I want to understand one part. First, we prove that if deg $ D geq 2g, $ then $ mathcal {O} _D $ is generated globally, i. H. for each $ x in X, exists f in H ^ 0 (X, mathcal {O} _D) $ so that
ord$ _x (f) = – D (x). $ Now we choose a divisor $ D $ on $ X $ with deg $ D geq 2g + 1. $ Therefore $ D $ the map $ X rightarrow mathbb {P} (H ^ 0 (X, mathcal {O} _D) ^ *) $ should be one. Now this part is not so clear to me.

Let's start anyway. So let's say we have two points $ x neq y in X. $ We look at the divider $ D-x $, It has deg $ geq 2g $, so generated globally. So, $ exists for $ With $ (f) geq -D + x $ so that ord$ _y (f) = – D (y) $ and ord$ _x (f) geq -D (x) + 1. $ Is that enough to say that the card is injective? If so, why can someone better describe the map and clear my doubts. Thank you in advance.

Reference Request – Example of an algebraic system that introduces infinite sets but not infinite numbers, allowing inversions of infinite sets.

When expanding real numbers, infinite sets usually do not cause much trouble, but if we consider infinitesimal numbers, they are more problematic.

This is because many things that were classically alike cease to be equal when we introduce infinitesimal numbers. For example, the sums of convergent series that are classically equal are suddenly differentiated by an infinitesimal number.

On the other hand, if we introduce infinite sets, everything that is the same will remain the same (except for entities that were equal only in terms of regularization).

So my question is, is there a well-known algebra on real numbers that introduces infinite sets and allows their (nonzero) inverses, but somehow avoids the introduction of infinitesimal numbers?

For example, can an inverse of infinite size be infinite or something else?

Algebraic Topology – Characteristics of a continuous map with colon

By the phrase of Borsuk-Ulam we know that every continuous card $ M: S ^ n to mathbb {R} ^ n $ has a point in its area that is the image of two points in the domain. When I consider that, I'm curious to see how much this duplicate point would exist for such maps with some minor limitations and what we can say about the amount of points in the area that make up the image of just one point in the domain all cards are.

It is obvious that for each contiguous non-one-element area map $ f_1: S ^ 1 to mathbb {R} $ There are exactly two points $ a, b in mathbb {R} $ for the both of $ f_1 ^ {- 1} (a), f_1 ^ {- 1} (b) $ are an element set. Also for every map there are infinitely many points in Codomain $ mathbb {R} $ who modeled on 2 or more elements.

It also seems intuitive that in the case of individual rolling cards $ f_2: S ^ 2 to mathbb {R} ^ 2 $ which the dimension of $ f_2 (S ^ 2) $ is $ 2 $ (the same as codomain $ mathbb {R} ^ 2 $ Dimension), there is exactly one closed loop $ mathcal {C} $ in his area, which is the injecting image of just a loop $ mathcal {S} $ from $ S ^ 2 $ and each of the rest points of the area is the image of several points of the domain $ S ^ 2 $,

In other words:

$ exists mathcal {C} in mathbb {R} ^ 2 and exists mathcal {S} in S ^ 2: f_2 ^ {- 1} ( mathcal {C)} = mathcal {S } and f_2 | _ mathcal {S} is Injective \ and \ forall p in mathbb {R} ^ 2 Backslash mathcal {C} : f_2 ^ {- 1} {p } has more than an element . $

ask:

First, does my intuition apply to the latter case (can someone prove or disapprove the result by a counterexample)?

Second, is there an invariant of the consecutive maps related to the aggregation of features (structure, size, dimension, …) of $ n $-Point ($ forall n in mathbb {N} $) and Unzählpunkt Phrases of the card?

Note: $ n $-Point Set for a card $ f: X to Y $ is the set of all points, such as $ p in Y $ that's the role model $ f ^ {- 1} ( {p }) $ exists exactly $ n $ shows in $ X $if the model $ f ^ {- 1} ( {p }) $ consist of innumerable points in the domain, then we call it one Unzählpunkt,

Agal Algebraic Geometry – Do infinite products oscillate with trivial cofibrations to obtain simple quantities?

I read the book by Voevodsky and Morel$ mathbb {A} ^ 1 $Homotopy Theory of Schemes & # 39 ;. In Remark 3.1.15 this is stated for every simple fiber color $ F $ and open sentences $ U subseteq V $. $ F (V) to F (U) $ is a vibration.

Prove by definition. We have a bifunktor
$$ begin {array} {ccccc} sSet & times & Shv (Sm / k) & to & sShv (Sm / k) \ (S &, & F) & mapsto & S times F end {array}, $$
from where $ (S times F) (X) _n = S_n times F (X) $, Consider the coequalizer
$$ lambda ^ n_k times U rightarrows lambda ^ n_k times V coprod triangle n times U to C. $$
Then there is a card $ i: C to triangle n times V $ and the question is reduced to the RLP of $ F $ w.r.t $ i $, So I want to prove it $ i $ is a trivial co-calibration.

It's obviously a co-calibration, but I'm determined to prove it's a weak equivalence. Suffice it to prove that the functor $ – times F: sSet to sShv (Sm / k) $ is a left-hand quill functor because we could then use the pushout diagram of $ C $, So we will prove that trivial cofibrations are transformed into infinite products by transferring them to stems …

I think we have to prove that geometric realization functor commutes with infinite products, at least to a weak equivalence. Is that true?

Thanks a lot!

Ag. Algebraic Geometry – Associativity of Feed Quotients

For a reduced group $ G $ to influence a projective variety $ X $, the chow quotient $ X // G $ is defined as follows. To let $ U subseteq X $ a sufficiently small open subset of $ X $ with the property that the homology class of $ overline {G.x} $ is independent of $ x in U $, Then we have a morphism $ U to Chaw (X) $ Sending a point to its orbit closure. We define $ X // G $ to close the image of this map (maybe you have to assume that $ X $ is smooth, but I'm not sure).

Question: Let $ H times G $ acts on a projective variety $ X $ and take it if necessary $ X // G $ and $ X // H $ are smooth. Is it true that $ (X // G) // H cong X // (G times H) $? This question arises from my attempt to understand the correspondence of Gelfand MacPherson, as described by Kapranov in the journal
https://arxiv.org/abs/alg-geom/9210002
more systematic, by applying this hypothetical identity to the action of a product of torus and torus $ GL_k $ for projecting the room of $ n times $ Matrices.

Ag. Algebraic Geometry – Formal Schemes Methods: Applications

Maybe this question is a bit too broad, but so far I could not find a satisfactory answer.

To let $ X $ to be present
Noetherian scheme and $ X & $ 39; subset X $ be a closed subschema of $ X $ which is defined by an ideal $ mathcal {I} subset mathcal {O} _X $, Again, one designates $ X_n = Spec mathcal {O} _X / mathcal {I} _ {n + 1} $ and get a chain of thinking
$ X_ • = (X_0 to X_1 to …) $, Take the Colimit from $ X_ • $ we get a formal scheme $ has {X} $ from where
$ | has {X} | = | X & # 39; | $ and $ mathcal {O} _ { has {X}} = varprojlim_n mathcal {O} _ {X_n} $, also as a formal conclusion of $ X $ wrt $ X & # 39; $, My main reference is Doan Trung Cuong's excellent mini-course.

My question is simple, where in algebraic geometry this concept of formal schemata is used fruitfully. The only "big" sentence based on this sentence is theSentence about formal functions".

Comment: By "fruitful" I mean that we can use this theory as a "new" toolkit to get new conclusions about common sense schemes (note that formal schemes are annuli, not schemata in the usual sense). An excellent prototype of such an interplay with the concept of formal schemata is again the theorem on formal functions, as we can conclude, for example, from the stone factorization, a variant of the main theorem of Zariski.

Unfortunately the TofF was until now during my research the only "big" result that this formalism provides. Are there any more such remarkable caliber results?

What is the philosophy of formally completing conventional schemes? Or what is the motivation, so what kind of "new" information about the schema do you want to find out when applying a concept that does not seem to be extractable without it?