In order to define any algebraic geometry structures (a sheaf, for example), one has to first understand what an algebraic variety is. And thus:
Observation I. It’s damn-near impossible to find someone who gives a straightforward definition of an algebraic variety straight off the bat.
Instead, most authors tend to define an affine algebraic variety – first as the common zero set of a collection 
This of course requires knowledge of category theory, the Zariski topology, etc. etc.
As of now, this 30+ minutes of searching has gotten me through about 3/4 of a page in Chris Elliott’s online manuscript concerning 
le sigh
being monogenic: My original claim was that 
for monogenic functions, but today, I miscalculated the partials for an example that led me to believe that 
was actually the criteria. That’s not correct at all.
and 
, respectively, to be a single sum ranging from 
instead of a term for 
and 
, respectively, plus a sum for 
. Later, when there were two parameters 
in the sum, I claimed that 
implies that 
; this, of course, is false, since 
is identified with 1 so that 
. On the other hand, with a proper bit of rigor, the proof is still essentially correct. 
, an equivalent representation of which is 
where 
and where 
represent the scalar and vector parts of 
to be the derivative of ![D_n[\mathbf{sc}(f)]=0](https://s0.wp.com/latex.php?latex=D_n%5B%5Cmathbf%7Bsc%7D%28f%29%5D%3D0&bg=ffffff&fg=111111&s=0&c=20201002)
and ![D_n[\mathbf{vec}(f)]=0](https://s0.wp.com/latex.php?latex=D_n%5B%5Cmathbf%7Bvec%7D%28f%29%5D%3D0&bg=ffffff&fg=111111&s=0&c=20201002)
. With regards to the scalar part of 
.
, and so each component must be zero. Hence, 
for 
.![\begin{array}{rcl}0 & = & \displaystyle\left(\frac{\partial}{\partial x_0}+\sum_{l=1}^n e_l\frac{\partial}{\partial x_l} \right)\circ\left(\sum_{k=1}^n u_k(x)e_k\right) \\[2em] & = & \displaystyle\sum_{k=1}^n \frac{\partial u_k}{\partial x_0}e_k + \sum_{k,l=1}^n e_l\,e_k\frac{\partial u_k}{\partial x_l}\,\,\,\,\,\,\,\,\,\,(1)\end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Brcl%7D0+%26+%3D+%26+%5Cdisplaystyle%5Cleft%28%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x_0%7D%2B%5Csum_%7Bl%3D1%7D%5En+e_l%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x_l%7D+%5Cright%29%5Ccirc%5Cleft%28%5Csum_%7Bk%3D1%7D%5En+u_k%28x%29e_k%5Cright%29+%5C%5C%5B2em%5D+%26+%3D+%26+%5Cdisplaystyle%5Csum_%7Bk%3D1%7D%5En+%5Cfrac%7B%5Cpartial+u_k%7D%7B%5Cpartial+x_0%7De_k+%2B+%5Csum_%7Bk%2Cl%3D1%7D%5En+e_l%5C%2Ce_k%5Cfrac%7B%5Cpartial+u_k%7D%7B%5Cpartial+x_l%7D%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%281%29%5Cend%7Barray%7D&bg=ffffff&fg=111111&s=0&c=20201002)
.
sum to zero precisely when each sum itself is equal to zero due to the linear independence of the basis elements 
, 
. In particular, then, the second sum in 
cited there should be 
matrices instead of 
. Recall also that the first equation in the system of equations shown in the original entry – the equation 
– is achieved by combining the equation of “mixed partials” of the form 
from the second sum in 
from above.
with 
0-cells, 
1-cells and 
are integers for which 
. The gist seems simple: Add on a cell in a precise, regimented way, and define a characteristic map which correctly “incorporates” the additional construction into the given construction. I get that. But where do I start?
riemannian hunger 