Intereting Posts

High school mathematical research
Injective Holomorphic Functions that are not Conformal?
Infimum over area of certain convex polygons
Is every field the field of fractions for some integral domain?
How to find a random axis or unit vector in 3D?
Prove that if $R$ is von Neumann regular and $P$ a prime ideal, then $P$ is maximal
Showing that $(A_{ij})=\left(\frac1{1+x_i+x_j}\right)$ is positive semidefinite
Could someone explain conditional independence?
Good 1st PDE book for self study
What's the relationship between a measure space and a metric space?
If$(ab)^n=a^nb^n$ & $(|G|, n(n-1))=1$ then $G$ is abelian
Can different tetrations have the same value?
Probability of guessing a PIN-code
Answer of $5 – 0 \times 3 + 9 / 3 =$
Lebesgue measure as $\sup$ of measures of contained compact sets

Let $X$ be a noetherian integral (separated) scheme which is regular in codimension one. Let $Y$ be a prime divisor and let $\eta$ be the generic point of $Y.$ It seems I am missing something easy but why $\mathcal{O}_{X, \eta}$ is a DVR with the quotient field the function field of $X?$

And when it is said, $X$ is regular (non-singular) of codimension one, does it follow from the definition that the local ring of a codimension one closed subscheme is regular in general? (otherwise, the terminology doesn’t make sense to me!)

- Why is the coordinate ring of a projective variety not determined by the isomorphism class of the variety?
- rational functions on projective n space
- Koszul complex of locally free sheaves
- What use is the Yoneda lemma?
- Does a non-empty locally closed subset of a $k$-scheme of finite type always contain a closed point?
- Homogeneous polynomials on a vector space $V$, $\operatorname{Sym}^d(V^*)$ and naturality

- Degree of a Cartier Divisor under pullback
- Quick question: Direct sum of zero-dimensional subschemes supported at the same point
- If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?
- $X \to Y$ flat $\Rightarrow$ the image of a closed point is also a closed point?
- Doing Complex Analysis on the Riemann Sphere?
- Weil conjectures - motivation?
- Graph of morphism , reduced scheme.
- Complements of hypersurfaces in a projective space is affine.
- Localizations of quotients of polynomial rings (2) and Zariski tangent space
- Pole set of rational function on $V(WZ-XY)$

$\mathcal{O}_{X,\eta}$ is a regular local $1$-dimensional noetherian domain. It is a Theorem in commutative algebra which says that this is precisely a DVR.

If $X$ is an arbitrary integral scheme and $x \in X$, then the quotient field of $\mathcal{O}_{X,x}$ is the function field of $X$. Namely, since this a local issue, we may assume $X=\mathrm{Spec}(A)$ for some integral domain $A$, and just have to observe that $\mathrm{Quot}(A_{\mathfrak{p}}) = \mathrm{Quot}(A)$ for every prime $\mathfrak{p} \subseteq A$.

As for the last question, you should look at the definitions. Nothing happens.

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