Reputation
649
Top tag
Next privilege 1,000 Rep.
Create new tags
Badges
5 14
Newest
 Yearling
Impact
~6k people reached

Apr
30
comment Show that R is isomorphic to a direct product of local rings.
@PetersonAuthor Yes that is true. Pick a basis $e_1,\ldots, e_n$ for $R$ over $k$. $k[X_1,\ldots,X_n]\to R$ given by $X_i\mapsto e_i$ is surjective, so modding out by the kernel establishes the isomorphism you want.
Apr
30
comment Calculating the intersection multiplicities of algebraic curves using Gröbner Basis
For the first question, you don't actually need that the Grobner basis consists of two curves but only that it cuts out the same ideal $(F,G)$
Apr
29
comment A Question about the Intersection Multiplicity
To address your second question, no it is not always finite dimensional (for instance what happens if $P$ is the origin and $f=x,g=x^2$), but it will always be if $f$ and $g$ are coprime to each other.
Apr
29
comment A Question about the Intersection Multiplicity
$\mathcal{O}_{\mathbb{A}^2,P}$ is a $K$ vector space (if a function is defined at $P$ then if you multiply it by any scalar it should still be defined). From this you it follows $\mathcal{O}_{\mathbb{A}^2,P}/(f,g)$ is $K$-vector space.
Apr
29
comment A Question about the Intersection Multiplicity
The dimension is as a $K$ vector space. Let $C$ be a curve in $\mathbb{A}^2$ and let $L$ be a line in $\mathbb{A}^2$ such that $C$ and $L$ intersect at $p$ and $L$ is tangent to $C$ at $p$. Then there intersection is not just a point but includes also a tangent direction thus the intersection multiplicity is at least 2 (try to workout an example and draw a picture).
Apr
22
comment Prove $sgn(π) = sgn(π^{-1})$?
Notice that $\pi=\pi^{-1}=id$ in the case you gave above, so they should have the same inversion count
Apr
22
revised Prove $sgn(π) = sgn(π^{-1})$?
Added some mathjax and removed proof theory tag
Apr
22
suggested approved edit on Prove $sgn(π) = sgn(π^{-1})$?
Apr
21
revised Constructing Incidence variety without using equations
added 113 characters in body
Apr
21
asked Constructing Incidence variety without using equations
Feb
29
accepted Classifying line bundles with basechange
Feb
14
asked Smooth divisors in an ample linear system
Jan
21
comment Sheafs of modules on Proj S
Try to define the morphism on distiguished affine opens and show they glue
Jan
16
comment Are rings $\mathbb Q[i]$ and $\mathbb Q[\sqrt{3}i]$ isomorphic?
Is there an element in $\mathbb{Q}(\sqrt{3}i)$ that squares to -1?
Dec
30
comment Pull back of canonical line bundle under a blow up
@Mohan That works. Thanks!
Dec
30
asked Pull back of canonical line bundle under a blow up
Dec
14
awarded  Yearling
Dec
14
revised Intersection theory question from Vakil's notes on Algebraic Geometry
deleted 37 characters in body
Dec
14
asked Intersection theory question from Vakil's notes on Algebraic Geometry
Dec
3
asked Unramified cocycles and the Selmer group of an ellptic curve