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 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