# Prologue {-} ## References - Gathmann's Algebraic Geometry notes[@AndreasGathmann515]. ## Notation - If a property $P$ is said to hold **locally**, this means that for every point $p$ there is a neighborhood $U_p \ni p$ such that $P$ holds on $U_p$. +--------------------------------------+------------------------------------------------------------------------------------------------------------------+ | Notation | Definition | +======================================+==================================================================================================================+ | $k[\vector x] = k[x_1, \cdots, x_n]$ | Polynomial ring in $n$ indeterminates | +--------------------------------------+---------------------------------------------------------------------+ | $k(\vector x) = k(x_1, \cdots, x_n)$ | Rational function field in $n$ indeterminates \ | | | $k(\vector x) = \ts{f(\vector x) = p(\vector x)/q(\vector x), \st p,q, \in \kx{n}}$ \ | +--------------------------------------+---------------------------------------------------------------------------------------+ | $\mathcal{U} \covers X$ | An open cover \ | | | $\mathcal{U} = \ts{U_j \st j\in J}, X = \Union_{j\in J}U_j$ \ | +--------------------------------------+---------------------------------------------------------------------+ | $\Delta_X$ | The diagonal \ | | | $\Delta_X \da \ts{(x, x) \st x\in X} \subseteq X\cross X$ \ | +--------------------------------------+---------------------------------------------------------------------+ | $\AA^n_{/k}$ | Affine $n\dash$space \ | | | $\AA^n_{/k} \da \ts{\vector a = \tv{a_1, \cdots, a_n} \st a_j \in k}$ \ | +--------------------------------------+-------------------------------------------------------------------------+ | $\PP^n_{/k}$ | Projective $n\dash$space \ | | | $\PP^n_{/k} \da \qty{k^n\smz}/x\sim \lambda x$ \ | +--------------------------------------+-------------------------------------------------------------------------------------+ | $V(J), V_a(J)$ | Variety associated to an ideal $J \normal \kx{n}$ \ | | | $V_a(J) \da \ts{\vector x\in \AA^n \st f(\vector x) = 0,\, \forall f\in J}$ \ | +--------------------------------------+-------------------------------------------------------------------------------+ | $I(S), I_a(S)$ | Ideal associated to a subset $S \subseteq \AA^n_{k}$ \ | | | $I_a(S) \da \ts{f\in \kx{n} \st f(\vector x) = 0\, \forall \vector x\in X}$ \ | +--------------------------------------+-------------------------------------------------------------------------------+ | $A(X)$ | Coordinate ring of a variety \ | | | $A(X) \da \kx{n}/I(X)$ \ | +--------------------------------------+---------------------------------+ | $V_p(J)$ | Projective variety of an ideal \ | | | $V_p(J) \da \ts{\vector x \in \PP^n_{/k} \st f(\vector x) = 0,\, \forall f\in J}$ \ | +--------------------------------------+-------------------------------------------------------------------------------------+ | $I_p(S)$ | Projective ideal (?) \ | | | $I_p(S) \da \ts{f\in \kx{n} \st f \text{ is homogeneous and } f(x) = 0\, \forall x\in S}$ \ | +--------------------------------------+---------------------------------------------------------------------------------------------+ | $S(X)$ | Projective coordinate ring \ | | | $S(X) \da \kx{n}/ I_p(X)$ \ | +--------------------------------------+-------------------------------+ | $f^h$ | Homogenization \ | | | $f^h \da x_0^{\deg f} f\qty{{x_1 \over x_0}, \cdots, {x_n \over x_0}}$ \ | +--------------------------------------+---------------------------------------------------------------------------+ | $f^i$ | Dehomogenization \ | | | $f^i \da f(1, x_1, \cdots, x_n)$ \ | +--------------------------------------+-------------------------------------+ | $J^h$ | Homogenization of an ideal \ | | | $J^h \da \ts{f^j \st f\in J}$ \ | +--------------------------------------+---------------------------------+ | $\bar X$ | Projective closure of a subset \ | | | $\bar X \da V_p(J^h) \da \ts{\vector x \in \PP^n \st f^h(\vector x) = 0\, \forall f\in X}$ \ | +--------------------------------------+-----------------------------------------------------------------------------------------------+ | $D(f)$ | Distinguished open set \ | | | $D(f) \da V(f)^c = \ts{x\in \AA^n \st f(x) \neq 0}$ \ | +--------------------------------------+-----------------------------------------------------------+ | $\mathcal{F}$ | Presheaf or a sheaf \ | +--------------------------------------+--------------------------+ | $f\in \mathcal{F}(U)$ | Section of a presheaf or sheaf \ | +--------------------------------------+-----------------------------------+ | $\underline{\mathbf{S}}$ | Locally constant functions valued in $S$ \ | | where $S$ is a set | \ | +--------------------------------------+--------------------------------------------+ | $\mathcal{F}_p$ | Stalk of a sheaf \ | | | $\mathcal{F}_p \da \ts{(U, \phi) \st p\in U \text{ open },\, \phi \in \mathcal{F}(U)}/\sim$ \ | | | where $(U, \phi) \sim (U', \phi') \iff \exists p\in W \subset U\intersect U' \text{ s.t. } \ro{\phi}{W} = \ro{\phi'}{W}$ \ | +--------------------------------------+------------------------------------------------------------------------------------------------+ | $f\in \mathcal{F}_p$ | Germs at $p$ \ | +--------------------------------------+---------------+ | $\OO_X$ | Structure sheaf \ | | | $\OO_X \da \ts{f:U\to k \st U \subseteq X \text{ is open}, f \in k(\vector x) \text{ locally}}$ \ | +--------------------------------------+----------------------------------------------------------------------------------------------------+ | $\OO_X(U)$ | Regular functions on $U$ \ | | | $\OO_X(U) \da \ts{f:U\to k \st f \in k(\vector x) \text{ locally}}$ \ | +--------------------------------------+-----------------------------------------------------------------------+ | $\OO_{X, p}$ | Germs of Regular functions? \ | +--------------------------------------+-----------------------------------------------------------------------+ ## Summary of Important Concepts - What is an affine variety? - What is the coordinate ring of an affine variety? - What are the constructions $V(\wait)$ and $I(\wait)$? - What is the Nullstellensatz? - What are the definitions and some examples of: - The Zariski topology? - Irreducibility? - Connectedness? - Dimension? - What is the definition of a presheaf? - What are some examples and counterexamples? - What is the definition of sheaf? - What are some examples? - What are some presheaves that are not sheaves? - What is the definition of $\OO_X$, the sheaf of regular functions? - How does one compute $\OO_X$ for $X = D(f)$ a distinguished open? - What is a morphism between two affine varieties? - What is the definition of separatedness? - What are some examples of spaces that are and are not separated? - What is a projective space? - What is a projective variety? - What is the projective coordinate ring? - How does one take the closure of an affine variety $X$ in projective space? - What is completeness? - What are some examples and counterexamples of complete spaces? \newpage ## Useful Examples ### Varieties - $V(x-p)$ a point. - $V(x)$ a coordinate axis - $V(xy) \subseteq \AA^2$ the coordinate axes - $V(xy-1) \subseteq \AA^2$ a hyperbola - $V(x_1^2 - x_2^2 - 1) \subseteq \AA^2_{/\CC}$ - $\AA^2\smz$ is **not** an affine variety or a distinguished open ### Presheaves / Sheaves - $C^\infty(\wait, \RR)$, a sheaf of smooth functions - $C^0(\wait, \RR)$, a sheaf of continuous functions - $\underline{\RR}(\wait)$, the constant sheaf associated to $\RR$ (locally constant real-valued functions) - $\Hol(\wait, \CC)$, a sheaf of holomorphic functions - $K_p$ the skyscraper sheaf: \[ K_p(U) \da \begin{cases} k & p\in U \\ 0 & \text{else}. \end{cases} \] - $\OO_X(\wait)$, the sheaf of regular functions on $X$ ## The Algebra-Geometry Dictionary Let $k=\bar k$, we're setting up correspondences: Algebra Geometry ----------------------------------------------------------- ------------------------------ $\kx{n}$ $\AA^n_{/k}$ Maximal ideals $\mathfrak{m}={x_1 - a_1, \cdots, x_n - a_n}$ Points $\vector a \da \tv{a_1, \cdots, a_n} \in \AA^n$ Radical ideals $J = \sqrt{J} \normal \kx{n}$ $V(J)$ the zero locus Prime ideals $\mathfrak{p}\in \spec(\kx{n})$ Irreducible closed subsets Minimal prime ideals of $A(X)$ Irreducible components of $X$ $I(S)$ the ideal of a set $S \subseteq \AA^n$ a subset $I + J$ $V(I) \intersect V(J)$ $\sqrt{I(V) + I(W)}$ $V\intersect W$ $I \intersect J, IJ$ $V(I) \union V(J)$ $I(V) \intersect I(W), \sqrt{I(V)I(W)}$ $V \union W$ $I(V) : I(W)$ $\bar{V\sm W}$ $\kx{n}/I(X)$ $A(X)$ (Functions on $X$) $A(X)$ a domain $X$ is irreducible $A(X)$ indecomposable $X$ is connected $k\dash$algebra morphism $A(X)\to A(Y)$ Morphisms of varieties $X\to Y$ Krull dimension $n$ (chaints of primes) Topological dimension $n$ (chains of irreducibles) Integral domains $S(X)$ Irreducible projective varieties $X$