Artin-Wedderburn Theorem
Structure theorem for semisimple rings and unique decomposition into matrix rings
Recent posts
Posts sorted by last_modified_at (most recently updated first).
Intersection Theory of Virtual Classes
Refined Gysin homomorphisms, excess intersection, and diagonal intersection of virtual classes
Perfect Obstruction Theory
Perfect obstruction theory, virtual fundamental class, intrinsic normal cone, localized Euler class
Integration on Deligne–Mumford Stacks
Rational Chow groups, proper pushforward, and integration on Deligne–Mumford stacks
Proper Stacks
Proper Deligne–Mumford stacks, their valuative criterion, and coarse moduli spaces
Moduli Space of Stable Maps
Moduli of stable maps, Deligne-Mumford stack, virtual dimension, convex targets
Deformations of Nodal Curves
Node smoothing, the local deformation sheaf, and the count 3g-3+n
Semisimple Modules
Equivalent characterizations of semisimple modules, isotypic decomposition, and uniqueness of decomposition
Deformation Theory and the Cotangent Complex
Square-zero extensions, first-order deformations, obstructions, and the cotangent complex
Moduli Spaces
Moduli functors, fine and coarse moduli spaces, and why the moduli stack is the right object
Algebraic Stacks
Artin and Deligne-Mumford stacks via atlases, and quotient stacks
Stacks
Groupoids, categories fibered in groupoids, descent, and the definition of a stack
Grothendieck Topologies
Grothendieck topologies, sites, sheaves on a site, and standard examples
Group Schemes
Group schemes, Hopf algebras, comodules, and torsors
Faithfully Flat Descent
Descent data, the cocycle condition, fpqc topology, and effective descent
Functor of Points
Functor of points, Yoneda embedding, representability, fiber products
Smooth and Étale Morphisms
Smooth, unramified, and étale morphisms; Jacobian and infinitesimal lifting criteria
Divisors and Linear Systems
Cartier and Weil divisors, the sheaf O_X(D), linear systems, and ampleness
Cyclotomic Extensions
Irreducibility of cyclotomic polynomials and Galois groups of cyclotomic fields
Classifying Spaces
Classification of principal G-bundles and construction of the classifying space BG
The Divergence Theorem and Stokes' Theorem
Divergence theorem, Stokes’ theorem, irrotational and conservative fields, unification of integral theorems
Green's Theorem
Green’s theorem, area formulas, circulation and divergence forms, simple connectivity, conservative fields
Surface Integrals and Flux
Parametric surfaces, normal vectors, surface area, scalar surface integrals, flux
Line Integrals
Scalar and vector line integrals, work, fundamental theorem, path independence, conservative fields
Vector Fields
Vector fields, gradient and conservative fields, potentials, divergence and curl, differential identities
Curves and Vector-Valued Functions
Vector-valued functions, parametric curves, velocity, tangents, arc length, curvature, and acceleration decomposition
Multiple Integrals
Multiple integrals, Fubini’s theorem, change of variables, and the Jacobian determinant
Functions of Several Variables and Partial Derivatives
Partial derivatives, gradient, differentiability, multivariable chain rule, extrema
Minimal Polynomial
Cayley-Hamilton theorem and minimal polynomial
Quadratic Forms and Sylvester's Law of Inertia
Classification of real symmetric bilinear forms
Singular Value Decomposition
Orthogonal decomposition of arbitrary real matrices
The Fundamental Theorem of Calculus
Fundamental theorem, existence of antiderivatives, Leibniz rule, termwise integration of power series
Improper Integrals
Improper integrals over infinite intervals, singular integrals, comparison test, absolute convergence
Integration
Antiderivatives, indefinite integrals, Riemann sums, definite integrals, properties, and mean value theorem
Taylor's Theorem
Taylor polynomials, Lagrange remainder, Maclaurin expansions, approximation and limits
Complex Spectral Theorem
unitary diagonalization of normal operators
Spectral Theorem
Orthogonal diagonalization of self-adjoint operators
Mean Value Theorem
Mean value theorem and applications: monotonicity, extrema, convexity tests, L’Hospital’s rule, optimization
Complex Inner Product Spaces
Hermitian inner products over complex numbers
Differentiation
Termwise power series differentiation, elementary function derivatives, product, quotient, and chain rules
Power Series
Power series, radius of convergence, elementary function expansions, and analytic functions
Infinite Series
Partial sums, convergence, geometric and p-series, convergence tests, absolute and conditional convergence
Limits of Sequences
Convergence of sequences, limit laws, standard limits and e, monotone convergence
Quasi-Coherent Sheaves
Sheaves of modules, equivalence on affine schemes, and quasi-coherence
Sheaf Cohomology of Schemes
Cohomology of quasi-coherent sheaves, Serre vanishing, the cohomological criterion for ampleness, and Hilbert polynomials
Differentials and Cotangent Sheaves
Kahler differentials, cotangent sheaves, tangent sheaves, Euler sequences, and canonical sheaves
Division Rings
Division rings, quaternions, Wedderburn’s little theorem, and Schur’s lemma
Central Idempotents and Ring Decomposition
Central idempotents, ring product decomposition, and the Chinese remainder theorem
Units and Zero Divisors
Units, regular elements, and their equivalence in finite commutative rings
Continuous Functions
Definition and properties of continuous functions: extreme and intermediate value theorems