Tags: #study-guides

Definitions

Undergrad

• What is a prime ideal?
• What is a primitive polynomial?
• What is the content of a polynomial?
• What is a separable extension
• What is a Galois field extension?
  • What is a normal field extension?
  • What is a separable field extension?
• What is a divisible group?
• What are Ext and Tor?

Rings

• What is the module-theoretic definition of a zero-divisor?

• What is the annihilator \(\operatorname{Ann}(x)\) of an element \(x\)?

• What is the ideal generated by a subset \(S\) of a ring \(R\)?

• What is a reduced ring?

• What is a local ring?

• What is a regular ring?

• What is a graded ring?

  • What is a filtered ring? What is the difference?

• What is a Dedekind domain?

• What is a DVR?

• What is a Cohen-Macaulay ring?

• What is a complete ring?

• What is a normal ring?

• What is a Noetherian ring?

• What is a valuation ring?

• What is a Jacobson ring?

• What is an Artin ring?

• What is a finite ring morphism?

• What is a finite type morphism of rings?

• What is a finitely generated ring?

• What is a regular local ring? (Several characterizations)

• What is a semilocal ring?

• What is an excellent?

• What is a Henselian ring?

• What is a Gorenstein ring?

• What is the localization of a ring?

  • What is a saturated multiplicative subset of a ring?
  • What are the p-local integers \({\mathbf{Z}} \left[ { \scriptstyle { {p}^{-1}} } \right]\)?

• What is the adic completion of a ring?

  • What are the p-complete integers \({\mathbf{Z}}{ {}_{ \widehat{p} } }\)? What is the description as a set involving denominators?

• What is a local property of a ring?

• What is a flat morphism of rings?

• What is a local morphism of local rings?

• For \(R,S\) local rings, what does it mean for \(R\) to dominate \(S\)?

• For a local ring \((R, {\mathfrak{m}})\), what is \(G_{\mathfrak{m}}(R)\)?

• What is a connected ring?

• What is \(\operatorname{Pic}(R)\) for \(R\) a ring.

Ideals

• What is a homogeneous ideal?
• What is the contraction (ideals) \(I_c\) of an ideal \(I\)?
• What is the extension \(I^e\) of an ideal \(I\)?
• What does it mean for two ideals to be coprime?
• What is a fractional ideal?
  • What is a principal fractional ideal?
  • What is a colon ideal?
  • What does it mean for a fractional ideal to be invertible?
• What is the Unsorted/class group?
• What does it mean for an ideal to belong to another ideal?
• What is a factorial ring?
• What is an irreducible ideal?
• What is a nilpotent ideal?
• What is a factorization of ideals?
• What is a primary ideal?
• What is the primary decomposition of an ideal?
• What is the Rees algebra of an ideal?
• What is an invertible ideal?
• What is the group of ideals in a Dedekind domain?
• For \(I{~\trianglelefteq~}R\), what is the associated graded ring \(G_I(R)\)?
• What is a complete set of orthogonal idempotents?

Modules and Algebras

• What is an invertible module?
• What is an algebra?
• What is the tensor algebra of a module?
  • What is the alternating algebra of a module?
  • What is the symmetric algebra of a module?
• What does it means for elements to generate an algebra?
• What is an etale algebra?
• What is a semisimple algebra?
• What is a separable algebra?
• What is a finitely generated \(A{\hbox{-}}\)algebra?
• What is a finite \(A{\hbox{-}}\)algebra?
• What is a finitely presented \(A{\hbox{-}}\)algebra?
• What is a faithful module?
• What is the radical of a submodule?
• What is a primary submodule?
• What does it mean for a module \(M\) to be module-finite over another module \(M'\)?
• What is a finite type module \(M\) over a commutative algebra \(A\)?
• What does it mean to be finitely generated as a module? As a field extension?
• How does a ring morphism induce an algebra structure?
• What is an integral \(A{\hbox{-}}\)algebra?
• What is a reduced \(k{\hbox{-}}\)algebra?
• What is the algebra structure on a tensor product of \(A{\hbox{-}}\)algebras?
• What is base change or extension of scalars?
• What is restriction of scalars?
• What is extension of scalars?
• What is the tensor product in ${}_{R}{\mathsf{Mod}} $?
• What is the tensor algebra in ${}_{R}{\mathsf{Mod}} $?
• What is the symmetric algebra in ${}_{R}{\mathsf{Mod}} $?
• What is the alternating algebra in ${}_{R}{\mathsf{Mod}} $?
• What is the exterior algebra in ${}_{R}{\mathsf{Mod}} $?
• What is a faithfully flat module (resp. morphism) in ${}_{R}{\mathsf{Mod}} $?
• What is a quotient morphism of modules?
• What is a regular element of an algebra?
• What is the Cayley-Hamilon theorem for modules?
• What does it mean for a \(k{\hbox{-}}\)algebra to be formally smooth over \(k\)?

Dimension Theory and Numerics

• What is the height of an ideal?
• What is the depth of an ideal?
• What is the codimension of an ideal?
• What is the Krull dimension of a ring?
• What is the global dimension of a ring?
• What is the irrelevant ideal?
• What is the length of a module?
• What is an additive function on a class of modules?
• What is the Poincare series of a graded module?
• What is a Hilbert-Samuel polynomial?
• What is a Hilbert function?
• What is the characteristic polynomial of an \({\mathfrak{m}}{\hbox{-}}\)primary ideal \(I\)?
• What is a system of parameters?

AG

• What is \(k[x]\)? \(k(x)\)? \(k[[x]]\)? \(k((x))\)?
• What is the homogenization of a polynomial?
  • The dehomogenization?
• For \(R\) a graded ring, what is \(\mathop{\mathrm{Proj}}R\)?
• What is a constructible subset of a Noetherian space?
• What is the coordinate ring \(A(V)\) for \(V\) an irreducible algebraic variety over \(k=\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\)?
• For \(k \coloneqq\operatorname{ff}(A)\), what is the rational function field \(k(V)\) on \(V\)?
• What is the local dimension of an algebraic variety at a point?
• What is a flat family?
• What is faithfully flat base change?
• What is a regular sequence?
• What is the Plucker embedding?

Integrality

• What is an integral extension of rings?
• What is the integral closure of a subring?
• What does it mean for an integral domain to be integrally closed?

NT

• What is a normal integral domain?
• What is an algebraic number field?
• What is the ring of integers?
• What is a uniformizer?
• What is a valuation?
• What is the transcendence degree of a field extension?
• For \(G\in {\mathsf{Ab}}{\mathsf{Grp}}\) totally ordered, what is the valuation ring of \(G\)?
  • What is the value group?
• What is a Cauchy sequence in a topological group?
• What is a monogenic field extension?