Users Guide to Hartshorne Algebraic Geometry/Chapter 0

References Algebraic Geometry - Hartshorne Arithmetic - Serre https://people.ucsc.edu/~weissman/Math222A/SerreAnn.pdf http://web.mit.edu/18.705/www/12Nts-2up.pdf (for basic commutative algebra) https://arxiv.org/pdf/1605.04832.pdf (for more advanced commutative algebra) Algebraic Number Theory, A Computational Approach - Stein Basic Commutative Algebra Categories of Commutative Rings and Algebras The starting point for this section is the definition of a commutative ring: a unital ring with commutative multiplication. In this book you can assume that all rings are commutative, so we will omit the 'commutative' adjective. The most basic rings include Fields Polynomial rings We can relate rings to one another using a morphism of rings. A function between rings is a morphism of rings if the following two axioms are satisfied (Additivity) (Multiplicativity) we could have stated this succinctly as a function which respects the ring structure. It turns out that rings with ring morphisms form a category . As an important technical note, there is no zero-ring given by a single element in our category. This category has an initial object given by the ring of integers because given a ring morphisms the ring morphism axioms forces , , and Recall that the category of -algebras has objects given by ring morphisms and morphisms given by commutative diagrams If we consider only algebras, the category is equivalent to the category . Note that it is common to consider the categories , , . The motivation for why will be readily apparent when considering categories of schemes. Ideals One of the ways to construct new rings is by taking quotient rings. An ideal of a ring is a subset which is An abelian group under addition Then, we can take the quotient of abelian groups and use the multiplicative structure on to construct one on . The second axiom of ideals guarantees that this is well-defined. This is called a quotient ring. Some typical examples of quotient rings are given by Playing with Presentations As we have seen, there are many ways to construct polynomial ring; but, another interesting technique for creating new polynomial rings is to attach variables which have relations between them. For example, consider . We can relabel the elements we've attached, so we consider the ring , but there are a couple relations between these variables: note that these two relations can be used to show others such as and . Hence Some other examples include Prime Ideals There are a special class of ideals called prime ideals: an ideal in a UFD is prime if For example, is the first known example of a prime ideal. It should be apparent that is not a prime ideal since but . Now, given an irreducible polynomial the ideal will be prime. A simple non-example of a prime ideal is given by . This can be generalized to . Some other examples of prime ideals include If you take the quotient ring of a prime ideal in a UFD you get an integral domain. This means your ring has the following multiplicative property: if or For example, in you will never be able to multiply two non-zero elements together to get zero. The two key non-examples of a ring being an integral domain are since since In general, an ideal of a ring is called prime if is an integral domain. If is also a field, then we call a maximal ideal. One useful exercise is to check that for a morphism and a prime ideal the inverse image is a prime ideal. The second example motivates the operation of taking radicals of an ideal. Given an ideal we define its radical asFor example, the radical of the ideal is . Given a quotient ring we call the ring its reduction; sometimes this is denoted . We define the nilradical of a ring as . The nonzero elements in the nilradical are called nilpotents. Eisenstein's Criterion and Constructing Prime Ideals There is a generalization of Eisenstein's criterion for integral domains: given a ring and a polynomial which can be written as then it cannot be written as a product of polynomials if the following conditions are satisfied: Suppose there exists a prime ideal such that then cannot be written as a product of polynomials . For example, consider the integral domain and the polynomial given by Using the prime ideal we have that , , and . Hence is a prime ideal. This example can be extended to show that generates a prime ideal. Nullstellensatz Now we are in the right place to discuss the foundational theorem of algebraic geometry: Hilbert's nullstellensatz. Here we fix as an algebraically closed field. Theorem: The maximal ideals of are in bijection with the set . For example, the kernel of is the ideal . This allows one to interpret quotient rings give by ideals as algebraic subsets of because an evaluation morphismis well-defined only if is a maximal ideal. For example, consider the following example and non-example: is a well defined morphism sinceThis implies that is not a well-defined morphism because ; there is no quotient ring . Hence Now we can interpret rings which are not integral. For example, we saw that is not an integral domain. Geometrically, this is the union of the and axes. The other main case of a non-integral ring is a non-reduced ring. For example, is the -axis but there is extra algebraic information from the left over. The way you should interpret this ring as is a fat line. Basic Scheme Theory We can now confidently define an affine scheme: it is a functorfor some fixed commutative ring . Localization The next basic construction in commutative ring theory is localization. This defines a generalization of inverting the non-zero integers and getting the rational numbers. Let be a multiplicatively closed subset with unity, meaning and . For example, for a fixed element consider the subset . We define a commutative ring as follows. First, consider the set where(don't worry, we will given a motivating example for this seemingly random ). It is an exercise to verify that this indeed defines an equivalence relation — it is standard to write these equivalence classes as . These equivalence classes have a well-define commutative ring structure given bySome basic examples of localization include The subset gives the ring . Notice that if we localized by the set then this gives the ring . But, because we could write as , these two rings are isomorphic. For brevity, we could just say that we localized by . Try localizing by some other non-zero integers and see why you find. An important geometric example is given by localizing by some non-zero polynomial . Given an integral domain , we can take the set . Then, is called the field of fractions of the integral domain. (It is an exercise to check that this is a field) Given a ring and a prime ideal , we can consider the set . This is multiplicatively closed because of the properties of primality of an ideal. The localization of by is typically denoted . For example, consider . The localization can be described as The last example is special because it motivates a definition: a ring is local if it has a unique maximal ideal. The pair is a local ring. Basic Module Theory A -module is defined as an abelian group with a fixed ring morphism . We will use the notation where for the ring action on . A morphism of -modules is defined by a commutative diagram We can use this construction to build a category of -modules which is abelian. This means that it has a zero object, kernels and cokernels, products and coproducts, and images/co-images agree. Please note that we've had to enlarge our category of commutative rings to all rings since the endomorphism ring of an abelian group is generally non-commutative; This is one of the only cases where we use non-commutative unital rings in this book. Typical examples of -modules includes the zero object ideals direct sums, such as a morphism of rings gives the structure of an -module on the underlying abelian group of Another useful technique for constructing new modules is taking the cokernel of a morphism . For example, the cokernel ofis . We can generalize this example using exact sequences. A sequence of objects in an abelian category is called exact if each in the last example, we had the exact sequence In general, if there is an exact sequence for finite integers , then we say that the module is of finite-type. If there is just a sequence then we say that the module is finite. For example, the module is finite but not finite-type since the kernel of the non-trivial morphism is the ideal Tensor Products construct tensor products for modules construct tensor products of algebras show that tensor products of integral domains are integral show that show Finiteness, Chain Conditions If we have an -algebra we say that it is a finite if it is finite as a module. We say that it is of finite-type if there exists a surjective morphism , implying that There are a couple other notions of "finiteness" which appear in commutative algebra called chain conditions. We say call a sequence of -modules an ascending chain and a descending chain. They satisfy the ascending chain condition or descending chain condition if there is some such that , . If there exist chains where or then we say is Noetherian or Artinian, respectively. One can show that every Artinian ring is Noetherian. The basic examples of Noetherian rings include Fields Finite algebras over fields Quotients of Noetherian rings. A simple non-example is given by the ring where is a field. There is a fundamental theorem in algebra called Hilbert's Basis Theorem stating:Theorem: If is Noetherian, then is NoetherianHence all rings of the formare Noetherian. Artinian rings are much simpler than Noetherian rings:Theorem: Every Artin ring is a finite product of Artin local rings.All we have to analyze is the structure of an Artin local ring . Notice that we have a descending chainwhich eventually stabilizes at some ; this is the zero ideal . We can use this to show the underlying -vector space of is finite dimensional. Some examples of artin local rings are Integrality Given a morphism of commutative rings we say an element is integral over if there is a monic polynomial and a morphismsending . For example, is integral over sinceAdjoining all of the integral elements is called the integral closure of in . An integral domain is called integrally closed if every element in its fraction field is integral over . For example, we can compute the integral closure offairly easily. Since it is isomorphic to the ring we should see immediately that is not contained in . Adjoining this element to gives a ring isomorphic to . As an exercise, try and unpackTODO: - hyperelliptic curves - quotient fields of curves - https://math.stackexchange.com/questions/2304521/why-is-this-coordinate-ring-integral-over-kx - rings of integers Chapter 0 Basic Commutative Algebra Structures tensor products of modules/algebras... localization of modules categories of commutative algebras support of modules graded rings colimits and stalks limits, completions, p-adics valuations - https://en.wikipedia.org/wiki/Valuation_(algebra)#P-adic_valuation_on_a_Dedekind_domain dimension transcendence degree functor formalism categorical structures such as pullbacks and pushforwards kahler differentials/ basic differential algebra smooth algebras/smooth morphisms complete intersection/local complete intersection morphisms etale algebras/etale morphisms galois theory with useful terminology... same w/ algebraic number theory terminology... basic homological algebra, ext, tor koszul complexes derived categories grothendieck group hilbert polynomial grobner bases/elimination theory - https://mathoverflow.net/questions/60957/how-to-determine-whether-an-ideal-is-prime-or-not-by-an-algorithm Theorems Eisenstein's Criterion Primary Decomposition Noether Normalization Going up and down Basic Differential/Complex Geometry Constructions Smooth manifolds Morphisms Vector Bundles Topological K-theory de-Rham Cohomology Complex manifolds and sheaves hodge decomposition of complex manifolds Theorems Whitney embedding theorem Submersion theorem Sard's theorem