Let $B,C$ be elements of a partially ordered set such that $B \subseteq C$. Then if we have an element $A \subseteq B$ we get $A \subseteq C$ or if we have an element $C \subseteq D$ then we can get $B \subseteq D$. This demonstrates that we can extend an interval in two ways: by making the predecessor smaller or making the successor larger. We will demonstrate an analagous concept exists in terms of the images of hom classes produced by functors.
Let $C$ and $D$ be categories, and $F : C \to D$ be a functor. Then we have the following I/O relationship pertaining to the morphism part of the functor: the hom class of the input morphism determines the hom class of the output morphism.
\[ f : Mor_C(a,b) \to Mor_D(F(a), F(b)) \]
Let $S$ be the class of all ordered pairs of objects $Ob(C)^2$. Then $f$ induces an equivalence relation $E$ on the ordered pairs of objects so that $E \subseteq (Ob(C)^2)^2$. Two ordered pairs of objects of objects in $C$ are equal if their hom classes are mapped to the same hom class in the output category by the action of the functor. This naturally leads to the hom class comparison preorder of a functor.
Definition. let $F: C \to D$ be a functor, $E$ be the equivalence relation induced by the functor on ordered pairs of objects, and $C$ an equivalence class in $E$. Then the hom class comparison preorder defined on $C$ is defined so that $(a,b) \leq (c,d)$ when $F(Mor_C(a,b)) \leq F(Mor_C(c,d))$.
It is immediate that the hom class comparison preorders of a functor are trivial for any functor which is object mono. The functor that maps any category to its underlying preorder by the hom class congruence, for example, is hom class comparison trivial because it is object mono. Therefore, from now on we can implicitly assume that $F$ is a faithful set-valued functor associated to a concrete category.
The sense that a structure preserving map is an inclusion can now be described functorially. Let $C$ be the category of preorders and monotone maps and suppose that $f : (A,\sqsubseteq) \to (B,\sqsubseteq)$ is a monotone map. Then if we make $(A,\sqsubseteq)$ a smaller preordering or we make $(B,\sqsubseteq)$ a larger preordering then the underlying function is still a monotone map, because it is still preserved as an image of the set-valued functor.
Proposition. let $f : (A,\sqsubseteq_A) \to (B,\sqsubseteq_B)$ be a monotone map then $f$ is still a monotone map for any smaller preorder then $\sqsubseteq_A$ or a larger preorder then $\sqsubseteq_B$.
A map $f$ is monotone iff $\sqsubseteq_A \subseteq map_{f,2}^{-1}(\sqsubseteq_B)$ this is an inclusion relation in a preorder so any transition which makes the predecessor smaller or the successor larger preserves the truth value of the statement. Inverse images are monotone so larger values of $\sqsubseteq_B$ produce larger values of $map_{f,2}^{-1}(\sqsubseteq_B)$ and smaller values of $\sqsubseteq_A$ produce smaller values of itself. $\square$
Recall that the family of all topological spaces on a set form a lattice. By the backwardness of continuous maps, we have that for any continuous map $f: S \to T$ any enlargement of the input topology on $S$ or any reduction of the output topology $T$ preserves the continuity of the function $f$. In particular, every map from a discrete topology or to an indiscrete topology is continuous.
This technique has been used to describe the inverse process: creating the largest/smallest topology on a given set that makes a family of continuous maps from a family of topological spaces into it continuous. The kernel topology is the smallest topology on the input of a common family, and the hull topology is the largest topology on the output of a common family. This sort of technique should be applicable to any category whose elements have structural lattices.
Monday, February 22, 2021
Friday, February 19, 2021
Radical ideals in Artitian rings are boolean
The lattice of ideals of a commutative ring $R$ form a modular lattice as does the sublattice of intermediate normal field extensions of a Galois extension. The lattice of radical ideals forms a distributive lattice as does the lattice of subfields of a finite field. Finally, the lattice of radical ideals of Artinian rings form a boolean algebra. And those are only the subalgebra lattices associated to commutative rings.
Corollary. $Spec(R)$ in an Artinian ring forms a finite discrete topology
Let $R$ be an Artinian ring, then $R$ is krull dimension zero, which means all prime ideals are maximal [1]. Additionally, there are finitely many such maximal ideals [1]. The first means that $Spec(R)$ is T1 and the second means that it is finite. All finite T1 spaces are discrete. $\square$
Corollary. the lattice of radical ideals of an Artitian ring forms a boolean algebra
The closed sets of $Spec(R)$ are a finite discrete topology, which is a power set. By basic set theory, the power sets of sets form a complete atomic boolean algebra. $Spec(R)$ is order dual to the lattice of radical ideals, so that means that the lattice of radical ideals of an Artinian ring also forms a boolean algebra. $\square$
A final comment is worthwhile on the subject of direct product decompositions. Artinian rings are the direct product of local Artinian rings [1], which only have a single maximal ideal. Finite boolean algebras on the other hand are direct products of ordered pairs. The two types of structure therefore are direct products.
Source:
[1] Zariski commutative algebra volume one part four
Corollary. $Spec(R)$ in an Artinian ring forms a finite discrete topology
Let $R$ be an Artinian ring, then $R$ is krull dimension zero, which means all prime ideals are maximal [1]. Additionally, there are finitely many such maximal ideals [1]. The first means that $Spec(R)$ is T1 and the second means that it is finite. All finite T1 spaces are discrete. $\square$
Corollary. the lattice of radical ideals of an Artitian ring forms a boolean algebra
The closed sets of $Spec(R)$ are a finite discrete topology, which is a power set. By basic set theory, the power sets of sets form a complete atomic boolean algebra. $Spec(R)$ is order dual to the lattice of radical ideals, so that means that the lattice of radical ideals of an Artinian ring also forms a boolean algebra. $\square$
A final comment is worthwhile on the subject of direct product decompositions. Artinian rings are the direct product of local Artinian rings [1], which only have a single maximal ideal. Finite boolean algebras on the other hand are direct products of ordered pairs. The two types of structure therefore are direct products.
Source:
[1] Zariski commutative algebra volume one part four
The affine scheme of Spec(R)
The construction of the affine scheme on Spec(R) hinges on the fact that you can you localize at the points of the spectrum: the irreducible radical ideals. This is possible because irreducible radical ideals are prime, which means that their complement is multiplicatively closed. This provides a link between the lattice of radical ideals and the lattice of multiplicative sets, which isn't entirely obvious. As a preliminary, I will therefore explain why irreducible radical ideals are prime ideals, and why you can indeed localize over them. This doesn't require much more then the proof that the nilradical is the intersection of all prime ideals and the lattice theorem.
Radical ideals are the complement of multiplicative-iteration closed sets and prime ideals are the complements of multiplicatively closed sets, so prime ideals are quite obviously radical. Prime ideals are also clearly irreducible (not the intersection of any other ideals) because if they are reducible then we can form elements from each set not in P whose product is in P. It remains only to show that irreducible radical ideals are prime. The proof is based upon Zorn's lemma, which is applicable because the lattice of ideals is a complete lattice.
Lemma. the nilradical $N$ is the intersection of all prime ideals
The nilradical is the smallest radical ideal, so it is contained in all prime ideals. We will show the converse. Let $x \not\in N$ be a non-nilpotent. Let $F$ be the family of all ideals not containing $x^n$ for $n \ge 0$. We can apply Zorn's lemma to get a maximal element $I_{max} \in F$. If $g,h \not\in I_max$ and $gh \in I_{max}$ we have $I_{max} + (g), I_{max} + (h) \not\in F$. Which implies that there exists $n,m \in \mathbb{N}$ such that $x^n \in I_{max} + (g)$ and $x^m \in I_{max} + (h)$ so $x^{n+m} \in I_{max}$ which contradicts the supposition that $I_{max}$ contains no powers of $x$. Non-nilpotents are all contained in prime ideals, so elements contained in all prime ideals are nilpotent. $\square$
The proof that the nilradical is the intersection of all prime ideals pays off, because everything else to do with the intersection between prime and radical ideals follows from it. The rest of the proofs are suprisingly easy.
Lemma. let $R$ be a commutative ring then if $(0)$ is an irreducible radical ideal it is prime.
Proof. If $(0)$ is a radical ideal it is the nilradical and the intersection of all prime ideals. By the fact that $(0)$ is irreducible it is contained in this set of all prime ideals which means it is a prime ideal. $\square$
Theorem. irreducible radical ideals are prime
Proof. let $I$ be an irreducible radical ideal of a commutative ring $R$. By the lattice theorem $(0)$ is an irreducible radical ideal in $\frac{R}{I}$ which means it is prime. The inverse image of prime ideal is prime, so this $(0)$ ideal in $\frac{R}{I}$ can be reflected back to get a prime ideal $I$. $\square$
With these preliminaries out of the way, we can now construct the affine scheme associated with a commutative ring $R$. In summary, the lattice of radical ideals is dual to a locale whose points are prime ideals. That the cotopology $Spec(R)$ is order-dual to the lattice of radical ideals is relevant, as for example coatomic ideals (which are called maximal ideals for some reason, although I have already gotten used to it) are translated into atomic sets in the cotopology which are of course singletons $\{x\}$.
The stalk of the affine scheme is defined by the localisation, which is a morphism from the lattice of multiplicative sets $Sub(R,*,1)$ to the objects of the category of rings $Ob(Rings)$. In this case of integral domains, the larger the zero free multiplicative set the larger the resulting localisation, as the localisation approaches the field of fractions, which means in certain cases localisation is a functor on a thin category. \[ \ell : Sub(R,*,1) \to Ob(Rings) \] By their very definition, the complement of a prime ideal is a multiplicative. This is formalized by the map $f : PrimeIdeals(R) \to Sub(R,*,1)$. The input action of this map on the localisation map $\ell$ produces a new morphism in the category of sets: \[ \ell : PrimeIdeals(R) \to Ob(Rings) \] We can now apply the image functor, to turn this into a map from sets of prime ideals (including open sets of our topology $Spec(R)$) to sets of rings. \[ \ell : \wp(PrimeIdeals(R)) \to \wp(Ob(Rings)) \] We now have a set of rings associated to any open set in the spectrum $Spec(R)$, but we want to get a single ring. There is a very obvious way to get a single object from a set of objects in a category: the product. This is precisely what we are going to use to construct the presheaf of rings. \[ \times : \wp(Ob(Rings)) \to Ob(Rings) \] We now have an adjusted localisation map $\ell_*$ which assigns a ring to any set of prime ideals by the product of localisations. \[ \ell_* : \wp(PrimeIdeals(R)) \to Ob(Rings) \] This is functorial on open sets because given any product of a set of objects we can form restriction morphisms to any product of a subset of objects, so the restriction maps merely restrict products of localisations to their subsets, which forms a ring homomorphism as it would in any category. So we have a presheaf of rings $\ell_*$ which we constructed from the localisation. This isn't necessarily a sheaf, so all that remains is to use sheafification to get a sheaf $\ell_*^{\#}$.
The sheafification, which is adjoint to the inclusion functor from the category of presheaves to the category of sheaves, is very convenient because it is much easier to construct presheafs then it is sheafs. Presheafs are abound in mathematics and in category theory, and all we need to do turn them into sheafs is to apply a single functor. This sheaf is a scheme because it is defined on the spectrum $Spec(R)$ of a topological space.
It is amazing that this scheme construction is even possible, which is why I spent so much time at the beginning describing why it is. There are obviously limitations of the traditional set-theoretic approach to algebraic geometry, such as that it doesn't take care of multiplicities which is responsible for the utility of this scheme construction. I will examine some of the details of that later.
Source:
Algebraic geometry by Robin Hatshorne
Radical ideals are the complement of multiplicative-iteration closed sets and prime ideals are the complements of multiplicatively closed sets, so prime ideals are quite obviously radical. Prime ideals are also clearly irreducible (not the intersection of any other ideals) because if they are reducible then we can form elements from each set not in P whose product is in P. It remains only to show that irreducible radical ideals are prime. The proof is based upon Zorn's lemma, which is applicable because the lattice of ideals is a complete lattice.
Lemma. the nilradical $N$ is the intersection of all prime ideals
The nilradical is the smallest radical ideal, so it is contained in all prime ideals. We will show the converse. Let $x \not\in N$ be a non-nilpotent. Let $F$ be the family of all ideals not containing $x^n$ for $n \ge 0$. We can apply Zorn's lemma to get a maximal element $I_{max} \in F$. If $g,h \not\in I_max$ and $gh \in I_{max}$ we have $I_{max} + (g), I_{max} + (h) \not\in F$. Which implies that there exists $n,m \in \mathbb{N}$ such that $x^n \in I_{max} + (g)$ and $x^m \in I_{max} + (h)$ so $x^{n+m} \in I_{max}$ which contradicts the supposition that $I_{max}$ contains no powers of $x$. Non-nilpotents are all contained in prime ideals, so elements contained in all prime ideals are nilpotent. $\square$
The proof that the nilradical is the intersection of all prime ideals pays off, because everything else to do with the intersection between prime and radical ideals follows from it. The rest of the proofs are suprisingly easy.
Lemma. let $R$ be a commutative ring then if $(0)$ is an irreducible radical ideal it is prime.
Proof. If $(0)$ is a radical ideal it is the nilradical and the intersection of all prime ideals. By the fact that $(0)$ is irreducible it is contained in this set of all prime ideals which means it is a prime ideal. $\square$
Theorem. irreducible radical ideals are prime
Proof. let $I$ be an irreducible radical ideal of a commutative ring $R$. By the lattice theorem $(0)$ is an irreducible radical ideal in $\frac{R}{I}$ which means it is prime. The inverse image of prime ideal is prime, so this $(0)$ ideal in $\frac{R}{I}$ can be reflected back to get a prime ideal $I$. $\square$
With these preliminaries out of the way, we can now construct the affine scheme associated with a commutative ring $R$. In summary, the lattice of radical ideals is dual to a locale whose points are prime ideals. That the cotopology $Spec(R)$ is order-dual to the lattice of radical ideals is relevant, as for example coatomic ideals (which are called maximal ideals for some reason, although I have already gotten used to it) are translated into atomic sets in the cotopology which are of course singletons $\{x\}$.
The stalk of the affine scheme is defined by the localisation, which is a morphism from the lattice of multiplicative sets $Sub(R,*,1)$ to the objects of the category of rings $Ob(Rings)$. In this case of integral domains, the larger the zero free multiplicative set the larger the resulting localisation, as the localisation approaches the field of fractions, which means in certain cases localisation is a functor on a thin category. \[ \ell : Sub(R,*,1) \to Ob(Rings) \] By their very definition, the complement of a prime ideal is a multiplicative. This is formalized by the map $f : PrimeIdeals(R) \to Sub(R,*,1)$. The input action of this map on the localisation map $\ell$ produces a new morphism in the category of sets: \[ \ell : PrimeIdeals(R) \to Ob(Rings) \] We can now apply the image functor, to turn this into a map from sets of prime ideals (including open sets of our topology $Spec(R)$) to sets of rings. \[ \ell : \wp(PrimeIdeals(R)) \to \wp(Ob(Rings)) \] We now have a set of rings associated to any open set in the spectrum $Spec(R)$, but we want to get a single ring. There is a very obvious way to get a single object from a set of objects in a category: the product. This is precisely what we are going to use to construct the presheaf of rings. \[ \times : \wp(Ob(Rings)) \to Ob(Rings) \] We now have an adjusted localisation map $\ell_*$ which assigns a ring to any set of prime ideals by the product of localisations. \[ \ell_* : \wp(PrimeIdeals(R)) \to Ob(Rings) \] This is functorial on open sets because given any product of a set of objects we can form restriction morphisms to any product of a subset of objects, so the restriction maps merely restrict products of localisations to their subsets, which forms a ring homomorphism as it would in any category. So we have a presheaf of rings $\ell_*$ which we constructed from the localisation. This isn't necessarily a sheaf, so all that remains is to use sheafification to get a sheaf $\ell_*^{\#}$.
The sheafification, which is adjoint to the inclusion functor from the category of presheaves to the category of sheaves, is very convenient because it is much easier to construct presheafs then it is sheafs. Presheafs are abound in mathematics and in category theory, and all we need to do turn them into sheafs is to apply a single functor. This sheaf is a scheme because it is defined on the spectrum $Spec(R)$ of a topological space.
It is amazing that this scheme construction is even possible, which is why I spent so much time at the beginning describing why it is. There are obviously limitations of the traditional set-theoretic approach to algebraic geometry, such as that it doesn't take care of multiplicities which is responsible for the utility of this scheme construction. I will examine some of the details of that later.
Source:
Algebraic geometry by Robin Hatshorne
Friday, February 12, 2021
Classification of sheaves
The concept of a sheaf requires classification, towards that end I have produced a tentative classification of sheafs and related concepts. One term which not be completely familiar is that of a "categorical morphism" or a morphism of categories. I simply introduced that term to create a class that includes both covariant and contravariant functors. Covariant and contravariant functors are similar in almost everyway, so they should belong to a single class.
The basic category-theoretic classification of a sheaf is that it is a special case of contravariant functor. The next concepts we ought to introduce is that of a presheaf to the topoi of sets, which is dual to the concept of a set-valued functor. From the perspective of topoi theory, it is sufficient to consider set-valued functors and presheafs because the functor categories constructed from either of them form elementary topoi.
After the class of presheafs, the classification splits up a bit as presheafs can be defined either on a site or the lattice of open sets of a topology. Site presheafs are more general then topological ones, as you can make a lattice of open sets into a site by defining the family of covers to be all common output morphism systems whose inputs join into the output. Sites generally just give selected categories more of a topological character which allows us to define sheaves over them.
The two different covering conditions (1) locality and (2) gluing complete our classification. At this point, I should briefly comment on concrete categories, which are defined by faithful functors to the category of sets. Although the basic concept of a presheaf is a contravariant functor to the topoi of sets, defining sheafs and presheafs to arbitrary concrete categories is also necessary for any real application, so presheafs are not always classified as set-valued contravariant functors. The underlying presheaf can clearly be derived by composition.
Topoi theory essentially splits up into the study of elementary topoi and of Grothendieck topoi. Elementary topoi are generalizations of the topoi of sets and Grothendeick topoi are functor categories of sheaves on a site. The monomorphisms in the topoi of sheaves, are precisely the natural transformations whose component morphisms in the output category object common to both functors are all monomorphisms. A key realisation is that the input action of these mono natural transformations induces a lattice of subobjects in the category of sheaves.
After the class of presheafs, the classification splits up a bit as presheafs can be defined either on a site or the lattice of open sets of a topology. Site presheafs are more general then topological ones, as you can make a lattice of open sets into a site by defining the family of covers to be all common output morphism systems whose inputs join into the output. Sites generally just give selected categories more of a topological character which allows us to define sheaves over them.
The two different covering conditions (1) locality and (2) gluing complete our classification. At this point, I should briefly comment on concrete categories, which are defined by faithful functors to the category of sets. Although the basic concept of a presheaf is a contravariant functor to the topoi of sets, defining sheafs and presheafs to arbitrary concrete categories is also necessary for any real application, so presheafs are not always classified as set-valued contravariant functors. The underlying presheaf can clearly be derived by composition.
Topoi theory essentially splits up into the study of elementary topoi and of Grothendieck topoi. Elementary topoi are generalizations of the topoi of sets and Grothendeick topoi are functor categories of sheaves on a site. The monomorphisms in the topoi of sheaves, are precisely the natural transformations whose component morphisms in the output category object common to both functors are all monomorphisms. A key realisation is that the input action of these mono natural transformations induces a lattice of subobjects in the category of sheaves.
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