Introduction to derived algebraic geometry
... k is just the category of cell cdgas. Let A be one. Then MB (A) should be the space of (B ⊗k A)-dg modules M such that M is projective of finite type over A, i.e. M is a direct summand of some Ap in D(A). We make MB (A) into a functor by, for a map A → A0 of cell cdgas, defining MB (A) → MB (A0 ) by ...
... k is just the category of cell cdgas. Let A be one. Then MB (A) should be the space of (B ⊗k A)-dg modules M such that M is projective of finite type over A, i.e. M is a direct summand of some Ap in D(A). We make MB (A) into a functor by, for a map A → A0 of cell cdgas, defining MB (A) → MB (A0 ) by ...
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... Example 2. For any topological space X, let Example 3. Let X be a smooth differentiable manifold. Let DX be the presheaf on X, with values in the category of real vector spaces, defined by setting DX (U ) to be the space of smooth real–valued functions on U , for each open set U , and with the restr ...
... Example 2. For any topological space X, let Example 3. Let X be a smooth differentiable manifold. Let DX be the presheaf on X, with values in the category of real vector spaces, defined by setting DX (U ) to be the space of smooth real–valued functions on U , for each open set U , and with the restr ...
Intersection homology
... The only allowable one-cycles in X are linear combinations of the ones shown in green in the diagram. (All other 1-cycles pass through the singular point of X and are therefore not allowable.) The green cycles are both boundaries of faces and hence they are trivial in homology, giving IH1 (X) = 0. ...
... The only allowable one-cycles in X are linear combinations of the ones shown in green in the diagram. (All other 1-cycles pass through the singular point of X and are therefore not allowable.) The green cycles are both boundaries of faces and hence they are trivial in homology, giving IH1 (X) = 0. ...
Profinite Groups - Universiteit Leiden
... with ci ∈ Z, 0 ≤ ci ≤ p − 1, called the digits of γ. This ring has a topology given by a restriction of the product topology—we will see this below. The ring Zp can be viewed as Z/pn Z for an ‘infinitely high’ power n. This is a useful idea, for example, in the study of Diophantine equations: if suc ...
... with ci ∈ Z, 0 ≤ ci ≤ p − 1, called the digits of γ. This ring has a topology given by a restriction of the product topology—we will see this below. The ring Zp can be viewed as Z/pn Z for an ‘infinitely high’ power n. This is a useful idea, for example, in the study of Diophantine equations: if suc ...
A Problem Course on Projective Planes
... planes, some methods of constructing them, the introduction of coordinates, collineations, and the basics of the relationships between the geometry of the plane, the algebraic properties of possible coordinate systems, and the properties of its collineation group. In keeping with the modified Moore- ...
... planes, some methods of constructing them, the introduction of coordinates, collineations, and the basics of the relationships between the geometry of the plane, the algebraic properties of possible coordinate systems, and the properties of its collineation group. In keeping with the modified Moore- ...
4.2 Every PID is a UFD
... Recall that an ideal I of a commutative ring with identity R is principal if I = hai for some a ∈ R, i.e. I = {ra : r ∈ R}. An integral domain R is a principal ideal domain if all the ideals of R are principal. Examples of PIDs include Z and F [x] for a field F . Definition 4.2.1 A commutative ring ...
... Recall that an ideal I of a commutative ring with identity R is principal if I = hai for some a ∈ R, i.e. I = {ra : r ∈ R}. An integral domain R is a principal ideal domain if all the ideals of R are principal. Examples of PIDs include Z and F [x] for a field F . Definition 4.2.1 A commutative ring ...
Some structure theorems for algebraic groups
... endomorphisms x 7→ xn of Gm . Moreover, µn is reduced if and only if n is prime to char(k). Example 2.1.9. Given a vector space V , the general linear group GL(V ) is the group functor that assigns to any scheme S, the automorphism group of the sheaf of OS -modules OS ⊗k V . When V is of finite dime ...
... endomorphisms x 7→ xn of Gm . Moreover, µn is reduced if and only if n is prime to char(k). Example 2.1.9. Given a vector space V , the general linear group GL(V ) is the group functor that assigns to any scheme S, the automorphism group of the sheaf of OS -modules OS ⊗k V . When V is of finite dime ...
Polynomial closure and unambiguous product
... This paper is a contribution to the algebraic theory of recognizable languages. The main topic of this paper is the polynomial closure, an operation that mixes together the operations of union and concatenation. Formally, the polynomial closure of a class of languages L of A ∗ is the set of language ...
... This paper is a contribution to the algebraic theory of recognizable languages. The main topic of this paper is the polynomial closure, an operation that mixes together the operations of union and concatenation. Formally, the polynomial closure of a class of languages L of A ∗ is the set of language ...
Chapter 3
... Remark: In an algebraic context equivalence classes are often called cosets. For example, lines and planes in Euclidean geometry (affine subspaces) are cosets of the underlying linear algebra, the equivalence relation on the vectors being that their difference belongs to the true subspace (line or p ...
... Remark: In an algebraic context equivalence classes are often called cosets. For example, lines and planes in Euclidean geometry (affine subspaces) are cosets of the underlying linear algebra, the equivalence relation on the vectors being that their difference belongs to the true subspace (line or p ...
GEOMETRIC CONSTRUCTIONS AND ALGEBRAIC FIELD
... the maximal ideals of a polynomial ring look like. The example we considered was a question about whether a polynomial had roots in a given field - namely, whether we could factor p(x) into linear factors. More generally, however, we can consider irreducible polynomials to be polynomials in F [x] th ...
... the maximal ideals of a polynomial ring look like. The example we considered was a question about whether a polynomial had roots in a given field - namely, whether we could factor p(x) into linear factors. More generally, however, we can consider irreducible polynomials to be polynomials in F [x] th ...
Chap 0
... Then F (x0 ⇥ K) ✓ U . By the Hot Dog Lemma, there are open nbhs V, W of x0 , K in X, Z resp. so that F (V ⇥ W ) ✓ U . This implies that (F )(V ) ✓ hK, U i. Therefore, (F ) : X ! M ap(Z, Y ) is continuous (without any assumptions on X, Y, Z). ...
... Then F (x0 ⇥ K) ✓ U . By the Hot Dog Lemma, there are open nbhs V, W of x0 , K in X, Z resp. so that F (V ⇥ W ) ✓ U . This implies that (F )(V ) ✓ hK, U i. Therefore, (F ) : X ! M ap(Z, Y ) is continuous (without any assumptions on X, Y, Z). ...
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... into statements about ideals in the ring of integers. In Z, for example, there is a one-to-one correspondence between numbers and ideals because Z is a principal ideal domain. This is not true for a ring that is not a principal ideal domain. This trouble aside, we will see in Section 3 that ideals o ...
... into statements about ideals in the ring of integers. In Z, for example, there is a one-to-one correspondence between numbers and ideals because Z is a principal ideal domain. This is not true for a ring that is not a principal ideal domain. This trouble aside, we will see in Section 3 that ideals o ...
Tannaka Duality for Geometric Stacks
... By combining the above methods, one can deduce that φ is an equivalence whenever X is given globally as a quotient of a separated algebraic space by the action of a linear algebraic group (and S is proper). The main motivation for this paper was to find a more natural hypothesis on X which forces φ ...
... By combining the above methods, one can deduce that φ is an equivalence whenever X is given globally as a quotient of a separated algebraic space by the action of a linear algebraic group (and S is proper). The main motivation for this paper was to find a more natural hypothesis on X which forces φ ...
High School: Geometry » Introduction
... Together, the Laws of Sines and Cosines embody the triangle congruence criteria for the cases where three pieces of information suffice to completely solve a triangle. Furthermore, these laws yield two possible solutions in the ambiguous case, illustrating that Side-Side-Angle is not a congruence cri ...
... Together, the Laws of Sines and Cosines embody the triangle congruence criteria for the cases where three pieces of information suffice to completely solve a triangle. Furthermore, these laws yield two possible solutions in the ambiguous case, illustrating that Side-Side-Angle is not a congruence cri ...
2-5 / 2-6 Class Notes
... always, sometimes, or never true. Explain. If plane T contains contains point G, then plane T contains point G. ...
... always, sometimes, or never true. Explain. If plane T contains contains point G, then plane T contains point G. ...
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... OK is finitely generated over Z–the result then being immediate from the structure theorem for finitely generated abelian groups. We will omit the proof of this, but it can be found on pages 12-13 of Neukirch. However, using this fact, we are able to prove the statement we set out to prove at the be ...
... OK is finitely generated over Z–the result then being immediate from the structure theorem for finitely generated abelian groups. We will omit the proof of this, but it can be found on pages 12-13 of Neukirch. However, using this fact, we are able to prove the statement we set out to prove at the be ...
0 1 0 0 0 0 1 0 0 0 0 1
... vector) into another point (or vector). • An affine transformation is a transformation that maps lines to lines. Why are affine transformations "nice"? We can define a polygon using only points and the line segments joining the points. To move the polygon, if we use affine transformations, we only m ...
... vector) into another point (or vector). • An affine transformation is a transformation that maps lines to lines. Why are affine transformations "nice"? We can define a polygon using only points and the line segments joining the points. To move the polygon, if we use affine transformations, we only m ...
Manifolds and Varieties via Sheaves
... connected and there exists a finite open cover {Ui } such that each (Ui , OX |Ui ) is isomorphic, as a k-space, to an affine variety. A morphism of prevarieties is a morphism of the underlying k-spaces. This is a “prevariety” because we are missing a Hausdorff type condition. Before explaining what ...
... connected and there exists a finite open cover {Ui } such that each (Ui , OX |Ui ) is isomorphic, as a k-space, to an affine variety. A morphism of prevarieties is a morphism of the underlying k-spaces. This is a “prevariety” because we are missing a Hausdorff type condition. Before explaining what ...
Algebraic and Transcendental Numbers
... Back to the theorem: We want to show that A is countable. For each height, put the algebraic numbers of that height in some order Then put these lists together, starting with height 1, then height 2, etc., to put all of the algebraic numbers in order The fact that this is possible proves that A is c ...
... Back to the theorem: We want to show that A is countable. For each height, put the algebraic numbers of that height in some order Then put these lists together, starting with height 1, then height 2, etc., to put all of the algebraic numbers in order The fact that this is possible proves that A is c ...
On phylogenetic trees – a geometer`s view
... proving one of the main results of the present paper, 3.26, which asserts that models of trees with the same number of leaves are deformation equivalent, that is they are in the same connected component of the Hilbert scheme of the projective space in question (hence they have the same Hilbert polyn ...
... proving one of the main results of the present paper, 3.26, which asserts that models of trees with the same number of leaves are deformation equivalent, that is they are in the same connected component of the Hilbert scheme of the projective space in question (hence they have the same Hilbert polyn ...
(January 14, 2009) [08.1] Let R be a principal ideal domain. Let I be
... [08.1] Let R be a principal ideal domain. Let I be a non-zero prime ideal in R. Show that I is maximal. Suppose that I were strictly contained in an ideal J. Let I = Rx and J = Ry, since R is a PID. Then x is a multiple of y, say x = ry. That is, ry ∈ I. But y is not in I (that is, not a multiple of ...
... [08.1] Let R be a principal ideal domain. Let I be a non-zero prime ideal in R. Show that I is maximal. Suppose that I were strictly contained in an ideal J. Let I = Rx and J = Ry, since R is a PID. Then x is a multiple of y, say x = ry. That is, ry ∈ I. But y is not in I (that is, not a multiple of ...
Néron Models of Elliptic Curves.
... We say that the fibered surface is regular if Y is a regular scheme. Let C be a normal, connected, projective curve over K. Definition. A (projective) model of C over B is a normal fibered surface M → Spec(B) together with an isomorphism MK ∼ = C. We say that the model is regular if M is regular. Th ...
... We say that the fibered surface is regular if Y is a regular scheme. Let C be a normal, connected, projective curve over K. Definition. A (projective) model of C over B is a normal fibered surface M → Spec(B) together with an isomorphism MK ∼ = C. We say that the model is regular if M is regular. Th ...
Algebraic variety
In mathematics, algebraic varieties (also called varieties) are one of the central objects of study in algebraic geometry. Classically, an algebraic variety was defined to be the set of solutions of a system of polynomial equations, over the real or complex numbers. Modern definitions of an algebraic variety generalize this notion in several different ways, while attempting to preserve the geometric intuition behind the original definition.Conventions regarding the definition of an algebraic variety differ slightly. For example, some authors require that an ""algebraic variety"" is, by definition, irreducible (which means that it is not the union of two smaller sets that are closed in the Zariski topology), while others do not. When the former convention is used, non-irreducible algebraic varieties are called algebraic sets.The notion of variety is similar to that of manifold, the difference being that a variety may have singular points, while a manifold will not. In many languages, both varieties and manifolds are named by the same word.Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry among the other subareas of geometry.