Andr´e-Quillen (co)Homology, Abelianization and Stabilization
... Definition: Quillen Homology is the total left derived functor of abelianization. For B ∈ C, LAb(B) gives the Quillen Homology of B. Examples: C = sSets Ab(X ) = Z[X ] =⇒ LAb(X ) = Z[X ] since X is cofibrant. π∗ LAb(X ) H∗ (X ) usual homology C = T op Ab(X ) = Sp ∞ (X ) =⇒ LAb(X ) = Sp ∞ (cX ). π∗ ...
... Definition: Quillen Homology is the total left derived functor of abelianization. For B ∈ C, LAb(B) gives the Quillen Homology of B. Examples: C = sSets Ab(X ) = Z[X ] =⇒ LAb(X ) = Z[X ] since X is cofibrant. π∗ LAb(X ) H∗ (X ) usual homology C = T op Ab(X ) = Sp ∞ (X ) =⇒ LAb(X ) = Sp ∞ (cX ). π∗ ...
Étale Cohomology
... Grothendieck was the first to suggest étale cohomology (1960) as an attempt to solve the Weil Conjectures. By that time, it was already known by Serre that if one had a suitable cohomology theory for abstract varieties defined over the complex numbers, then one could deduce the conjectures using th ...
... Grothendieck was the first to suggest étale cohomology (1960) as an attempt to solve the Weil Conjectures. By that time, it was already known by Serre that if one had a suitable cohomology theory for abstract varieties defined over the complex numbers, then one could deduce the conjectures using th ...
A Common Recursion For Laplacians of Matroids and Shifted
... between the Laplacian eigenvalues of matroids and shifted complexes: they satisfy the exact same recursion, which we call the spectral recursion, equation (2). This recursion is stated in terms of the spectrum polynomial, a natural generating function for Laplacian eigenvalues, defined in equation ( ...
... between the Laplacian eigenvalues of matroids and shifted complexes: they satisfy the exact same recursion, which we call the spectral recursion, equation (2). This recursion is stated in terms of the spectrum polynomial, a natural generating function for Laplacian eigenvalues, defined in equation ( ...
Discrete-Time Sequences and Systems
... The next several sections follow the progression of topics in this brief introduction: We start by formally defining the various types of sequences we discussed above. Section 2.3 considers linear discrete-time systems, especially of the shift-invariant kind, which correspond to difference equations ...
... The next several sections follow the progression of topics in this brief introduction: We start by formally defining the various types of sequences we discussed above. Section 2.3 considers linear discrete-time systems, especially of the shift-invariant kind, which correspond to difference equations ...
On the homology and homotopy of commutative shuffle algebras
... zeroth level consists precisely of the unit of the underlying category. This generalizes Quillen’s result [Qu69, Remark on p. 223] in the characteristic zero setting. Brooke Shipley showed [S07] that there is a Quillen equivalence between the model categories of Hk-algebra spectra and differential g ...
... zeroth level consists precisely of the unit of the underlying category. This generalizes Quillen’s result [Qu69, Remark on p. 223] in the characteristic zero setting. Brooke Shipley showed [S07] that there is a Quillen equivalence between the model categories of Hk-algebra spectra and differential g ...
Higher regulators and values of L
... Q (~))cKj(X)~Q be the eigenspace of weight i relative to the Adams operator [2]; then ch~ defines a r e g u l a t o r - a morphism r~:HJs~(X,Q(i))-+I-I~(X,Q(i)). [It is thought that for any schemes there exists a universal cohomology theory H~(X, Z(i)), satisfying Poincare duality and related to Qui ...
... Q (~))cKj(X)~Q be the eigenspace of weight i relative to the Adams operator [2]; then ch~ defines a r e g u l a t o r - a morphism r~:HJs~(X,Q(i))-+I-I~(X,Q(i)). [It is thought that for any schemes there exists a universal cohomology theory H~(X, Z(i)), satisfying Poincare duality and related to Qui ...
A Bousfield-Kan algorithm for computing homotopy
... remains hypothetical. An analogous work can be done with the Whitehead tower. Using the new concept of effective homology [7], a process fundamentally different from Edgar Brown’s, the Postnikov and Whitehead towers have on the contrary easily been implemented, allowing us to access a few homotopy g ...
... remains hypothetical. An analogous work can be done with the Whitehead tower. Using the new concept of effective homology [7], a process fundamentally different from Edgar Brown’s, the Postnikov and Whitehead towers have on the contrary easily been implemented, allowing us to access a few homotopy g ...
Math 256B Notes
... The second property is referred to as the Leibniz rule. Note also that by ∂(b), we really mean map b into A using the given ring map, and then apply ∂. The set of B-linear derivations of A into M is denoted DerB (A, M ), or often just Der(M ) if the ring map B → A is clear from context. Note also th ...
... The second property is referred to as the Leibniz rule. Note also that by ∂(b), we really mean map b into A using the given ring map, and then apply ∂. The set of B-linear derivations of A into M is denoted DerB (A, M ), or often just Der(M ) if the ring map B → A is clear from context. Note also th ...
Math 8211 Homework 2 PJW
... Math 8211 Homework 2 PJW Date due: Monday October 15, 2012. In class on Wednesday September 17 we will grade your answers, so it is important to be present on that day, with your homework. As practice, but not part of the homework, make sure you can do questions in Rotman apart from the ones listed ...
... Math 8211 Homework 2 PJW Date due: Monday October 15, 2012. In class on Wednesday September 17 we will grade your answers, so it is important to be present on that day, with your homework. As practice, but not part of the homework, make sure you can do questions in Rotman apart from the ones listed ...
Homological algebra
... MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to do this first so that grad students will be more familiar with the ideas when they are applied to a ...
... MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to do this first so that grad students will be more familiar with the ideas when they are applied to a ...
Spectral measures in locally convex algebras
... In a Hausdorff space, every normal cone is proper. Let E be an ordered topological vector space, and let ~ be a family of bounded subsets of E such that E = U (S: S E ~ } . K is said to be an Q-cone (strict ~-cone) in E if for each S E ~ , there exists an S ' E ~ such that S c S' fl K - S ' N K ( S ...
... In a Hausdorff space, every normal cone is proper. Let E be an ordered topological vector space, and let ~ be a family of bounded subsets of E such that E = U (S: S E ~ } . K is said to be an Q-cone (strict ~-cone) in E if for each S E ~ , there exists an S ' E ~ such that S c S' fl K - S ' N K ( S ...
Continuous cohomology of groups and classifying spaces
... interplay between topology and algebra. The examples range through geometry and into the study of differential equations. Indeed when Sophus Lie began to look at continuous groups around 1870 [35], he was particularly interested in those respecting a geometric structure and those respecting the solu ...
... interplay between topology and algebra. The examples range through geometry and into the study of differential equations. Indeed when Sophus Lie began to look at continuous groups around 1870 [35], he was particularly interested in those respecting a geometric structure and those respecting the solu ...
The certain exact sequence of Whitehead and the classification of
... define a topological map α : X → Y from the algebraic map f ∗ : H ∗ ( X , Z) → H ∗ (Y , Z) in the ladder. We need a recursive condition about the compatibility of α with Whitehead’s Γ -groups, and call strong morphism, denoted by ( f ∗ , γ∗ ), such a ladder of maps between Whitehead’s exact sequences ...
... define a topological map α : X → Y from the algebraic map f ∗ : H ∗ ( X , Z) → H ∗ (Y , Z) in the ladder. We need a recursive condition about the compatibility of α with Whitehead’s Γ -groups, and call strong morphism, denoted by ( f ∗ , γ∗ ), such a ladder of maps between Whitehead’s exact sequences ...
The structure of Coh(P1) 1 Coherent sheaves
... negative. In general, in s, there are thus no terms with more than one exponent negative. Next, consider the xi0 -exponent-negative terms in si0 and si1 . Because si0 i1 i2 = 0, these terms must be equal (recall the powers of -1 in the differential of the Cech complex). It follows that we can pick a ...
... negative. In general, in s, there are thus no terms with more than one exponent negative. Next, consider the xi0 -exponent-negative terms in si0 and si1 . Because si0 i1 i2 = 0, these terms must be equal (recall the powers of -1 in the differential of the Cech complex). It follows that we can pick a ...
as a PDF
... respectively. These spectral sequences have Adams style differentials, and in all cases except the third, each of these spectral sequences can be realized as an Adams spectral sequence in an appropriate category of module spectra (to ensure equality on E1 , we choose a minimal resolution of our modu ...
... respectively. These spectral sequences have Adams style differentials, and in all cases except the third, each of these spectral sequences can be realized as an Adams spectral sequence in an appropriate category of module spectra (to ensure equality on E1 , we choose a minimal resolution of our modu ...
Clock-Controlled Shift Registers for Key
... clock control technique is a forward clock control (as opposed to feedback clock control). A comprehensive survey on clock-controlled shift registers can be found in [17]. In order to ensure security of a key-stream generator against the BerlekampMassey algorithm, its output sequence should have lar ...
... clock control technique is a forward clock control (as opposed to feedback clock control). A comprehensive survey on clock-controlled shift registers can be found in [17]. In order to ensure security of a key-stream generator against the BerlekampMassey algorithm, its output sequence should have lar ...
Lexlike sequences - Oklahoma State University
... we will work over the ring B = k[x1 , · · · , xn ]/(x21 , · · · , x2n ). (Since we consider only Hilbert functions of monomial ideals, we can replace an exterior algebra by B.) We have numbered the results in this section not sequentially but in a way that emphasizes the analogy with the correspond ...
... we will work over the ring B = k[x1 , · · · , xn ]/(x21 , · · · , x2n ). (Since we consider only Hilbert functions of monomial ideals, we can replace an exterior algebra by B.) We have numbered the results in this section not sequentially but in a way that emphasizes the analogy with the correspond ...
MATH1023 Calculus I, 2016
... For even k, we have cos x ≥ cos a2 > 0 on [ak , bk ]. For odd k, we have cos x ≤ − cos a2 < 0 on [ak , bk ]. Since the arithmetic sequence a, 2a, 3a, · · · has increment a, which is the length of [ak , bk ], we must have nk a ∈ [ak , bk ] for some natural number nk . Then cos(n2k a) is a subsequence ...
... For even k, we have cos x ≥ cos a2 > 0 on [ak , bk ]. For odd k, we have cos x ≤ − cos a2 < 0 on [ak , bk ]. Since the arithmetic sequence a, 2a, 3a, · · · has increment a, which is the length of [ak , bk ], we must have nk a ∈ [ak , bk ] for some natural number nk . Then cos(n2k a) is a subsequence ...
The Prime Spectrum and the Extended Prime
... spectrum of such a ring is spectral. Belluce [1] generalized Kaplansky’s theorem by introducing quasi-commutative rings and proving that the prime spectrum of such a ring is spectral. The question of whether the quasi-commutative rings are precisely those rings for which the prime spectrum is spectr ...
... spectrum of such a ring is spectral. Belluce [1] generalized Kaplansky’s theorem by introducing quasi-commutative rings and proving that the prime spectrum of such a ring is spectral. The question of whether the quasi-commutative rings are precisely those rings for which the prime spectrum is spectr ...
1. Outline of Talk 1 2. The Kummer Exact Sequence 2 3
... Proof. Looking at the Weil-divisor exact sequence, it suffices to show that H r (X, DivX ) = 0 and H r (X, g∗ Gm,K ) = 0 for r > 0, where K is the function field of X. For the first statement, note that DivX is the direct sum of pushforward of sheaves on closed points. Thus, the Grothendeick spectra ...
... Proof. Looking at the Weil-divisor exact sequence, it suffices to show that H r (X, DivX ) = 0 and H r (X, g∗ Gm,K ) = 0 for r > 0, where K is the function field of X. For the first statement, note that DivX is the direct sum of pushforward of sheaves on closed points. Thus, the Grothendeick spectra ...
3 Lecture 3: Spectral spaces and constructible sets
... j ≥ i to be the identity map, which is obviously continuous with respect to the respective topologies. The limit X = lim Xi is N endowed with the discrete topology, which is not qc! Note that the Xi are ...
... j ≥ i to be the identity map, which is obviously continuous with respect to the respective topologies. The limit X = lim Xi is N endowed with the discrete topology, which is not qc! Note that the Xi are ...
Structured Stable Homotopy Theory and the Descent Problem for
... Thus, the K-theory spectrum of any algebraically closed field is equivalent to the connective complex K-theory spectrum away from the characteristic of the field. The key question now is how to obtain information about the K-theory of a field which is not algebraically closed. An attempt to do this ...
... Thus, the K-theory spectrum of any algebraically closed field is equivalent to the connective complex K-theory spectrum away from the characteristic of the field. The key question now is how to obtain information about the K-theory of a field which is not algebraically closed. An attempt to do this ...
foundations of algebraic geometry class 38
... (3) We’ll later see that this will show how cohomology groups vary in families, especially in “nice” situations. Intuitively, if we have a nice family of varieties, and a family of sheaves on them, we could hope that the cohomology varies nicely in families, and in fact in “nice” situations, this is ...
... (3) We’ll later see that this will show how cohomology groups vary in families, especially in “nice” situations. Intuitively, if we have a nice family of varieties, and a family of sheaves on them, we could hope that the cohomology varies nicely in families, and in fact in “nice” situations, this is ...
The symplectic Verlinde algebras and string K e
... of the Verlinde algebra, and in particular, perhaps, construct a self-dual topological field theory in some sense? The purpose of this paper was to investigate this question. What we found was partially satisfactory, partially not. First of all, it turns out that twisted K-theory of loop spaces is qu ...
... of the Verlinde algebra, and in particular, perhaps, construct a self-dual topological field theory in some sense? The purpose of this paper was to investigate this question. What we found was partially satisfactory, partially not. First of all, it turns out that twisted K-theory of loop spaces is qu ...
Group cohomology - of Alexey Beshenov
... If one takes f(g, h) = 1 for all g, h ∈ G, then (L/K, f) ' Mn (K). The cross product algebras up to isomorphism correspond to H 2 (Gal(L/K), L× ). Later on we will see how to calculate such cohomology groups. In the example with quaternions actually H 2 (Gal(C/R), C× ) ' Z/2, so there are two algebr ...
... If one takes f(g, h) = 1 for all g, h ∈ G, then (L/K, f) ' Mn (K). The cross product algebras up to isomorphism correspond to H 2 (Gal(L/K), L× ). Later on we will see how to calculate such cohomology groups. In the example with quaternions actually H 2 (Gal(C/R), C× ) ' Z/2, so there are two algebr ...