Commutative monads as a theory of distributions
... allows us to talk about partial T -linear maps, as well as T -bilinear maps, as we shall explain. An example of a commutative monad, with E the category of sets, is the functor T which to a set X associates (the underlying set of) the free real vector space on X. In this case, the algebras for T are ...
... allows us to talk about partial T -linear maps, as well as T -bilinear maps, as we shall explain. An example of a commutative monad, with E the category of sets, is the functor T which to a set X associates (the underlying set of) the free real vector space on X. In this case, the algebras for T are ...
Invariant differential operators 1. Derivatives of group actions: Lie
... We want an intrinsic approach to existence of differential operators invariant under group actions. The translation-invariant operators ∂/∂xi on Rn , and the rotation-invariant Laplacian on Rn are deceptivelyeasily proven invariant, as these examples provide few clues about more complicated situatio ...
... We want an intrinsic approach to existence of differential operators invariant under group actions. The translation-invariant operators ∂/∂xi on Rn , and the rotation-invariant Laplacian on Rn are deceptivelyeasily proven invariant, as these examples provide few clues about more complicated situatio ...
256B Algebraic Geometry
... we define the stalk Fx as the colimit of the F (U ) over all open sets U containing x. We then define F + (U ) to be the set of all assignments σx , where x ∈ U and σx ∈ Fx , such that for all x ∈ U there is x ∈ V ⊆ U and σ ∈ F (V ) such that σ restricts to σy for all y ∈ V . Generally speaking, the ...
... we define the stalk Fx as the colimit of the F (U ) over all open sets U containing x. We then define F + (U ) to be the set of all assignments σx , where x ∈ U and σx ∈ Fx , such that for all x ∈ U there is x ∈ V ⊆ U and σ ∈ F (V ) such that σ restricts to σy for all y ∈ V . Generally speaking, the ...
Differential Manifolds
... • It materializes on components by eliminating pairs of co- and contra- variant indices: ...
... • It materializes on components by eliminating pairs of co- and contra- variant indices: ...
SIMPLE AND SEMISIMPLE FINITE DIMENSIONAL ALGEBRAS Let
... Such a sequence exists: indeed the length k of such a strictly decreasing sequence is bounded by dimF A. Given a simple A-module M , fix φ : A A → M a surjective A-module map—see the previous paragraph—and let r be the least integer such that φ|Nr = 0. Then r ≥ 1 and M ' Nr−1 /Nr . Thus any simple m ...
... Such a sequence exists: indeed the length k of such a strictly decreasing sequence is bounded by dimF A. Given a simple A-module M , fix φ : A A → M a surjective A-module map—see the previous paragraph—and let r be the least integer such that φ|Nr = 0. Then r ≥ 1 and M ' Nr−1 /Nr . Thus any simple m ...
1 Equivalence Relations
... In the spirit of the slogan, we now give an equivalent formulation for an equivalence relation on a set X. In the end, we will see that giving an equivalence relation on X is the same as specifying a partition of the set X. Recall that a relation is called an equivalence relation if it is reflexive, ...
... In the spirit of the slogan, we now give an equivalent formulation for an equivalence relation on a set X. In the end, we will see that giving an equivalence relation on X is the same as specifying a partition of the set X. Recall that a relation is called an equivalence relation if it is reflexive, ...
Gal(Qp/Qp) as a geometric fundamental group
... a Yoneda-style construction. The category PerfC of perfectoid spaces over C has a pro-étale topology, [Sch13], and one has a sheaf of sets Z on PerfC , namely the sheafification of X 7→ Hom(X, D̃∗ )/Q× p . Thus Z belongs to the category of sheaves of sets on PerfC which admit a surjective map from ...
... a Yoneda-style construction. The category PerfC of perfectoid spaces over C has a pro-étale topology, [Sch13], and one has a sheaf of sets Z on PerfC , namely the sheafification of X 7→ Hom(X, D̃∗ )/Q× p . Thus Z belongs to the category of sheaves of sets on PerfC which admit a surjective map from ...
Descent and Galois theory for Hopf categories
... data. An elegant formulation of the theory was given by Brzeziński in [5], based on the theory of corings. In this formalism, classical descent data (as introduced in [11] for schemes, in [12] for extensions of commutative rings, and in [8] for extensions of non-commutative rings) as well as Hopf-G ...
... data. An elegant formulation of the theory was given by Brzeziński in [5], based on the theory of corings. In this formalism, classical descent data (as introduced in [11] for schemes, in [12] for extensions of commutative rings, and in [8] for extensions of non-commutative rings) as well as Hopf-G ...
Finite flat group schemes course
... this is a “locally small category” but no matter] with a bit of extra structure and some axioms. The example to bear in mind if you’re worrying about set-theoretic issues is the category of sets: there is no “set of all sets” because its subset, the set of all sets that don’t contain themselves as e ...
... this is a “locally small category” but no matter] with a bit of extra structure and some axioms. The example to bear in mind if you’re worrying about set-theoretic issues is the category of sets: there is no “set of all sets” because its subset, the set of all sets that don’t contain themselves as e ...
Categories, functors, and equivalences
... precisely one morphism from a to b. An object b is called terminal if for each a ∈ Ob(C ) there exists precisely one morphism from a to b. We can now define our class of ‘nice’ categories that we wish to compare to module categories. D EFINITION 4.4.3. An additive category is a category C in which ( ...
... precisely one morphism from a to b. An object b is called terminal if for each a ∈ Ob(C ) there exists precisely one morphism from a to b. We can now define our class of ‘nice’ categories that we wish to compare to module categories. D EFINITION 4.4.3. An additive category is a category C in which ( ...
MATH 436 Notes: Ideals.
... for all r1 , r2 ∈ R. To show it is well-defined assume r1 +I = r1′ +I and r2 +I = r2′ +I. Then r1 = r1′ +α and r2 = r2′ +β where α, β ∈ I. Then r1 r2 = r1′ r2′ +(αr2′ +r1′ β+αβ). The term in brackets is in I since I is a two-sided ideal of R. Thus indeed r1 r2 + I = r1′ r2′ + I and the multiplicati ...
... for all r1 , r2 ∈ R. To show it is well-defined assume r1 +I = r1′ +I and r2 +I = r2′ +I. Then r1 = r1′ +α and r2 = r2′ +β where α, β ∈ I. Then r1 r2 = r1′ r2′ +(αr2′ +r1′ β+αβ). The term in brackets is in I since I is a two-sided ideal of R. Thus indeed r1 r2 + I = r1′ r2′ + I and the multiplicati ...
The structure of Coh(P1) 1 Coherent sheaves
... This object leads to some results we might expect from the analogy with R-modules. Definition 1.12. For a locally free sheaf F, define its dual F ∨ = homOX (F, OX ). Proposition 1.13. O(d)∨ ' O(−d). Proof. Simply check that the morphism O(d) ⊗ homOX (O(d), O) → O is an isomorphism on the level of st ...
... This object leads to some results we might expect from the analogy with R-modules. Definition 1.12. For a locally free sheaf F, define its dual F ∨ = homOX (F, OX ). Proposition 1.13. O(d)∨ ' O(−d). Proof. Simply check that the morphism O(d) ⊗ homOX (O(d), O) → O is an isomorphism on the level of st ...
2.3 Quotient topological vector spaces
... Hausdorff t.v.s.. If in addition X is Hausdorff, then Corollary 2.2.4 guarantees that {o} = {o} in X. Therefore, the quotient map φ : X → X/{o} is also injective because in this case Ker(φ) = {o}. Hence, φ is a topological isomorphism (i.e. bijective, continuous, open, linear) between X and X/{o} whic ...
... Hausdorff t.v.s.. If in addition X is Hausdorff, then Corollary 2.2.4 guarantees that {o} = {o} in X. Therefore, the quotient map φ : X → X/{o} is also injective because in this case Ker(φ) = {o}. Hence, φ is a topological isomorphism (i.e. bijective, continuous, open, linear) between X and X/{o} whic ...
On bimeasurings
... Definition 2.1. If N and T are bialgebras and A an algebra, then a map : N ⊗ T → A is a bimeasuring if N measures T to A and T measures N to A, i.e. (nm, t) = (n, t1 )(m, t2 ), (1N , t) = ε(t), (n, ts) = (n1 , t)(n2 , s), (n, 1T ) = ε(n) for n, m ∈ N and t, s ∈ T . Definition 2.2. Let T be a ...
... Definition 2.1. If N and T are bialgebras and A an algebra, then a map : N ⊗ T → A is a bimeasuring if N measures T to A and T measures N to A, i.e. (nm, t) = (n, t1 )(m, t2 ), (1N , t) = ε(t), (n, ts) = (n1 , t)(n2 , s), (n, 1T ) = ε(n) for n, m ∈ N and t, s ∈ T . Definition 2.2. Let T be a ...
On the homology and homotopy of commutative shuffle algebras
... of Σn with coefficients in V ⊗n vanishes. Therefore, quite often arguments work in this context that otherwise only hold in characteristic zero. A second way to interpret Stover’s result is that the difference between commutative monoids and commutative monoids with divided power structures disappea ...
... of Σn with coefficients in V ⊗n vanishes. Therefore, quite often arguments work in this context that otherwise only hold in characteristic zero. A second way to interpret Stover’s result is that the difference between commutative monoids and commutative monoids with divided power structures disappea ...
Sans titre
... (7) mb(t@t ) 2 mV 1 (DX ) (V -filtration with respect to t), (8) mts b(s) 2 mts+1 DX [s]. Proof of Proposition 9.2.1. (1) Let M be a coherent DX -module which is R-specializable along H. Let us first show the coherence of M(⇤H). This is a local problem; moreover, by induction on the cardinal of a ge ...
... (7) mb(t@t ) 2 mV 1 (DX ) (V -filtration with respect to t), (8) mts b(s) 2 mts+1 DX [s]. Proof of Proposition 9.2.1. (1) Let M be a coherent DX -module which is R-specializable along H. Let us first show the coherence of M(⇤H). This is a local problem; moreover, by induction on the cardinal of a ge ...
Lectures on Modules over Principal Ideal Domains
... Theorem 2.3 Let R be a PID and M be a finitely generated free module over R, of rank n. Then every submodule of M is also free of rank ≤ n. Proof: Let {e1 , e2 , · · · , en } be a basis of M so that M ∼ = Rn . We prove the theorem by induction on n. If n = 1, then M ∼ = R. Since R is a PID, every id ...
... Theorem 2.3 Let R be a PID and M be a finitely generated free module over R, of rank n. Then every submodule of M is also free of rank ≤ n. Proof: Let {e1 , e2 , · · · , en } be a basis of M so that M ∼ = Rn . We prove the theorem by induction on n. If n = 1, then M ∼ = R. Since R is a PID, every id ...
The structure of the classifying ring of formal groups with
... the flat cohomology of the moduli stack of formal A-modules. I have also referred to “the moduli stack of formal A-modules” several times in this paper. This is slightly ambiguous for the following reason: formal A-modules as defined below in terms of power series, as a formal group law equipped wit ...
... the flat cohomology of the moduli stack of formal A-modules. I have also referred to “the moduli stack of formal A-modules” several times in this paper. This is slightly ambiguous for the following reason: formal A-modules as defined below in terms of power series, as a formal group law equipped wit ...
DEFORMATION THEORY
... A. Since there is an obvious equivalence between modules and trivial extensions, we obtain: Theorem 1.16. The category (Ass /A)ab is isomorphic to the category of Amodules. Exercise 1.17. Prove analogous statements also for (Com /A)ab and (Lie /L)ab . Exercise 1.18. The only property of abelian grou ...
... A. Since there is an obvious equivalence between modules and trivial extensions, we obtain: Theorem 1.16. The category (Ass /A)ab is isomorphic to the category of Amodules. Exercise 1.17. Prove analogous statements also for (Com /A)ab and (Lie /L)ab . Exercise 1.18. The only property of abelian grou ...
A quasi-coherent sheaf of notes
... Now suppose the converse, i.e. a system of maps for sets U, V as above. Then we define the map F → f∗ G as follows. For V open, we need to define F(V ) → (f∗ G)(V ) = G(f −1 (V )) in a way commuting with the restriction maps. But we can use f −1 (V ), V as the two sets U, V and use the given map F( ...
... Now suppose the converse, i.e. a system of maps for sets U, V as above. Then we define the map F → f∗ G as follows. For V open, we need to define F(V ) → (f∗ G)(V ) = G(f −1 (V )) in a way commuting with the restriction maps. But we can use f −1 (V ), V as the two sets U, V and use the given map F( ...
M.4. Finitely generated Modules over a PID, part I
... to pay particular attention to the case that R is a Euclidean domain, for two reasons. First, in applications we will be interested exclusively in the case that R is Euclidean. Second, if R is Euclidean, Gaussian elimination is a constructive process, assuming that Euclidean division with remainder ...
... to pay particular attention to the case that R is a Euclidean domain, for two reasons. First, in applications we will be interested exclusively in the case that R is Euclidean. Second, if R is Euclidean, Gaussian elimination is a constructive process, assuming that Euclidean division with remainder ...
Fell bundles associated to groupoid morphisms
... We first discuss a class of examples considered by Fell. Let G be a discrete group, let K be a normal subgroup and let H = G/K with π : G → H the canonical morphism. Then we get a Fell bundle E over H with the fiber Cr∗ (K) over the identity element, such that Cr∗ (G) ∼ = Cr∗ (E). It is equivalent t ...
... We first discuss a class of examples considered by Fell. Let G be a discrete group, let K be a normal subgroup and let H = G/K with π : G → H the canonical morphism. Then we get a Fell bundle E over H with the fiber Cr∗ (K) over the identity element, such that Cr∗ (G) ∼ = Cr∗ (E). It is equivalent t ...
ABELIAN VARIETIES A canonical reference for the subject is
... Proof. Choose a nonempty open affine U ⊂ X and let D = X \ U with the reduced structure. We claim that D is a Cartier divisor (that is, its coherent ideal sheaf is invertible). Since X is regular, it is equivalent to say that all generic points of D have codimension 1. Mumford omits this explanation ...
... Proof. Choose a nonempty open affine U ⊂ X and let D = X \ U with the reduced structure. We claim that D is a Cartier divisor (that is, its coherent ideal sheaf is invertible). Since X is regular, it is equivalent to say that all generic points of D have codimension 1. Mumford omits this explanation ...
2. Cartier Divisors We now turn to the notion of a Cartier divisor
... Example 2.4. The quadric cone Q, given by xy − z 2 = 0 in A3k is not factorial. The line l, given by x = z = 0, is a Weil divisor which is not Cartier (one needs to check that the ideal hx, zi inside OQ,0 is not principal). The hyperplane x = 0 cuts out the double line 2l. Definition-Lemma 2.5. Let ...
... Example 2.4. The quadric cone Q, given by xy − z 2 = 0 in A3k is not factorial. The line l, given by x = z = 0, is a Weil divisor which is not Cartier (one needs to check that the ideal hx, zi inside OQ,0 is not principal). The hyperplane x = 0 cuts out the double line 2l. Definition-Lemma 2.5. Let ...
Two-dimensional topological field theories and Frobenius - D-MATH
... embedding j : m → ∂M. Since we labeled all components, we clearly have i (n) t j(m) = ∂M. This finally specifies a cobordism. We argue that the cobordism class is independent of both the choice of diffeomorphisms and orientation of M. Assume that ik and ik0 are two such orientation reversing diffeom ...
... embedding j : m → ∂M. Since we labeled all components, we clearly have i (n) t j(m) = ∂M. This finally specifies a cobordism. We argue that the cobordism class is independent of both the choice of diffeomorphisms and orientation of M. Assume that ik and ik0 are two such orientation reversing diffeom ...
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.