Equivalence Relations and Partial Orders ()
... So the equivalence classes modulo n are precisely: nZ, 1 + nZ, 2 + nZ, . . . , (n − 1) + nZ. This is exactly what we saw when we were working mod n = 4. When working with equivalence relations a new problem arises. Often we want to define functions and operations in terms of representatives for our ...
... So the equivalence classes modulo n are precisely: nZ, 1 + nZ, 2 + nZ, . . . , (n − 1) + nZ. This is exactly what we saw when we were working mod n = 4. When working with equivalence relations a new problem arises. Often we want to define functions and operations in terms of representatives for our ...
EXERCISES IN MA 510 : COMMUTATIVE ALGEBRA
... the kernel of f is P and hence P is a prime ideal. Show that I(V (P )) = P. Is R integrally closed in its quotient field ? Find f ∈ R/P which is transcendental over k such that R/P is a finite k[f ]-algebra. (26) Let R = k[x, y]/(y 2 − x3 + x). Find an algebraically independent f ∈ R such that R is ...
... the kernel of f is P and hence P is a prime ideal. Show that I(V (P )) = P. Is R integrally closed in its quotient field ? Find f ∈ R/P which is transcendental over k such that R/P is a finite k[f ]-algebra. (26) Let R = k[x, y]/(y 2 − x3 + x). Find an algebraically independent f ∈ R such that R is ...
On the topological Hochschild homology of bu. I.
... where the first factor of S ⊗ S op acts on S by multiplication on the left and the second factor by multiplication on the right [8, page 169]. The reader may perhaps wonder why one uses this definition for the homology of S instead of the “obvious” definition TorS (k, k). For our purposes, the answe ...
... where the first factor of S ⊗ S op acts on S by multiplication on the left and the second factor by multiplication on the right [8, page 169]. The reader may perhaps wonder why one uses this definition for the homology of S instead of the “obvious” definition TorS (k, k). For our purposes, the answe ...
The support of local cohomology modules
... Local cohomology is a powerful tool introduced by Alexander Grothendieck in the 1960’s ([Har67]) and it has since yielded many geometric and algebraic insights. From an algebraic point of view, given an ideal I in a commutative ring R, local cohomology modules HiI (−) (i ≥ 0) arise as right-derived ...
... Local cohomology is a powerful tool introduced by Alexander Grothendieck in the 1960’s ([Har67]) and it has since yielded many geometric and algebraic insights. From an algebraic point of view, given an ideal I in a commutative ring R, local cohomology modules HiI (−) (i ≥ 0) arise as right-derived ...
Locally cartesian closed categories and type theory
... closed iff for all/,/* has a right adjoint Hf. For any C with finite limits, each/* of C has a left adjoint 2y (defined by composition), and the Beck condition for C is satisfied (it says the composite of two pullback diagrams is a pullback diagram, which is always true). 2-5. In Seely[10] it is sho ...
... closed iff for all/,/* has a right adjoint Hf. For any C with finite limits, each/* of C has a left adjoint 2y (defined by composition), and the Beck condition for C is satisfied (it says the composite of two pullback diagrams is a pullback diagram, which is always true). 2-5. In Seely[10] it is sho ...
Outline notes
... Problem sheets. There will be 8 marked problem sheets; the first will be due in on Thursday 11th October. To encourage you to work hard throughout the term, each problem sheet is worth 1.25% of your overall grade. Note that this mark is awarded for any reasonable attempt at the sheet. (There is a li ...
... Problem sheets. There will be 8 marked problem sheets; the first will be due in on Thursday 11th October. To encourage you to work hard throughout the term, each problem sheet is worth 1.25% of your overall grade. Note that this mark is awarded for any reasonable attempt at the sheet. (There is a li ...
A note on closedness of algebraic sum of sets
... Proof. (Necessity.) If X is a reflexive Banach space then we can take τ to be equal the weak topology on X. (Sufficiency.) Suppose that every closed convex and bounded subset of X is compact in the topology τ. Then for any closed convex and bounded set A the algebraic sum A + B, (where B is a closed ...
... Proof. (Necessity.) If X is a reflexive Banach space then we can take τ to be equal the weak topology on X. (Sufficiency.) Suppose that every closed convex and bounded subset of X is compact in the topology τ. Then for any closed convex and bounded set A the algebraic sum A + B, (where B is a closed ...
Refinement by interpretation in a general setting
... the usual algebraic structures, resorting to algebras endowed with a set of k-tuples. Formally, a k-data structure over a signature Σ is a pair A = hA, F i where A is a Σ-algebra and F is just a subset of Ak . The set F , of designated elements of A, can be regarded as the set of truth values on A: ...
... the usual algebraic structures, resorting to algebras endowed with a set of k-tuples. Formally, a k-data structure over a signature Σ is a pair A = hA, F i where A is a Σ-algebra and F is just a subset of Ak . The set F , of designated elements of A, can be regarded as the set of truth values on A: ...
Affine Varieties
... as a category with regular maps as the morphisms, then Proposition 3.5 shows that this is the same as the category of domains C[x1 , ..., xn ]/P with C-algebra homomorphisms (going in the opposite direction!) via X 7→ C[X] and Φ 7→ Φ∗ . That is, any property of affine varieties translates into an eq ...
... as a category with regular maps as the morphisms, then Proposition 3.5 shows that this is the same as the category of domains C[x1 , ..., xn ]/P with C-algebra homomorphisms (going in the opposite direction!) via X 7→ C[X] and Φ 7→ Φ∗ . That is, any property of affine varieties translates into an eq ...
N AS AN AEC Very Preliminary We show the concept of an Abstract
... Definition 1.2. (1) ⊥ N = {A : Ext(A, N ) = 0} (2) For A ⊆ B both in ⊥ N , A≺N B if B/A ∈ ⊥ N . This notion generalizes the concept of a Whitehead group; that is the special case: ⊥ Z. There is a great deal of work on such classes and on their duals, so-called cotorsion pairs in [GT06] and [EM02]. A ...
... Definition 1.2. (1) ⊥ N = {A : Ext(A, N ) = 0} (2) For A ⊆ B both in ⊥ N , A≺N B if B/A ∈ ⊥ N . This notion generalizes the concept of a Whitehead group; that is the special case: ⊥ Z. There is a great deal of work on such classes and on their duals, so-called cotorsion pairs in [GT06] and [EM02]. A ...
A GUIDE FOR MORTALS TO TAME CONGRUENCE THEORY Tame
... Theorem 1.9 (P.P. Pálfy). [TCT 4.7, 4.6] Every minimal algebra M with |M | ≥ 3 and having a polynomial operation which depends on more than one variable, is polynomially equivalent with a vector space. Proof. First we explore the consequences of M being minimal and having at least 3 elements. Claim ...
... Theorem 1.9 (P.P. Pálfy). [TCT 4.7, 4.6] Every minimal algebra M with |M | ≥ 3 and having a polynomial operation which depends on more than one variable, is polynomially equivalent with a vector space. Proof. First we explore the consequences of M being minimal and having at least 3 elements. Claim ...
Equations and Dot-Depth One By: Francine Blanchet
... of letters in w. A word u is a prefix of w if there exists a word v such that uv = w. A word u is a suffix of w if there exists a word v such that vu = w. A word u is a factor (or segment) of a word v if there exists words x and y such that v = xuy. A word u = a1…an (where a1,… ,an are letters) is a ...
... of letters in w. A word u is a prefix of w if there exists a word v such that uv = w. A word u is a suffix of w if there exists a word v such that vu = w. A word u is a factor (or segment) of a word v if there exists words x and y such that v = xuy. A word u = a1…an (where a1,… ,an are letters) is a ...
201A Homework
... n. (This smallest positive element must exist because H 6= h0i and H cannot contain only negative numbers since it is a subgroup and must have inverses.) If n = 1 we get that H = Z since Z = h1i. In this case H = 1Z = Z, so suppose that n > 1. We clearly have that nZ ⊂ H since n, −n ∈ H and anything ...
... n. (This smallest positive element must exist because H 6= h0i and H cannot contain only negative numbers since it is a subgroup and must have inverses.) If n = 1 we get that H = Z since Z = h1i. In this case H = 1Z = Z, so suppose that n > 1. We clearly have that nZ ⊂ H since n, −n ∈ H and anything ...
![Rings of constants of the form k[f]](http://s1.studyres.com/store/data/021729650_1-3a6201c0eec615e02140355abcc4b661-300x300.png)
Rings of constants of the form k[f]
... Lemma 2.3. If h ∈ k[X] \ k, then k[h] is a maximal element in the family M if and only if the algebra k[h] is integrally closed in k[X]. In particular, if f ∈ k[X] \ k, then the integral closure of k[f ] in k[X] is of the form k[g], for some g ∈ k[X] \ k. Note also the following obvious lemma. Lemma ...
... Lemma 2.3. If h ∈ k[X] \ k, then k[h] is a maximal element in the family M if and only if the algebra k[h] is integrally closed in k[X]. In particular, if f ∈ k[X] \ k, then the integral closure of k[f ] in k[X] is of the form k[g], for some g ∈ k[X] \ k. Note also the following obvious lemma. Lemma ...
ORTHOGONAL BUNDLES OVER CURVES IN CHARACTERISTIC
... G-bundle EG over X a reduction EP of EG to a parabolic subgroup P ⊂ G, the so-called canonical reduction — see e.g. [Ra], [Be], [BH] or [H] for its definition. We only mention here that in the case G = GL(n) the canonical reduction coincides with the Harder-Narasimhan filtration of the rank-n vector ...
... G-bundle EG over X a reduction EP of EG to a parabolic subgroup P ⊂ G, the so-called canonical reduction — see e.g. [Ra], [Be], [BH] or [H] for its definition. We only mention here that in the case G = GL(n) the canonical reduction coincides with the Harder-Narasimhan filtration of the rank-n vector ...