Derivations and differentials
|
|
- Margaret Franklin
- 5 years ago
- Views:
Transcription
1 Derivations and differentials Johan Commelin April 24, 2012 In the following text all rings are commutative with 1, unless otherwise specified. 1 Modules of derivations Let A be a ring, α : A B an A algebra, and M a B-module. 1.1 Definition. An A-derivation is an A-linear map d : B M, satisfying the Leibniz rule, i.e., d( f g) = f d(g) + gd( f ) for all f, g B. «Observe that the set of derivations Der A (B, M) is an B-module in a natural way. We write Der A (B) for Der A (B, B). We are used to the fact that the derivative of a constant is 0. An analogues statement is true in this setting. 1.2 Lemma. For all c α(a) and d Der A (B, M) we have d(c) = 0. Proof. d(c) = cd(1) = cd(1) + 1 d(c) d(c) = d(1 c) d(c) = Exercise. Other facts that we are used to are also true. 1. Prove that, for d Der A (B, M) we have d(a n ) = na n 1 d(a) and therefore, in a ring of characteristic n, we have d(a n ) = Prove that, for d Der A (B) we have a Leibniz formula for powers of d, d n (ab) = n i=0 ( ) n d i (a) d n i (b); i and if A is of characteristic n, this reduces to d n (ab) = ad n (b) + bd n (a). «1
2 Let A be a ring, and let a commutative diagram C/I q β B C of A-algebras be given, where I is an ideal of C, and q is the reduction map. A map γ : B C is called a lift of β if the obvious triangle commutes. 1.4 Lemma. Suppose I 2 = 0. Let S be the set of lifts B C. Let γ be a fixed element of S. Then S γ = Der A (B, I). Proof. Let γ S be given. Clearly γ γ maps into I, since q (γ γ) = β β = 0. Also γ γ is clearly A-linear. It satisfies the Leibniz rule, since 0 = (γ γ)( f )(γ γ)(g) = γ ( f g) + γ( f g) γ( f )γ (g) γ ( f )γ(g), and therefore (γ γ)( f g) = γ( f )γ (g) + γ(g)γ ( f ) 2γ( f g) = γ( f )(γ γ)(g) + γ(g)(γ γ)( f ). Finally observe that I inherits a B-module structure via γ, and that this does not depend on the choice of γ, since I 2 = 0. This shows that S γ Der A (B, I). It is left as an easy exercise to verify that for d Der A (B, I) the map d + γ is a lift of β. 1.5 Lemma. The functor M Der A (B, M) is representable. Proof. We want to show that there exists a B-module Ω B/A together with an A-derivation d B/A : B Ω B/A, such that Der A (B, M) = Hom B (Ω B/A, M) in a functorial way. Define µ : B A B B by f g f g. Put I = ker µ, Ω B/A = I/I 2 and C = (B A B)/I 2. Then µ induces a map µ : C B, and 0 Ω B/A C µ B 0 is an exact sequence of B-modules. For i {1, 2} define λ i : B C by λ 1 ( f ) = f 1 and λ 2 ( f ) = 1 f. Observe that these are splittings of the exact sequence. Observe that as (Ω B/A ) 2 = 0 as an ideal of C. Define d = d B/A = λ 1 λ 2. Indeed, d is a derivation, by lemma 1.4. We claim that said isomorphism is given by Der A (B, M) = Hom B (Ω B/A, M) φ d B/A φ. δ ( f g f δg) It is left as an exercise to prove that the maps are each others inverses. 2
3 The module Ω B/A is called the module of Kähler differentials. We also give another construction of Ω B/A. For all b B, we denote with db an abstract symbol. Write Ω for the free A-module generated by {db : b B}. We define Ω B/A to be the quotient of Ω by the submodule generated by d(c f + c g) cd f + c dg c, c A, f, g B d( f g) f dg gd f f, g B. Finally, we define d B/A : B Ω B/A by f d f. By definition this module represents the functor Der A (B, _), and therefore it is isomorphic to the previous construction in lemma Lemma. If the map A B is surjective, then Ω B/A = 0. Proof. Immediate from the construction of Ω B/A, and the result of lemma Exercise. Prove that, if B is generated by S B as an A-algebra, then Ω B/A is generated by d(s), using the Leibniz rule repeatedly. In particular, if B is finitely generated as A algebra, then Ω B/A is finitely generated as B-module. «1.8 Exercise. If B = A[x 1,..., x n ], then Ω B/A = n i=1 Adx i. Use this to show that if B = A[x 1,..., x n ]/I with I = ( f 1,..., f m ) then Ω B/A = coker(d : I/I 2 n i=1 Bdx i. If we precompose d with m i=1 Se i I/I 2 : e i f i, then Ω B/A is the cokernel of the jacobian matrix f 1 x f n x... 1 f 1 x m. f n x m. «2 Formally smooth, unramified, etale We return to the situation before, about lifts of ring maps. Let A be a ring, and let a commutative diagram C/I β B q α C A of A-algebras be given, where I is an ideal of C, and q is the reduction map. Let S be the set of lifts B C of β. 2.1 Definition. If for all pairs (C, I), where I C is an ideal satisfying I 2 = 0, we have #S 1 we say that α is formally smooth; 3
4 #S 1 we say that α is formally unramified. If α is both formally smooth and formally unramified, then we say that α is formally etale. «2.2 Lemma. The map α is formally unramified if and only if Ω B/A = 0. Proof. The implication to the left is clear from the definition and lemma 1.4. For the other implication take consider the construction of Ω B/A via C = (B A B)/I 2 and Ω B/A = I/I 2. Then (Ω B/A ) 2 = 0. Also we had two lifts λ 1 and λ 2. If α is formally unramified, we have λ 1 = λ 2, and hence d = λ 1 λ 2 = 0. Since d(b) generates Ω B/A we conclude that Ω B/A = Exercise. Let A be a ring, and S A a set. Prove that the localization A S 1 A is formally etale. «3 Two exact sequences Let a commutative diagram α φ B B α ψ A A of ring maps be given. 3.1 Exercise. Show that an A -derivation d : B M induces an A-derivation d ψ. «By the universal property of Ω B/A we see that φ induces a B-linear map Ω B/A Ω B /A. Explicitly the map is given by f dg φ( f )dφ(g). 3.2 Lemma. If φ is surjective, so is the induced map Ω B/A Ω B /A, and its kernel is generated by {d f φ( f ) A }. Proof. Note that Ω B/A is generated by d(b), while Ω B /A is generated by d(b ). Clearly d(b ) = d(φ(b)), and hence the image of the induced map generates Ω B /A. But then it is surjective, since it is B-linear. If f dg is mapped to 0, then φ( f )dφ(g) = 0. This shows that the kernel is generated by elements of the form id f with i ker φ and f B, together with elements of the form d f satisfying φ( f ) A. But since di f = id f + f di, we see that id f = di f f di. Hence the elements of the second form suffice. All these preperations lead to two important exact sequences. 3.3 Proposition. Let A B C be ring maps. Then there is a canonical exact sequence of C-modules Ω B/A B C Ω C/A Ω C/B 0. 4
5 Proof. Observe that this sequence is exact if for every C-module M the sequence Hom C (Ω B/A B C, M) Hom C (Ω C/A, M) Hom C (Ω C/B, M) 0 is exact. But that sequence actually is Now the exactness is clear. Der A (B, M) Der A (C, M) Der B (C, M) Exercise. In the above situation, if B C is formally smooth, then the sequence actually is short exact (i.e., the first map is injective) and split. «We now consider the case where the second map is surjective. In the above sequence we would have Ω C/B = 0, by lemma 1.6. Therefore we are interested in the kernel of the first map of the sequence. 3.5 Proposition. Let A B C be ring maps, where the second map is surjective, with kernel I B. Then there exists a canonical exact sequence of C-modules I/I 2 Ω B/A B C Ω C/A 0, where the first map is given by f d f 1. Proof. We leave it as an exercise to the reader to show that the first map is well-defined. By proposition 3.3 it is clear that we only need to show exactness at the second term. Since I maps to 0 in C, it follows that for f I, d f maps to 0 in Ω C/A. Therefore the composition of the first two maps is zero. Again let M be an arbitrary C-module. Then Hom C (I/I 2, M) Der A (B, M) Der A (C, M) is exact, since if δ Der A (B, M) maps to 0, this means that δ(i) = 0. But then δ comes from a derivation C = B/I M. This proves the exactness of the sequence. 3.6 Exercise. In the above situation, if the compostion A C is formally smooth, then the sequence actually is short exact (i.e., the first map is injective) and split. «4 Colimits Let A be ring. 4.1 Lemma. Let B, A be A-algebras. Put B = B A A. Then Ω B /A = ΩB/A A A. I.e., formation of differentials commutes with arbitrary change of base. Proof. Observe that d 1: B Ω B/A A A is an A -derivation, which gives a map φ : Ω B /A Ω B/A A A, by the universal property. On the other hand, the composite map B = B A A B A A d ΩB /A is an A-derivation, which induces a map ψ : Ω B/A Ω B /A. Since the target is a B -module, we get another induced map, which is the inverse of φ. 5
6 The following fact we shall not prove. It is an analogue to the fact that for two manifolds X and Y we have the identity T (x,y) X Y = T x X T y Y. Observe that in the following lemma, instead of taking a product of algebras, we take a coproduct, since moving between geometric categories (manifolds, varieties, schemes) and the category of rings is contravariant. 4.2 Lemma. Let B i be A-algebras, for i I. Let T denote the coproduct A B i. Then Ω T/B = i (Ω Bi /A Bi T). Proof. See Eisenbud, Remark. In Eisenbud, 397, is proven that formation of differentials commutes with arbitrary colimits. «As a consequence we state the following lemma, that we also do not prove. 4.4 Lemma. Let A B be a map of rings. Let S B be a subset. Then Ω S 1 B/A = Ω B/A B S 1 B. Proof. See Eisenbud, 397. Finally, we prove that formation of differentials commutes with finite products. 4.5 Lemma. Let B 1 and B 2 be A-algebras. Write B = B 1 B 2. Then Ω B/A = Ω B1 /A Ω B2 /A. Proof. Let e 1 denote (1, 0) B and e 2 = (0, 1). Let M be an arbitrary B-module, and δ : B M an A-derivation. Since e i is idempotent, δe i = 0. Hence by the Leibniz rule δ(e i f ) = e i δ f. But then δ maps B i = e i B to e i M. Thus δ corresponds to a map Ω Bi /A e i M. It follows that Ω B1 /A Ω B2 /A satisfies the universal property for Kähler differentials. 4.6 Exercise. Let A B be a map of rings. Let δ : B M be an A-derivation. Let e B be an idempotent. Prove that δe = 0. «6
7 5 Differential forms Let B be a ring, and M an B-module. 5.1 Definition. Let k be an integer. Let M k denote the k-fold tensor product of M over B. Let N denote the submodule generated by m 1 m 2... m k, i, j : i = j, m i = m j. The k-th exterior power of M is defined as M k /N, and is denoted Λ k M. An element of Λ k is written as a wedge product: m 1... m k with m i M. «Observe that Λ 0 M = B. Let α : A B be an A-algebra. Recall that Ω B/A is a B-module. We write Ω k B/A for Λk Ω B/A. We define maps d i : Ω i B/A Ωi+1 B/A f ω 1... ω i d f ω 1... ω i. 5.2 Exercise. Prove that the maps d i are well-defined. «Observe that d i+1 d i = 0. Therefore we have an (algebraic) de Rham complex associated to α : A B Ω B/A : 0 Ω0 B/A d Ω 1 B/A d 1 Ω i B/A We can therefore define de Rham cohomology groups associated to B/A. H i dr (B/A) = ker di / im d i 1 d i... Ω i+1 B/A Remark. It can be proven (see e.g., Hartshorne) that if A = C and B is the ring of global sections of a smooth affine variety X over C, then HdR i coincides with the usual singular homology group H sing i (X, C). «6 Tangent spaces In view of the previous it makes sense to view Der A (B) as the tangent space, since it is the dual module to Ω B/A. 6.1 Exercise. The map Der A (B) Der A (B) Der A (B) defined by [d 1, d 2 ] = d 1 d 2 d 2 d 1 is a Lie bracket. In particular if A is a field, this turns Der A (B) into a Lie algebra. Proof. We verify that the map is well-defined. Clearly [d 1, d 2 ] is an A-linear map B B. Also [d 1, d 2 ]( f g) = d 1 d 2 ( f g) d 2 d 1 ( f g) = d 1 ( f d 2 (g) + gd 2 ( f )) d 2 ( f d 1 (g) + gd 1 ( f )) = f d 1 d 2 (g) + d 1 ( f )d 2 (g) + gd 1 d 2 ( f ) + d 1 (g)d 2 ( f ) f d 2 d 1 (g) d 2 ( f )d 1 (g) gd 2 d 1 ( f ) d 2 (g)d 1 ( f ) = f [d 1, d 2 ](g) + g[d 1, d 2 ]( f ) 7
8 shows that [d 1, d 2 ] is a derivation. Clearly [_, _] is anti-symmetric. To show that [_, _] is a Lie bracket, we must prove that it satisfies the Jacobi identity. [d, [e, f ]] + [ f, [d, e]] + [e, [ f, d]] = de f d f e e f d + f ed+ f de ed f de f + d f e e f d f ed f de + ed f = 0 Consequently, Der A (B) is a Lie algebra. Let (A, m) be a local ring containing a field k isomorphic to its residue field A/m. 6.2 Lemma. The map is an isomorphism. d : m/m 2 Ω A/k A k Proof. We have a sequence k A k, where the second map is surjective. Therefore, by proposition 3.5 we see that m/m 2 Ω A/k A k Ω k/k 0 is an exact sequence. The surjectivity follows immediately. (Since k k is formally smooth, we also get injectivity from this sequence. We will give another proof below.) To prove the injectivity we pas to the dual modules, and prove surjectivity there. d : Hom k (Ω A/k A k, k) Hom k (m/m 2, k) Observe that the left hand side is isomorphic to Hom A (Ω A/k, k) = Der k (A, k). Note that for any derivation δ : A k we have δ(m 2 ) = 0 by the Leibniz rule. Let h Hom k (m/m 2, k) be given. Note that f A can be uniquely written as c + f, with c k and f m. Define δ f = h( f ). We claim that δ is a k- derivation, and that d (δ) = h. Clearly, δ is k-linear. Also, if we write f = c + f and g = c + g, then δ( f g) = h(c f + cg ) = c h( f ) + ch(g ) = gδ f + f δg. Thus δ is indeed a k-derivation. Also, since δ(m 2 ) = 0, we see that the restriction of δ to m yields h, i.e., d (δ) = h. Hence d is surjective, and therefore d is injective. As a consequence of the provious lemma we see that the notions of tangent space in algebraic geometry and differential geometry coincide. 8
De Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1)
II. De Rham Cohomology There is an obvious similarity between the condition d o q 1 d q = 0 for the differentials in a singular chain complex and the condition d[q] o d[q 1] = 0 which is satisfied by the
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar
More informationConnecting Coinvariants
Connecting Coinvariants In his talk, Sasha taught us how to define the spaces of coinvariants: V 1,..., V n = V 1... V n g S out (0.1) for any V 1,..., V n KL κg and any finite set S P 1. In her talk,
More informationSmooth morphisms. Peter Bruin 21 February 2007
Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to define smooth morphisms of schemes, which are one of the main ingredients in Néron s fundamental theorem [BLR, 1.3,
More informationEXT, TOR AND THE UCT
EXT, TOR AND THE UCT CHRIS KOTTKE Contents 1. Left/right exact functors 1 2. Projective resolutions 2 3. Two useful lemmas 3 4. Ext 6 5. Ext as a covariant derived functor 8 6. Universal Coefficient Theorem
More informationLECTURE 3: RELATIVE SINGULAR HOMOLOGY
LECTURE 3: RELATIVE SINGULAR HOMOLOGY In this lecture we want to cover some basic concepts from homological algebra. These prove to be very helpful in our discussion of singular homology. The following
More informationMatsumura: Commutative Algebra Part 2
Matsumura: Commutative Algebra Part 2 Daniel Murfet October 5, 2006 This note closely follows Matsumura s book [Mat80] on commutative algebra. Proofs are the ones given there, sometimes with slightly more
More informationA TALE OF TWO FUNCTORS. Marc Culler. 1. Hom and Tensor
A TALE OF TWO FUNCTORS Marc Culler 1. Hom and Tensor It was the best of times, it was the worst of times, it was the age of covariance, it was the age of contravariance, it was the epoch of homology, it
More information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.
More informationEILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY
EILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY CHRIS KOTTKE 1. The Eilenberg-Zilber Theorem 1.1. Tensor products of chain complexes. Let C and D be chain complexes. We define
More information3.1. Derivations. Let A be a commutative k-algebra. Let M be a left A-module. A derivation of A in M is a linear map D : A M such that
ALGEBRAIC GROUPS 33 3. Lie algebras Now we introduce the Lie algebra of an algebraic group. First, we need to do some more algebraic geometry to understand the tangent space to an algebraic variety at
More informationLOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT
LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral D-module, unless explicitly stated
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationFORMAL GLUEING OF MODULE CATEGORIES
FORMAL GLUEING OF MODULE CATEGORIES BHARGAV BHATT Fix a noetherian scheme X, and a closed subscheme Z with complement U. Our goal is to explain a result of Artin that describes how coherent sheaves on
More informationTCC Homological Algebra: Assignment #3 (Solutions)
TCC Homological Algebra: Assignment #3 (Solutions) David Loeffler, d.a.loeffler@warwick.ac.uk 30th November 2016 This is the third of 4 problem sheets. Solutions should be submitted to me (via any appropriate
More informationGeneralized Alexander duality and applications. Osaka Journal of Mathematics. 38(2) P.469-P.485
Title Generalized Alexander duality and applications Author(s) Romer, Tim Citation Osaka Journal of Mathematics. 38(2) P.469-P.485 Issue Date 2001-06 Text Version publisher URL https://doi.org/10.18910/4757
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationHARTSHORNE EXERCISES
HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing
More informationIndCoh Seminar: Ind-coherent sheaves I
IndCoh Seminar: Ind-coherent sheaves I Justin Campbell March 11, 2016 1 Finiteness conditions 1.1 Fix a cocomplete category C (as usual category means -category ). This section contains a discussion of
More informationON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF
ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF MATTHEW H. BAKER AND JÁNOS A. CSIRIK Abstract. We give a new proof of the isomorphism between the dualizing sheaf and the canonical
More informationGood tilting modules and recollements of derived module categories, II.
Good tilting modules and recollements of derived module categories, II. Hongxing Chen and Changchang Xi Abstract Homological tilting modules of finite projective dimension are investigated. They generalize
More informationA Primer on Homological Algebra
A Primer on Homological Algebra Henry Y Chan July 12, 213 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section Before telling you what a module is, you probably
More informationLecture 9 - Faithfully Flat Descent
Lecture 9 - Faithfully Flat Descent October 15, 2014 1 Descent of morphisms In this lecture we study the concept of faithfully flat descent, which is the notion that to obtain an object on a scheme X,
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23 6.4. Homotopy uniqueness of projective resolutions. Here I proved that the projective resolution of any R-module (or any object of an abelian category
More informationINTRO TO TENSOR PRODUCTS MATH 250B
INTRO TO TENSOR PRODUCTS MATH 250B ADAM TOPAZ 1. Definition of the Tensor Product Throughout this note, A will denote a commutative ring. Let M, N be two A-modules. For a third A-module Z, consider the
More informationFormal power series rings, inverse limits, and I-adic completions of rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More information3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection
3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection is called the objects of C and is denoted Obj(C). Given
More information2. Intersection Multiplicities
2. Intersection Multiplicities 11 2. Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes.
More information10. Smooth Varieties. 82 Andreas Gathmann
82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It
More informationAN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES
AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES MATTHEW H. BAKER AND JÁNOS A. CSIRIK This paper was written in conjunction with R. Hartshorne s Spring 1996 Algebraic Geometry course at
More informationTHEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)
4 Vector fields Last updated: November 26, 2009. (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M
More informationHungry, Hungry Homology
September 27, 2017 Motiving Problem: Algebra Problem (Preliminary Version) Given two groups A, C, does there exist a group E so that A E and E /A = C? If such an group exists, we call E an extension of
More informationChern classes à la Grothendieck
Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces
More informationMATH 101B: ALGEBRA II PART A: 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
More informationMATH 233B, FLATNESS AND SMOOTHNESS.
MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)
More informationConformal blocks for a chiral algebra as quasi-coherent sheaf on Bun G.
Conformal blocks for a chiral algebra as quasi-coherent sheaf on Bun G. Giorgia Fortuna May 04, 2010 1 Conformal blocks for a chiral algebra. Recall that in Andrei s talk [4], we studied what it means
More information1 Flat, Smooth, Unramified, and Étale Morphisms
1 Flat, Smooth, Unramified, and Étale Morphisms 1.1 Flat morphisms Definition 1.1. An A-module M is flat if the (right-exact) functor A M is exact. It is faithfully flat if a complex of A-modules P N Q
More informationare additive in each variable. Explicitly, the condition on composition means that given a diagram
1. Abelian categories Most of homological algebra can be carried out in the setting of abelian categories, a class of categories which includes on the one hand all categories of modules and on the other
More informationALGEBRAIC GROUPS JEROEN SIJSLING
ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined
More informationLie Algebra Cohomology
Lie Algebra Cohomology Carsten Liese 1 Chain Complexes Definition 1.1. A chain complex (C, d) of R-modules is a family {C n } n Z of R-modules, together with R-modul maps d n : C n C n 1 such that d d
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationCohomology and Base Change
Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is half-exact if whenever 0 M M M 0 is an exact sequence of A-modules, the sequence T (M ) T (M)
More informationHOMEWORK SET 3. Local Class Field Theory - Fall For questions, remarks or mistakes write me at
HOMEWORK SET 3 Local Class Field Theory - Fall 2011 For questions, remarks or mistakes write me at sivieroa@math.leidneuniv.nl. Exercise 3.1. Suppose A is an abelian group which is torsion (every element
More informationThe Gauss-Manin Connection for the Cyclic Homology of Smooth Deformations, and Noncommutative Tori
The Gauss-Manin Connection for the Cyclic Homology of Smooth Deformations, and Noncommutative Tori Allan Yashinski Abstract Given a smooth deformation of topological algebras, we define Getzler s Gauss-Manin
More informationTRIANGULATED CATEGORIES, SUMMER SEMESTER 2012
TRIANGULATED CATEGORIES, SUMMER SEMESTER 2012 P. SOSNA Contents 1. Triangulated categories and functors 2 2. A first example: The homotopy category 8 3. Localization and the derived category 12 4. Derived
More informationand this makes M into an R-module by (1.2). 2
1. Modules Definition 1.1. Let R be a commutative ring. A module over R is set M together with a binary operation, denoted +, which makes M into an abelian group, with 0 as the identity element, together
More informationWIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES
WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels,
More informationCommutative algebra 19 May Jan Draisma TU Eindhoven and VU Amsterdam
1 Commutative algebra 19 May 2015 Jan Draisma TU Eindhoven and VU Amsterdam Goal: Jacobian criterion for regularity 2 Recall A Noetherian local ring R with maximal ideal m and residue field K := R/m is
More informationNotes on the definitions of group cohomology and homology.
Notes on the definitions of group cohomology and homology. Kevin Buzzard February 9, 2012 VERY sloppy notes on homology and cohomology. Needs work in several places. Last updated 3/12/07. 1 Derived functors.
More information5 Dedekind extensions
18.785 Number theory I Fall 2016 Lecture #5 09/22/2016 5 Dedekind extensions In this lecture we prove that the integral closure of a Dedekind domain in a finite extension of its fraction field is also
More informationIntroduction to Chiral Algebras
Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy - a version of the Eckmann Hilton argument
More informationABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY
ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ARDA H. DEMIRHAN Abstract. We examine the conditions for uniqueness of differentials in the abstract setting of differential geometry. Then we ll come up
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationREPRESENTATION THEORY, LECTURE 0. BASICS
REPRESENTATION THEORY, LECTURE 0. BASICS IVAN LOSEV Introduction The aim of this lecture is to recall some standard basic things about the representation theory of finite dimensional algebras and finite
More informationAn introduction to Algebra and Topology
An introduction to Algebra and Topology Pierre Schapira http://www.math.jussieu.fr/ schapira/lectnotes schapira@math.jussieu.fr Course at University of Luxemburg 1/3/2012 30/4/2012, v5 2 Contents 1 Linear
More informationLie algebra cohomology
Lie algebra cohomology Relation to the de Rham cohomology of Lie groups Presented by: Gazmend Mavraj (Master Mathematics and Diploma Physics) Supervisor: J-Prof. Dr. Christoph Wockel (Section Algebra and
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationDeligne s Mixed Hodge Structure for Projective Varieties with only Normal Crossing Singularities
Deligne s Mixed Hodge Structure for Projective Varieties with only Normal Crossing Singularities B.F Jones April 13, 2005 Abstract Following the survey article by Griffiths and Schmid, I ll talk about
More informationDuality, Residues, Fundamental class
Duality, Residues, Fundamental class Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu May 22, 2011 Joseph Lipman (Purdue University) Duality, Residues, Fundamental class
More informationHomework 2 - Math 603 Fall 05 Solutions
Homework 2 - Math 603 Fall 05 Solutions 1. (a): In the notation of Atiyah-Macdonald, Prop. 5.17, we have B n j=1 Av j. Since A is Noetherian, this implies that B is f.g. as an A-module. (b): By Noether
More informationMONDAY - TALK 4 ALGEBRAIC STRUCTURE ON COHOMOLOGY
MONDAY - TALK 4 ALGEBRAIC STRUCTURE ON COHOMOLOGY Contents 1. Cohomology 1 2. The ring structure and cup product 2 2.1. Idea and example 2 3. Tensor product of Chain complexes 2 4. Kunneth formula and
More informationHomological Methods in Commutative Algebra
Homological Methods in Commutative Algebra Olivier Haution Ludwig-Maximilians-Universität München Sommersemester 2017 1 Contents Chapter 1. Associated primes 3 1. Support of a module 3 2. Associated primes
More informationMath 210B. Artin Rees and completions
Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show
More informationEilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )
II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups
More informationMath Homotopy Theory Hurewicz theorem
Math 527 - Homotopy Theory Hurewicz theorem Martin Frankland March 25, 2013 1 Background material Proposition 1.1. For all n 1, we have π n (S n ) = Z, generated by the class of the identity map id: S
More informationLECTURES ON DEFORMATIONS OF GALOIS REPRESENTATIONS. Mark Kisin
LECTURES ON DEFORMATIONS OF GALOIS REPRESENTATIONS Mark Kisin Lecture 5: Flat deformations (5.1) Flat deformations: Let K/Q p be a finite extension with residue field k. Let W = W (k) and K 0 = FrW. We
More informationHomological Algebra and Differential Linear Logic
Homological Algebra and Differential Linear Logic Richard Blute University of Ottawa Ongoing discussions with Robin Cockett, Geoff Cruttwell, Keith O Neill, Christine Tasson, Trevor Wares February 24,
More informationLie algebra cohomology
Lie algebra cohomology November 16, 2018 1 History Citing [1]: In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the
More informationALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES
ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open
More informationGeometry 9: Serre-Swan theorem
Geometry 9: Serre-Swan theorem Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have
More informationEquivalence Relations
Equivalence Relations Definition 1. Let X be a non-empty set. A subset E X X is called an equivalence relation on X if it satisfies the following three properties: 1. Reflexive: For all x X, (x, x) E.
More informationCOMMUTATIVE ALGEBRA LECTURE 1: SOME CATEGORY THEORY
COMMUTATIVE ALGEBRA LECTURE 1: SOME CATEGORY THEORY VIVEK SHENDE A ring is a set R with two binary operations, an addition + and a multiplication. Always there should be an identity 0 for addition, an
More informationEtale cohomology of fields by Johan M. Commelin, December 5, 2013
Etale cohomology of fields by Johan M. Commelin, December 5, 2013 Etale cohomology The canonical topology on a Grothendieck topos Let E be a Grothendieck topos. The canonical topology T on E is given in
More informationNOTES ON BASIC HOMOLOGICAL ALGEBRA 0 L M N 0
NOTES ON BASIC HOMOLOGICAL ALGEBRA ANDREW BAKER 1. Chain complexes and their homology Let R be a ring and Mod R the category of right R-modules; a very similar discussion can be had for the category of
More informationNOTES IN COMMUTATIVE ALGEBRA: PART 2
NOTES IN COMMUTATIVE ALGEBRA: PART 2 KELLER VANDEBOGERT 1. Completion of a Ring/Module Here we shall consider two seemingly different constructions for the completion of a module and show that indeed they
More informationDerived Algebraic Geometry I: Stable -Categories
Derived Algebraic Geometry I: Stable -Categories October 8, 2009 Contents 1 Introduction 2 2 Stable -Categories 3 3 The Homotopy Category of a Stable -Category 6 4 Properties of Stable -Categories 12 5
More informationHolomorphic line bundles
Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank
More informationHomotopy-theory techniques in commutative algebra
Homotopy-theory techniques in commutative algebra Department of Mathematical Sciences Kent State University 09 January 2007 Departmental Colloquium Joint with Lars W. Christensen arxiv: math.ac/0612301
More informationRecall: a mapping f : A B C (where A, B, C are R-modules) is called R-bilinear if f is R-linear in each coordinate, i.e.,
23 Hom and We will do homological algebra over a fixed commutative ring R. There are several good reasons to take a commutative ring: Left R-modules are the same as right R-modules. [In general a right
More information3 The Hom Functors Projectivity and Injectivity.
3 The Hom Functors Projectivity and Injectivity. Our immediate goal is to study the phenomenon of category equivalence, and that we shall do in the next Section. First, however, we have to be in control
More informationA Leibniz Algebra Structure on the Second Tensor Power
Journal of Lie Theory Volume 12 (2002) 583 596 c 2002 Heldermann Verlag A Leibniz Algebra Structure on the Second Tensor Power R. Kurdiani and T. Pirashvili Communicated by K.-H. Neeb Abstract. For any
More informationSTABLE MODULE THEORY WITH KERNELS
Math. J. Okayama Univ. 43(21), 31 41 STABLE MODULE THEORY WITH KERNELS Kiriko KATO 1. Introduction Auslander and Bridger introduced the notion of projective stabilization mod R of a category of finite
More informationn P say, then (X A Y ) P
COMMUTATIVE ALGEBRA 35 7.2. The Picard group of a ring. Definition. A line bundle over a ring A is a finitely generated projective A-module such that the rank function Spec A N is constant with value 1.
More informationA Version of the Grothendieck Conjecture for p-adic Local Fields
A Version of the Grothendieck Conjecture for p-adic Local Fields by Shinichi MOCHIZUKI* Section 0: Introduction The purpose of this paper is to prove an absolute version of the Grothendieck Conjecture
More informationNon-commutative Algebra
Non-commutative Algebra Patrick Da Silva Freie Universität Berlin June 2, 2017 Table of Contents 1 Introduction to unital rings 5 1.1 Generalities............................................ 5 1.2 Centralizers
More informationHOMOLOGICAL DIMENSIONS AND REGULAR RINGS
HOMOLOGICAL DIMENSIONS AND REGULAR RINGS ALINA IACOB AND SRIKANTH B. IYENGAR Abstract. A question of Avramov and Foxby concerning injective dimension of complexes is settled in the affirmative for the
More informationSECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS
SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS In this section we will prove the Künneth theorem which in principle allows us to calculate the (co)homology of product spaces as soon
More informationUNIVERSAL DERIVED EQUIVALENCES OF POSETS
UNIVERSAL DERIVED EQUIVALENCES OF POSETS SEFI LADKANI Abstract. By using only combinatorial data on two posets X and Y, we construct a set of so-called formulas. A formula produces simultaneously, for
More informationDirect Limits. Mathematics 683, Fall 2013
Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry
More informationLecture 4 Super Lie groups
Lecture 4 Super Lie groups In this lecture we want to take a closer look to supermanifolds with a group structure: Lie supergroups or super Lie groups. As in the ordinary setting, a super Lie group is
More informationON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS
ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical
More informationThe most important result in this section is undoubtedly the following theorem.
28 COMMUTATIVE ALGEBRA 6.4. Examples of Noetherian rings. So far the only rings we can easily prove are Noetherian are principal ideal domains, like Z and k[x], or finite. Our goal now is to develop theorems
More informationLectures on Grothendieck Duality II: Derived Hom -Tensor adjointness. Local duality.
Lectures on Grothendieck Duality II: Derived Hom -Tensor adjointness. Local duality. Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu February 16, 2009 Joseph Lipman (Purdue
More informationCOHEN-MACAULAY RINGS SELECTED EXERCISES. 1. Problem 1.1.9
COHEN-MACAULAY RINGS SELECTED EXERCISES KELLER VANDEBOGERT 1. Problem 1.1.9 Proceed by induction, and suppose x R is a U and N-regular element for the base case. Suppose now that xm = 0 for some m M. We
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationarxiv: v1 [math.kt] 18 Dec 2009
EXCISION IN HOCHSCHILD AND CYCLIC HOMOLOGY WITHOUT CONTINUOUS LINEAR SECTIONS arxiv:0912.3729v1 [math.kt] 18 Dec 2009 RALF MEYER Abstract. We prove that continuous Hochschild and cyclic homology satisfy
More informationLectures on Grothendieck Duality. II: Derived Hom -Tensor adjointness. Local duality.
Lectures on Grothendieck Duality II: Derived Hom -Tensor adjointness. Local duality. Joseph Lipman February 16, 2009 Contents 1 Left-derived functors. Tensor and Tor. 1 2 Hom-Tensor adjunction. 3 3 Abstract
More informationRELATIVE HOMOLOGY. M. Auslander Ø. Solberg
RELATIVE HOMOLOGY M. Auslander Ø. Solberg Department of Mathematics Institutt for matematikk og statistikk Brandeis University Universitetet i Trondheim, AVH Waltham, Mass. 02254 9110 N 7055 Dragvoll USA
More informationHOMOLOGY AND COHOMOLOGY. 1. Introduction
HOMOLOGY AND COHOMOLOGY ELLEARD FELIX WEBSTER HEFFERN 1. Introduction We have been introduced to the idea of homology, which derives from a chain complex of singular or simplicial chain groups together
More information