Michael Artin Algebra Pdf 14

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This course is intended as an introduction to modern abstract algebra and the way of algebraic thinking in advanced mathematics. The course focuses on basic algebraic concepts which arise in various areas of advanced mathematics, and emphasizes on the underlying algebraic structures which are common to various concrete mathematical examples.

I know that having the class on zoom made it difficult. Thank you for sticking it out.Have a nice summer. Reminder: Information Subject Description (pdf)Subject Outline (pdf)Summaries Summary, Feb 17 (pdf) Summary, Feb 19 (pdf) Summaries, Feb 22 and 24 (pdf) Summaries, Feb 26 and Mar 1 (pdf) Summaries, Mar 3 and 5 (pdf) Summaries, Mar 9 and 10 (pdf) Summaries, Mar 12 and 15 (pdf) Summaries, Mar 17 and 19 (pdf) Summaries, Mar 24 and 26 (pdf) Summaries, Mar 29 and 31 (pdf) Summaries, Apr 2 and 5 (pdf) Summaries, Apr 7 and 9 (pdf) Summaries, Apr 12 and 14 (pdf) Summary, Apr 16 (pdf) Summaries, Apr 21, 23, 26 (pdf) Summaries, Apr 28 and 30 (pdf) Summaries, May 3 and 5 (pdf) Summaries, May 10 and 12 (pdf) Summaries, May 17 and 19 (pdf)Problem Sets Problem Set 1 (pdf) Problem Set 2 (pdf) Problem Set 3 (pdf) Problem Set 4 (pdf) Problem Set 5 (pdf) Problem Set 6 (pdf) Problem Set 7 (pdf) Problem Set 8 (pdf) Problem Set 9 (pdf) Problem Set 10 (pdf) Problem Set 11 (pdf)Quizzes Quiz 1 (pdf) Quiz 2 (pdf) Quiz 3 (pdf) Quiz 4 (pdf) Quiz 5 (pdf) Quiz 6 (pdf)Handouts Comments on Problem Set 1 (pdf) Comments on Problem Set 2 (pdf) Comments on Problem Set 3 (pdf) Comments on Problem Set 4 (pdf) Comments on Problem Set 5 (pdf) Comments on Problem Set 7 (pdf) Comments on Problem Set 8 (pdf) Comments on Problem Set 9 (pdf) Comments on Problem Set 10 (pdf) Notes for an algebraic geometry class 18.721 (pdf) Other Information Course StaffMike Artin, artin@math.mit.edu, x3-3689, Room 2-274 office hours: MWF after class, or by appointment. TAs: Murilo Corato Zanarella, muriloz@mit.edu office hours: M 1-1, W 7-8 Xzavier Herbert, xherbert@mit.edu office hours: Tu 3-4, Th 5-6Anqi Li, anqili@mit.edu office hours: M 8-9, Tu 8-9 You are encouraged to make use of the office hours. If youcan't make the times listed, email one of us toset up an appointment. Please arrive to office hours during the firsthalf hour. We may sign off after a half hour if no one is there. Web address: www.math.mit.edu/classes/18.702/

In a nutshell, algebraic geometrystudies systems of polynomial equations and the geometry of their solutionsets. It is one of the oldest branches of mathematics, with many connections toother areas such as number theory, complex geometry, combinatorics, ortheoretical physics.

In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves M g , n {\displaystyle {\mathcal {M}}_{g,n}} and the moduli stack of elliptic curves. Originally, they were introduced by Grothendieck[1] to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are smooth. But, through many generalizations the notion of algebraic stacks was finally discovered by Michael Artin.[2]

One of the motivating examples of an algebraic stack is to consider a groupoid scheme ( R , U , s , t , m ) {\displaystyle (R,U,s,t,m)} over a fixed scheme S {\displaystyle S} . For example, if R = μ n × S A S n {\displaystyle R=\mu _{n}\times _{S}\mathbb {A} _{S}^{n}} (where μ n {\displaystyle \mu _{n}} is the group scheme of roots of unity), U = A S n {\displaystyle U=\mathbb {A} _{S}^{n}} , s = pr U {\displaystyle s={\text{pr}}_{U}} is the projection map, t {\displaystyle t} is the group action

It turns out using the fppf-topology[4] (faithfully flat and locally of finite presentation) on ( S c h / S ) {\displaystyle (\mathrm {Sch} /S)} , denoted ( S c h / S ) f p p f {\displaystyle (\mathrm {Sch} /S)_{fppf}} , forms the basis for defining algebraic stacks. Then, an algebraic stack[5] is a fibered category

For the fppf topology, having an immersion is local on the target.[7] In addition to the previous properties local on the source for the fppf topology, f {\displaystyle f} being universally open is also local on the source.[8] Also, being locally Noetherian and Jacobson are local on the source and target for the fppf topology.[9] This does not hold in the fpqc topology, making it not as "nice" in terms of technical properties. Even though this is true, using algebraic stacks over the fpqc topology still has its use, such as in chromatic homotopy theory. This is because the Moduli stack of formal group laws M f g {\displaystyle {\mathcal {M}}_{fg}} is an fpqc-algebraic stack[10]pg 40.

Note that many stacks cannot be naturally represented as Deligne-Mumford stacks because it only allows for finite covers, or, algebraic stacks with finite covers. Note that because every Etale cover is flat and locally of finite presentation, algebraic stacks defined with the fppf-topology subsume this theory; but, it is still useful since many stacks found in nature are of this form, such as the moduli of curves M g {\displaystyle {\mathcal {M}}_{g}} . Also, the differential-geometric analogue of such stacks are called orbifolds. The Etale condition implies the 2-functor

Using other Grothendieck topologies on ( F / S ) {\displaystyle (F/S)} gives alternative theories of algebraic stacks which are either not general enough, or don't behave well with respect to exchanging properties from the base of a cover to the total space of a cover. It is useful to recall there is the following hierarchy of generalization

The structure sheaf of an algebraic stack is an object pulled back from a universal structure sheaf O {\displaystyle {\mathcal {O}}} on the site ( S c h / S ) f p p f {\displaystyle (Sch/S)_{fppf}} .[19] This universal structure sheaf[20] is defined as

Many classifying stacks for algebraic groups are algebraic stacks. In fact, for an algebraic group space G {\displaystyle G} over a scheme S {\displaystyle S} which is flat of finite presentation, the stack B G {\displaystyle BG} is algebraic[2]theorem 6.1.

This book is an introduction to the geometry of complex algebraic varieties. It is intended for students who have learned algebra, analysis, and topology, as taught in standard undergraduate courses. So it is a suitable text for a beginning graduate course or an advanced undergraduate course. The book begins with a study of plane algebraic curves, then introduces affine and projective varieties, going on to dimension and constructibility. \(\mathcal{O}\)-modules (quasicoherent sheaves) are defined without reference to sheaf theory, and their cohomology is defined axiomatically. The Riemann-Roch Theorem for curves is proved using projection to the projective line. Some of the points that aren't always treated in beginning courses are Hensel's Lemma, Chevalley's Finiteness Theorem, and the Birkhoff-Grothendieck Theorem. The book contains extensive discussions of finite group actions, lines in \(\mathbb{P}^3\), and double planes, and it ends with applications of the Riemann-Roch Theorem.

Course materials: Lecture Topics coveredLecture notes/slidesHomeworks/Tutorials/Solutions 1 (30/09) Introduction. Vectors in 2D and 3D, addition, re-scaling, scalar product (dot product). L1 [PDF] 2 (01/10) Properties of scalar products. Vector product (cross product). L2 [PDF] 3 (01/10) Multilinearity. Areas and volumes. L3 [PDF] HW1 [PDF] 4 (07/10) First tutorialT1 [PDF], T1 solutions [PDF] 5 (08/10) Applications of scalar and vector products. Quaternions. Lines and planes in 2D and 3D. Systems of linear equations. L4 [PDF] 6 (08/10) Gauss-Jordan elimination. Elementary row operations. Reduced row echelon form. Towards matrix arithmetic. L5 [PDF]HW1 solutions [PDF]HW2 [PDF] 7 (14/10) Matrix operations. Three definitions of matrix product. L6 [PDF] 8 (15/10) Elementary matrices. Invertible matrices. Only a square matrix can be invertible. Computing inverses using elementary row operations. L7 [PDF] 9 (15/10) Permutations. Odd and even permutations. Determinants. Elementary row operations on determinants. L8 [PDF]HW2 solutions [PDF] HW3 [PDF] 10 (21/10) Second tutorialT2 [PDF], T2 solutions [PDF] 11 (22/10) Determinants and invertibility. Determinant of the product of matrices. Minors and cofactors. Row expansion of determinants (statement). L9 [PDF] 12 (22/10) Row expansion of determinants (proof). Adjoint / adjugate matrix. A closed formula for the inverse matrix. Cramer's formula. Summary of results on systems with the same number of equations and unknowns.L10 [PDF]HW3 solutions [PDF] HW4 [PDF] 13 (28/10) Finite dimensional Fredholm's alternative. An application: the discrete Dirichlet problem. Vandermonde determinant. Lagrange interpolation formula.L11 [PDF] 14 (29/10) Row expansions, 3D volumes, and cross products. Coordinate vector spaces. Linear independence. Spanning property. Bases. L12 [PDF] 15 (29/10) Linear maps and matrices. Subspaces of Rn. Two main examples: solution sets to systems of linear equations and linear spans. Relating these two examples. L13 [PDF]HW4 solutions [PDF] HW5 [PDF] 16 (04/11) Third tutorial. T3 [PDF], T3 solutions [PDF] 17 (05/11) Abstract vector spaces. Examples. L14 [PDF] 18 (05/11) Consequences of properties of vector operations. Fields. L15 [PDF]HW5 solutions [PDF] HW6 [PDF] Reading week, no classes 19 (18/11) No linear algebra class, an analysis lecture by Prof O'Donovan instead 20 (19/11) "Coin weighing problem" as an example of field change. Linear independence, span, bases in abstract vector spaces.L16 [PDF] 21 (19/11) Finite-dimensional and infinite-dimensional spaces. Dimension. Coordinates.L17 [PDF] HW6 solutions [PDF] HW7 [PDF] 22 (25/11) Fourth tutorial. T4 [PDF], T4 solutions [PDF] 23 (26/11) Change of coordinates. Transition matrix of coordinate change. Linear maps and transformations. Examples.L18 [PDF] 24 (26/11) Linear maps. Matrix of a linear map relative to given bases. Examples.L19 [PDF]HW7 solutions [PDF] HW8 [PDF] 25 (02/12) Composition of linear maps corresponds to the matrix product. Change of the matrix of a linear map under coordinate changes. Linear transformations and the corresponding changes of matrices.L20 [PDF] 26 (03/12) Invariants of linear transformations. Example of change of coordinates. Making matrices of linear transformations diagonal: examples and counterexamples.L21 [PDF] 27 (03/12) Computing Fibonacci numbers using linear algebra.L22 [PDF]HW8 solutions [PDF] HW9 [PDF] 28 (09/12) Tutorial classT5 [PDF], T5 solutions [PDF] 29 (10/12) Eigenvectors and eigenvalues. Example: "porridge problem". L23 [PDF] 30 (10/12) An application of linear algebra: Google PageRank algorithm.HW9 solutions [PDF] HW10 [PDF] Practice exam [PDF] 31 (16/12) No class on December 16 32 (17/12) Revision of the module 33 (17/12) Revision of the module HW10 solutions [PDF] Exam materials: sample papers, actual papers, solutionsSee course materials, mainly. Work through solutions to home assignments and tutorials, since exam papers will rely on fluency in the respective methods. Generally, check out past papers from the previous 4-5 years here. TextbooksThis module does not follow any particular textbook. When needed, consult ``Elementary Linear Algebra'' by Anton and Rorres, and ``Algebra'' by Michael Artin. There will also be online lecture notes for many of the lectures. For more examples and exercises for the 1111 part (very useful to get a grip of the material), check out the free online linear algebra book by Jim Hefferon available at this http URL. Assessment The final mark is 80%*final exam mark + 20%*home assignments result. 2b1af7f3a8