# Math 894 | Algebra II

M. Roth | Winter 2018
Tue: 13:00–14:30, Jeff 319
Thurs: 11:30–13:00, Jeff 319

40% Homework (Best 10 out of 12)
30% Take home exam
30% Written Final

Algebra is a subject with extremely concrete origins: it is the language of the everyday manipulation of symbols, and the solution of equations. It is the language of computation in coordinates, of differential operators, and of symmetries. At the same time, it can be quite abstract; modern algebra is also the language of relations between objects, sometimes divorced from the objects themselves.

This course is the second part of the graduate core algebra sequence. Its aim is to introduce the algebraic concepts which are common knowledge for the working mathematician, and to make the link between the abstract definitions and their concrete incarnations.

The emphasis in the course will be on understanding ideas rather than speeding through the material. The following outline of topics is therefore likely to change.

### 1. Basic Notions of Category Theory

 Jan. 9 Categories; definitions by diagrams. 11 Universal properties; products and coproducts. H1 (Due Jan 18) 16 Infinite products; fibre products; push-outs. 18 Adjoint functors; preservation of products and coproducts. H2 (Due Jan 25)

### 2. Multilinear Algebra

#### A. Tensor Products

 Jan. 23 Universal mapping properties; bifunctoriality. 25 Tensor products of sums and free modules. H3 (Due Feb 1) 30 Right-exactness of the tensor product. Feb. 1 Homomorphism identities; change of scalars. H4 (Due Feb 8)

#### B. Symmetric and Alternating Products

 Feb. 6 Algebras; tensor product of algebras; tensor algebra of a module. 8 Functorial properties of the tensor algebra; change of rings. H5 (Due Feb 15) 13 Symmetric algebra; symmetric products; Symmetric product and direct sums; 15 Alternating algebra; alternating products; determinants. H6 (Due Mar 1)

### 3. Representations of finite groups

 Feb. 27 Group representations. Examples. Preview of main theorems. Mar. 1 HomG and HomG; Maschke's theorem; Schur's lemma. H7 (Due Mar 8) 6 Inner product on representations; Class functions. Mar. 8 Regular representation and its universal property; proof of remaining theorems. H8 (Due Mar 15) 13 Using the orthogonality relations, representations of S4. 15 Representations of the symmetric group; isotypic components; Schur functors. H9 (Due Mar. 22)

### 4. Introduction to Homological Algebra

 Mar. 20 Chain complexes; homology 22 Flat, Projective, and Injective modules H10 (Due Mar. 29) 27 Projective resolutions 29 The snake lemma H11 (Due Apr. 5) Apr. 3 Definition of Tor and properties 5 Spectral sequences H12 (Due Apr. 12)