Hom, Tensor and Limits

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Explanations 2

Pontryagin duals: see Poonen's notes on Tate's thesis, for instance.
By Victor Wang
Flatness to-do Equivalent definitions of flatness, see e.g. Qing Liu... when is locally free a good way to think about it? Use both algebra (e.g. number theory) and geometry int…
By Victor Wang

Problems 10

Let AAA be a group, and let BBB be a torsion-free group. What can you say about the set Hom⁡(A,B)\operatorname{Hom}(A,B)Hom(A,B)?
Let AAA be a local ring, with maximal ideal m\mathfrak{m}m and residue field k=A/mk = A/\mathfrak{m}k=A/m. What can you say about the set of ring homomorphisms Hom⁡(k,A)\operato…
Where does the term OX,x\mathcal{O}_{X,x}OX,x come from?
Which of the following abelian groups have commutative endomorphism rings?
Let GGG be a group and let G2G^2G2 denote the subset of squares of GGG. What can you say about G2G^2G2?
Let V,WV,WV,W be vector spaces over a field KKK. When is the natural map V∗⊗KW→Hom⁡K(V,W)V^\ast \otimes_K W \to \operatorname{Hom}_K(V,W)V∗⊗KW→HomK(V,W) an isomorphism? Give t…
If A→BA\to BA→B is a flat ring map, and ttt is a non-zerodivisor of AAA, must the image of ttt in BBB be a non-zerodivisor?
Which of the following are flat ring maps?
What is the data of a morphism G→π∗F\mathcal{G} \to \pi_\ast \mathcal{F}G→π∗F of sheaves over YYY equivalent to? Select all that apply.
Let I,JI,JI,J be ideals in a (commutative, unital) ring AAA. Compute the AAA-algebra tensor product A/I⊗AA/JA/I\otimes_A A/JA/I⊗AA/J.