By A.N. Parshin, I.R. Shafarevich, I. Rivin, V.S. Kulikov, P.F. Kurchanov, V.V. Shokurov

The 1st contribution of this EMS quantity on advanced algebraic geometry touches upon a number of the important difficulties during this enormous and extremely energetic sector of present learn. whereas it's a lot too brief to supply entire assurance of this topic, it presents a succinct precis of the parts it covers, whereas offering in-depth assurance of convinced vitally important fields.The moment half offers a short and lucid advent to the hot paintings at the interactions among the classical quarter of the geometry of complicated algebraic curves and their Jacobian kinds, and partial differential equations of mathematical physics. The paper discusses the paintings of Mumford, Novikov, Krichever, and Shiota, and will be an outstanding significant other to the older classics at the topic.

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**Extra info for Algebraic geometry III. Complex algebraic varieties. Algebraic curves and their Jacobians**

**Example text**

2. Let f : H → G be a homomorphism of groups. Then any H-graded vector space is naturally G-graded (by pushforward of grading). Thus we have a natural monoidal functor f∗ : VecH → VecG . If G is the trivial group, then f∗ is just the forgetful functor VecH → Vec. 3. Let k be a ﬁeld, let A be a k-algebra with unit, and let C = A−mod be the category of left A-modules. 29) F : M → (M ⊗A −) : A−bimod → End(C). This functor is naturally monoidal. A similar functor F : A−bimod → End(C) can be deﬁned if A is a ﬁnite dimensional k-algebra, and C = A−mod is the category of ﬁnite dimensional left A-modules.

Any essentially small locally ﬁnite abelian category C over a ﬁeld k is equivalent to the category C−comod for a unique pointed coalgebra C. In particular, if C is ﬁnite, it is equivalent to the category A−mod for a unique basic algebra A (namely, A = C ∗ ). 10. The Coend construction Let C be a k-linear abelian category, and F : C → Vec an exact, faithful functor. 9) Coend(F ) := (⊕X∈C F (X)∗ ⊗ F (X))/E where E is spanned by elements of the form y∗ ⊗ F (f )x − F (f )∗ y∗ ⊗ x, x ∈ F (X), y∗ ∈ F (Y )∗ , f ∈ Hom(X, Y ); in other words, ∗ Coend(F ) = − lim → End(F (X)) .

Such functors have an obvious monoidal structure. An example especially important in this book is the forgetful functor Rep(G) → Vec from the representation category of a group to the category of vector spaces. More generally, if H ⊂ G is a subgroup, then we have a forgetful (or restriction) functor Rep(G) → Rep(H). Still more generally, if f : H → G is a group homomorphism, then we have the pullback functor f ∗ : Rep(G) → Rep(H). All these functors are monoidal. 2. Let f : H → G be a homomorphism of groups.