On the closed subschemes of a scheme (Q6056665)

From previous articles, like [\textit{M. Temkin} and \textit{I. Tyomkin}, Eur. J. Math. 2, No. 4, 960--983 (2016; Zbl 1375.14012)], we know that the pushout of any two closed immersions of schemes with a fixed source exists in the category of schemes. Enlightened by the question of the existence of pushouts in the category of schemes, the author is dedicated to the closed subschemes of a given scheme. The main results are that using a bijection between the class of closed subschemes of a scheme \(X\) and the collection \(M(X)\) of isomorphism classes of closed immersions of schemes with target \(X\), the author discovered that the collection of closed subschemes of a scheme can be viewed as a set, then we can add the operations of multiplication and addition on it, and thus eventually construct contravariant functors from the category of schemes to that of commutative monoids. Of course, the main manipulating object pivots from the family of closed subschemes of a scheme to \(M(X)\), so are the supplemental object of the two operations mentioned above, for the concern of convenience. The preliminaries are quite simple. The reader only needs to know the very basics of modern algebraic geometry. For example, affine schemes, sheaves of rings, closed subschemes, closed immersions, and functors between categories. Surprisingly, the author sufficiently utilizes diagrams that are commutative to perform spectacular reasoning. The reviewer really appreciates those expressions and writing, especially that in Theorem 3.6. As a by-product, the author proved an easily-remembered yet pivotal proposition that every closed immersion of schemes is a monomorphism, which the reviewer benefits much from. The contribution of the paper will improve the understanding of the closed subschemes of a scheme from the perspective of the category of commutative monoids. Reviewer's remark: After careful reading, the reviewer would like to pinpoint a typo, \(\eta\circ \lambda\) (which should be replaced by \(\delta\circ \lambda\)), in the last lines \(4\) and \(3\) in page \(8\) in the proof of the Theorem~3.6.