# Sheaf of Modules over $F_1$-schemes

## Meeting Details

Abstract: Using Connes and Consaniâ€™s deï¬nition of F1 -schemes, we deï¬ne and study the category of coherent sheaves over an F1 -scheme. We show that exact sequences of locally free modules are well deï¬ned in the category of coherent sheaves over an F1 -scheme. We then apply Q-construction to deï¬ne algebraic K-theory of F1 -schemes. In partic- ular, we show that the algebraic K-groups of S pec(F1 ) are the stable homotopy groups of the sphere $S^0$ , which is generally believed to be true. If time permits, we deï¬ne algebraic K-theory of not necessarily commutative monoids. In particular, we discuss the homotopy invari- ance property of algebraic K-theory of monoids and F1 -schemes.