On The Algebraic $K$-Theory of Double Points

Abstract

In this paper, we use trace methods to study the algebraic $K$-theory of rings of the form $R[x_1,\dots ,x_d]/(x_1,\dots ,x_d{)}^2$. We compute the relative $p$-adic $K$ groups for $R$ a perfectoid ring. In particular, we get the integral $K$ groups when $R$ is a finite field, and the integral relative $K$ groups $K_{\ast }(R[x_1,\dots ,x_d]/(x_1,\dots ,x_d{)}^2,(x_1,\dots ,x_d))$ when $R$ is a perfect ${\mathbb{F}}_p$-algebra. We conclude the paper with some other notable computations, including some rings which are not quite of the above form.

24 pages. Added the case p=2. Also added section 4 to fix a mistake pointed out to me by Martin Speirs. Changed the numbering of theorems to be more readable. Comments welcome

References in corpus (4)

• On the K-theory of pullbacks
• The Wirthmuller isomorphism revisited
• On the K-theory of coordinate axes in affine space
• Algebraic $K$-theory of planar cuspidal curves

Cited by in corpus (1)

• Algebraic $K$-theory of $\text{THH}({\mathbb{F}}_p)$