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I want to use the technicalities of structured ring spectra for the first time in my life, and I am not really familiar with the relevant literature. I am looking for a reference that defines algebraic K-theory for associative unital algebras in symmetric spectra.

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Tom, not precisely sure what you want. I'm guessing you want to think of a commutative symmetric ring spectrum $R$ and then an $R$-algebra $A$. Without the extra layer, just using an $S$-algebra $R$ (not necessarily commutative), the original source for the algebraic $K$-theory of $R$ is Chapter VI of EKMM, but of course in terms of the $S$-algebras there.

The essential point is to have a homotopically well-behaved symmetric monoidal category of spectra in which to work. We now have several: symmetric spectra, orthogonal spectra, and EKMM $S$-modules. Since we have Quillen equivalences that preserve smash products among them, for many foundational purposes it doesn't matter in which category you work. Since that was clear early on, there are many things in EKMM that have not been repeated in full detail in later sources. The differences in detail are not especially significant and comparison theorems are not hard to come by showing that it doesn't matter where you work. (That is emphatically not true when doing multiplicative infinite loop space theory, which is relevant but I think not what you are asking.) Probably there is a good more recent source focusing on symmetric spectra, but I don't know of one.

Here is a relevant quote from a nice paper of Blumberg and Mandell (The localization sequence for the algebraic $K$-theory of topological $K$-theory): "We work in the context of EKMM $S$-modules, $S$-algebras, and $R$-modules. Since other contexts for the foundations of a modern category of spectra lead to equivalent $K$-theory spectra, presumably the arguments presented here could be adjusted to these contexts, but the EKMM categories have certain technical advantages that we exploit and that affect the precise form of the statements below."

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  • $\begingroup$ Thanks, Peter. The reason I said symmetric spectra is that Ching and Harper have written down nice "higher Blakers-Massey" results in that setting. If the best $K$-theory references are in the EKMM setting, then I will rely on the fact that the two settings are interchangeable for my purposes. $\endgroup$
    – Tom Goodwillie
    Commented Jul 27, 2014 at 21:53

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