Learn Swedish: Vi kunde ha köpt det vackra objektet på auktionen igår. - We could have bought the beautiful item at the auction yesterday.

Yes, it is very common in spoken and everyday Swedish to drop ha after modal verbs in the past tense. You will frequently hear or read vi kunde köpt instead of vi kunde ha köpt. Both are grammatically accepted, though keeping ha is better for formal writing.

Since objekt is an ett word, shouldn't the adjective be vackert?

If it were indefinite, yes, you would say ett vackert objekt. However, this sentence uses the double definite construction. Because of the definite article det, the adjective must take the -a ending to become vackra, matching the definite noun objektet.

Does objektet refer to any everyday item?

Not generally. While it translates to object or item, ett objekt is frequently used in specific commercial contexts in Swedish, such as items sold at an auction or properties in real estate. For a standard everyday item, you would be more likely to use saken or föremålet.

Why use the preposition på for an auction?